EUROPEAN SYMPOSIUM ON COMPUTER-AIDED P R O C E S S E N G I N E E R I N G - 15
COMPUTER-AIDED CHEMICAL ENGINEERING Advisory Editor: R. Gani Volume 1: Volume 2: Volume 3: Volume 4: Volume 5: Volume 6: Volume 7: Volume 8: Volume 9: Volume 10: Volume 11: Volume 12: Volume 13: Volume 14: Volume 15: Volume 16: Volume 17: Volume 18: Volume 19: Volume 20:
Distillation Design in Practice (L.M. Rose) The Art of Chemical Process Design (G.L. Wells and L.M. Rose) Computer Programming Examples for Chemical Engineers (G. Ross) Analysis and Synthesis of Chemical Process Systems (K. Hartmann and K. Kaplick) Studies in Computer-Aided Modelling. Design and Operation Part A: Unite Operations (1. Pallai and Z. Fony6, Editors) Part B: Systems (1. Pallai and G.E. Veress, Editors) Neural Networks for Chemical Engineers (A.B. Bulsari, Editor) Material and Energy Balancing in the Process Industries - From Microscopic Balances to Large Plants (V.V. Veverka and F. Madron) EuropeanSymposium on Computer Aided Process Engineering-10 (S. Pierucci, Editor) EuropeanSymposium on Computer Aided Process Engineering-11 (R. Gani and S.B. Jorgensen, Editors) European Symposium on Computer Aided Process Engineering-12 (J. Grievink and J. van Schijndel, Editors) Software Architectures and Tools for Computer Aided Process Engineering (B. Braunschweig and R. Gani, Editors) Computer Aided Molecular Design: Theory and Practice (L.E.K. Achenie, R. Gani and V. Venkatasubramanian, Editors) Integrated Design and Simulation of Chemical Processes (A.C. Dimian) European Symposium on Computer Aided Process Engineering-13 (A. Kraslawski and I. Turunen, Editors) Process Systems Engineering 2003 (Bingzhen Chen and A.W. Westerberg, Editors) Dynamic Model Development: Methods, Theory and Applications (S.P. Asprey and S. Macchietto, Editors) The Integration of Process Design and Control (P. Seferlis and M.C. Georgiadis, Editors) European Symposium on Computer-Aided Process Engineering-14 (A. Barbosa-P6voa and H. Matos, Editors) Computer Aided Property Estimation for Process and Product Design (M. Kontogeorgis and R. Gani, Editors) European Symposium on Computer-Aided Process Engineering-15 (L. Puigjaner and A. Espufia, Editors)
COMPUTER-AIDED CHEMICAL ENGINEERING, 20B
EUROPEAN SYMPOSIUM ON COMPUTER-AIDED P R O C E S S E N G I N E E R I N G - 15 38th European Symposium of the Working Party on Computer Aided Process Engineering ESCAPE-15, 29 May- 1 June 2005, Barcelona, Spain
Edited by
Luis Puigjaner UPC-ETSEIB Barcelona, Spain
Antonio Espufia UPC-ETSEIB Barcelona, Spain
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C o n t e n t s - Part B Process Operation and Control A Framework for the Mixed Integer Dynamic Optimisation of Waste Water Treatment Plants using Scenario-Dependent Optimal Control J. Busch, M. Santos', J. Oldenburg, A. Cruse a n d W. M a r q u a r d t ............................. 955 On-line Fault Diagnosis Support for Real Time Evolution applied to MultiComponent Distillation S. Ferrer-Nadal, I. Ydlamos-Ruiz, M. Graells and L. Puigjaner .............................. 961
Real-Time Feasibility of Nonlinear Predictive Control for Semi-batch Reactors H. Arellano-Garcia, T. Bar,,, M. Wendt a n d G. Wozny ............................................ 967
Control of Integrated Process Networks - A Multi-Time Scale Perspective 973
M. Baldea and P. Daoutidis . .....................................................................................
Minimum-Cost Operation in Heat-Exchanger Networks 979
A. H. Gonzdlez and J. L. Marchetti ...........................................................................
An Online Decision Support Framework for Managing Abnormal Supply Chain Events M. Bansal, A. A dhitya, R. Srinivasan and i. A. Karimi ............................................. 985 Novel Scheduling of a Mixed Batch/Continuous Sugar Milling Plant using Petri nets M. Ghaeli, P. A. Bahri and P. L. Lee ........................................................................
991
Improving Short-Term Planning by incorporating Scheduling Consequences P. Hei./nen, I. B o u w m a n s and Z. Verwater-Lukszo ...................................................
997
Multi-scale Planning and Scheduling in the Pharmaceutical Industry H. Stefansson and N. Shah ......................................................................................
1003
Initiation and Inhibiting Mechanisms for Multi-tasking Control in Discrete Event Systems S. Macchietto, N. J. Alsop, R. J. B a i r d Z. P. Feng a n d B. H. Chen ....................... 1009 Model Based Parametric Control in Anaesthesia P. Dua, V. Dua and E. N. Pistikopoulos .................................................................
1015
Anti-Slug Control Experiments on a Small-Scale Two-Phase Loop H. Sivertsen a n d S. S k o g e s t a d .................................................................................
1021
Using CLP and MILP for Scheduling Commodities in a Pipeline L. Magat6o, L. V R. Arruda and F. Neves Jr .........................................................
1027
Scheduling of a Pipeless Multi-Product Batch Plant using Mixed-Integer Programming Combined with Heuristics S. Panek, S. Engell a n d C. Lessner ......................................................................... 1033 On the State-Task Network Formulation: Time Representations C. T. Maravelias ...................................................................................................... Optimization of Biopharmaceutical Experiences from the Real World
Manufacturing
with
1039
Scheduling Tools -
C. A. Siletti, D. Petrides and A. Koulouris ..............................................................
1045
vi Advances in Robust Optimization Approaches for Scheduling under Uncertainty S. L. Janak and C. A. Floudas ................................................................................. 1051 Proactive Approach to address Robust Batch Process Scheduling under Short-Term Uncertainties A. Bonfill, A. Espu~a and L. Puigjaner ................................................................... 1057 A Rigorous MINLP for the Simultaneous Scheduling and Operation of Multiproduct Pipeline Systems R. Rejowski Jr. and J. M. Pinto ............................................................................... 1063 Multicommodity Transportation and Supply Problem with Stepwise Constant Cost Function Z. Lelkes, E. Rev, T. Farkas, Z. Fonyo, T. Kovacs and I. Jones .............................. 1069 Design and Planning of Supply Chains with Reverse Flows M. I. Gomes Salema, A. P. Barbosa-P6voa and A. Q. Novais ................................ 1075 Heterogeneous Batch Distillation Processes: Real System Optimization S. Pommier, S. Massebeuf V. Gerbaud, O. Baudouin, P. Baudet andX. Joulia .... 1081 Modelling and Optimisation of Distributed-Parameter Batch and Semi-batch Reactor Systems X. Zheng, Robin Smith and C. Theodoropoulos ...................................................... 1087 Optimal Start-up of Micro Power Generation Processes P. I. Barton, A. Mitsos and B. Chachuat ................................................................. 1093 Performance Monitoring of Industrial Controllers Based on the Predictability of Controller Behavior R. A. Ghraizi, E. Martinez, C. de Prada, F. Cifuentes and J. L. Martinez ............. 1099 A Systematic Approach to Plant-Wide Control Based on Thermodynamics L. T. Antelo, I. Otero-Muras, J. R. Banga and A. A. Alonso ................................... 1105 A Multiple Model, State Feedback Strategy for Robust Control of Nonlinear Processes F. E Wang, P. A. Bahri, P. L. Lee and I. T. Cameron ............................................ 1111 A Robust Discriminate Analysis Method for Process Fault Diagnosis D. Wang and J. A. Romagnoli ................................................................................. 1117 Learning in Intelligent Systems for Process Safety Analysis C. Zhao and V. Venkatasubramanian ..................................................................... 1123 Multivariate Decision Trees for the Interrogation of Bioprocess Data K. Kipling, G. Montague, E. B. Martin and A. J. Morris ........................................ 1129 On a New Definition of a Stochastic-Based Accuracy Concept of Data Reconciliation-Based Estimators M. Bagajewicz ......................................................................................................... 1135 The Integration of Process and Spectroscopic Data for Enhanced Knowledge Extraction in Batch Processes C. W. L. Wong, R. E. A. Escott, A. J. Morris and E. B. Martin ............................... 1141 A Systematic Approach for Soft Sensor Development B. Lin, B. Recke, P. Renaudat, J. Knudsen and S. B. Jorgensen ............................. 1147
vii Application of Multi-Objective Optimisation to Process Measurement System Design D. Brown, F. MarOchal, G. Heyen a n d J. Paris ...................................................... 1153 Utilities Systems On-Line Optimization and Monitoring: Experiences from the Real World D. Ruiz, J. Mamprin, C. Ruiz, D. Nelson and G. R o s e m e ....................................... 1159 A Continuous-Time Formulation for Scheduling Multi-Stage Multi-product Batch Plants with Non-identical Parallel Units L. Fu and L A. Karimi .............................................................................................. 1165 Optimal Scheduling of Supply Chains: A New Continuous-Time Formulation A. C. S. A m a r o a n d A . P. B a r b o s a - P 6 v o a ...............................................................
1171
Effect of Pricing, Advertisement and Competition in Multisite Capacity Planning M. Bagq/ewicz ......................................................................................................... 1177 Multi-objective Optimization of Curds Manufacture N. G. Vaklieva, A. Espu~a, E. G. Shopova, B. B. Ivanov and L. Puigianer ............ 1183
Global Supply Chain Network Optimization for Pharmaceuticals R. T. Sousa, N. Shah a n d L. G. Papageorgiou ........................................................
1189
Linear Quadratic Control Problem in Biomedical Engineering L E Sanchez Chdvez, R. Morales-Mendndez a n d S. O. Martinez Chapa ............... 1195
Using Structured and Unstructured Estimators for Distillation Units: A Critical Comparison F. Bezzo, R. Muradore a n d M. Barolo .................................................................... 1201 Modeling of Complex Dynamics in Reaction-Diffusion-Convection Model of CrossFlow Reactor with Thermokinetic Autocatalysis T. TrdvniOkovd, L Schreiber and M. KubiOek ......................................................... 1207 A Design and Scheduling RTN Continuous-time Formulation P. M. Castro, A. P. B a r b o s a - P 6 v o a a n d A. Q. Novais ............................................
1213
Use of Perfect Indirect Control to Minimize the State Deviations E. S. Hori, S. S k o g e s l a d and W. H. K w o n g .............................................................
1219
Constraints Propagation Techniques in Batch Plants Planning and Scheduling M. T. M. Rodrigues a n d L. Gimeno .........................................................................
1225
Information Logistics for Supply Chain Management within Process Industry Environments M. Vegelti, S. Gonnet, G. H e n n i n g a n d H. Leone ................................................... 1231 Plant Structure Based Equipment Assignment in Control Recipe Generation Considering Conflicts with Other Batches T. Fuchino and H. Watanabe .................................................................................. 1237 IMC Design of Cascade Control M. R. Cesca and J. L. Marchetti ..............................................................................
1243
Robust Model-Based Predictive Controller for Hybrid System via Parametric Programming A. M. Manthanwar, V. Sakizlis, V. Dua and E. N. Pistikopoulos ............................ 1249
viii Model Based Operation of Emulsion Polymerization Reactors with Evaporative Cooling: Application to Vinyl Acetate Homopolymerization S. Arora, R. Gesthuisen and S. Engell ..................................................................... 1255 Event-Based Approach for Supply Chain Fault Analysis R. Sarrate, F. Nejjari, F. D. Mele, J. Quevedo and L. Puigjaner ........................... 1261
Back-off Application for Dynamic Optimisation and Control of Nonlinear Processes S. 1. Biagiola, A. Bandoni and J. L. Figueroa ......................................................... 1267
Operational Planning of Crude Oil Processing Terminals A. M. Blanco, A. B. Morales Diaz, A. Rodriguez Angeles and A. Sdnchez ............. 1273
A Hierarchical Approach to Optimize LNG Fractionation Units H. E. Alfadala, B. M. A h m a d andA. F. Warsame ................................................... 1279
First Principles Model Based Control M. Rodriguez and D. Pdrez ..................................................................................... 1285
On-line Oxygen Uptake Rate as a New Tool for Monitoring and Controlling the SBR Process S. Puig, Ll. Corominas, J. Colomer, M. D. Balaguer and J. Colprim .................... 1291 On-Line Dynamic Monitoring of the SHARON Process for Sustainable Nitrogen Removal from Wastewater K. Villez, C. Rosen, S. Van Hulle, C. Yoo and P. A. Vanrolleghem ........................ 1297 Robust Controller Design for a Chemical Reactor M. Bako~ovd, D. Puna and A. Mdsz~ros .................................................................
1303
A M1NLP/RCPSP Decomposition Approach for the Short-Term Planning of Batch Production N. Trautmann and C. Schwindt ............................................................................... 1309 A Framework for On-line Full Optimising Control of Chemical Processes P. A. Rolandi and J. A. Romagnoli .......................................................................... 1315
Wavelet-Based Nonlinear Multivariate Statistical Process Control A. H. S. Maulud, D. Wang and J. A. Romagnoli ..................................................... 1321
Anaerobic Digestion Process Parameter Identification and Marginal Confidence Intervals by Multivariate Steady State Analysis and Bootstrap G. Ruiz, M. Castellano, W. Gonzdlez, E. Roca and J. M. Lema .............................. 1327 An Efficient Real-Time Dynamic Optimisation Architecture for the Control of NonIsothermal Tubular Reactors M. R. Garcia, E. Balsa-Canto, C. Vilas, J. R. Banga andA. A. Alonso ................. 1333 Model Based Control of Solidification B. Furenes and B. Lie .............................................................................................. 1339
h-Techsight: A Knowledge Management Platform for Technology Intensive Industries A. Kokossis, R. Ba~ares-Alc(mtara, L. Jimdnez Esteller and P. Linke ................... 1345 Modelling for Control of Industrial Fermentation J. K. Rasmussen, H. Madsen and S. B. Jorgensen .................................................. 1351
ix
System-Dynamics Modelling to Improve Complex Inventory Management in a Batch-Wise Plant Z. I,'erwater-Lukszo and 7". S. Christina .................................................................. 1357 Dynamic Modeling and Nonlinear Model Predictive Control of a Fluid Catalytic Cracking Unit R. Roman, Z. K. Nag3,, F. AllgOwer and S. Agachi ................................................. 1363 Improving of Wavelets Filtering Approaches R. I.~. Tona, .4. Espu~a ~lncl L. Pui~/aner . ................................................................
1369
Supply Chain Monitoring: A Statistical Approach F. D. Me/e, E. Musulin and L. P u i ~ / a n e r . ...............................................................
1375
Closing the Intbrmation Loop in Recipe-Based Batch Production E. Mztsulin, M. d. Arbiza, A. Bon/il[ and L. Puigffaner ............................................
1381
Agent-Based Intelligent System Development for Decision Support in Chemical Process Industry ): Oao and A. Kokossis ........................................................................................... 1387 Enhanced Modelling of an [industrial Fermentation Process through Data Fusion Techniques S. Triadaphillou, E. B. Martin, G. Montague, P. Je~jkins, S. Stimpson and A. Nordon ..................................................................................................................... 1393
Implementation of Multi-Kalman Filter to Detect Runaway Situations and Recover Control R. Nomen, d. Sem/gere, E. Serra and d. Cano .......................................................... 1399 Supply Chain Management through a Combined Simulation-Optimisation Approach F. D. Mele, A. Espu~a and L. Pui~janer .................................................................
1405
Data-Based Internal Model Controller Design for a Class of Nonlinear Systems A. G. Kalnlukale and M.-S. Chiu .............................................................................
1411
Measurement-Based Run-to-run Optimization of a Batch Reaction-Distillation System A. Marchetti, B. Srinivasan, D. Bonvin, S. Elgue, L. Prat and M. C a b a s s u d ......... 1417 Nonlinear Behaviour of a Low-Density Polyethylene Tubular Reactor-SeparatorRecycle System M. H~/ele, I. Disli-Uslu, A. Kienle, I': M. Krishna, S. P u s h p a v a n a m and C.-U. Schmidt .................................................................................................................... ! 423 Control and Optimal Operation of Simple Heat Pump Cycles J. B. Jensen and S. Skogestad ..................................................................................
1429
Advanced Process Control of Pantolactone Synthesis using Nonlinear Model Predictive Control (NMPC) ('. Cormos and S. A,~zachi ........................................................................................ 1435 Design and Analysis of a Classical Controller to the Residual Oil in a Small Scale Semibatch Extractor A. F. Cust6dio, D. F. Rezende, M. R. Wolj:Maciel and R. M. Filho ....................... 1441 Optimal Sensor Network Design and Upgrade using Tabu Search M. C. Carnero, ,1. L. Hern(mdez and M. C. Sdmchez ...............................................
1447
Multiperiod Planning of Multiproduct Pipelines D. C. Cafaro andJ. Cerdd ...................................................................................... 1453
Statistical Performance Monitoring Using State Space Modelling and Wavelet Analysis A. Alawi, A. J. Morris and E. B. Martin .......................................... , ....................... 1459 Predictve Functional Control Applied to Multicomponent Batch Distillation Column D. Zumoffen, L. Garyulo, M. Basualdo and L. Jimdnez Esteller ............................ 1465 Fault Tolerant Control with Respect to Actuator Failures. Application to Steam Generator Process A. A ~touche and B. Ould Bouamama ....................................................................... ! 471
Open/Closed Loop Bifurcation Analysis and Dynamic Simulation for Identification and Model Based Control of Polymerization Reactors M. P. Vega and M. R. C. Fortunato ........................................................................ 1477 Effect of Recycle Streams on Energy Performance and Closed Loop Dynamics of Distillation Sequences S. Herndndez, J. G. Segovia-Herndndez, J. C. Cdrdenas and V. Rico-Ramirez ..... 1483 Expert System for the Control of Emergencies of a Process Plant M. L. Espasa and F. B. Gibert .................................................... , ............................ 1489 An Expert System for a Semi-Batch Pilot Scale Emulsion Copolymerisation Facility R. Chew, B. Alhamad, V. G. Gomes and J. A. Romagnoli ...................................... 1495
Integrating Data Uncertainty in Multiresolution Analysis M. S. Reis and P. M. Saraiva ................................................................................... 1501
Integrated Approaches in CAPE Integrated Process and Product Design Optimization: A Cosmetic Emulsion Application F. P. Bernardo and P. M. Saraiva ........................................................................... 1507 Design Synthesis for Simultaneous Waste Source Reduction and Recycling Analysis in Batch Processes I. Halim and R. Srinivasan ...................................................................................... 1513
Design and Control Structure Integration from a Model-Based Methodology for Reaction-Separation with Recycle Systems E. Ramirez and R. Gani ........................................................................................... ! 519
Modelling and Optimisation Collaborative Research Project
of Industrial
Absorption
Processes:
An
EC
P. Seferlis, N. Dalaouti, E. Y. Kenig, B. Huepen, P. Patil, M. Jobson, J. Kleme~, P. Proios, M. C. Georgiadis, E. N. Pistikopoulos, S. Singare, C. S. Bildea, J. Grievink, P. J. T. Verheijen, M. Hostrup, P. Harper, G. Vlachopoulos, C. Kerasidis, J. Katsanevakis, D. Constantinidis, P. Stehlik and G. Fernholz ................................. 1525
An Integrated Approach to Modelling of Chemical Transformations in Chemical Reactors T. Salmi, D. Yu. Murzin, J. Wdrngt, M. Kangas, E. Toukoniitty and V. Nieminen .. 1531
xi
An MILP Model for the Optimal Design of Purification Tags and Synthesis of Downstream Processing E. Simeonidis, J. M. Pinto and L. G. Papageorgiou ............................................... 1537 An Upper Ontology based on ISO 15926 R. Batres, M. West, D. Leal, D. Price and }I. Naka ................................................. 1543
Multi-agent Systems for Ontology-Based Information Retrieval R. Ba~ares-Alc~ntara, L. Jim~nez Esteller and A. Aldea ....................................... 1549
An Agent-Based Approach for Supply Chain Retrofitting under Uncertainty G. GuillOn, F. D. Mele, F. Urbano, A. Espu~a and L. Puigjaner ........................... 1555
Pharmaceutical Informatics: A Novel Development and Manufacture
Paradigm for Pharmaceutical
Product
C. Zhao, G. Joglekar, A. Jain, V. Venkatasubramanian and G. V. Reklaitis .......... 1561
A Web Service Based Framework for Information Integration of the Process Industry Systems Xiangyu Li, Xiuxi Li and E Qian ............................................................................ 1567 A Library for Equation System Processing based on the CAPE-OPEN ESO Interface G. Schopfer, J. Wyes, W. Marquardt and L. von Wedel .......................................... 1573 On the Optimal Synthesis of Micro Polymerase Chain Reactor Systems for DNA Analysis T. Zhelev .................................................................................................................. 1579 An Agent-oriented Approach to Integrated Process Operations in Chemical Plants M. Nikraz and P. A. Bahri ....................................................................................... 1585
Entire Supply Chain Optimization in Terms of Hybrid in Approach T. Wada, E Shimizu and J. Yoo ............................................................................... 1591 A Computer Architecture to Support the Operation of Virtual Organisations for the Chemical Development Lifecycle A. Conlin, P. English, H. Hiden, A. J. Morris, Rob Smith and A. Wright ............... 1597 An Approach for Integrating Process and Control Simulation into the Plant Engineering Process M. Hoyer, R. Schumann and G. C. Premier ............................................................ 1603 Process Integration and Optimization of Logistical Fuels Processing for Hydrogen Production F. T. Eljack, R. M. Cummings, A. F. A bdelhady, M. R. Eden and B. J. Tatarchuk 1609 A Systematic Approach for Synthesis of Optimal Polymer Films for Radioactive Decontamination and Waste Reduction T. L. Mole, M. R. Eden, 7". E. Burch, A. R. Tarrer and J. Johnston ........................ 1615 Integration of Planning and Scheduling in Multi-site Plants- Application to Paper Manufacturing S. A. Munawar, M. D. Kapadi, S. C. Patwardhan, K. P. Madhavan, S. Pragathieswaran, P. Lingathurai and R. D. Gudi .................................................. 1621
Review of Optimization Models in the Pollution Prevention and Control E. Kondili .................................................................................................................
1627
Models for Integrated Resource and Operation Scheduling A. Ha~t, M. TrOpanier and P. Baptiste ..................................................................... 1633
xii Automated Process Design Using Web-Service based Parameterised Constructors T. Seuranen, T. Karhela and M. Hurme .................................................................. 1639
Integrated Design of Optimal Processes and Molecules: A Framework for SolventBased Separation and Reactive-Separation Systems A. L Papadopoulos and P. Linke ............................................................................. 1645 A Computer-Aided Methodology for Optimal Solvent Design for Reactions with Experimental Verification M. Foli~, C. S. Adjiman and E. N. Pistikopoulos .................................................... 1651 Development of Information System for Extrusion Forming Process of Catalyst Pastes A. V. Jensa, A. A. Polunin, V. V. Kostutchenko, l. A. Petropavlovskiy and E. M. Koltsova ................................................................................................................... 1657
Integrating Short-Term Budgeting into Multi-site Scheduling G. Guill~n, M. Badell, A. Espu~a and L. Puig/aner ................................................ 1663
An Integrated Modelling Framework for Asset-Wide Lifecycle Modelling S. Sundaram and K. Loudermilk ............................................................................. 1669
AUTHOR INDEX ................................................................................................. 1675
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 ElsevierB.V. All rights reserved.
955
A Framework for the Mixed Integer Dynamic Optimisation of Waste Water Treatment Plants using Scenario-Dependent Optimal Control Jan Busch a, Marcella Santos b, Jan Oldenburg c, Andreas Cruse d and Wolfgang Marquardt a* aLehrstuhl ffir Prozesstechnik, RWTH Aachen, D-52056 Aachen, Germany bSchool of Chemical Engineering, UNICAMP, Campinas- S P - Brazil CBASF Aktiengesellschaft, D-67056 Ludwigshafen dUhde GmbH, Friedrich-Uhde-Str. 15, D-44141 Dortmund
Abstract in many real life processes, operational objectives, constraints and the process itself may change over time. This is due to changing product requirements, market demands and other external or internal influences, which constitute a certain scenario. Modelbased techniques can provide optimal solutions to the corresponding scheduling and process control problems. This paper focuses on those situations, where the objectives and constraints of plant operation depend on the scenario and therefore change over time. A framework is developed, which enables the modelling and scheduling of different operational strategies on the optimisation horizon. A case study involving a waste water treatment plant is used to demonstrate the approach. Existing expert knowledge is used to relate certain operational objectives and constraints to corresponding scenarios. It is shown that easily interpretable optimisation results are obtained. Also, the results are significantly improved as compared to a weighted average approach only approximating sequential strategies.
Keywords: online optimisation, plant scheduling, scenario-dependent optimal control, waste water treatment, mixed integer dynamic optimisation 1. Introduction With recent advances in the fields of process modelling, optimisation algorithms, computing power and practical experience, model-based techniques like online optimisation and plant scheduling have moved from being a purely academic challenge towards industrial relevance. Model-based control and scheduling require process models and models for the operational constraints and objectives. In the following, the operational constraints and objectives are defined as the operational strategy. Two typical operational strategies are to produce a certain product grade at minimum cost or at maximum throughput. Secondly, the scenario is defined to be the whole of those internal and ex-
Author to whom correspondence should be addressed:
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956 ternal factors, which determine a suitable operational strategy, e.g. stock hold-ups, predicted market demand etc. This paper focuses on those situations, where the operational strategy depends on the present scenario, which may change over time. If such a change occurs on the time horizon which is considered in the optimisation, two or more periods (stages) with different operational strategies have to be scheduled in a way to yield optimal overall plant performance. Therefore, a framework will be proposed, which allows for the accurate modelling and scheduling of such problems, introducing the notion of scenario-dependent optimal control. a)
.
I I ]
stage
0
stage
stage
b)
stage 3
I [ 1 i st. 2 i
I
I
stage 3
I stage 1 i st. 2 i stage 3 ]
I time
time
Figure l a,b. Moving Horizon Control and Plant Scheduling
2. State of the art The types of processes considered in this work are continuous processes, for which optimal base controller set point trajectories are to be determined using dynamic real time optimisation. Disturbances on a fast time scale are not accounted for on this control level, but long-term, predictable disturbances are considered. The optimisation is iteratively performed on a moving horizon, as depicted in Fig. l a. In the following, two approaches will be introduced, which have been extensively treated in literature and which are valid for certain problem specifications. Based on this, the idea of scenariodependent optimal control will be developed. If the operational objectives and constraints are assumed to be time invariant, the horizon can be interpreted as one stage, on which the continuous optimisation problem is solved (single stage formulation). A plant scheduling problem evolves, when different products or product grades are to be produced sequentially, as e.g. in modem multipurpose plants (Papageorgiou et al., 1996). Here not only the individual production campaigns have to be optimised, but also their number, sequence and lengths. The optimisation horizon is split into stages, which correspond to the campaigns. On each stage, the process model and the operational constraints may be different. However, the operational objective is the same on all stages. Although usually performed offiine, an iterative moving horizon implementation is shown in Fig. lb.
3. Scenario-dependent optimal control 3.1 Motivation
In contrast to these approaches, this paper considers situations where the operational constraints and the objectives change over time. Abel and Marquardt (2000) have treated the case that at every point in time a critical event could take place, e.g. the burst of a rupture disc, after which it must still be possible to meet certain safety constraints
957 using a suitable operational strategy. Since the moment of failure is expected at every point in time, this leads to a high problem complexity. Here the case is considered that the sequence of operational strategies depends on predictable scenarios. The motivation to do so originates from research on the optimisation-based control of the waste water treatment plant discussed in the case study. When trying to formulate a suitable model-based operational strategy, it was found that plant operators employ two different strategies. At certain times the objective is to maximise the throughput in order to increase flexibility, at other times it is to minimise the energy consumption. The choice of a strategy mainly depends on the predicted inflow and therefore changes over time. There are several possibilities to approximate the use of scenario-dependent operational strategies, e.g. by considering a weighted average of the individual objectives. This weighted average can be expressed in a single objective function, which is valid at all times and which can be optimised using a single stage formulation. The main difficulty is assigning suitable values to the weights, which means e.g. to assign a monetary value to flexibility that is valid in all possible scenarios. However, to the authors' knowledge, no exact modelling and scheduling methodology for scenario-dependent operational strategies has been reported so far. This paper proposes such a rigorous approach, which can be interpreted as a variation of regular plant scheduling on a moving horizon. The campaigns are substituted by time periods, during which a certain operational strategy is employed. The main difference is that not only the constraints, but also the operational objectives may change from stage to stage, while the process model is always the same. The expected benefits of the are: 1. Expert knowledge concerning the strategies and the heuristics, when to employ which strategy, can be exactly represented. This might enhance the overall optimisation results as compared to approaches only approximating sequential strategies. 2. Since the optimisation results correspond to well known strategies, which are separated in time, they are more easily interpretable by plant operators. This is a key factor in enhancing the acceptance of model-based techniques in industrial settings.
3.2 Formulation of the optimisation problem K
rain Y', w/<:/< (x(t f ,k ),z(t./.,/,. ),u(t /.,/< ),d(t f ,/<), p,t f,l <) u,tf,k /<=1
l(x,x,z,u,a,p,t)=o,
x(to,~)= xo, t e[to,k,t/k],
k {1..... K),
g (x.~.z...u.~,.t)<_ o. s.t.
uk(x.x.z...a.p.t)<_o.
k {1..... K},
,, ), l,,t.:.,. )- o, hk, (.~(t/. ,. ), x(t i. ,. ), z(t j. ,. ),#(tj., ),a(t/., ), l,,t./., )<- o,
(P1)
v,~,, (x(t :. ,. ), x(t:.,. ,, ), ~(t/. ,. ), u(t/. ,~ ),a(t:.
r(,,(t :., ))- ,,(to..,+~ ),
,
{1..... K - 1 1 , {1.....
The proposed problem formulation is given by (P 1). The process model, which is valid on all stages, is determined by f and g*. The disturbances are given by d, u are the controis, x are the differential and z are the algebraic state variables, and p are the parame-
958 ters. The sequence of operational strategies is reflected by a sequence of K stages on the optimisation horizon. Each stage k is assigned one operational strategy, which is modelled in detail by a set of constraints gk and an objective function ~bk.The length of stage k is defined by the interval [t0.k,t~k]. Between the stages, K- 1 switching conditions q~sand hs and stage transition conditions F~ for the differential states have to be defined. The initial conditions x0 have to be determined from measurements or from state estimation. In the overall objective function, the weighted objective functions of the stages are summed up. For two reasons, these weights are much easier to choose than in weighted average approaches. In case there is only one strategy used on the optimisation horizon, no weighting is considered at all. If several strategies are employed, the weighting only has the role of setting priorities among the objectives in a certain scenario, as will be demonstrated in the case study. Since the number, type and order of strategies are not known beforehand, they represent discrete degrees of freedom. There are two possibilities to determine the values for the discrete variables: either they are determined by an appropriate optimisation algorithm together with the continuous optimisation problem, or they are determined beforehand by some decision algorithm or expert knowledge. If they can be determined beforehand, the problem reduces to a so-called multistage problem. Otherwise, a mixed integer dynamic optimisation (MIDO) problem has to be solved. Due to limited space, only the formulation of the multistage problem is presented here. The formulation of the MIDO problem can be derived from Oldenburg et al. (2003). Multistage and especially MIDO problems are computationally more demanding than single stage problems. However, since planning problems are usually solved on longer time horizons, the increase in computation time is not expected to be a limiting factor. The software tool DyOS (DyOS, 2002), which has been developed at the LPT, is an efficient tool for solving problems of type (P 1) as well as MIDO problems and is used in the following case study (Schlegel et al., 2004).
4 Case study: Waste water treatment plant The proposed methodology will be exemplified in a case study involving a simulated municipal waste water treatment plant (WWTP). A simplified flowsheet of the process is shown in Fig. 2. Incoming waste water is stored in a buffer tank. From there, it is transferred to a biological stage, consisting of a denitrification and a nitrification tank, which are connected by a recycle stream. Air can be blown into the nitrification tank and sludge can be withdrawn from it. The biomass and the purified water (product) are separated by membrane filtration. The relevant operational constraints are the upper limits of the buffer tank volume and of the permeate ammonia concentration. The operational cost of the process is mainly caused by the air flow. There is a maximum power consumption level, above which the cost increases dramatically. In the following, this is related to an air flow threshold. From discussions with plant operators, it is understood that there are two operational strategies, whose employment mainly depends on the predicted inflow: 1. Regular inflow: The aim is to bring the buffer tank volume to a certain, usually low target value, while keeping the air flow below the threshold. This is called the flexibility strategy, as it increases flexibility for unforeseen high inflow situations.
959 2.
High inflow: If staying below the air flow threshold and meeting all constraints cannot be achieved due to high inflow, the air flow is allowed to exceed the threshold, but as little as possible. This is called the e c o n o m i c s strategy. These two strategies are modelled with appropriate constraints and objective functions. The stage transition conditions /-" enforce continuity of all x between all stages. No switching conditions h or (p are required. The process m o d e l f g* is given by biological kinetics (Gujer et al., 1999) and mass balances. The membrane is assumed to be an ideal splitter. The resulting model consists of 42 differential and 236 algebraic equations. The size of the problem mainly results from the 13 components considered in each tank. The overall length of the optimisation horizon is 3 days, for which a perfect inflow prediction is assumed to be available. In this case study, a new optimisation mn is performed once per day. The sequence of operational strategies on the horizon is determined manually by inspection of the predicted inflow. The weights wk are chosen such that the minimisation of cost has priority over the increase of flexibility. The degrees of freedom are the controls u and the endpoints of the stages t/:,. waste water
btlffel
tank
denitrification
nitri-
1Fleli1-
fication
brahe
!rec3'cleflow l i t
airflow " sludge withdrawal
Figure 2. Schematic representation o[the waste water treatment 1)/ant
Some results of an extensive optimisation study are depicted in Fig. 3. The inflow history is characterised by a strong rain event on day 7 and a smaller one on day 10, superposed by the regular daily inflow. From day I to day 4, the flexibility strategy is applied. This implies that the air flow does not exceed the threshold of 55 1/day and that the objective of the optimisation is to lower the volume in the buffer tank. On day 5, the strong rain event on day 7 is predicted. As now the flexibility strategy would become infeasible, the economics strategy is employed. The air flow increases as much as necessary. This in turn allows the permeate flow to increase such that the upper limit of the buffer tank volume of 3000 m 3 is met. At the end of day 7, the flexibility strategy is employed again. The interpretation of the results is straightforward, because at every point in time only one operational strategy is employed, the choice of which is deduced from the operational context. This enables simple relations between the operational objective and constraints and the resulting input and state trajectories. Fig. 3 also shows the computed trajectories in the same setting, but approximating the problem with a weighted average single stage formulation. The deviation from the air flow threshold and from the buffer volume target value are weighted and summed up in a single objective function. This implies that both objectives are considered at every point in time. Several weights have been tried and the most successful combination is presented here. While the computed permeate flow and the resulting buffer tank volume are very similar to the results of the multistage optimisation, a different recycle stream and a much higher air flow are computed, leading to an increase of cost by a factor of 170%. This shows that separating the strategies and sequencing them, as suggested by
960 practitioners, leads to substantially better optimisation results. Also, the interpretation of the single stage results is not straightforward (consider e.g. the peak of the air flow on day 7), since at every point in time two competing objectives are considered. Inflow .
~" 6ooo ~4000
.
.
.
.
.
Buffer kink volume
.
.
.
.
.
.
.
.
.
.
3000
.
""'I!2 ''r'''g''
~*""g"
2000
.
- - I~KilKsfage
.
50
.... .
. ]
.
. .
. .
. .
~. . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
l'hlme[~.tysl Recycle stream
~_ so .
.
1 2 3 d 5 6 7 8 O 10 1| I2 13 14
Air flow
.
.
I 72°°°'s ~ lo0o
1 2 3 4 5 6 7 8 9 10 |1 112 13 ld Tin~ [daysl
"~ 60
.
.
.
,,, .... .
.
.
.
,, . . . . . .
I :2 3 4 5 6 7 8 9 1011 12 13 t4
Time l~y~l
--
~"
"= s000 =~ 4000 0
......................
1 2 3 4
5 6
7 8 9
t011 12
13 14
Time Iday,q
Figure 3. Optimisation results (multistage a n d single stage f o r m u l a t i o n )
5 Conclusions
Scenario-dependent optimal control allows to model and schedule sequences of different operational strategies, each of which is employed in a certain operational scenario. It leads to easily interpretable and, in the presented case study, to significantly improved optimisation results. Future research should analyse for which processes improved optimisation results can be expected and concentrate on applications. Reliable disturbance forecasting and automatic sequencing are important aspects in this context. References
Abel, O., Marquardt, W., 2000, Scenario-Integrated Modeling and Optimization of Dynamic Systems, AIChE Journal 46, 4. DyOS, 2002, DyOS User Manual, Release 2.1, Lehrstuhl for Prozesstechnik, RWTH Aachen, Aachen. Gujer, W., Henze, M., Mino, T. and van Loosdrecht, M., 1999, Activated Sludge Model No. 3, Water Science and Technology 39, 1. Oldenburg, J., Marquardt, W., Heinz, D., Leineweber, D. B., 2003, Mixed-Logic Dynamic Optimization Applied to Batch Distillation Process Design, AIChE Journal 49, 11. Papageorgiou, L. G., Pantelides, C. C., 1996, Optimal campaign planning scheduling of multipurpose batch semicontinuous plants. 1. Mathematical formulation, Ind. Eng. Chem. Res. 35, 2. Schlegel, M., Stockmann, K., Binder, T., Marquardt, W., 2004, Dynamic optimization using adaptive control vector parameterization, submitted to: Computers & Chemical Engineering. Acknowledgements
The financial support by the DFG (German Research Foundation) in the project "Optimisation-based process control of chemical processes" is gratefully acknowledged.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjaner and A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
961
On-line Fault Diagnosis Support for Real Time Evolution applied to Multi-Component Distillation Sergio Ferrer-Nadal, Ignacio Y61amos-Ruiz, Mois~s Graells and Luis Puigjaner* Chemical Engineering Department, Universitat Politbcnica de Catalunya Av. Diagonal 647, E-08028, Barcelona, Spain
Abstract In this paper, the Real Time Evolution algorithm (Sequeira et al., 2002) is applied to the on-line optimization of a debutanizer distillation column. A fault diagnosis system (FDS) implemented within a supervisory module is responsible for handling incidences (faults and disturbances) happening in the plant by taking the appropriate corrective actions, including the activation of the RTE system. Thus, a more robust on-line performance is achieved. The implementation of the RTE scheme has been performed using Matlab© and the commercial simulation package HYSYS.Plant©, taking advantage of their communication capabilities (COM technology). Different possible plant incidences are addressed, involving different sources and types of disturbances. Results of RTE are compared with those obtained using the standard Real Time Optimization approach, showing better performance in most of the cases.
Keywords: On-line optimization, Real-time evolution, Multi-component distillation, Fault diagnosis.
1° Introduction On-line optimization is a very important issue in chemical and petrochemical plants since it allows immediate response to internal or external disturbances in order to continuously maximize the economic and environmental performance of the whole process. Real Time Evolution (RTE) introduces a new approach to on-line optimization that overcomes the drawbacks reported for the classical RTO approach. According to Friedman (1995), the steady state data required in a common dynamic environment needs heavy filtering and leads to long waiting. Furthermore, detailed process models are demanded whereas a proper trajectory for the implementation of the manipulated variables to reach the optimum is usually not provided. The aim of this work is to show the concept of Real Time Evolution supervised by a fault diagnosis system in a multi-component distillation column. This system operates under a more robust framework, able to distinguish between incidences susceptible to be optimized (disturbances) and incidences that require alternative actions to keep the
Author to whom correspondence should be addressed:
[email protected]
962 process stability (faults). This new structure decides continuously whether to operate under RTE or to activate alternative actions depending on the fault diagnosed.
2. Real Time Evolution RTE responds quickly to disturbances (both, external or internal) by a continuous adjustment of the set-point values based on a steady state model. The plant parameters evolve smoothly, but continuously, towards the objectives without performing formal optimization. RTE is able to deal with continuous disturbances because it does not need to reach the steady state to trigger the improvement mechanism, leading to a better overall process performance. Plant model updating is only carried out when the steady state is reached. RTE relies on four main aspects: improvement algorithm, RTE neighbourhood, time between successive executions and steady state model. The improvement algorithm is based on the way that the new points in the surroundings of the current set point values are selected. This neighbourhood is defined as the maximum allowed changes in the process variables to be evaluated. A successful RTE application requires an appropriate parameter tuning and selection of these elements (Sequeira et al, 2002).
3. Supervisory Control The optimization technique described in the previous section runs under a supervisory fault diagnosis system. This supervisory module is the core of the on-line response system against deviations from the normal behaviour of the plant making the overall process more reliable and safe. The supervisory module organizes the global performance of the plant, in such a way that differentiates the incidences that can arise in the plant, classifying them into two main groups. Incidences affecting the profit of the plant are classified as disturbances and are susceptible to operate under on-line optimization by the RTE, while incidences that affect the safety and good performance of the plant are named and classified as faults that may generate different responses of the supervisory module, i.e. activating a set of corrective actions to prevent more critical situations by monitoring different alarms and advising operators in the way they should act against this abnormal events. Incidence detection is carried out through a MSPCA (Multiscale principal component analysis) (Bakshi, 1998), which is based on PCA (principal component analysis), one of the most applied multivariate statistical techniques (Kourti and MacGregor, 1995). Through the structure of MSPCA, the capacity of PCA to extract the relationship between variables is combined with the capacity of wavelets to separate deterministic features from stochastic processes. It approximately decorrelates the auto-correlation among the measurements. Incidence isolation and diagnosis is carried out by a feed forward neural network trained in order to classify the signals received from the plant. Most possible disturbances and faults are tested, to extract enough data for properly training the artificial neural network. Then, it is prepared to classify new data from an abnormal behaviour of the plant.
963
4. System Architecture The supervisory control decides at each moment if data received from plant correspond to normal behaviour or an abnormal situation (figure 1). When an abnormal event arises, it decides if this incidence is critical for the good performing of the plant (fault) or the plant is susceptible to be optimized (disturbance). In this case, RTE is automatically activated by the supervisory control. The time for diagnosing the fault must be minimized to take advantage of the benefits gained by an early reaction of RTE. First, current conditions from the plant are collected and applied to the steady state model. With this model, the improvement algorithm explores the surroundings of the current set-point values evaluating the resulting objective function. The combination improving the objective function value is chosen and immediately applied to the plant through the control system. OUTPUTS
INPUTS
PLANT
E
T Disturbances
I
1
I
I
__
CONTROL
T
i I
~ ....
Faults
I
SUPERVISORY I I I
CONTROL I
FDS
ACTIONS
II _!
L--
Figure 1. R TE supported by fault diagnosis system
When the plant reaches its optimum operation conditions and no disturbances occur, the value of the proposed set-points will not change. Therefore, no action will be done in the plant. in case a fault is diagnosed, the supervisory control shows warnings or advices to the operator. Depending on the nature of the fault, it also can execute a previously designed protocol consisting of corrective actions to be applied to the process. The implementation has been done with a Matlab function. It acts as a data manager establishing the communication between the plant (Hysys in dynamic mode), the model (Hysys in steady state mode), the optimization algorithm (Matlab), and the supervisory module (Matlab). Communication between Matlab and Hysys.Plant® is executed by mean of COM technology.
5. Case Study: D e b u t a n i z e r C o l u m n The debutanizer column provided by Aspentech© at their World Wide Site documentation (http://support.aspentech.com) has been chosen for this paper in order to make the results more reproducible. This multi-component distillation column has fifteen stages and is fed by two streams consisting of a mixture of light hydrocarbons. Firstly, the column has been simulated in Hysys.Plant® in steady state mode. This will
964 be the steady state model for the RTE algorithm. Secondly, the necessary modifications have been made in order to build the dynamic simulation including the control mechanism (Table 1) given by Figure 2. This dynamic simulation will be used as the real plant to be online optimized. Vent
PIC-IO0 Cond Duty Butane Product
Butanes
Feed2
••DReubty
.-.,.
Feed1
LIC-101
C5+
VLM- 101
Liquid Product
Figure 2. Debutanizer Flowsheet Table 1. Parameters for the PI controllers
Controller Kc Ti, min
LIC- 100 1.80 10.00
LIC- 101 2.00 10.00
PIC- 100 2.00 2.00
TIC- 100 5.00 20.00
An objective function IOF (instant objective function) has been proposed in order to evaluate the plant performance at every moment: IOF (t) = (Amount of C4 & lighter in Butane Product) • (Price of Butane Product) + (Amount of C5+ in Liquid Product) • (Price of Liquid Product) - (Flow of Feed l) • (Price of Feed l) - (Flow of Feed2) • (Price of Feed2) - (Condenser Heat Duty) • (Price of Condenser Heat) - (Reboiler Heat Duty) • (Price of Reboiler Heat), m.u./time (1) Table 2. Prices of the feed and product streams
Feed1,
Feed2,
Butane P.,
Liquid P.,
Cond. Heat,
Reb.Heat,
m.u./kg
m.u./kg
m.u./kg
m.u./kg
m.u./kW
m.u./kW
2.00
2.00
3.00
3.00
0.00009
0.0005
As RTO and RTE are not comparable in terms of an instant objective function, a mean objective function is employed taking into account the accumulative produced profit:
MOF
(t) -
I' IOF o
(t)dt
( t - to)
, m.u./time
(2)
965
6. Results A case study is considered for which the decision variables are the set points of one of the controllers (6 th stage temperature) and the reflux rate of the column. RTE requires a parameter tuning depending on the studied scenario, in this case, a maximum allowed change in set points of 0.3% around the old value and 50 seconds between consecutive executions have been considered showing satisfactory results. In a first situation, a step rise (+20%) in the mass flow of Feedl is diagnosed and automatically the RTE optimization is activated. While RTE reacts immediately against this disturbance, RTO has to wait until steady state is reached. This faster RTE response is translated in a better value of the mean objective function compared to those obtained by RTO. Figure 3 shows RTO has reacted later (11000 seconds approx.), losing benefits during the transition to steady state. Feed1
2.9
,",,,,
._
2.8
\
~1
2.7
3.1
~
3
=
Step
Fall to 2.232
kg/s
2.6 .-, 2.5
,,. . . . . . , . - " ' -
iQ~;~
2.3
No Optimiza{ion
................
R T O
0
0.2
0.4
0.6
0.8
Simulation
1
1.2
Ti
me,
1.4
s
1.6
1.8
2 x
10 4
Figure 3. Mean Objective Function for a step disturbance in the Feedl temperature
In case of a continuous disturbance arises in the mass flow of Feedl (-0.0002 kg/s), the optimization by RTO shown in Figure 4 is not possible since the plant never reaches a steady state. This RTO's weak point is where RTE shows better performance, since for these situations RTO may not be applied. Therefore, the obtained results with RTE can be only compared when no optimization is carried out. Figure 4 shows the instant objective function for this case while Figure 5 represents the mean objective function for the simulation time considered. In this distillation column, an example of possible fault is a fall of 20% in the mass flow of Feed2. When the supervisory control diagnoses this fault, a message advises the operator to check the valves involved. In addition, the supervisory control will perform a protocol consisting of a by-pass of the feed flows and an operation change to total reflux until the break-down is mended.
7. Conclusions and Future W o r k A supervisory control based in a fault diagnosis system integrated to the RTE algorithm has been implemented for a debutanizer column. Further work will be orientated to
966
adapt the fault diagnosis system to the new conditions of the plant after an optimization, since current FDS is only prepared to detect disturbances from the base normal operation. It will be also possible to modify critical parameters from the optimization model obtaining for each case diagnosed a more accurate model. Feed1
Ramp
Fall-0.0002
kg/s
3
E 2.5
xx
¢
2
>
1.5
RTE
No
1
'",
Optimization
5000
10000
S i m u l a t i o n
Ti
me,
15000
s
Figure 4. Instant Objective Function for a continuous disturbance in the Feedl temperature Feed1
Ramp
Fall
-0.0002
kg/s
2.8
2.6
S ]~
2.4
2.2
1.8
No
Optimization
"''''"--.
:.
5000
10000
S i m u l a t i o n
Time,
15000
s
Figure 5. Mean Objective Function for a continuous disturbance in the Feedl temperature
References Sequeira, S., M. Graells, and L. Puigjaner, 2002, Real-Time Evolution for On-line Optimization of Continuous Processes, Ind.Eng.Chem.Res., Vol.41, 1815-1825 Friedman, Y., 1995, What's wrong with unit closed loop optimization?, Hydrocarbon Processing, 107
Bakshi, B.R., 1998, Multiscale PCA with Application to Multivariate Statistical Process Monitoring, AIChE J., Vol. 44, No. 7, 1596-1610 Kourti, T. and J.F. MacGregor, 1995, Process analysis, monitoring and diagnosis, using multivariate projection methods, Chemometrics and Intelligent Laboratory Systems 28, 3-21 Acknowledgements Financial support received from "Generalitat de Catalunya" (FI program) and Spanish "Ministerio de Educaci6n y Ciencia" (grant FPI) is fully appreciated.
European Symposium on Computer Aided Process Engineering- 15 L. Puigjaner and A. Espufia (Editors) © 2005 Elsevier B.V. All rights reserved.
967
Real-Time Feasibility of Nonlinear Predictive Control for Semi-batch Reactors Harvey Arellano-Garcia*, Tilman Barz, Moritz Wendt and Gtinter Wozny Department of Process Dynamics and Operation, Technical University of Berlin TU Berlin, Sekr. KWT-9, D-10623 Berlin, Germany
Abstract In this work a nonlinear model predictive control scheme for on-line optimization of semi-batch reactors is presented. Since the true process optimum occasionally lies on a boundary of the feasible region defined by one or more active constraints, the process is forced into an infeasible region due to the uncertainty in the parameters and measurement errors. An alternative new approach is proposed to assure both robustness and feasibility with respect to input and output constraints. This approach is based on the backing off of the constrained output variable bound along the moving horizon. With this strategy, a trajectory of mathematical constraint limits will be formed. The trajectory of these bounds is dependent on the amount of measurement error and parameter variation including uncertainty. The perl%rmance of the proposed approach is assessed via an application to a semi-batch reactor under safety constraints, where a strongly exothermic series reaction conducted in a non-isothermal batch reactor is considered to show the analytical steps of the approach, and to demonstrate the efficiency of the proposed online framework.
Keywords: NMPC, Output-Constraints, Dynamic Real Time Optimization, Batch Processes, Safety. 1. Introduction Batch processing provides greater flexibility in the production of specialty and pharmaceutical chemicals. The trend in the chemical industry towards high value products has increased interest in the optimal, model-based control of batch processes. These control problems are generally posed as tracking problems for time-varying reference trajectories defined over a finite time interval. However, during the course of a typical batch, process variables swing over wide ranges and process dynamics go through significant changes due to nonlinearity. Furthermore, batch processes are characterized by significant uncertainties, a certain number of noisy measurements and the fact that the controlled properties are usually not measured on-line. Therefore, the potential advantages of a model based control system are likely to lead to significant tracking errors. With the aim of dealing with model uncertainties and process disturbances, the optimal control problem can also be solved on-line. The on-line
Author to whom correspondence should be addressed:
are i
lano-garcia@tu-be rlin. d e
968 optimization problem is generally non-convex for NMPC. Thus, practical implementation of NMPC becomes difficult for any reasonably nontrivial nonlinear system (Mayne D. Q., 2000). However, critical issues are robustness and the feasibility of the optimization problem, i.e. the presence of an input profile that satisfies the constraints. In this contribution an alternative new approach is proposed to assure both robustness and feasibility with respect to input and output constraints.
2. Problem Formulation A strongly exothermic series reaction conducted in a non-isothermal batch reactor is considered to demonstrate the efficiency of the proposed online framework. The reaction kinetics are second-order for the first reaction producing B from A, and an undesirable consecutive first-order reaction converting B to C. The intermediate product B is deemed to be the desired product. Since the heat removal is limited, the temperature is controlled by the feed rate of the reactant A and the flow rate of the cooling liquid in the nominal operation. The reactor is equipped with a jacket cooling system. At the start, the reactor partly contains the total available amount of A. The remainder is then fed and its feed flow rate is optimized to maximize the yield. However, the accumulation of A at the start of the batch time must be prevented, otherwise, as the batch proceeds; exhaustion of the cooling system capacity can not be avoided. Furthermore, whilst the reaction proceeds, the reactor's volume diminishes so that the computation of the corresponding cooling capacity is adapted according to the remaining cooling jacket area. The developed model takes in both the reactor and the cooling jacket energy balance. Thus, the dynamic performance between the cooling medium flow rate as manipulated variable and the controlled reactor temperature is also included in the model equations. Thereby, it can be guaranteed that later the computed temperature trajectory can be implemented by the controller. Moreover, by this means the limitations of the cooling system (pump capacity) can explicitly be taken into account for the optimization. The resulting model comprises 5 differential states, 2 algebraic state variables, and 3 time-varying operational degrees of freedom. The objective is to maximize the production of CB while minimizing the total batch time subject to diverse operational, quality as well as safety related constraints during the batch and at its final time. The objective function then reads: min (-C B (t r ) + ~ . t r ) %oo,,Fe~d,At There are open-loop final time operation,
with
~=
1 70
(1)
path and end point constraints for the reaction process to be fulfilled by the optimal recipes. First, a limited available amount of A to be converted by the is fixed to NAtot(tf)=500mol. Furthermore, so as to consider the shut-down the reactor temperature of at the final batch time must not exceed a limit.
Tr ( t f ) ~ 303 K
(2)
There are path constraints for the reactor temperature and the adiabatic end temperature r r ( t ) < 356 K
(3)
T~ (t) _<500 K
(4)
969 •~. 500
.~. 3 5 5 L
m
L
335
450
e,~
e'~
E 40O
E 315 .=,=,
'1o ¢..
o ,,,,=, o 295 L
®
350
f .Q 3 0 0 1500
500
2500
3500
4500
."o
0
900
time [s]
1800
2700
3600
4500
time [s]
Figure 1. Path constraints: Reactor temperature (left) and adiabatic end temperature (right). Tad is used to determine the temperature after failure (Abel and Marquardt, 2000). Additionally, the cooling flow rate changes from interval to interval are restricted to an upper bound abs[
(k)] o,o
...., (k +
(5)
The decision variables u(t) are the feed flow rate into the reactor, the cooling flow rate V,,,,, and the different time intervals. A multiple time-scale strategy based on the orthogonal collocation method in finite elements is applied for both discretization and implementation of the optimal policies according to the controller's discrete time intervals (6-12s; 600-700 intervals). The resulting trajectories of the reactor temperature and the adiabatic end temperature for which constraints have been formulated are depicted in Fig. 1. It can be observed that during a large part of the batch time both states variables evolve along the upper limit. The adiabatic end temperature, in particular, is an active constraint over a large time period. Although operation at this optimum is desired, it typically cannot be achieved with simultaneous satisfaction of all constraints, because of the influence of external disturbances (Loeblein and Perkins, 1999). Thus, an NMPC based approach is proposed to implement such an optimal strategy despite disturbances.
3. Nonlinear Model Predictive Control NMPC provides a systematic methodology to handle constraints on manipulated and controlled variables not being limited to a certain model structure. However, for the online optimization of the semi-batch process, the momentary criteria on the restricted controller horizon with regard to the entire batch operation are insufficient. Therefore, the original objective (eq. 1) must be substituted by an appropriate alternative which can be evaluated on the local MPC prediction horizon. N~
min ~'oo]
Ntj
J (N,,N~,N~,)-Zd(j).[.~(t+ j It)-w(t+j)~ + ~ ' 2 ( j ) . [ A u ( t + j - 1 ) ~ -
'
j=N I
(6)
,/'=1
The optimization objective is reduced to a tracking problem. The first term stands for the task of keeping as close as possible to the calculated open loop optimal trajectory of the reactor temperature, whereas the second term corresponds to control activity under the consideration of the systems restriction's described above. Both reactor temperature and adiabatic end temperature are defined as hard-constraints. However, it is well known that hard output constraints can cause problems for two reasons: the optimization can become infeasible, and some of the constraints must then be relaxed or
970
m.~
!
:
i
i
hmcl constraints :.; ,
upl:~, r b o u ! n d
:leterval
n:
deviation by measurement (
iii y
- ri~al
:;
i
;
i
i..................;..................!....................hOr, .........izon ...........................;..................i...................i..................i..................
t=%
future
_'-
Figure 2. Back-off strategy within moving horizon. eliminated. In the latter both approaches rely on relaxation which is, in fact, inapplicable for safety restrictions. In this work an alternative new approach is proposed to assure both robustness and feasibility with respect to input and output constraints. This approach is based on the backing-off of the bound of the hard-constrained output variable along the moving horizon (8intervals). Thus, a trajectory of mathematical constraint limits will be formed. For the near future time points within the horizon, these limits (bounds) are more severe than the real physical constraints and will gradually be eased (logarithmic) for further time points (see Figure 2). The trajectory of these bounds is dependent on the amount of measurement error and parameter variation including uncertainty. The back-off strategy is introduced into the optimization to guarantee that the safety restrictions are not violated at all when the calculated trajectory is applied to the batch process. The basic idea is shown in Fig. 3. The true process optimum lies on the boundary of the feasible region defined by the active constraint. Due to the uncertainty in the parameters and the measurement errors, the process optimum and the set-point trajectory would be infeasible. By introducing a back-off from the active constraints in the optimization, the region of the set-point trajectory is moved inside the feasible region of the process to ensure, on the one hand, feasible operation, and to operate the process, on the other hand, still as closely to the true optimum as possible.
rain u .
.....
.
.
Y <_ymaX ~ack-O~
opiimaibacked off~ ; ~
feasibler e g m n •
.
de/ inistic
. .~%~ . . . . ., e .r ~ . . .
J(u)
point
H 1
Figure 3. Back-offfrom active constraints
971 Table 1. Objective function parameter and hard-constraints back-off.
Tp prediction horizon Tc control horizon MV variation weighting factor offset weighting factor T,,,a., maximum allowable Reactor temperature T,......=356K T,a maximum allowable adiabatic end temperature T,,,ox=508.5K
8 intervals 8 intervals 3000 o~{rrP-i) with ~=0, 7 T ..... ( j = 2...8) = T"~..... - ~,,~x.a'(j--~' with 7~.... =4K, T ..... ( ~ - 2 . . . s ) :
will,
~ ..... =3K,
a~=0.5 ~5;" .... - ~, .... .o, ''--~'
a=0.5
For the formulation of the NMPC-based online optimization, the objective function parameters (eq. 6) and the hard-constraints back-off are presented in Table 1, respectively. The decision variable is again the cooling flow rate. In order to compare the performances of the open-loop nominal solution and the nominal NMPC with the proposed on-line framework under uncertainty, different disturbances have been considered, namely: catalyst activity mismatch and fluctuations of the reactor jacket cooling fluid temperature. Additionally, all measurements are corrupted with white noise e.g. component amount 8%, Temperature 2%.
4. Dynamic Real-Time Optimization The size of the dynamic operating region around the optimum (see Figure 3) is affected by fast disturbances. These are, however, efficiently rejected by the proposed regulatory NMPC-based approach. On the other hand, there are, in fact, slowly time-varying nonzero mean disturbances or drifting model parameters which change the plant optimum with time. Thus, a re-optimization i.e. dynamic real-time optimization (D-RTO) may be indispensable lbr an optimal operation. When on-line measurement gives access to the system state, on-line re-optimization promises considerably improvement. Moreover, additional constraints can be formulated. In this work, we assume that the state information is available.
n1parameters_
Feed
Y
reference trajectory
~ ~ V
(Model)
,,
disturbances
reference trajectory
ad [ Constraints /
I
cSonstrainafety
L
Feasibility
re-optimization D-RTO
__q ~,,~a,,o. s,.,e'I" 'Tri.,.r I ~°-°~'
Figure 4. On-line Framework: Integration of NMPC and dynamic re-optimization
972
(9
1
~
',
,
i
350
335 (9
E (9
320
,4,=,1 L_
0
,!==1
305
refei'ence trajectory ~:
to
(9 '- 290
f
900
I
1800
2700
I
3600
I
4500
time [s] Figure 5. On-line re-optimization." reactor temperature
In order to compensate slow disturbances, the on-line re-optimization problem is automatically activated three times along the batch process time according to a trigger defined as the bounded above difference between the reactor temperature and the temperature reference trajectory (see Figure 5). New recipes resulting from this are then updated as input to the on-line framework. Due to the different trigger time-points the current D-RTO problem progressively possesses a reduced number of variables within a shrinking horizon (Nagy and Braatz, 2003). As a result of this, and a catalyst contamination the total batch time increases. But, despite the large plant mismatch and the absence of kinetic knowledge nearly perfect control is accomplished.
5. Conclusions Feasibility and robustness with respect to input and output constraints have been achieved by the backing-off strategy. The resulting NMPC scheme embedded in the online re-optimization framework is viable for the optimization of the semi-batch reactor recipe while simultaneously guaranteeing the constraints compliance, both for nominal operation as well as for cases of large disturbances e.g. failure situation. The proposed scheme yields almost the same profit as the one of the off-line optimization operational profiles. Currently, the approach is extended to a probabilistically constrained nonlinear model predictive controller.
References Abel O. and W. Marquardt, 2000, AIChE J., 46, 4, 803-823. Loeblein C. and J. D. Perkins, 1999, AIChE J., 45, 5, 1018-1029. Mayne D. Q., 2000, Automatica, 36, 789-814. Nagy, Z.K., and R.D. Braatz, 2003, AIChE J., 49, 1776-1786. Acknowledgements The authors gratefully acknowledge the financial support of the German Research Foundation Deutsche Forschungsgemeinschaft (DFG) under the contract WO 565/12-2.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
973
Control of Integrated Process N e t w o r k s - A Multi-Time Scale Perspective M. Baldea a and P. Daoutidis a.* a
Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455, USA
Abstract In this paper, we analyze the dynamics of integrated process networks featuring large recycle streams and small purge streams. We consider a prototype network comprising of a reactor and a condenser, and, within the framework of singular perturbations, we establish that the dynamics of such a network exhibits three time scales. We describe a model reduction procedure which leads to an explicit nonlinear description of the dynamics in each time scale, and we outline a controller synthesis procedure that rigorously accounts for this time-scale multiplicity.
Keywords: model reduction, singular perturbations, plant-wide control
1. Introduction Process integration, through material and energy recycle streams, represents the rule, rather than an exception, in the process industries. While offering significant economic benefits, tightly integrated process networks pose distinct challenges, as the feedback interactions among the process units, induced by the recycle, give rise to complex, overall network dynamics, in addition to the dynamics of the individual units. Yet, most studies on control of process networks with recycle streams (e.g. Luyben, 1993; Yi and Luyben, 1997) are within a multi-loop linear control framework. The strong coupling between the control loops in such an approach has been recognized as a major issue that must be addressed in a plant-wide control setting (Price and Georgakis, 1993; Luyben et. al., 1997; Ng and Stephanopoulos, 1998, Larsson and Skogestad, 2000; Skogestad 2004). In our previous work (Kumar and Daoutidis, 2002), we considered process networks with large material recycle. Within the framework of singular perturbations, we established that the large recycle induces a time scale separation, with the dynamics of individual processes evolving in a fast time scale, and the dynamics of the overall network evolving in a slow time scale. We proposed a model reduction method for deriving nonlinear low-order models of the slow dynamics and a controller design framework comprising of properly coordinated fast and slow controllers. In (Baldea et.
Author to whom correspondence should be addressed. Email:
[email protected]. P. Daoutidis currently at Aristotle University of Thessaloniki, Thessaloniki, Greece
974 al., 2004), we focused on process networks with recycle, in which small quantities of inert components are present and a small purge stream is used for their removal. Adopting again a singular perturbation perspective, we established the presence of a slow, core dynamics associated with the presence of the inert and outlined a framework for rationally addressing the control of inert levels in the network. In the present paper, we analyze the dynamics of integrated process networks featuring large recycle streams and inert components purged through a small stream. Specifically, we consider a prototype network comprising of a gas phase reactor and a separation system with a gas recycle stream. Within the framework of singular perturbations, we establish that such a network exhibits three time scales: the fastest time scale, in the order of magnitude of the time constants of the individual units, a fast one, in which the dynamics of the entire network evolves, and a slow time scale associated with the presence of the inert and the small purge flowrate. Furthermore, we describe a model reduction procedure which leads to an explicit nonlinear description of dynamics in each time scale and we outline a controller synthesis procedure that rigorously accounts for the time-scale multiplicity feature of such networks. Finally, we provide illustrative numerical simulation results.
2. Modeling of Process Networks with Recycle and Purge We consider the network of Figure. 1, consisting of a gas-phase reactor and a condenser.
R
Fo~yAo~ylo ~
P
BKym
L
Figure 1. Process network with large recycle and purge The reactant A is fed at a constant molar flowrate Fo to the reactor, where a first order, irreversible reaction takes place with reaction constant kl. The outlet stream from the reactor is fed to a partial condenser that separates the light unconverted reactant A from the heavy product B. The gas phase is then recycled at a high rate back to the reactor. A very volatile impurity I is also present in the feed stream in small quantities, and a small purge stream P is used to remove this impurity from the recycle loop. The interphase mole transfer rates for the components A, B, I in the condenser are governed by rate expressions of the form NJ = K ja(yj-P~)~v xy)V')~,,, where Kja represents a mass transfer coefficient, yj the mole fraction in the gas phase, the mole fraction in the liquid phase,/~j the saturation vapour pressure of the component j, and P the pressure in the condenser. Assuming isothermal operation, the dynamic model of the network has the form:
975 M e -(+R-F 1
•
YI,R -- ~
[F,(Y,
,o
- v . I,R
-
I
-y,.)]
H
~VI,. - F -
R-
N-
P
1
)[,/ = ~
1 [F(,,',~-, )'/ ) - N , + y , N ]
M~=N-L "~[t - ~-]-L
•
1
N]
(1)
MR, Mv, Mr. d e n o t e the m o l a r h o l d u p s in the r e a c t o r and c o n d e n s e r v a p o u r a n d
where
liquid phase, respectively, and N=N~+N~+Nz. We consider F
to be 0(1), and define
c z=Fo/R,, as the ratio of the steady state values of the feed and recycle flowrates. Note
that, due to the assumption that the recycle flowrate is much higher than the throughput, e'~ << 1. Also, we define c~ - P / F << 1 (under steady-state conditions, the purge flowrate is much smaller than the feed), the scaled (possibly manipulated) inputs uR=R/R,, u c = F / F , and u p = P / p , and the O(1) ratio k = F / R , . We assume that the mole fraction of the inert in the feed is very small, i.e. y j . , - % c , where % = O(1), that the inert is very volatile, i.e. p / / p = a: / x, with a 2 - O(1), and that its mass transfer rate is very small, i.e. K ~ a = a ~ x ~-, c~ =O(1), the latter two assumptions implying that a negligible amount of inert leaves the recycle loop and exits through the liquid stream at the bottom of the condenser. Based on the above assumptions, the model of the process network takes the form: 1 ~,~ = F +-- F, (,~ - k.,. ) g1
:i'.,.z, = ~
'
1
i',,R = - ~
M~.
F(I-azc~ -Y,.e )- knA//z,Y.,.R+--F,.,.ue()[ig,-.Y.,je)
'
1
1
F, ( ~,c~ - v ,,~ ) + --e,&'" ~(y~ - >'''~ )
- N , N B -(a~e~_y~ + a, a2c~x z) Mz PL
1 gl
'fi
j:, =--~. --F,kz,, (y ~.,~- y , ) - N, + y~(N , + N,)+ y.,(a,c~y, +a, os2e=xt) ML ] Pl. I = 1 F kzL (yz.R-yz)-(crv~ vz + rztrz_~2xz)M,. +yz(N~ + NB)+),z(al~:v~ +oflc~2~2xz) ML M~ = N ~+ N B + ( u ~ )'~ + a~m_e'~_.,<~) M~ _ L PL
k t =I[N,MI.
G~x l) M z-] x I(N t + N e) -.r,(ofl~:,'l+oc, ~ .
.
-fz =--~ (a,e;)q +a, u2e=xz)
.
.
.
P~. I
""-xz(N, + Ne)-xz(a,e~y I +a,a:e=xl) Mc PL PL I
976 Evidently, the above model contains terms of O(1), O(1/e) and O(e), which suggest potentially a three time scale behaviour. In what follows, we document this feature within the framework of singular perturbations.
3. Model Reduction and Control We begin by obtaining a description of the fastest dynamics of the network. To this end, we define the fastest, stretched time scale r~=t/el. In the limit ~ ~ 0, corresponding to an infinitely large recycle flowrate, we obtain the following description of the network dynamics in this fastest time scale: MR .- F . (u R. kttF . ) dr I ....
Y..I.R dr 1
M~___L=F.,.(kUF_UR ) dr~
YI___L=F., kUF(Y.,,a _ y.,) dr~ M r
F"' Yl.R - =1 MR UR(YA - Y..~.R) -dr
F °" uR(Yl-YI.R) MR
(3)
y__L_~=F,. kur(y~, _ y,) dr~ M~.,
M~ = 0 x.,~ _ 0 X--L-~= 0 dr I dr I dr~ Note that not all the nontrivial algebraic equations that correspond to the equilibrium of the fastest dynamics are linearly independent. Specifically, the last three nontrivial equations can be expressed as functions of the first three. This implies that the steadystate condition associated to the fastest dynamics does not specify isolated equilibrium points, but rather a six-dimensional manifold in which a slower dynamics evolves Also, note that the control objectives in this time scale must be addressed using the large inputs u 1, as the inputs u' and d' have no effect on the fastest dynamics. Turning now to the dynamics on this equilibrium manifold, multiplying Eq. (1) by e~ and considering the limit e 1 ~ 0, the linearly independent constraints: 0 = u~ - k u r , 0 - t g k ( y A --YA.k)' 0 - u R ( y ~ --Y~,R)are obtained, which must be satisfied in the fast time scale. In this limit,
the terms which involve division of the above functions with the small parameter e~ become indeterminate. Let z l denote these finite, but unknown terms. Then the model of the dynamics after the fastest boundary layer becomes a Differential Algebraic Equation (DAE) system of nontrivial index (owing to the unspecified variables z!). Setting the large flowrates u ~with appropriate feedback laws (in the prototype network considered, we use ue=l-k,,R(Mv.sp-Mv) and ur=l-k,,F(MR..v,-Mn)), the algebraic variables zl can be computed after one differentiation of the algebraic constraints (Kumar and Daoutidis, 2002). Thus, the index of the DAE system is exactly ~wo. Employing a coordinate change which involves the total material holdup of the recycle [oop. and the holdups of the individual components in the recycle loot'., i.e. ~, - MR + M y ' ~2 - MRYA.R + M v Y A ' ~3 -- MRY,.R + M, ," ),~ ' ~=~ - M ~ 46 - x , , r l ,
- u~(x)
+ ku~(x),rh
- u~(x)(y~
- y.~,~),rl~ .... u ~ ( x ) ( ) , ~
"~
:= - x ~'
(4)
- ),,.~ )
the resulting underlying ODE model becomes: ¢, = Yo - U ~ - X .
- e2~,
~2--"-(g2Fo., ,k k,y~2
Up +
- ~Fo...
k, ~2 k,r MR..,.~,- k F Y.4,o ~ k,r + k k,r N A ~1
Jr- ~2 kuR e2 Hp F,s)/(~ 1 (k kuF -.[- kuR))
-
kkl~2
977 ~4 -
NA + Ne + g2NI - L
~5 - N4 - ~ (N.4 + NR + g2~, ) with ~ - ( a ~ g ~ / , ~
(5)
+a~a~_~6)~a/PL" Notice that the above model is still stiff, due to the
presence of the small singular perturbation parameter 6' 2 . Repeating the model decomposition procedure followed above, we can derive separate representations for the fast and slow dynamics. Specifically, considering the limit 6"2-----+0 (corresponding to the absence of the inert component from the feed and a zero purge flowrate) in a slow, compressed, time scale re=tee we obtain the following description of the slow dynamics d ~3 _( _ g ue Fo, + F a, ,~, ) / ~:, dr 2
(6)
The control objectives at the network level (such as product purity, control of total material holdup, etc) can be achieved in the fast time scale using the manipulated inputs u', as neither the large inputs z/, nor the purge stream have any effect in this time scale. Similarly, the control of the inert levels in the network can only be accomplished in the slow time scale using the purge stream d'.
4. Simulation Study We carried out simulation studies on a specific process network of the type presented in Figure 1, with the parameter values and nominal steady state given in Table 1. Table l. Nominal process parameters MR, tool
4007.0
p~, , m o l / m 3
15000.0
KA,mOl/m 2 rain
1140.0
My, mol
2710.33
3' ~.,,
0.98
K s , m o l / m e min
1200.0
Mz ,mol
1069.82
y/.~,
0.020
K ~ , m o l / m 2 min
60-0"10-6
F,,mol/min
100
pi'p a
1.16-10 s
a, m2/m -~
17.5
R, m o l / m i n
1002.73
P~,Pa
1.67"105
kr
0.05
F, m o l / m i n
1102.73
P[,Pa
5.67.107
k, R
0.01
P, mol/rain
2.28
~], m 3
3.0
k,L
0.04
/tl, 1/ rain
0.06
T~,K
373.0
T o,,j,K
273.0
Specifically, we controlled the liquid level in the condenser (and, thereby, the total holdup of the network) using the effluent flowrate and a proportional controller, L=L~(1-kL(M~.v,-ML)). Subsequently, we designed an input-output linearizing feedback controller with integral action (Daoutidis and Kravaris, 1992) for the product purity xB = 1-xA-x~, using the reactor holdup setpoint MR..v, as a manipulated input and requesting the critically damped second order response xe + fl~ ~dxB _ + f12 d2xu dt 2 - x~.~.~with fl1=60 min, fl2=900 min 2. Finally, we employed the derived reduced-order model of the slow dynamics (Eq. 6) to design an input-output linearizing controller with integral action for the total inert holdup, requesting the first order response ~3 + Y d~ 3 / dt - ~'3.v, with
978 7" = 1200rain. Figure 2 shows the closed-loop behaviour of the process network for a step change in the product purity setpoint xB,=p and in the presence of unmeasured disturbances. Clearly, the proposed controller exhibits a very good setpoint tracking and disturbance rejection performance. :~::~ ~.
a:,~[~
..................................................................
:~ 2 ~
.
.
.
.
.
.
.
.
.
.~.~.,)
...................................................................................
.............................................
/ .
.
.
.
.
.
.
.
a
.
b
Figure 2. Closed-loop responses of the reactor-condenser network for (a) a step change in the purity setpoint to 0.33 and (b) for a 20% estimation error of the mass transfer coefficient KB
5. Conclusions In this paper, we analyzed the dynamic behaviour of integrated process networks with large recycle flows and small purge streams. Using a prototype network comprising of a reactor and a separator, we showed that such systems possess a dynamic behaviour featuring three distinct time scales and we outlined a controller design framework that naturally accounts for this time scale separation.
References Baldea, M., P. Daoutidis and A. Kumar, 2004, Dynamics of process networks with recycle and purge: time scale separation and model decomposition, In: Preprints of ADCHEM'03, Hong Kong, pp. 645-650. Daoutidis, P. and C. Kravaris, 1992, Chem. Eng. Sci., 47(4), 837-849. Kumar, A. and P. Daoutidis, 2002, J. Proc. Contr., 12, 475-484. Larsson, T and S. Skogestad, 2000, Mod. Ident. Contr., 21(4), 209-240. Luyben, M.L., B.D. Tyreus and W.L. Luyben, 1997, AIChE J., 43, 3161-3174. Luyben, W.L., 1993, Ind. Eng. Chem. Res., 32, 466-486, 1142-1162. Ng, C. and G. Stephanopoulos, 1998, Plant-wide control structures and strategies, In: Proceedings of DYCOPS-5, Corfu, Greece, pp. 1-16. Price, R.M. and C. Georgakis, 1993, Ind. Eng. Chem. Res., 32, 2693-2705. Skogestad, S. 2004, Comp. Chem. Eng. 28, 219-234. Yi, C.K. and W. L. Luyben, 1997, Comp. Chem. Eng. 21 (1), 25-68.
Acknowledgements Partial financial support for this work by ACS-PRF 38114-AC9 and NSF-CTS 0234440 is gratefully acknowledged.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) (~?)2005 Elsevier B.V. All rights reserved.
979
Minimum-Cost Operation in Heat-Exchanger Networks Alejandro H. Gonzfilez a and Jacinto L. Marchetti a a institute of Technological Development for the Chemical Industry (INTEC) Guemes 3450, 3000 Santa Fe, Argentine
Abstract This report revises the use of the maximum energy recovery criterion as objective function for real-time optimisation of heat-exchanger networks. Though in general this criterion leads to minimum total heat exchanged in the service units, it is not sufficient to achieve the actual minimum operation cost. This analysis discusses the characteristics of the network structure for which the last statement is applicable, and proposes an alternative performance index to address more directly the final economic objective for which these heat-recovery systems are created. An application example demonstrates the significant differences in operating conditions that may result from using one or the other criterion, something of outmost importance when defining online optimisation for these systems.
Keywords" heat-exchanger networks, optimal operation, utility cost, heat integration 1. Introduction Marselle et al. (1982) was one of the firsts articles to discuss the optimal operation problem of heat-exchanger networks (HEN), where simultaneous regulation and optimisation are considered. More closely, important results on these topics were presented in Mathisen thesis (1994), Uzturk and Akman (1997), Aguilera and Marchetti (1998) and Glemmestad et al. (1999). Most of these works provide specific representation models of HENs that can be used for searching the convenient operating point attending to the full network capacity under given inlet stream conditions and temperature targets. When an optimal operation is regarded, it means that at least the following two goals are pursued: i) outlet temperature targets for the process streams, and ii) minimum operating cost. This second goal, however, has been frequently understood as attaining the operation condition that yields the greatest possible heat integration. However, this report objective is to demonstrate and analyse the existence of an additional optimisation space going beyond maximum energy recovery. The presented results show that though maximum heat integration in HENs implies minimum overall heat exchanged in service units, in many cases it is not sufficient for determining minimum operating cost. The following section describes the structural network conditions that are necessary for the existence of an extra degree of freedom associated to the service units. Then, the third section shows how the optimisation problem must be formulated when minimum utility costs are pursued. The numerical results of running an example problem with
980 different objective functions are shown in Section 4, and the final section gives the conclusions of this work.
2. Conditions for Extra Degrees of Freedom of the Service Variables Based on the energy balance of the total network, Aguilera and Marchetti (1998) analysed the degrees of freedom of a HEN as structural possibilities of optimisation once process stream targets and input conditions are fixed. However, once maximum heat integration is achieved in a flexible HEN, there may be additional degrees of freedom for the service variables depending not only on the network structure but also on conditions of temporary operating points. To determine when these characteristics are present, let us analyse first the relationships describing the tasks to be accomplished on each process stream with a temperature target. To reach the temperature target a hot process stream i has to release the energy Qi and a cold process stream j has to receive the energy Q;. These energies are released and received through different process-to-process exchanges q~ and service tasks qci and qhj along each stream path, namely
Q~=-~-'~qk-qc,,
ieH,
Q/-
kcK i
~~qk +qhj,
j~C
(1)
kEKj
where H stands for all the hot streams, C represents all the cold streams, and Ki and Kj are the subset of heat-exchangers on streams i and j respectively. Because the sum of all the process-to-process heat-exchanger duties represents the same integrated energy in both equations, a simple relationship between cold and hot service units can be obtained
Zqh,-Zqc, j~C
=ZQ/+ZQ,
i~H
j~C
•
(2)
icH
Furthermore, notice that for fixed temperature targets and defined inlet conditions, the right side of (2) is constant. Calling global network duty D to this constant, the following expression is reached:
~-~qh~ -~-~qo, +D" jcC
(3)
ieH
Let us consider now a performance index typically used when maximum heat integration is required, or equivalently, an index expressing the total heat flow handled by the service units
J-~-'q~ j~C
+~-'qc,
(4)
icH
Combining (3) and (4) gives the following relationship:
J - Z~-" qo + D icH
- 2~-" qh - D
(5)
jeC
According to (5), for any HEN having nh active heaters and no active coolers, under optimal heat integrating condition, an extra degree of freedom may appear when nh > 2+no, or no > 2+nh. For instance, if all the qo, = 0 (or qh, = 0), the index function J is constant despite of how many qhj (or qc, ) are in the network, how is the connecting
981 structure, or how is the total service duty distributed among the q~,, (or q~, ). However, performing the same heat duty in different exchanger units (with different UA) implies different utility flow rates. Thus, once the maximum heat integration is attained, different combinations of q/, (or q~: ) are still possible, which implies different use of utility streams and consequently different utility costs. This extra degree of freedom, which goes beyond heat integration, appears also when a HEN having any number of cooling and heating services goes to temporary operating points where the number of active service units complies with any of the above relationships.
3. The Optimal HEN Operation Problem Let us assume that the HEN structure and the heat exchanger areas are completely defined for a given case problem, where there are enough degrees of freedom as to perform steady-state optimisation. Assume also that all process stream targets are known, and that the convenient control structure for temperature regulation has been already defined. When the desired condition actually focus on minimum operating cost, the optimal solution can be obtained by solving the following minimization problem:
subject to - ~--'~qk - q~.(w ) - Q,
i E H,
(7)
j~C,
(8)
k E 1,he+s},
(9)
k~K i
qx +q,, (% ) - Q~ k~K~
-q, < 0
k e {1,ne + s} ,
q~. (w ) = e (w ) L (w. ) qh, ( % , ) = e,, ( % , ) L~,,
i~H, j ~ C,
(~o) (11) (12)
where ne stands for the total number of process-to-process heat-exchangers, s is the total number of serviced units, w~.i stands for the cold utility flow rate of the service unit on the hot stream i, and wl~/is the hot utility flow-rate of the service unit on the cold stream j. The supra index " in (9) indicates fully open control valve or fully closed bypass; the functions e and L are defined in the Appendix. Note that the utility costs per unit mass (c~i and chi) are included in the performance index, and that (11) and (12) are non-linear equations determining w~; and whj at the optimal operation condition. if the network structure does not include stream splits or multiple bypasses (bypass to more than one unit), excellent initialisations for the above NLP problem are obtained by first solving the associate LP for maximum heat integration; this LP uses heat flows as exclusive optimising variables and excludes the equality constraints (11) and (12).
982
4. Example Problem Figure 1 shows the sketch of a HEN composed by two process-to-process exchangers and three service units connected by one hot and two cold process streams. Table 1 gives the stream conditions, and Table 2 shows the UA values for two network designs A and B. The factor F, is assumed 1.0 in all the units, and the utility cost cci and chj are also set to 1. The first row in Table 3 shows the results obtained by solving the above optimisation problem when the design A is used. Though the problem of maximizing the utility expenses J makes no sense from the operating point of view, when this maximum is subject to the same energy recovery obtained before, an important reference solution is determined. For instance, the result in the second row of Table 3 is obtained by including the constraint q l + q2 = 115, or equivalently q~l + q~2 + q.~3 = 60, in the problem formulation. Thus, the results in rows 1 and 2 become the extremes of a set of infinite solutions maintaining the same energy recovery, but showing different levels of utility expenses. Any of these solutions in the set could be reached by minimizing the total service heat duty depending on the initialisation or the numerical method used to find the minimum. A similar numerical experience is repeated for design B, where the results show a greater cost function difference than in case A (see rows 3 and 4). Analysing the slack variables when there is an extra degree of freedom, it was noted that each extreme operation condition is associated with an active constraint (9) related to different units located in the path connecting $2 and $3. Notice that with the stream conditions in Table 1, the solutions avoid using the cooler S 1, creating the condition for an extra degree of freedom. However, when the output temperature of stream H1 is set to 50 °C, the cooler is activated and the extra degree of freedom is lost; see rows 5 and 6 where both, the min J and the constrained max J problems give the same solution for each design. Nevertheless, even though the designs A and B have the same total exchange area, they show different operation cost.
%
Figure 1: General structure of the HEN example
Csl
983 Table 1. Nominal process and service stream conditions
wc bTlet temp. ('C) Outlet temp. (°C)
H1
C1
C2
Csl
1.0 190 75
1.5 80 160
0.5 20 130
1.0 1.0 1.0 15 200 200 . . . . . .
Hs:
Hss
Table 2. UA values used in the above network
UA (case A) UA (case B)
ql
q:
5.0 5.0
2.0 2.0
q.,.1 3.0 3.0
q.,.2
q.,.s
3.0 1.0
1.0 3.0
Table 3. Effect of the extra degree offi'eedom on the operation cost.
Case A-minJ A - max J B- minJ B - max J A-min/max J B min/maxJ
q~
102.07 67.44 102.07 85.98 102.07 102.07
q:
q.,.1
12.93 47.56 12.93 29.02 31.50 31.50
0.0 0.0 0.0 0.0 6.43 6.43
q,:
17.93 52.56 17.93 34.02 17.93 17.93
q.,.s
x,t
x,.2
x,s
J
42.07 7.44 42.07 25.98 23.50 23.50
0.0 0.0 0.0 0.0 0.155 0.155
0.345 0.757 0.39 1.0 0.345 0.390
0.32 0.088 0.274 0.213 0.208 0.201
0.665 0.845 0.664 1.213 0.533 0.591
5. C o n c l u s i o n s Any HEN having nh active heaters and nc active coolers, which operates under optimal heat integration, may have an extra degree of freedom when nh > nc +2, or when nc > nh +2. The reference to active service units emphasizes that this additional optimisation space may turn on and off depending on the operating condition. The immediate consequence of this finding is that the problem formulation for HEN online optimisation has to be adapted to effectively address the minimum utility cost objective. In other words, the extra degree of freedom means that optimising service costs by directly minimising expenses associated to utility flow rates may yield additional benefits as compared to maximising heat-integration. Besides, minimising expenses associated to utility flow rates implies maximising heat-integration. Furthermore, these results also show that HEN designs based on maximum energy recovery combined with minimum total-exchange-area cost should be revised when operation cost is the main goal.
References Aguilera, N. and J. L. Marchetti, 1998, Optimising and Controlling the Operation of HeatExchanger Networks, AIChE Journal, 44 (5), 1090. Glemmestad, B., S. Skogestad, and T. Gundersen, 1999, Optimal Operation of Heat Exchanger Networks, Comput. Chem. Eng., 23,509. Marselle, D.F., M. Morari, and D.F. Rudd, 1982, Design of Resilient Processing Plants: II. Design and Control of Energy Management System, Chem. Eng. Sci., 37 (2), 259. Mathisen, K.W., 1994, Integrated Design and Control of Heat Exchanger Networks, Doctoral Thesis, University of Trondheim, Norway. UztOrk, D. and Akman, U., 1997, Centralized and Descentralized Control of Retrofit HeatExchanger Networks, Comput. Chem. Eng., 21, $373.
984 Acknowledgements
This work has received the support from Universidad Nacional del Litoral and CONICET. Appendix" M o d e l of a H e a t E x c h a n g e r in a N e t w o r k
A convenient way of writing the main equations modeling the stationary condition of a single heat exchanger is obtained from the steady energy balance and the constitutive equation for the heat transferred, namely q
= wici(Ti 0 -Ti)
- wjcj(Tj.
-T O) -
(A1)
UA(FrATn,,)
In this expression, the supra index 0 stands for inlet conditions, A is the heat-exchanger area, U is the overall heat transfer coefficient, and F r is a factor correcting the logarithmic mean temperature difference ATml to account for deviations from pure counter-current pattern. Algebraic rearranges combining the above equalities show the convenience of defining the following parameters: wjcy .__ (T~°-T~) R ___..__:_ WiC i (Tj - V°) '
Xvu
- - UA WjCj
and
B-exp{UruFv(R-l)}
- -( T_ j __ T.O )
(A2)
These three parameters help the computation of the heat-exchanger efficiency defined by 1- B 1 - RB
( T j - T ° ) for R:/:B:/:I, or ( Ti° - T° )
e
Nru for R l + Ur~:
B
1
(A3)
Combining Eqs. A1 to A3 allows writing the heat flow rate as q = q(wi, wj) = e(wi, w s ) L ( w j ) , with
L(Wj)= W/cs(Ti ° - T° )
(A4)
where L(wj) may be interpreted as the virtual amount of exchangeable energy and the efficiency e(wi,wj) defines the amount of heat transferred when a finite exchange area is available between the two fluids. Thus, the equations representing any heat-exchanger k being part of a network at the stream match (ij) are similar to Eqs. (A4), the main difference is that the heat-exchanger inlet temperatures T~° and T° are the results of previous exchanges, namely Ti° ( k ) = riiin
1 Z qli' Win i Ci
TO
(k)-
[i ~ prei(k),
i~H
(A5)
1i
Ti n 1 ~ + w~"cj ~ qtj,
l~ ~ pre~(k),
j ~ C
(A6)
lj
where the superscript in stands for inlet to the network, and prei(k) and prej(k) represent exchanges previous to k. Notice that w/or wj may be different from the nominal inlet stream flow rate because of the control valve position. Thus, the heat duty upper bound is determined by (A4) for fully open control valve or fully closed bypass, namely q< _ q° =e(wi" ,wji.~) L,twji~) - eOLo .
(A7)
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
985
An Online Decision Support Framework for Managing Abnormal Supply Chain Events Mukta Bansal, Arief Adhitya, Rajagopalan Srinivasan*, and I.A. Karimi Laboratory for Intelligent Applications in Chemical Engineering Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576, Singapore
Abstract Enterprises today have acknowledged the importance of supply chain management to achieve operational efficiency, and cutting costs while maintaining quality. Uncertainties in supply, demand, transportation, market conditions, and many other factors can interrupt supply chain operations, causing significant adverse effects. These uncertainties motivate the development of simulation models and decision support system for managing disruptions in the supply chain. In this paper, we propose a new agent-based online decision support framework for disruption management. The steps for disruption management are: monitoring the KPIs, detecting the root cause for the deviation of KPIs, identifying rectification strategies, finding the optimal rectification strategy and rescheduling operates as necessary in response to the disruption. The above framework has been implemented as a decision support system and tested on a refinery case study.
Keywords: Risk Management, Optimization, Agent-Based, Uncertainty 1. Introduction In the face of highly competitive global markets, companies are pressurized to reduce costs and increase efficiency. As a consequence, they are employing new strategies which result in complex supply chains. These strategies include outsourcing, single sourcing, and centralised distribution. An efficient supply chain requires transparency among its constituent entities. Complex and lengthy supply chains lack visibility and this leads to disruptions. Unhindered and timely material, information and finance flow between different entities of supply chain is another important element. Blockage in any of these would lead to undesirable events like process shutdown, financial loss, under-supply or over-supply, etc. Hence there is a greater need for risk and disruption management. A disruption management system should be capable of detecting abnormal situations before they occur, diagnose the root cause, and propose corrective actions as required. While a complete rectification is desirable, in cases where the effect of disruption is
Author to whom correspondence should be addressed:
[email protected]
986 manifested very late, it may only be feasible to effect a partial recovery. A robust system should be capable of handling both complete and partial rectification. The challenges involved in developing disruption management system are: 1) Supply chain entities are dynamic, disparate and distributed; and 2) they are tightly linked at intra- and inter-enterprise levels and affect the performance of one another. These make the detection of disruption and the root cause difficult. Further, complex rectification strategies are needed to partially or completely overcome the disruption. For example, the implementation of a rectification strategy in some cases will need rescheduling of operations. In this paper, we propose an agent-based framework for disruption management. Intelligent agents measure key performance indicators in each supply chain entities. Disruptions are detected when these KPIs deviates from a pre-specified set points or when unplanned events are detected. Root causes are diagnosed using a causal model. Rectifications are proposed and optimised through a model of supply chain linkage. When necessary, rescheduling is performed to recover from a disruption. In this paper, we present the details of the framework and its implementation and illustrate it using a refinery supply chain. 1.1
Literature Review
There is limited literature in the area of disruption management; no general structured methodology exists to date. Sheffi et al. (2003) describe mechanisms which companies follow to assess terrorism-related risks, protect the supply chain from those risks and attain resilience. They provide classifications of disruptions, security measures and ideas to achieve resilience. They report various case studies and interviews with executives of companies. Wilson (2003) focuses only on transportation disruptions. Julka et al (2002; a, b) proposed an agent-based framework for modelling a supply chain. In their framework, the behaviour of every entity in the supply chain is emulated using an agent that imitates the behaviour of various departments in refinery. Mishra et al. (2003) reported an agent-based decision support system to manage disruptions in a refinery supply chain. In the event of a disruption, agents collaborate to identify a holistic rectification strategy using heuristic rules. In this paper, we generalize their approach and develop a model-based framework for managing supply chain.
2. Framework and Methodology for Disruption Management The proposed framework is described in Figure 1 and is capable of handling situations where 1) occurrence of a disruption is manifested only through deviations from set point and the root cause is not observable, as well as 2) cases when disruptions can be detected at source i.e., the root cause is observable. From control theory, the former needs feedback control while the latter requires a feedforward mechanism. In the general case, the steps for disruption management are carried out by the following components: KPI Monitors: To manage disruptions in supply chain, it is essential to measure key performance indicators (KPIs) and to identify their effect on the supply chain. We use stock inventory, throughput, and other similar indicators to monitor the state of each constituent of the supply chain. These can be measured at regular intervals and
987 monitored by comparing their day-to-day values against pre-specified limits. Alarms are generated when a sustained deviation in any KPI is detected. {
•
I*2PIs
,-
...... :
hffOHnatiOll of Deviation
Alarms
m
e
I) lsluptioll ( ( - ' O1 l"e c t i v e
Action 2,~e<essalT)
i
Possible Causes
I
Y os sible Re ctifi c atiOllS m
Ne~v N:hedule and ('oHecW~e Actkms k-PIs fOl each '~ • •
I ecHfi(
~ (-)primal RectifF. ation
atlOll
i
y
S Cell211 i 0
........S c ~ e d u ! e r
..........................~
"
A
[NNNN ...........
New Sdledule
Figure l. Frameworkjbr disruption management system 2.
Root Cause Identifier: Causal models are used to identify the possible causes for the alarms. Hypotheses are proposed to identify the root cause and confirmed if all expected consequences are manifested online. 3. Rectifications Proposer: The list of corrective actions to rectify the root cause is generated using a causal model, which accounts for the linkages among the supply chain entities. Each rectification strategy is simulated using a supply chain simulator and feasibility and KPIs for each scenario is evaluated. 4. Optimizer: One rectification strategy is selected based on feasibility and KPIs. 5. Scheduler: In a general case, disruption may make the existing operation schedule infeasible or sub-optimal. Optimal rectification strategy may require rescheduling of operations. Our rescheduling scheme is described in more detail in Section 2.1. 6. Coordinator: Numerous activities may be necessary to partially or completely rectify the disruption. The implementation of these rectification strategies can be coordinated by this agent. All the above steps are necessary for disruptions whose root cause is not observable. In cases where the root cause is observable, the occurrence of abnormal event triggers a comparison with KP|s. If corrective actions are found necessary steps 3 to 6 described above are implemented.
988
2.1 Methodology for rescheduling Our rescheduling approach is described in Figure 2. Implementation of the optimal rectification strategy may require changes which affect the original schedule, i.e. some operations may be cancelled or rescheduled and new operations may be necessary. These changes are reconciled with the original schedule by the scheduler agent. New schedules are generated based on heuristics taking into account relevant data from the plant hardware model. The evaluator calculates the profit of each new schedule so that the best one can be implemented.
Scheduler A~ent
n
Optimal Rectification
; ~
Strategy
1
blew schedules
:
i
'--
EvalUator
:]*-- [: :Heuristic: ~ : ] ............. ::::::::::: . . . . . . [: R e s c h e d u l e r : "~-
(.~riginal.~_~_~ ........ Schedul~~ ~ " Sales(Productiont a r g e t s ) / |
[
• Procurement (Crude
j/~/Plant [
procurementdata)
/
[.. • LogistiCs (Ships arrival)
J
"
i ,.,,o
Hardware ''" ..... , Model ""~:,
.........,. - ~ p p e ~ ~
~ p ~ °
i
".. transter rates etc,)
Figure 2. Scheduler agent of the disruption system
The rescheduler uses a heuristic multi-step block preservation strategy. An operation spanning one or more periods is considered a block if it involves no intervening change in configuration. As a corollary, adjacent blocks are separated by a change in configuration. Our approach seeks to minimize changes to blocks. First, feasibility of the disrupted schedule is checked. Second, a heuristic rescheduling strategy is employed to improve optimality of the disrupted schedule for each type of disruption. Five types of disruption have been considered: ship arrival delay, SBM/jetty unavailability, tank unavailability, equipment unavailability, and demand change. To deal with ship arrival delays, relative positions of the blocks of operation as mapped by the original schedule are maintained while adjusting the lengths of the blocks and the volumes involved in the blocks. In response to unavailability of equipment (e.g. pumps, tanks, CDUs, etc.), alternate processing strategies that can retain the blocks of operation from the original schedule are sought. To handle changes in demands, the relative positions of the blocks are kept while adjusting the volume involved in the blocks. The key here is to preserve the characteristic of the original schedule, which is the map of the blocks of operations scheduled. Finally, the objective value of the new schedule is evaluated.
3. Case Study We have implemented the above framework as a decision support system in Gensym's G2 expert system shell. In this section, we illustrate it using a case study derived from the supply chain of a Singapore refinery. Consider the scenario where there is a sudden increase in demand starting from the 41 st day. Operation therefore increases throughput
989 immediately. As a consequence, the KPI monitor finds that the crude oil inventory for the 42 ~d day has gone low and generates an alarm. The root cause identifier finds out the possible causes of the deviation: raw material delay, high demand of products, or the efficiency of process decreases such that more crude oil is required. Comparison of the actual and planned ship arrival indicates that there is no arrival delay. Similarly, a check with the operation department reveals that there is no problem with the efficiency of process. Further check with the sales department indicates that the demand is higher than previously projected by 330 kbbl. The diagnosis agent thus confirms the root cause for the low inventory alarm to be a new order on the 41 ~t day. The rectification proposer agent finds two possible rectifications: 1) postponement of other orders, and 2) emergency crude procurement. The optimizer agent evaluates the two options and flags emergency crude procurement of 330 kbbl to arrive on the 44 th day as the optimal rectification strategy. The flow of events in this case study is shown in Figure 3. The optimal rectification strategy is then passed to the scheduler agent who reschedules the operation as shown in Figure 4 and 5. The new schedule (Figure 5) includes the emergency crude and deals with the increase in demand by rescheduling operations. ...-.
['~1 " M°nit°ring~ I ow Inventory Deteot-~ L 1
~..
l 4" EmergencY ardval Crude
..."-.. ... /
"--.. \ \
", I
.. \
Day I~:--...,. __ I 41 "q.i:ii_-- ......... - - - ~ 4 _ e ,............ 2 Diagnosis .....: . . . . . '[,... High demand ,' .............
, I 44 /~.~Optimal Rectification"?'--.. 43 _----~-------------__
[ \
Emergency Crude
,]
,%. Procurement-Passed
.,M/
"--._._
to Scheduler _ _ _ j ~ -
Figure 3. Eventjlow./br the case study I
I
,i
c:~......... + :::; > ~
n
4
i
An-~ount ~
......."I ........ l
'i
. . . . . . . . . .
t
]
.........................................
A~"~ount
t::!~
L-£-;i;!:~;:~:!t 'i :::~,,i 3 i,.-; charging :':::~i.::~kbbl to C.i:::::;i..i2 on day ~::[i
Amou~'~t I
. ".
........ "
-----~J ,~_/v
. . .[ . :i~-~i:s is r.,:.i,,.i.~ :.;:i::ii::ikb~, f~o~ v~!!~:i~=!~:~,:i!:~.ion ,W 4S I <~;" Figure 4. Original operation schedule
990
[,,,,Resche,d,ul,ed operations ] Ta~,k: .
41 .Amo~n,.
42 Am'~'[],,,
Z, ~gVL~ ~~~~ 4~.
~,~:=0
4~
4S
L~S
Amoor, l
4~
A.t',~r,t
:f .... [>
~~%~i .%
Arno,Jn,
4:8
4g
Amour, t Amo,jnl
#~.............................. .,4
2
3
-so ul :-7o u~ J d " ~ 3 ~ : E ~ I d 6 % ] * - 7 o
:~:
iU:2::~::::O2:
u~ ,7:o u1-7o u~
2 ~ :P4:'
I Emergency Crude (EC) - prevents stock-out Figure 5. New operation schedule including emergency crude
4. Conclusions Disruptions are common in supply chains. In this paper, we proposed a systematic framework for disruption management. One key advantage of the proposed approach is that it can handle a wide variety of disruptions, whether due to supplier, transport, operation, or customer problems. The success of this method is predicated on the accuracy of the supply chain model. Our current work is directed towards improving robustness of the approach.
References Julka, N., R. Srinivasan and I. Karimi, 2002, Agent-based Supply Chain Management-l: Framework, Comput. Chem. Eng. 26(12), 1755-1769. Julka, N., R. Srinivasan and I. Karimi, 2002, Agent-based Supply Chain Management-2: A Refinery Application, Cornput. Chem. Eng. 26(12), 1771-1781. Mishra, M., R. Srinivasan, and I. A. Karimi, 2003, Managing disruptions in refinery supply chain using an agent-based decision support system, Presented in the AIChE annual meeting, San Francisco, Nov 16-21. Sheffi, Y., J.B. Rice, Jr., J.M. Fleck, and F. Caniato, 2003, Supply Chain Response to Global Terrorism: A Situation Scan, Eur OMA-POMS Conference. Wilson, M.C., 2002, Transportation Disruptions in the Supply Chain: Simulator as a Decision Support Tool, Proceedings of the 31 st Annual Logistics Educators Conference.
Acknowledgements This research work is funded by NUS and TLIAP.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) ,<32005 Elsevier B.V. All rights reserved.
991
Novel Scheduling of A Mixed Batch/Continuous Sugar Milling Plant Using Petri nets M. Ghaeli a, P.A. Bahri b*, P. Lee a aDivision of Engineering, Science and Computing, Curtin University Bentley, WA 6845, Australia bSchool of Engineering and Science, Murdoch University Murdoch, WA 6150, Australia
Abstract Scheduling of processes in mixed batch/continuous plants, due to their hybrid nature can become very complex. This paper presents the Timed Hybrid Petri net (THPN) as a suitable tool for modelling and scheduling of hybrid systems. One of the major benefits over traditional methods is a significant reduction in complexity during problem formulation. A sugar milling plant containing both batch and continuous processing units is considered as a case study by which the application of the proposed scheduling algorithm is illustrated.
Keywords: scheduling, mixed batch/continuous, Petri net 1. Introduction A scheduling method should formulate a problem by considering all the associated constraints to find the best sequence of operations that optimises a specific objective function. Although much research has been done on the development of different methods for scheduling of either continuous or batch chemical plants, there is a distinct absence of research in the scheduling of mixed batch/continuous systems (hybrid systems). In the past, scheduling of these plants has been accomplished by either discretising time or considering a continuous time model (Nott, 1998). Simulation models have often been used to schedule these plants. Djavdan relied on such a model with an emphasis on the importance of the size of intermediate storage (Djavdan, 1993). Many researchers have acknowledged that the scheduling of mixed-batch/continuous plants is a very important issue (Neville et al., 1982; Sicignanoet al., 1984) but still, a concrete method for solving these types of scheduling problems with less computational complexity remains. The Timed Petri net based formulation has proven to be a promising technique to solve many scheduling problems (Ghaeli et al., 2004). This paper considers a timed Hybrid Petri net (THPN), which is a class of Petri net, as a suitable tool for modelling and scheduling of mixed batch/continuous plants. A substantial reduction in the complexity of the problem formulations has also been achieved. This is because the THPN tool Author to whom correspondence should be addressed:
[email protected]
992 allows for effective illustration of the batch and continuous operations and storage between continuous units with the associated capacities and all other constraints. Initially, an introduction to THPN is presented. Then, in section 3 the model of a sugar milling case study, which is an example of a mixed batch/continuous plant is developed and the proposed approach applied. Finally, a brief conclusion is presented.
2. Timed Hybrid Petri net A Petri net is a particular kind of bipartite directed graph populated by three types of objects: places, transitions and directed arcs. The dynamics of the model are represented by the movement of tokens, which are located in places and illustrated by either small dots or a real number. The firing rules of transitions in a Petri net model make the consideration of the resource use, storage policy and product sequence decisions. The graphical and Mathematical properties of Petri net are also useful when it is used to model hybrid plants. Hybrid Petri net (HPN) is a class of Petri net made up of a "continuous part" (continuous places and transitions) and a "discrete part" (discrete places and transitions). Figure 1 shows the graphical notations for places and transitions in HPN as used in this paper.
Discrete Place
Discrete transition
O
II
Continuous Place
I" I
Continuous Transition
Figure 1. The representation o f places and transitions in H P N
Formally, HPN is a six-tuple H P N - { P , T ,
p r e , p o s t , m o , h } (David and Alla, 2001)
where P = {P1,P2 ,.-., f',, } and T = {T~,T2,..., Tn} are finite but not empty set of places and transitions respectively; p r e : P
× T - - ~ R + is the input incidence matrix while
p o s t : P x T - ~ R ÷ is the output incidence matrix. Note that P ~ T = ~b; m ° : P - ~ R +is
the initial marking. An incidence matrix is a matrix, which includes the weights of the arcs connecting places to transitions or transitions to places. A node is defined to be either discrete (D) or continuous (C) by the hybrid function: h ' P c~ T ~ {D, C} ; °P~( ° T ) and ~° (T; °) depict the set of input transitions (places) and the set of output transitions (places) respectively. The state equation is the same in a hybrid Petri net m = m o + U . V and the difference is that in HPN an integer value in vector V is related to the number of firings of a discrete transition while a nonnegative real number corresponds to a firing quantity of a continuous transition. To consider quantitative properties in a THPN, time is added to HPN. Based on how the places and transitions are defined in a THPN, time can be introduced to either places or transitions.
993 3. C a s e S t u d y
To show the details of the proposed method, a sugar milling case study (Nott, 1998) is considered. Sugar Milling is a hybrid system, which can be divided into two parts: the high-grade and the low-grade systems. The flowsheet for the sugar mill is given in Figure 2. Boxes with a solid outline represent batch operations, whereas continuous operations are represented with a broken outline. Triangular shapes represent storage facilities. The input and output of each pan are shown in the flowsheet. As the use and production of water does not have any effect on the scheduling, it is not considered in the flowsheet. The processing time of each pan is depicted in the parenthesis next to each pan in the flowsheet. The main assumption is that processing of fugal A and fugal B can not be done simultaneously. The maximum flowrate of super, csystems and dryer are 16.5, 32.5, 22 tonnes per half hour, respectively. The syrup flow into the system is 106 tonnes per hour. The maximum capacity as well as the initial amount for each storage are given in Table 1.
Hi(5)
]25 i
12~1 3
j ~ 8o
~
0
Pan H2 (6) I I Pan H4 (5)
I
"•
?,
3O >-] P~' H3 (6) I 200;
pan H5 (6) I -
~
IPanL3 (8)
44.221
80160
I
200¢
34.84 671
v
fu,,al ~(0 ~)
.
67
fugalB (0.~)
36.t3° I ° " -i
~
I ;
\
, Csystems ', , _ ~__.~. . . .
r .......
1
,
[ ~"J'Sr__i
//
......... ,
Figure 2. The flowsheet of the sugar milling case study Table 1. Storage Specoqcations (tonnes) Syrup
Magma
AMolasses
BMolasses
Ree3
Rec5
LRee
Wetsugar
Max
410
50
310
200
330
330
230
250
Initial value
1O0
20
140
110
200
200
70
25
'
994 In order to solve the scheduling problem of this case study, the model should first be created using THPN.
3.1 Model of the case Study The THPN model of the sugar mill system is depicted in Figure 3. In this model, it is assumed that there is a possible change of flowrates for the continuous processing units every half hour. While discrete transitions are used to show the batch operations, continuous units are represented by continuous transitions. In addition, discrete places between the batch operations show the availability of the materials in the previous batch processing unit and the continuous places illustrate the storages. The major benefit of the proposed model is the formulation part of the problem, which is much less complicated than traditional methods. Applying THPN for modelling and scheduling of hybrid systems, the formulation of the important constraints such as assignment constraints, material balance constraints and resource constraints will become much simpler as discussed below: • Assignment constraints: consideration of places having one token as an input to a discrete transition enforces the constraint that there should not be more than one task processed in a processing unit at any time. • Material balances constraints: the amount of inputs and outputs of a processing unit are considered as weights located on arcs connecting a transition (place) to a place (transition). • Resource constraints: for the storages with a limited capacity, a place is connected as an input to a transition (operation unit) producing the products to be stored in the associated storage. The arc connecting this place to the transition has a weight equal to the amount of the resulting products. The same amount is released as soon as the materials leave the associated processing unit. It should be noted that all the above constraints have been considered in the firing rules of transitions and there is no need to set any time consuming formulations or variables. The objective function of the system is to find a schedule with the maximum product (profit) or equivalently minimum idle time (A period during which a processing unit is not in use, but is available), within a horizon time of 16 hours. There is also a penalty of 0.1 for changes in the flowrates of continuous units. 3.2 Proposed algorithm In the proposed approach, there are two steps. First, based on some initial values for the flowrates of continuous units, the scheduling algorithm will find the optimal schedule with respect to the objective function. Second, the optimal schedule obtained is passed to the CFSQP optimisation algorithm (Lawrence et al., 1997). CFSQP will find the best flowrates for each continuous unit, which maximises the products (sugar and Cmolasses) within the obtained optimal schedule. It should be noted that before applying the proposed algorithm, the horizon time is divided into half hourly periods. This will allow the change in the flowrate of each continuous unit every half hour. The following shows the steps of the scheduling algorithm" SI" Begin with the initial marking and a large value for the lower_idle_time; set the flowrates of the continuous units to some initial values. $2" With the initial marking, check if the current transition is not enabled due to the lack of materials or space in the storage, then consider the associated idle time.
995 $3" With the initial marking, check if the current transition is enabled; determine the new marking and the usage time of each processing unit including its idle time and put them all with the old markings into one result matrix. $4: If there are more enabled transitions with the current initial marking, go to $2. $5: Check the last two rows of the result matrix and perform merging if the related transitions are not in conflict; in case of conflict check if the shared input place has enough markings (weights) to fire these two transitions simultaneously. If so perform the merging and repeat $5 for the next two rows. $6: Check the time of all the batch processing units and if there is at least one with the time of less than the horizon time go to $2. $7: If the latest total idle time of the batch processing units is less than the lower_idle_time update loweridle_time to this time. $8: If all the rows of the result matrix have been assessed, the search is complete. Output the lower_idle_time and the feasible schedule to this lower_idle_time. $9: Start checking from the last row of the result matrix upward and find the row in which the associated transition is enabled and has not been fired yet" set the marking of this row as the initial marking and go to $2. The scheduling algorithm was implemented in the C programming language and the optimal solution yields the Gantt chart shown in Figure 4. As was mentioned previously, based on the optimal schedule, CFSQP gives the best flowrate per half hour for each continuous processing unit (Table 2).
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Figure 3. Timed Hybrid Petri net (THPN) model of the sugar milling case study
996 Applying the proposed method to solve the sugar mill case study reduces the computational time from 140 seconds in the previous method (Nott, 1998), which used a sets implementation technique, to 1 second in the current study. This confirms the power of THPN for modelling and scheduling of hybrid systems. Units
12 h
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Figure 4. Gantt chart of sugar milling case study Table 2. Optimal flowrates (tonnes/half hour)
Time 0-2 hours 2-14 hours 14-16 hours
fl (super) 2.04375 2.06875 2.04375
f2 (dryer) 7.0125 7.0375 7.0125
f3 (Csystem) 2.16875 2.19375 2.16875
4. Conclusions The THPN has been introduced as a suitable tool for systems with both discrete and continuous behaviour in which many of the constraints can be shown graphically. The great potential of this model for handling complicated operations in the mixed batch/continuous processes has been illustrated through a sugar milling case study, which is an example of a hybrid system. An algorithm, which is a mixture of scheduling and optimisation, is proposed to solve the scheduling problem of the case study. A substantial reduction in the computation time is achieved thereof. Division of the horizon time into a smaller period is currently being researched.
References David, R and H. Alla, 2001, On Hybrid Petri net, Discrete Event Dynamic Systems: Theory and Application, vol. 11, pp.9-40. Djavdan, P., 1993, Design an on-line scheduling strategy for a combined batch/continuous plant using simulation, Computers and Chemical Engineering, vol. 17, pp.561-567. Ghaeli, M., P.A. Bahri, P. Lee and T. Gu, 2004, Petri-Net Based Formulation and Algorithm for Short Term Scheduling of Batch Plants, Computers and Chemical Engineering, Accepted. Lawrence, C., J.L. Zhou and A.L. Tits, 1997, User's Guide for CFSQP Version 2.5: A C Code for Sloving (Large Scale) Constrained Nonlinear Satisfying All Inequality Constraints. Neville, J.M., R.Ventker and T.E. Baker, 1982, An interactive process scheduling system, the American Institute of Chemical Engineers. Nott, H.P., 1998, Modelling alternatives for scheduling mixed batch/continuous process plants with variable cycle time, Doctor of Philosophy Thesis, Murdoch University. Sicignano, A., J.D. McKeand, S.F. LeMasters, 1984, IBM Pc Schedules batch processes, cuts inventories at Houston refinery, Oil and Gas Journal, pp.63-67.
European Symposiumon Computer Aided Process Engineering- 15 L. Puigjaner and A. Espufia (Editors) © 2005 Elsevier B.V. All rights reserved.
997
Improving short-term planning by incorporating scheduling consequences Petra Heijnen*, Ivo Bouwmans, Zofia Verwater-Lukszo Faculty of Technology, Policy and Management Delft University of Technology Delft, The Netherlands
Abstract Decisions in short-term planning and scheduling in multi-product multi-purpose plants are often taken in isolation from each other and may be based on conflicting objectives. This may result in decisions that are sub-optimal for the plant as a whole. A more integral perspective on the decision-making could lead to a better overall performance. This paper describes a mathematical reformulation of the short-term planning and scheduling decisions in a bi-level top-down program. It is based on a recursive formula and a smart definition of the cutting rules, which greatly reduces the complexity of the optimization. A practical case shows that the reliability of the planning results increases significantly.
Keywords: Bi-level programming, short-term planning, scheduling, optimisation, decision support
1. Introduction In a multi-product multi-purpose plant, decisions are usually taken at different locations and on different hierarchical levels, too often in isolation from each other (Shobrys et al. 2002). Together, however, these decisions influence the overall performance of the plant to a high extent. Ryu, Dua and Pistikopoulos (2003) address plant-wide planning and network decisions. This paper aims at the planning and scheduling level. Planners as well as schedulers can only decide about their own decision factors and have no control over the others, but the decisions can mutually interfere, causing far from optimal results. This happens in particular when the respective decisions are based on conflicting objectives. Therefore, a more integral perspective on short-term planning and scheduling decisionmaking in the plant is desirable to guarantee a better overall performance. Multi-level programming is a specific way of mathematically formulating such multi-level decision problems in a complex system, by incorporating the consequences of one decision on the other, aiming at a better performance of the system as a whole (e.g. Bialas et al. 1982).
Author to whom correspondence should be addressed:
[email protected]
998 In this paper, the short-term planning and scheduling decisions in a multi-product plant, aiming at different, conflicting objectives, are mathematically reformulated in a bi-level top-down program. The method is based on a recursive formula by which the scheduling objective function can be calculated. The use of this recursive formula and a smart definition of the cutting rules significantly reduce the complexity of the search for an optimal planning.
2. Short-term planning and scheduling in a multi-product plant In the short-term planning, customer orders are accepted or rejected based on a chosen criterion, which in our example is profitability. At the start of each week, the planners receive the set of customer orders from the sales department. Every customer order indicates the amount of product, the date the order should be delivered and the type of product. Since capacity is limited, only the most profitable set of orders from the order list is selected and passed on to the schedulers. In the scheduling, the orders accepted in the planning stage are sequenced minimizing the total tardiness, i.e. the total time of delivery after due date. To reduce complexity, the assumption is made here that all orders are to be produced in one reactor. The scheduling performance highly depends on the planning decision, since the schedulers have to schedule all the orders selected by the planners, whereas another selection of orders might have led to a better schedule with less tardiness. Moreover, if in the final schedule two consecutive orders are of different types of product, extra changeover costs are made, reducing the profit. The planners, however, only estimate the profit by correcting every order with a fixed sum of changeover costs, disregarding the real consequences of the scheduling process. The schedulers, in turn, have no influence on the order selection procedure. The problem situation described here might seem a little artificial, since planners and schedulers, working for the same company, should aim at a better overall performance and not only at their own objectives. Nevertheless, isolated decision making is not uncommon in practice for various reasons, such as habits, the complexity of integrated decision making, or the impossibility of unambiguously defining the influence of lowerlevel activities on strategic objectives.
2.1 Mathematical formulation of the planning and scheduling decisions Each week, the planners solve the following decision problem:
From the total set of customer orders, find the subset of orders that maximize the profit and for which the total available capacity is not exceeded. Let
C=[[cl,,di,Pi ] i E{1,2 ..... m}]
(1)
be the total set of customer orders received from the sales department, where ag is the ordered amount of product (in tons), di the due date (in working hours from the start of the week), and pi the type of product ordered. The planners should then find the binary vector x - [xl, x2, ..., Xm] for which the total profit
999 tn
P(x)-~-~(aif(pi)-c)x,
(2)
i=l
is maximized. Here, J(pi) is the profit that can be achieved with one ton of product of type p;. The decision vector x_is bounded by the constraint
~ a , t ( p i )x, <_T,
(3)
i 1
in which t(pi) is the time (in hours) needed to produce one ton of product of type p;. Also weekly, the schedulers solve the following optimization problem:
For the selected orders received fi'om the planners, find the production sequence that minimizes total tardiness. Let S(x) be the set of n selected orders received from the planners,
.... ,m}]
<4>
The schedulers should find the permutation.~ = HS(x) for which the total tardiness
T(y)-~max(O,c/(y)-d/),
(5)
in which d/is the due date of" the/-th order in £ and ./
c;
lp, 1
<6>
l=1
is the completion time (in working hours from the start of the week) of orderj in schedule ~ with Pl the type of product of the/-th order in X-
3. Solving the planning and scheduling problem The planning and the scheduling decision can be solved separately by a branch-andbound algorithm, first proposed by Land and Doig (1960). However, better results can be achieved when the consequences of the planning results on the scheduling level are incorporated in the planning process. Therefore, the planning and scheduling decisions are reformulated into a bi-level top-down decision-making problem here, assuming that the decision makers on the planning level have complete knowledge about the optimization problem solved on the scheduling level.
3.1 Bi-level programming In a bi-level program, two levels are distinguished where closely related decisions are taken, and where the decision makers at one level can only decide about a limited set of decision variables and have no power on the decision variables of the other level. A top-down program is a specific form of a bi-level program in which • the decisions on the upper level determine the parameter values on the lower level, constraining the feasible region of the lower-level solution space; ° the decision on the lower level influences the degree to which the objective on the upper level is achieved; • the decision on the upper level is taken before the decision on the lower level.
1000 The objectives on both levels are generally conflicting in that the optimal values of the decision variables of either level influence the solution of the other level negatively. As a consequence, to determine the best performance of the overall system the different optimisation problems of both levels should be integrated and solved jointly. However, in general this is unfeasible. The usual strategy to solve bi-level optimisation problems is to incorporate the resulting decisions of the lower level into the upper level decision and solve the optimisation problems of the lower level iteratively. This is done under the assumption that the decision makers on the upper level have complete knowledge about the objective functions and constraints of the lower level.
3.2 Planning and scheduling bi-level program For the decisions on the planning level, a better performance of the system as a whole can be achieved when the consequences of the planning on the scheduling level are incorporated in advance. On the scheduling level, the orders selected on the planning level are sequenced to minimize total tardiness (Pinedo, 1995). However, every changeover between different types of products in the chosen schedule will reduce the profit. Although the planners cannot decide about the final production schedule, they can expect that within the limited solution space caused by the planners' selection of orders, the schedulers will choose the optimal schedule with respect to their own objective of minimal tardiness. The planning decision problem is therefore reformulated to incorporate scheduling results: II Find the binary vector X=[Xl,X2,...,Xm]for which the total profit m
P(x__)-Z aif (Pi)xi-qC(L)
(7)
i=1
is maximized, in which C(X*) is the number of changeovers in the optimal schedule y*, belonging to the planned orders, and q is the fixed costs for one changeover. As before, the decision vector x_ is bounded by the constraint (1.3). The formulation of the scheduling problem remains the same.
3.3 Top-down solving of the planning and scheduling problems The new top-down formulation of the planning and scheduling decisions makes solving of the planning problem much more complex, since for the evaluation of the profit expected from a planning, the scheduling problem must be solved as well. The planning problem is solved by a branch-and-bound algorithm. The complexity of a branch-and-bound search strongly depends on the efficiency of cutting branches from the solution tree. For the planning problem, the tree starts with an empty planning, [ ]. Every node in the tree, corresponding with a binary vector xl of length smaller than or equal to m, is split into two branches, indicating that the next order is added to the planning, resulting in node [xl, 1], or that it is not, resulting in node [xl,0]. A branch of the planning tree can be excluded from further analysis if either • the total completion time of the orders selected so far is larger than the available capacity T, or
1001 •
the potential profit, i.e. the profit obtained with the selected orders plus the total profit of all orders not considered so far, is smaller than a certain lower bound. However, for the second cutting rule the maximum potential profit of an incomplete planning in a certain node can only be determined by solving the scheduling problem for every complete planning branching from that node. This procedure will be far from efficient. Under the assumption that the profit achieved from a customer order will always be more than the costs for one extra changeover, q, the maximum profit achievable in a planning that branches from a certain node corresponding with x~ is at most
N([xl,l,1,...,1])-l) is the number of different product types in the planned orders corresponding with xl and in the remaining orders. Since N([xl,l,1,...,1])-l) is a lower bound for the total number of changeovers in a schedule belonging to the planning [x~,l,1 .... ,1], Equation (1.8) will yield a lower bound for the potential profit of an incomplete planning x~. By using this lower bound for the profit, the scheduling problem only needs to be solved when a complete planning is achieved in one of the leaves of the planning tree. The actual profit can then be computed taking into account the real changeover costs that result from the optimal schedule for that planning. Still the complexity of the planning algorithm is high and the reuse of previous scheduling results is not that simple. Let schedule I and schedule II be the optimal schedules belonging to planning I and planning II, respectively. Even if planning I consists of the orders of planning II plus just one order extra, the orders in schedule I and schedule II may have a totally different sequence. The complexity of calculating the tardiness of a schedule, however, can be reduced significantly by using the following recursive formula. Let S(x) be the set of selected orders in a certain planning, represented by the binary vector x. Let S~ be an ordered subset of S(x) and let S_~ be the complement of S~, $2 = S(x_)\S~. Let T(S2,S~) be the tardiness of the orders in S~, when the orders in $2 are completed before the orders in S~ can start. Then
T(S2 \{kI,[k,S,l)-T(S2,Xl)+max(O,C(S2)-dk)
(9)
in which C($2) is the total completion time of all orders in set S 2. With this recursive formula, the planners can efficiently optimize the set of orders, which results in a better solution for the plant as a whole. The example in the next section illustrates this.
4. An illustrative example Assume that for the coming week a company has received the fourteen customer orders given in table 1. In this example the extra costs for one changeover amount to (~ 3000. The total available production time equals 90 h. The due date indicates the hour within this available time at which the order should be finished.
1002
Table 1. Customer orders for the following week. Order number Amount(t) Due date(h) Product type
1 85 52 5
2 55 26 5
3 58 58 2
4 50 25 2
5 44 83 4
6 36 56 2
7 31 70 4
8 29 61 4
9 28 75 4
10 26 41 2
11 21 27 5
12 19 32 5
13 16 48 3
14 15 40 4
If the planning and scheduling activities are executed independently, the following results are obtained. The optimal planning without incorporating the scheduling results is S = {3,4,6,7,10,13,14}, with an estimated total profit off~ 318,048. The optimal scheduling of the planned orders, minimizing total tardiness, is .2" = [4,14,10,13,6,7,3] (total tardiness 32 h). This schedule requires six changeovers and the actual profit is f~ 321,048. Incorporating the scheduling results in advance results in a higher profit, with S = {3,4,6,7,9,10}. The optimal scheduling for this planning, with a tardiness of only 19.2 h, is j = [4,10,6,3,7,9]. Implementation of this planning and schedule algorithm, including the required changeover, results in a total profit off~ 332,188. 5. C o n c l u s i o n s
The use of a bi-level top-down optimization algorithm significantly enhances the reliability of the planning results, in that the predicted profit is closer to the actual one that is obtained after the scheduling process. In addition, the set of orders handed over to the schedulers by the planners has the potential to yield a result that is better for the plant as a whole, in contrast with the sub-optimal result that can be expected when the planning and the scheduling are optimized separately. The mathematical reformulation that is presented in this paper effectively reduces the computational complexity. Using this process, that can be completed within reasonable time, the planners are able to anticipate the decision of the schedulers. References Bialas Wayne F., Mark H. Karwan, 1982, On two-level optimization, IEEE transactions on automatic control, Vol. AC-27, No. 1. Land A. H., A. G. Doig, 1960, An Automatic Method for Solving Discrete Programming Problems, Econometrica, Vol.28, pp. 497-520. Pinedo M., 1995, Scheduling: Theory, Algorithms and Systems, Prentice Hall. Ryu, Jun-Hyung, Vivek Dua, Efstratios N. Pistikopoulos, 2003, A bilevel programming framework for enterprise-wide process networks under uncertainty, Computers & Chemical Engineering. Shobrys, Donald E., Douglas C. White, 2002, Planning, scheduling and control systems: why cannot they work together. Computers & Chemical Engineering 26, 149-160. Acknowledgements The research is part of the program Next Generation Infrastructures. For more information: www.nginfra.nl
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1003
Multi-scale Planning and Scheduling in The Pharmaceutical Industry Hlynur Stefansson a and Nilay Shah a* aCentre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London SW7 2AZ. United Kingdom.
Abstract Most sophisticated planning and scheduling approaches for the process industry consider a fixed time horizon and assume that all data is given at the time of application. In this contribution we propose a planning and scheduling approach for a continuous and dynamic decision process where decisions have to be made before all data are available. As an inspiration we have a real world problem originating from a complex pharmaceutical production plant. The approach we propose is based on a hierarchically structured moving horizon framework. At each level we propose optimisation models to provide support for the relevant decisions. The levels are diverse regarding time scope, aggregation, update rate and availability of data at the time applied. The framework receives input data piece by piece and has to make decisions with only a partial knowledge of the required input. Solution procedures have been developed and the optimisation models have been tested with data from the real world problem. The solution procedures were able to obtain solutions of good quality within acceptable computational times.
Keywords: Production planning, Scheduling, Integrated multi-scale approach, Realtime optimisation, MIP
1. Introduction The pharmaceutical industry has become a very competitive and unpredictable industry where customers constantly demand low prices as well as high service levels and flexibility. Flexible multi-product production processes have become commonly used as they help companies to respond to changing customer demand and increase plant utilisation, but the greater complexity of these processes together with the altered market conditions have rendered the simple planning and scheduling techniques previously used insufficient. It is thus very important to improve production plans and schedules in order to strive for superior utilisation of resources, flexibility and response time at the same time as cutting down cost of production. There is an extensive body of literature dealing with planning and scheduling problems in the process industry. Recent reviews of planning and scheduling problems can e.g. be found in (Shah, 1999, Pinto and Grossmann, 1998, Shah, 2004, Kallrath, 2002).
Author to whom correspondence should be addressed:
[email protected]
1004 However most planning and scheduling approaches that can be found in the literature for the process industry consider a fixed time horizon and assume that all data is given or subjected to given stochastic uncertainties at the time of application. In this paper we propose a planning and scheduling approach for a continuous and dynamic decision process where decisions have to be made before all data are available. This has been motivated by a real world problem originating from a complex pharmaceutical enterprise.
2. Problem Description We focus on the single plant production planning and scheduling for a secondary pharmaceutical production facility with order-driven multistage, multi-product flowshop production. The plant consists of a large number of multi-purpose production equipment items on each production stage, operated in batch mode. When switching between batches containing products from different product families large sequence dependent set-up and cleaning times are required. These times are much lower when switching between batches containing different products within the same product family. The plant therefore uses campaign production where a campaign is an ordered set of batches containing products from the same product family produced consecutively by the same machine. Each product has a number of different feasible production routes through the plant. The overall goal of the problem is to determine campaign plan and to schedule customer orders within the campaigns. The customers request certain delivery dates for their orders and the production attempts to meet those requests.
3. Modelling Approach 3.1 Online and Dynamic Characteristics of The Problem
In order to develop optimization models and an overall approach that is realistic and can be applied in actual circumstances it is necessary to acknowledge and cope with the continuous dynamic decision process and the online characteristics of the problem we are considering. Most planning and scheduling approaches that can be found in the literature for the process industry can be classified as "offiine" approaches where the ideal case is assumed with all data given or subjected to a given stochastic uncertainties and a fixed time horizon is considered (Kallrath, 2002). A review of heuristics for theoretically oriented online scheduling problems can be found in (Sgall, 1998). Some practical online scheduling approaches are also described in the literature and for an example (Sand et al., 2000, Sand and Engell, 2004) describe online scheduling algorithms from a real world multi-product batch plant producing expandable polystyrene where demand as well as the production process is subjected to stochastic changes. An interesting telescopic decomposition approach with a number of layered sub-models of different degrees of temporal aggregation is suggested for solving online scheduling problems in multi-product batch plants by Engell et al. (2001). Order driven production planning and scheduling in the pharmaceutical industry is a critical example of a continuous and dynamic online optimisation problem. The production is an ongoing process that is affected by several uncertain inputs where the
1005 most important one is the demand from customers. The first task of the planning process is to create a campaign plan for long term planning purposes such as raw material procurement as the lead time is often very long. The campaign structure has to be decided quite a lot earlier than the orders become available and demand predictions are used instead of the actual orders. Then later the plant receives new customer orders each week with a requested delivery date and the orders need to be scheduled and feedback given to customers with a confirmed delivery date. When the orders are scheduled within the campaigns the information about other orders that will later be added to the same campaigns are not available. The decision maker does not have access to the whole input instance at the time decisions need to be made and instead he or she sees the input piece by piece and has to make new decisions with only a partial knowledge of the input. Once decisions have been made it can be difficult to change them later on. As more information become available the uncertainty of the decision problem decreases and the planning horizon is continuously moving as production of older orders is finished and the plan is built further ahead, step by step. 3.2 Hierarchically Structured Framework To cope with the continuous dynamic decision process we propose a multi-scale hierarchically structured framework. On each level of the hierarchically structured framework we propose optimisation models to provide support for the relevant accisions. The different levels are diverse regarding the scope and the availability of information at the time when they are applied. Aggregation / Available information ~ n Lev
Uncertainty /
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Figure 1. The.figure shows the hierarchical ~'amework drawn together with the representation of uncertainty and availahiliO: ((information with regard to time This is a continuously moving framework updated daily as the nature of the actual decision problem and the models are used with a different frequency depending on their level of aggregation and the update ;ate of their input factors. The most detailed decisions are made as late as possible when more information is available and the uncertainty has decreased.
3.3 Levels in The Framework 3.3.1 Campaign Planning at Level 1 At the top level in the hierarchical framework we propose an aggregated optimisation model to optimise the long-term campaign plan with the objective to fulfil demand and minimize cost of production. The input for this model is mainly based on a combination
1006 of sales forecasts and long-term customer orders as well as information regarding products, production process, performance and current status of production. The output from the model is a campaign plan that considers all machines on all production stages. The time horizon of this model is 12 months and it is used every 3 months or more often if new sales forecasts or other aggregated information becomes available. We use mathematical programming techniques to formulate the model and propose a novel and efficient mixed integer linear program with a discrete representation of time.
3.3.2 Simultaneous Campaign Planning and Order Scheduling at Level 2 At the middle level we propose an optimisation model to optimise campaign plan simultaneously with optimising allocation of orders and other production tasks, within the campaigns. The campaign plan is now based on more accurate input information compared to the earlier level as the customer orders are more or less available here. As output from this model we receive a revised campaign plan and a production schedule that specifies in which campaign each order is produced on every production stage and it also specifies the delivery date of the order. After the delivery dates have been confirmed it is very desirable to respect them although the production schedule can be changed in many other ways. The time horizon for this level is 3 months and the optimisation model is used once every week. For this level we also propose a novel and efficient mixed integer linear program with a discrete representation of time. 3.3.3 Detailed Scheduling at Level 3 At the lowest level in the hierarchical framework we propose an optimisation model for the detailed scheduling of production tasks. The optimisation is based on the confirmed customer orders together with the newest possible information each time it is used and the resulting production schedule specifies in which campaign each order is produced on each production stage, the production sequence of the orders, its precise starting time and duration and the setup tasks required between orders. The model accounts for sequence dependent setup times between the campaigns as well as between the orders within the campaigns. The time horizon is one month and the model should be used every day if some new information has become available that affects the feasibility or optimality of the last production schedule. For this level we propose a model unlike the two preceding ones regarding the representation of time as the model is based on a continuous time representation similar to that of Mendez et al. (2001) and the starting times of events are not restricted to the resolution of a time grid and can therefore be very detailed.
3.4 Integration of Levels The different levels interact with a directed flow of information between them. Results from the higher level models are transferred and used as input for the lower level models. From the first level the campaign plan is transferred and used as input for the second level. The second level is however not required to strictly respect the campaign plan and is allowed to revise it if it results in an improved schedule. The campaign plan and confirmed delivery times are transferred to the third level together with allocation of orders in the campaigns which the third level model can change to improve the schedule and react to more current and detailed information. When the flow of information between hierarchically structured levels is directed it is very important to ensure that the results transferred from higher levels provide feasible
1007
input for lower levels, if that is ensured, a mono directional integration should be sufficient. However if not, we may need to transfer information from the lower levels and develop bi directional integration of levels through feedback loops.
4. Solution A p p r o a c h e s Planning and scheduling problems, in particular scheduling problems, are known to be very difficult to model and solve in an efficient manner (Kallrath, 2002). Our objective is to provide realistic and accurate models solvable within acceptable computational time but that is very difficult for complex and comprehensive real-world problems such as the one we are working with.
4. 1 Decomposition Algorithm After standard solution methods failed to provide solution to the problems at the middle and top level we developed and implemented decomposition heuristic. The decomposition heuristic is inspired from the production performance problems found in the case we are working with. When we started analysing production data and schedules from the plant, it was obvious that there were two bottlenecks in the overall production process, at the first production stage and at the last production stage. Because of the limited capacity at the first and the last production stage we decomposed the problem into two main components: first to solve the problem for the first and last stage and then to solve the problem for the intermediate stages. The two components are integrated with a minimum expected flow time for the intermediate stages and the flow time is adjusted and the models resolved iteratively if feasible solutions are not obtained at first attempt.
4.2 Lower bounds improved with Valid Inequalities Resource constrained scheduling models often suffer from poor LP relaxations and in some cases this has made them very difficult or even impossible to solve with MILP methods. Resource constraints can easily be fulfilled with fractional values of the binary variables used in the discrete time formulations and as a result the lower bounds become very weak (Kallrath, 2002). To improve the lower bounds for our models we added two kinds of valid inequalities, valid inequalities for the minimum number of campaigns required and for the minimum unavoidable delays, that is the delay of the requested or confirmed delivery dates of customer orders. We used two methods to estimate the minimum number of campaigns, one was a simple data analysis routine and the other was a more complicated heuristic method. To obtain the value for the minimum unavoidable delays we used a two step algorithm based on first solving a relaxed model and then using its solution to provide a valid lower bound for the delays and solved the complete model with the valid inequalities for the delays.
5. Results The models that we have developed and briefly described in this paper have been implemented and tested with real world data from the problem under consideration. By
1008 using data collected in the pharmaceutical plant we created several full scale test cases that reflect the decisions problems at each level in the hierarchical framework. Table 1. Results from solving real world test cases with the proposed models for each level. Performed on Unix based machines with 2GB RAM, 1.8 GHz processor and CPLEX 9.0 solver. Level 3 3 3 2 2 2 1
Number of Orders 25 50 75 100 150 200 225
LP Objective S o l u t i o n Value 17.11 17.11 17.11 17.11 17.11 17.1I 13.15 14.05 20.17 20.39 28.91 29.51 29.87 32.84
SolutionTime [CPU seconds] 10 68 318 693 7234 32458 49621
Gap [%] 0 0 0 6.4 1.1 2.0 9.0
The results obtained so far indicate that the models and solution procedures proposed for the hierarchical framework are capable of obtaining solutions of good quality within acceptable computational time for very difficult real world problems.
6. Conclusions and Future W o r k Much research has been undertaken in planning and scheduling for the process industry however research considers a fixed time horizon with data being given. We believe that multi scale dynamic online procedures are more suitable for many specific types of planning and scheduling problems found in the process industry and should be further explored. A lot of work remains on improving the integration and flow of information between the levels in the hierarchical framework and different strategies need to be evaluated with the objective to capture the imperatives and the dynamics of the decisions problem. It is also of interest to introduce uncertainty in the models as there are several sources of uncertainty apart from the demand that affect the problem.
References Engell, S., Markert, A., Sand, G., Schultz, R. and Schutz, C. (2001) In Online Optimization of Large Scale Systems, SPRINGER-VERLAG BERLIN, Berlin, pp. 649-676. Kallrath, J. (2002) Or Spectrum, 24, 219-250 Mendez, C. A., Henning, G. P. and Cerda, J. (2001) Computers" & Chemical Engineering, 25, 701-711. Pinto, J. M. and Grossmann, I. E. (1998)Annals of Operations Research, 81,433-466. Sand, G. and Engell, S. (2004) Computers & Chemical Engineering, 28, 1087-1103. Sand, G., Engell, S., Markert, A., Schultz, R. and Schulz, C. (2000) Computers & Chemical Engineering, 24, 361-367. Sgall, J. (1998) In Online Algorithms, Vol. 1442, pp. 196-231. Shah, N. (1999) AIChE Symposium Series, 320, 75-90. Shah, N. (2004) Proc ESCAPE-14, Portugal.
European Symposium on Computer Aided Process Engineering- 15 L. Puigjaner and A. Espufia (Editors) © 2005 Elsevier B.V. All rights reserved.
1009
Initiation and inhibiting mechanisms for multi-tasking control in discrete event systems S. Macchietto ~, Nicholas J. Alsop, Ross J. Baird, Zhang P. Feng and Bing H. Chen Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Abstract A theoretical analysis is presented of sequential controller initiation (pre-check) and inhibition for concurrent operations within a formal Procedural Control Theory (PCT) framework. These aspects are essential to the safety and performance of multitasking sequential controllers at the design and operation stages. The pre-check mechanism ensures the correct initial state of the process when controller is synthesized. The inhibit function prevents nominated controllers potentially using shared devices from starting while another controller is already active. The ideas are explained via a small illustrative example.
Keywords: formal
methods, procedural control theory, discrete event systems, safety.
1. Introduction Formal techniques to support the design of automation software for discontinuous processes are grounded in logic and discrete systems theory. Their primary goal is the improvement in process safety and operation by eliminating errors in sequential control software that often occur during the design of control logic and its implementation in automation systems. Unlike the verification approach (Nimmo 1994, Moon et al 1992), which checks controlled process against specifications, the idea in sequential control synthesis methods is to develop "provably correct systems" that can be shown a-priori to meet all specifications and generate no undesired behaviours. The Procedural Control Theory (PCT) (Sanchez and Macchietto, 1995, Sanchez et al., 1999) is one of them. Industrial sequential controllers are programmed in control languages typically organized into sets of "operations" (in the ISA $88 standard sense, ISA 1995). Each operation is started by an operator or a supervisory control system. Control architectures are multitasking, i.e. support concurrent control by multiple operations. Operations use process related hardware resources (e.g. pumps) that are usually shared with other operations. Without adequate safeguards, an operation could be started when its resources are in an inappropriate state or already in use by other operations. In either case conflicts may arise and at the very least the proper execution of the operation cannot be guaranteed. At worst, unsafe states of the process could be reached. Therefore, suitable initiation and inhibit logic is required in such circumstances, but
Corresponding author, Tel: +44 20 5945575, Fax: +44 20 75945604, email:
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1010 these issues have been ignored in the literature. This work deals with them by formally introducing controller initiation (pre-check) and inhibition within the PCT framework.
2. Process Modelling within PCT In PCT, a Discrete Event Systems (DESs) model of a process is described as a Finite State Machine (FSM) comprising of a finite set of states and transitions, by 7 tuples:
M -{Q,V'Z",E,~,y, qo,Qm}
(l)
where Q is the set of states and Z, = Z,c u Z, the set of transitions (other symbols are defined in the nomenclature). Controllable transitions Ec, e.g. turning on/off a pump, are forced by a controller while uncontrollable Zu transitions, such as the increase/decrease of a level sensor signal in a tank, occur indirectly and may only be observed. In PCT, an elementary device (e.g. pump or sensor) is represented by a unique "state variable" which may take a number of discrete realizations (states) and by the set of "transitions" that take the device from one state to another. Two operations on FSMs generate the model of a composite system: the "Asynchronous Product" to interleave states from each FSM and the "Synchronous Product" to intersect two FSMs to generate strings common to both original FSMs. The FSM model M of a system represents every possible combination of feasible states and process trajectories in the open loop process. Desirable behaviour of the process is defined by imposing the existence or otherwise of states and transitions, or allowable strings of process events, via formal specifications.
3. The Closed Loop Controller Initiation with Pre-checks The closed loop behaviour obtained by a controller c X that implements given specifications on model M is denoted by (L(Cx/M) In general, controller performance .
is sensitive to any departure in the initial state of process M from its nominal initial state q0 '. Thus, additional functionality is required to handle possible variability in the initial state of the process. The simplest mechanism is a pre-check that ensures that an operator request for starting controller c x from a process state q is granted only if q is the nominal initial state q0' of the process. A sufficient condition for
q=qo' is
that
every elementary component of M is in its nominal initial state. We define a new transition ax to represent the actual start of controller c x . Strictly, this is not an elementary process event (i.e. o-X~ Y_,) and therefore does not alter the process state. Then the pre-check mechanism is to ensure that the process is in state q0 for event ax to occur (i.e. for the controller cx to start). Therefore, o-~ is included in the model of the process as a self-looped transition at each potential starting state q e Qx. The closed loop behaviour of a system M under control of cx with pre-checks is then formulated as:
L(Cx/M ) = E* "err'L(. sXwcr)nL(Sffi'M)z,, .
(2)
This can be read as: the closed loop language generated by c x on M with pre-checks is the set of physically possible strings made up from the concatenation of an open loop
1011 trajectory and the starting event o-, plus the closed loop behaviour. M is not under control from ~' until the event o-, at a potential starting state q e Qx- Similarly, the marked closed loop response is given by: . L,,,(C,/M)
-
(3)
. s X~,, " c , . ) A L m ( S ~ - ' M ) ,Y_,*-a,.L,,,{
4. C o n t r o l l e r I n h i b i t i n g A controller inhibit mechanism is defined to disable a controller from starting if the system does not satisfy a given set of conditions. In particular, this mechanism checks the state of nominated controllers in a multitasking control system. Checks are performed before a controller is started so that abortive action can be taken to avoid the concurrent operation of controllers that compete R)r the same resource. The approach is based on the concept of non-cooperation (Alsop, 1996) and requires the closed loop language generated by the process under parallel control by controllers c~ and c, each with pre-checks. The cooperative and non-cooperative properties are used as follows: if c, is non-cooperative with c,., then c, must inhibit c , , and if d, cooperates with c, then no inhibition mechanism is necessary.
4.1 Closed loop language with pre-checks for two controllers in parallel Using the closed loop language generated by a single controller (equation 2), the closed loop language generated by two controllers c, and c, on process M augmented with the self-loop or, at each possible initial state ~/e (2, and dr,, at each possible initial state qe (2,. with a pre-check mechanism is given by:
L(C v "~C,,/M)-7O,71{Z:'I.o,.L(S~",C,)IA~,:'{Z:I,.o,,.L(S~"IC,,)JAL(SQ'SQ'M)} •
"
"
"
"
"
.
.
.
.
u."
-
v
(4)
O'v
where Z = E , uZ,, and Z, hE,, _cZ,,. The first term of equation 4 is equivalent to the first term of equation 2 with the addition of the proiection operation /~,71 . This operation permits any event from (E-Z,)vodr,, , and thereby restricts no events generated b y e , . Similarly, the second term represents the set of strings generated by controller c',,, which permits any event generated by C," The third term is the set of physically possible strings (i.e. the process model), augmented with self-loops or, at each state re(J, and or, at each state qe(2,
The augmented self-looped transitions
allow both controllers to start when the process is appropriately initialised.
4.2 Inhibit design for controllers with shared controllable items In the lormulation of the combined closed loop language (equation 4), it is necessary for the two controller languages to not share controllable transitions (i.e. E, hE,, _c E,, ). At first this may appear to reject from the inhibit analysis the case in which c, and c, share elementary components generating controllable events (e.g. driving the same valve). Typically, inhibits are necessary between pairs of controllers which drive the same equipment item. However situations may arise in which two controllers cooperatively employ the same equipment item. A special modelling technique is
1012 employed to handle this case to ensure that controllable transitions will not be generated concurrently by both controllers. In other words, it is not possible for the controllers to synchronise on a common controllable event. Therefore two common controllable transitions generated by the two controllers are unique, and must be labelled accordingly. The labels identify an event with the controller from which it is generated, i.e. transitions in cx and cy need be relabelled to respect the condition that Y--x and :~y share only uncontrollable events. In this work, we use a 'convert' technique (Wonham, 1996) for re-labelling transitions. For convenience, the set ~:v,, is defined as all those controllable events in E,, which have an equivalent gx : Z.~:v = { G E E o , / V q E Q,d(cr, q ) ! , 3 c r ' e Ecx s.t. 6(o-',q)!Aa(o-',q)=6(cr,q)}
(5)
Events in E~, can occur in the open loop process or when generated by Cy, but not when controller cx is the only active controller. Similar occurrences apply to events in Ey X. A regulator R is therefore introduced to meet this requirement by interlocking shared controllable items. Its language is given by:
L(R) = 7__,*.o-~.[7__,-gxy . ]* .Oy .Z* w 7__,*. O y . [ g - g y x ]* .o-~. . Y_,*
(6)
The first term prohibits events not in [E-Exy] after starting c~ and before starting cy with the reverse for the second term. The closed loop language generated by process M under parallel control from c~ and cy with pre-checks where c X and Cy share elementary components generating controllable events is given by:
x x. /~,71 }m L(S~'S~"M)mL(R) , y L(C x "~Cv/M)=T)xl{gx.Crx.L(Sz,,'xCx)In . {g~,. . a y •L(Sz,~.~,C:,)
(7)
Two controllers are cooperative (can be concurrent without inhibits) if the condition L(Cx ~ Cy/m) is a subset of U.ax.~-~L(Sx) holds true (Alsop, 1966) and vice versa.
5. An Example A liquid ingredient is dosed in a tank (Figure 1) using Feed Valve (FV), Drain Valve (DV) and level sensors LV1 and LV2. These devices are modelled as two state FSMs (Figure 2). Nodes on the graph represent states and edges transitions. Controllable transitions of FV and DV are duplicated for the use of inhibitions. Uncontrollable transitions are shown as dashed arcs, while controllable ones as solid arcs. Level Sensor LVI
Level Sensor LV2
Feed Valve
20 ......
i'~
~:::~.::
,......
Figure 1. Dosing tank
1 ~
/ 2
3 ~
l6
Drain Valve
2
~ 4
8
~
208
Figure 2. Elementary components modelled as F S M s
We wish to design a procedure is to fill the tank to level 2, ready for the next stage of the process. The initial and goal (marked) states are: • Initial state: FV is closed, DV is closed, LV 1 is low and LV2 is low. • Goal state: FV is closed, DV is closed, LV 1 is low and LV2 is high.
1013 The overall process model (Figure 3), obtained by the asynchronous operator of the components of Figure 2, includes all possible combinations of state variables {FV,DV,LV1,LV2} and transitions. A standard controller (i.e. befl)re pre-checks and inhibits) for filling the tank up to level sensor 2 is obtained (Figure 4 (a)) using the lollowing specifications" Static constraints: 1. The process should never reach a state where LV1 is high 2. When LV2 is low, LV1 cannot be high Dynamic constraints: 3. If DV is closed it is impossible for LVI or LV2 to fall If FV is closed it is impossible for LV1 or LV2 to rise . 5. If FV is open, do not allow DV to open 6. If LV2 is low, do not allow FV to close 7. If FV and DV are closed and LV 1 and LV2 are low, then open FV 8. If FV and DV are closed, LV 1 is low and LV2 is high, do not open DV 9. If FV and DV are closed, LV 1 is low and LV2 is high, do not open FV 10. If LV2 is low, do not allow DV to open 1 1. If valve DV is closed, do not open it
~:i:~'::~:; :>::71:.~i::~ "*-:i~!! ~!-,:~ i~21:~::~,;*i ,:..............~.:.,,
..... ~...,,~
...........* . . ~ .
.:,..........~:.,.,....
~,.:...~.........
Figure 3. Asynchronous product for the Dosing tank, where C for 'closed', O for 'open', L for 'low', H for 'high'respectively
(a) (b) Figure 4. Controllers for filling tank up to level 2 (a) and level 1 (b) without pre-check
Adding pre-checks to ensure that the initial state is consistent with the nominal initial state gives the close loop language of Figure 5(a), where or. =111 is defined as the starting event of this controller, which we call c,.. A similar procedure to fill up to level 1 can be obtained (Figure 4 (b)) which uses the same valve and sensor devices. The closed loop language of two parallel controllers with pre-check is shown in Figure 5(a) and (b). The regulator language for the concurrent use of the two parallel controllers is shown in Figure 6. It can now be formally established that the two controllers are noncooperative and hence c,_ must inhibit c,, and vice versa. 1,2,3,4 1.2
i, 2 3,4
5,6,7,8 2{)5,206 207,208
)4
Cx O F F Cy O F F
5 . 6 , 7, 8 ,2
207, 2()8
C~ON CyOFF
1,2
\
117 ~ ~ ~ / I
/I IIII Cx~FV CyON"
1,2,3,4 ~ 205,206
Cy O N
207, 2O8
(a)
(b)
Figure 5. FSM with pre-checks for two controllers in parallel
Figure 6. Regulator language for controllers with shared controllable items
1014
6. Conclusion The novel theoretical d e v e l o p m e n t presented allows incorporating controller initiation and controller inhibiting features within a PCT framework. These two mechanisms guarantee essential safety properties of a multitasking sequential control system, prohibiting the start of control actions in inappropriate situations. The p r e - c h e c k m e c h a n i s m ensures that control is initiated only when the state of the process is compatible with predefined nominal initial state(s). Similarly, the inhibit function prevents nominated controllers from starting while a given controller is already active. The formal closed loop language generated by a controller with a p r e - c h e c k m e c h a n i s m on a process was derived. This formulation, combined with that for the closed loop language generated by two parallel controllers, yields a general formal definition of the closed loop behavior generated by two parallel controllers each with p r e - c h e c k s . A further extension in the form of a regulator language handles the special case of shared controllable items. Further extension to multiple parallel controllers is immediate. The example used here is illustrative only. Larger applications will be published elsewhere.
Nomenclature C~and C,, - Controllers - Closed loop language generated by controller Cx and process M Lm(C~/M) - Closed loop language generated by controller Cx and marked process
L(C:/M)
L ( C x "~ C y / M )
-Closed loop language
q'0 - Nominal initial state S x - The specification for controller C, S X,,, - The self-loop operation at wait state with Y'u
uncontrollable transition E,,
generated by parallel controllers C, and C,. M - Process model FSM Q - Set of states Qm - Set of marked states qo - Initial state
transition from o-x . V..... The set of states each represented by n state
Greek Symbols 7 - State variable transition partial function - State transition partial function - Controller state transition partial function E - Set of transitions or alphabet of process events
E* - The set of all process trajectories composed of transitions in E ~ - Process event
s.~,- The self-loop operation at state Q, with
variables
Reference Alsop, N. J. 1996, Formal Techniques for the Procedural Control of Industrial Processes, PhD thesis, University of London ISA, 1995, Batch Control Part 1: Models and Terminology. Moon, I., G.J. Powers, J.R. Butch and E.M. Clarke, 1992, Automatic verification of sequential control systems using temporal logic. AIChE, 38(1), 67. Nimmo, I., 1993, Start up plant safely, C h e m i c a l E n g i n e e r i n g p r o g r e s s 89(12), 66. Sanchez A and S. Macchietto, 1995. Design of procedural controllers for chemical process, Computers and Chemical Engineering, 19S, 381. Sanchez, A., G. Rotstein, N. Alsop and S. Macchietto, 1999, Synthesis and implementation of procedural controllers for event-driven operations, AIChE J, 45 (8), 1753. Wonham, W.M., 1996, N o t e s on C o n t r o l o f D i s c r e t e - E v e n t Systems, Systems Control Group, Dept. of Electrical & Computer Engineering, University of Toronto. ECE 1636F/1637S.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. gspufia(Editors) (O2005 Elsevier B.V. All rights reserved.
1015
Model Based Parametric Control in Anesthesia Pinky Dua ~, Vivek Dua b and Efstratios N. Pistikopoulos a°* aCentre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK bCentre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK
Abstract This work presents a compartmental model for delivery of three drugs (isoflurane, dopamine and sodium nitroprusside) for regulation of anesthesia. The key feature of this model is that mean arterial pressure, cardiac output and unconsciousness of the patient can be simultaneously regulated. This model is 'validated' by carrying out a number of dynamic state simulations and then used for designing model based parametric controllers.
Keywords: Anesthesia, Mean Arterial Pressure, Model Based Control, Parametric Controller
1. Introduction Safe and adequate anesthesia is characterized by muscle relaxation, hypnosis and analgesia while maintaining vital functions such as mean arterial pressure (MAP), heart rate and cardiac output (CO) within certain desired ranges. A tight control of these vital functions is very important; otherwise it may lead to fatal situations. The anesthesiologist maintains these vital functions by regularly infusing various anesthetic drugs and/or intravenous fluids. Automatic control of the vital functions can reduce undesirable situations and improve the safety of the patient, by monitoring conditions that can not be measured easily, allowing the anesthesiologist to focus on critical issues, and also reduce the cost of the drugs and the time spent in the post-operative care unit. The automatic control techniques rely on a model of the patient that can incorporate the dynamic response of the patient to drug infusions and disturbances (Yasuda et al., 1991; Rao et al., 2001; Mahfouf et al., 2003). Gentilini et al. (2001) observed that controlling MAP and hypnosis simultaneously with isoflurane was difficult. Yu et al. (1990) proposed a model for regulating MAP and CO using dopamine (DP) and sodium nitroprusside (SNP), but the control of hypnosis was not considered. In the next section, a compartmental model is presented, which allows the simultaneous regulation of the MAP, CO and the unconsciousness of the patients. The model is characterized by: (i) pharmacokinetics for the uptake and distribution of the drugs, (ii)
Author to whom correspondence should be addressed:
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1016 pharmacodynamics which describes the effect of the drugs on the vital functions and (iii) baroreflex for the reaction of the central nervous system to changes in the blood pressure. The model involves choice of three drugs, isoflurane, DP and SNP. In Section 3 model based controllers for the regulation of anesthesia are designed by using Model Predictive Control (MPC) Toolbox (1998). Note that MPC solves a quadratic program at regular time intervals. Pistikopoulos et al. (2002), have presented model based parametric controllers that partition the space of state variables into a number of regions and each region is characterized by a control law, which is an explicit function of the state variables. This parametric control technology is used to design an explicit controller for the infusion of isoflurane for regulation of MAP and BIS. This reduces implementation of model based controllers to simple function evaluations. Concluding remarks are presented in Section 4.
2. Modelling Anesthesia The model is based on the distribution of isoflurane in the human body (Yasuda et al., 1991). It consists of five compartments organized as shown in Figure 1. Respiratory System J" soflurane
Tl
U
Uptake
Injection of DP and
SNP
Figure 1. Compartmental Model
The compartments 1-5 represent Lungs, Vessel rich organs (e.g. liver), Muscles, Other organs and tissues and Fat tissues respectively. The distribution of the drugs occurs from the central compartment to the peripheral compartments by the arteries and from the peripheral to the central by the veins. The first compartment in Figure 1 is the central compartment and heart can be considered to be belonging to the central compartment, whereas compartments 2-5 are the peripheral compartments.
2.1 Pharmacokinetic Modelling The pharmacokinetic model is developed based on the works of Yasuda et al. (1991), Yu et al. (1990) and Gentilini et al. (2001). The uptake of isoflurane in central compartment via the respiratory system is modelled as: V dCinsp
dt
- Q i n C i n - (Qin - A Q ) C i n s p - f R (VT - A ) ( C i n s p - C o u t )
(1)
1017 where C~n.v,is the concentration of isoflurane inspired by the patient (g/mL), C;,, is the concentration of isoflurane in the inlet stream (g/mL), Co,,, is the concentration of isoflurane in the outlet stream (g/mL), Q;, is the inlet flow rate (mL/min), AQ is the losses (mL/min), V is the volume of the respiratory system (mL), j)~ is the respiratory frequency (1/min), Vr is the tidal volume (mL) and A is the physiological dead space (mL). For the central compartment, the concentration of isoflurane is given by:
VI - &
i=2
~
- Cl
11
+ fR (VT - A)(Ci,,.w - Ci )
(2)
where Ci is the concentration of the drug in compartment i (g/mL), R~ is the partition coefficient between blood and tissues in compartment i, Q~ is the blood flow in compartment i (mL/min). The concentration of DP and SNP in the central compartment is modelled as follows:
v, ~
--£-c,
+c,~,-~-c,v,
7=2
(3)
rl/~2
where C w is the flowrate of the drug infused (g/min), Vi is the volume of compartment i (mL) and "c~/2 is the half-life of the drug (min). The distribution of isoflurane in compartments 2 to 5 is given by Vi - ~
- Qi CI -
- kiC i, i = 2 .... ,5
(4)
The natural decay of DP and SNP in the body, for compartment 2 to 5, is given by: d C i ( C i )
v,7--e,
c,---£, -
1
(5)
c i v i , i= 2,..,5
2.2 Pharmacodynamic Modelling DP and SNP affect MAP indirectly. These drugs influence two of the heart's characteristic parameters: maximum elastance (E,,,,,) and systemic resistance (R,,,), which is given by: d E f f = k 1C N (E f f m ax - E l f ) - k 2 E f f dt
(6)
where JEff is the measure of the effect of drug on the parameters of interest,
'~max - '~max,O 0 + ~S.r,~-,~ ...... ),
'~<
- ~s~s,O O-'~SS,~,~-R,.,..,.-
~SS~<,~-R,.,..,.)' ~" k~
are the rate constants and N is the non-linearity constant. MAP can be expressed as a function of E,,,~,x and R,,,, as:
1
MAP 2 ---7+ 2K2 MAP - 2K2VLvEmax R~,~
- 0
(7)
The relationship between MAP and CO is given by: MAP = R>~x CO where MAP is the mean arterial pressure (mmHg) and CO is cardiac output (L/min).
(8)
1018 Isoflurane affects MAP as follows:
MAP =
Q1 ~-'~(gi,o (1 +biCi) ) i=2
(9)
where, gi, o is the baseline conductivities (mL/(min.mmHg)) and bi is the variation coefficient of conductivity (mL/g). There is experimental evidence that a transportation delay exists between the lungs and the site of effect of isoflurane on the unconsciousness of the patient. In order to model this, an effect compartment is linked to the central compartment. The concentration of isoflurane within this compartment is related to the central compartment, which is given by: dCe ~=keo(C
dt
(10)
1 -C~)
where Ce is the concentration of isoflurane in the effect compartment (g/mL), and ke0 is the kinetics in the effect compartment (min-1). The action of isoflurane can be then expressed as follows"
Cey
A B I S - SaNIS MA X
(11)
c; + ECho where A B I S - B I S - B I S 0 , A B I S M A X - B I S M A x - B I S o , BISo is the baseline value of BIS (assumed to be 100), BISMAx is the maximum value of BIS (assumed to be 0), ECso is the patient's sensitivity to the drug and 7 is the measure of the degree of non-linearity.
2.3 Baroreflex Baroreflex is obtained from a set of transfer functions relating the mean arterial pressure to the maximum elastance and the systemic resistance and is given by:
ec(MAP-MAPO) bfc = 1 + ec(MAP-MAP0)
(12)
where c is the empirical constant (mmHg). For further details on the model and the parameters used see Dua et al., 2005.
3. Control of Anesthesia The model presented in the previous section was validated by carrying out a number of dynamic simulations for different amounts of drug dosages and disturbances using gPROMS (2003). This model was then used for designing model based and parametric controllers. For designing model based controllers, the model was linearized at the nominal values of inputs: 0.6% vol. of isoflurane, 2 ~tg/kg/min of DP and 4 gg/kg/min of SNP and outputs: 57.38 mmHg of MAP, 61.1 BIS and 1.21 L/min of CO to obtain a state-space model consisting of 23 states, 3 outputs and 3 inputs. This state-space form of the model is then adapted for designing model predictive controller by using the MATLAB Model Predictive Control Toolbox (1998). For designing the MPC controller, the following input: 0 < D P < 7 ¢tg/kg.min, 0 < S N P < 10 ¢tg/kg.min, 0 <
1019 lsq#urane < 5% vol., and output constraints: 40 <_ MAP< 150 mmHg, 40
<
BIS < 65, 1 <
CO < 6.5 L/min are used. A prediction horizon of 5, control horizon of 3 and sampling
time of 0.5 minutes are considered. A set point of [20 -10 1]' deviation from the nominal point of the output variables is given and the performance of the controller is shown in Figure 2 where the control variables are also deviations from the nominal values. It is observed that the MPC tracks the set point quite well. Ou-l:~>uts
20-
' •
~':)
....
P.s
co
:
0
.. 10 .
.
0
2
.
4
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Figure 2. MPC Performance for Anesthesia
xlO 7'
.,,&
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Figure 3. Parametric Controller: State Profiles
1020 For the design of parametric controller (Pistikopoulos et al., 2002) a reduced form of the model, presented in section 2, corresponding to the infusion of isoflurane regulating MAP and BIS is considered. The model has 7 states, xl to xT, representing the concentration of isoflurane in the 5 compartments and its effect on Emax and Rs,,s and the input variable is given by the inlet concentration of isoflurane. Prediction and control horizons of 3 and equal weightings on state and control variables are used resulting in 48 regions in the space of the state variables. The performance of the parametric controller was tested for a given input and the profile of the state variables is shown in Figure 3. It was observed that the profiles match closely with those obtained from the simulation of the nonlinear model by using gPROMS.
4. Concluding Remarks Automatic regulation of anesthesia can provide tighter control allowing anesthesiologist to focus on more critical issues which will result in less time spent by the patients in the post-operative care unit, reduction in the amount of drugs used and side-effects and above all a much safer platform for surgery under anesthesia. A compartmental model for anesthesia based upon the infusion of three drugs for the simultaneous regulation of mean arterial pressure and unconsciousness of the patient has been presented. This model was validated and then used for designing model based and parametric controllers. The performance of the controllers was tested and observed to be very good. From the results obtained it can be inferred that the model based and parametric control are promising technologies for automatic control of MAP, CO and unconsciousness of the patients being operated under anesthesia.
References Dua, P., V. Dua and E.N. Pistikopoulos, 2005, accepted for publication in the l6 th IFAC World Congress, Prague. Gentilini, A., C.W. Frei, A.H. Glattfelder, M. Morari, T.J. Sieber, R. Wymann, T.W. Schnider and A.M. Zbinden, 2001, IEEE Engineering in Medicine and Biology, 20(1), 39-53. gPROMS, Introductory User' s Guide, 2003, Process Systems Enterprise Ltd., London UK. Mahfouf, M., A.J. Asbury and D.A. Linkens, 2003, Control Engg. Practice, 11(12), 1501-1515. Morari, M. and N.L. Ricker, Model Predictive Control Toolbox User's Guide, 1998. Pistikopoulos, E.N., V. Dua, N.A. Bozinis, A. Bemporad and M. Morari, 2002, Comput. Chem. Engg., 26, 175-185. Rao, R.R., C.C. Palerm, B. Aufderheide and B.W. Bequette, 2001, IEEE Engineering in Medicine and Biology, 20(1), 24-38. Yasuda, N., S.H. Lockhart, E.I. Eger, R.B. Weiskopf, J. Liu, M. Laster, S. Taheri and N.A. Peterson, 1991, Anesthesia and Analgesia, 72, 316-324. Yu, C., R.J. Roy and H. Kaufman, 1990, Med. Prog. Technol., 16, 77-88.
Acknowledgements PD would like to acknowledge financial support from the Overseas Research Student Award Scheme (ORS) and the Industrial Consortium of the Centre for Process Systems Engineering. A. P. Muller and S. Chrysafi's contribution to this work is also acknowledged.
European Symposiumon Computer Aided Process Engineering- 15 L. Puigjaner and A. Espufia (Editors) © 2005 Elsevier B.V. All rights reserved.
1021
Anti-slug control experiments on a small-scale two-phase loop Heidi Sivertsen and Sigurd Skogestad* Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway
Abstract Anti-slug control applied to two-phase flow provides a very challenging and important application for feedback control. It is important because it allows for operation that would otherwise be impossible, and challenging because of the presence of both RHP-poles and RHP-zeros. To conduct experiments on pipeline-riser anti-slug control, a small-scale slug-loop has been build. The loop has been modeled and analyzed using a simplified model by Storkaas. The results from this analysis and experimental results using a PI-controller is presented in this paper.
Keywords: feedback control, riser slugging, controllability analysis 1. Introduction Some of the problems in the offshore oil industry that have received increasingly interest in the last years are related to multiphase flow. In multiphase flow different flow regimes can develop, depending on parameters such as flow rates, fluid properties and pipeline geometry. Slug flow is a flow regime which can cause a lot of problems for the production facilities. The slug flow is characterized by alternating bulks of gas and oil, and can be further divided into hydrodynamic and terrain induced slugging. Hydrodynamic slugs are caused by velocity differences between the phases and occur in near horizontal pipelines. These slugs are usually short and appear frequently. Terrain induced flow however, can contain a lot of liquid and therefore induce large pressure variations in the system. This flow is induced by low points in the pipeline geometry. When the low-point is realized by a downsloping pipe terminating in a riser, we get what is known as riser slugging. Because of the large and abrupt fluctuations in pipe pressure and gas and liquid flow rates at the outlet, these slugs cause huge problems for the processing equipment. Unwanted variations in the separator level give rise to poor separation and possible overfilling. The pressure fluctuations wear and tear on the equipment and can sometimes result in unplanned process shutdowns. *e-mail:
[email protected]:phone: +47-7359-4154; fax: +47-7359-4080
1022
T 4
4----
3
Figure 1. Illustration of the cyclic behavior (slug flow) in pipeline-riser systems The behavior of pipeline-riser slug flow is illustrated in Figure 1. Liquid accumulates in the lowpoint of the riser, blocking the gas (1). As more gas and liquid enters the system, the pressure will increase and the riser will be filled with liquid (2). After a while the amount of gas that is blocked will be large enough to blow the liquid out of the riser (3). After the blow-out, a new liquid slug will start to form in the lowpoint (4). Several solutions for eliminating or reducing these problems have been proposed (Sarica and Tengesdal, 2000), but they usually come at a price. Choking the valve at the top of the riser is one example of this. The slugging will disappear, but the increased pressure drop over the valve will lead to a lower production rate. Stabilizing the flow using active control has been proposed earlier and also tested out both on experimental rigs (Hedne and Linga, 1990) and on offshore installations (Havre et al., 2000) and (Courbot, 1996). It has been proved that it is possibly to stabilize the flow at a pressure drop that would lead to slug flow if left uncontrolled. However, there is still a lot that can be done on deciding which measurements and control configuration gives the best results. Some measurements, like the inlet pressure, can even be hard to implement and maintain. A small-scale loop (the Miniloop) was build in order to test out and analyze different control strategies in a cheap and easy way. The loop is very simple with a flow consisting of only two phases, air and water. We still get the same slugging phenomenon as expirienced offshore, with pressure fluctuations and varying flow rates. This makes it possible to screen different ideas before testing them on larger and more expensive experimental rigs and a lot of money can be saved.
2. Experimental setup To test different control configurations, a small-scale two-phase flow loop with a pipelineriser arrangement was build. The flow consists of water and air, which are mixed at the inlet of the system. Both the pipeline and the riser was made of a 20mm diameter transparent rubber hose, which makes it easy to change the shape of the system. A schematic diagram of the test facilities is shown in Figure 2. From the mixing point the flow goes trough the low-point at the bottom of the riser and depending on different conditions, slug flow may occur. At the top of the riser there is a separator, which leads the water to a reservoir. From there the water is pumped back into the system through the mixing point. The air is being let out through a small hole at the
1023
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i
J
ii;
i
!:
i ? i:i !
i!: .,2,,,; . . . . . .
Figure 2. Exlwrimental settq~ top of the separator. For slugging to appear there must be enough air in the system to blow the water out of the 1,5 meter long riser. This requires a certain amount of volume, which is accounted for by a buffer tank (BT) between the air supply and the inlet. The volume of the gas can be changed by partially filling this tank with water. The flow rates of gas (Qair) and water (Qw) determines whether we will have slug flow in open loop operation or not. These flow rates are being measured upstream the inlet. Typically flow rates during an experiment are 1 l/rain for the gas and 3 l/rain for the water. So far there are three pressure sensors located at different places in the loop. One is located at the inlet (P1) while the two others are topside measurements, located at the top of the riser (P2) and at top of the separator (P3). The latter is used for measuring the flow of air out of the separator. Fiber optic sensors (S1, $2) give a signal depending on the amount of water in the hose where they are located. They can easily be mowed around to measure the holdup at different locations in the loop. A control valve is placed at the top of the riser. A signal from the control panel sets the opening percentage of the valve. The control panel converts the analog signals from the sensors into digital signals. These signals are then sent to a computer. The signals are continuously displayed and treated using Labview software. Depending on the control configuration, some of the measurements are used by the controller to determine the opening percentage for the control valve.
3. Controllability Analysis Storkaas et al. (2003) have developed a simplified macro-scale model to describe the behavior of pipeline-riser slugging. The model has three states; the holdup of gas in the feed section (;~;,c;1), the holdup of gas in the riser (';~J¢72), and the holdup of liquid (~I~,L). Using this model we are able to predict the variation of system properties such as pressure, densities and phase fractions. In order for the model to fit the MiniLoop, it needs to be tuned. To do this we compare the bifurcation diagrams for the model and the Miniloop, plotted in Figure 3. The upper lines shows the maximum pressure at a particular valve opening and the lower line shows the
1024 minimum pressure. The two lines meet at around 20% valve opening. This is the point with the highest valve opening which gives stable operation when no control is applied. When Storkaas' model is properly tuned, the bifurcation point from the model will match the one from the experimental data. The dashed line in the middle shows the unstable steady-state solution, which is the desired operating line with closed-loop operation. 1.25
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L 20
1 0
i 40
i 60 Valve opening %
i 80
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Figure 3. Bifurcation diagrams from experimental data (dotted line) and Storkaas' model (solid line) When the model is tuned it can be used to perform a controllability analysis on the system. This way we can predict which measurements are suitable for control, thus avoiding slug flow. The analysis shows that the system consists of the poles given in Table 1. Table 1. Poles of the system for valve openings z=O.12 and z=0.25 z
0.12 -20.3411 -0.0197 ± 0.1301i
0.25 -35.2145 0.0071 + 0.1732i
Since all poles of the system are in the LHP when using a valve opening of 12%, this valve opening results in stable flow in the pipeline. However, when the valve opening is set to 25% we get a pair of RHP poles leading to riser slugging. This could also be predicted from the bifurcation diagram in Figure 3. To stabilize the flow we have available several measurements. Four of these are topside measurements; pressure P2, density p, volume flow Fq and mass flow Fw. The fifth measurement is the inlet pressure, P1. The zeros of the system using different measurements are given in Table 2.
Table 2. Zeros of the system using different measurements at valve opening z=0.25
~'~
P2
p
Fq
p~
- 1.285
46.984 0.212
0.092 -0.0547
-3.958 -0.369 ± 0.192i
-65.587 -0.007 ± 0.076i
It is well known that stabilization (shifting of poles from RHP to LHP) is fundamentally difficult if the plant has a RHP-zero close to the RHP-poles. From this, we expect no
1025 particular problems using P1 as the measurement. Also, Fq and Fw could be used for stabilization, but we note that the steady-state gain is close to zero (due to zeros close to the origin), so good control performance can not be expected. On the other hand, it seems difficult to use O or Pg_ for stabilization because of presence of RHP-zeros. From the controllability analysis we therefore can draw the conclusion that when using only one measurement for control, the inlet pressure/:'1 is the only suitable choice.
4. Experimental results The analysis showed that using the inlet pressure P 1 w a s the only possibility when using only one measurement for control. Based on this, a PI-controller was used to control the system using this measurement. The MiniLoop was first run open loop for two minutes, with a valve opening of 30%. This is well inside the unstable area, as the bifurcation diagram shows. The result is the pressure oscillations plotted in Figure 4, which illustrates how the pressure and valve opening varies with time. Both experimental and simulated values using the simplified model are plotted. When the controller is activated after two minutes, the control valve starts working. The flow is almost immediately stabilized, even though the average valve opening is still within the unstable area. It remains that way until the controller is turned of again after 8 min. When the controller is turned off, the pressure starts oscillating again. Pl
z
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A ~.18
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0 2
4 6 Time [min]
8
10
0
2
4 6 Time [min]
8
Figure 4. Experin~ental and simulated results using a Pl-controller
From Figure 4 we see that the controller efficiently stabilizes the flow, confirming the results from the analysis. However, this measurement can be difficult to use in offshore installations because of its location.
1026 Using other control configurations or measurements other than the ones analyzed in this paper might be the solution if there are only topside measurements available. The plan is to test out different ways to do this in the near future. The first possibility that will be explored, is using a cascade configuration involving the topside pressure/92 and one of the flow measurements F~ or Fq. Storkaas and Skogestad (2003) have proved theoretically that this works for another case of riser slugging.
5. C o n c l u s i o n From the controllability analysis it was found that using the bottom hole pressure was the only measurement of the five measurements analyzed, that could be used for controlling the system. The experiments confirmed that the model used for the analysis was good, and that using this measurement we where able to control the flow without problems. We are, however, looking for other ways to control the flow because of the problems related to down hole measurements. When using some of the other measurements analyzed, we must use combinations of measurements in order to avoid the problems related to the zeros introduced.
References Courbot, A. (1996). Prevention of Severe Slugging in the Dunbar 16" Multiphase Pipeline. Offshore Technology Conference, May 6-9, Houston, Texas. Havre, K., Stornes, K. and Stray, H. (2000). Taming Slug Flow in Pipelines. ABB review, 4:pp. 55-63. Hedne, E and Linga, H. (1990). Supression of Terrein Slugging with Automatic and Manual Riser Choking. Advances in Gas-Liquid Flows, pp. 453-469. Sarica, C. and Tengesdal, J. (2000). A new teqnique to eliminating severe slugging in pipeline/riser systems. SPE Annual Technical Conference and Exibition, Dallas, Texas. SPE 63185. Storkaas, E. and Skogestad, S. (2003). Cascade control of Unstable Systems with Application to Stabilization of Slug Flow. Storkaas, E., Skogestad, S. and Godhavn, J. (2003). A low-dimentional model of severe slugging for controller design and analysis. In Proc. of MultiPhase '03, San Remo, Ita(v, 11-13 June
2003.
European Symposiumon Computer Aided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1027
Using CLP and MILP for Scheduling Commodities in a Pipeline Leandro Magatfio*, L.V.R. Arruda, Flfivio Neves-Jr. The Federal Center of Technological Education of Paranfi (CEFET-PR) Graduate School in Electrical Engineering and Industrial Computer Science (CPGEI) Av. Sete de Setembro, 3165, 80230-901, Curitiba, PR, Brazil Tel: +55 41 310-4707 - Fax: +55 41 310-4683
[email protected] arruda@cpgei, cefetpr.br
[email protected]
Abstract This paper addresses the problem of developing an optimization structure to aid the operational decision-making in a real-world pipeline scenario. The pipeline connects an inland refinery to a harbor, conveying different types of commodities (gasoline, diesel, kerosene, alcohol, liquefied petroleum gas, etc). The scheduling of activities has to be specified in advance by a specialist, who must provide low cost operational procedures. The specialist has to take into account issues concerning product availability, tankage constraints, pumping sequencing, flow rate determination, and a series of operational requirements. Thus, the decision-making process is hard and error-prone, and the developed optimization structure can aid the specialist to determine the pipeline scheduling with improved efficiency. Such optimization structure has its core in a novel mathematical approach, which uses Constraint Logic Programming (CLP) and Mixed Integer Linear Programming (MILP) in an integrated CLP-MILP model. Moreover, a set of high-level modeling structures was created to straightforward formulate the CLPMILP model. The scheme used for integrating techniques is double modeling (Hooker, 2000), and the CLP-MILP model is implemented and solved by using a commercial tool. Illustrative instances have demonstrated that the optimization structure is able to define new operational points to the pipeline system, providing significant cost saving.
Keywords: Optimization, Scheduling, Constraint Logic Programming (CLP), Mixed Integer Linear Progralmning (MILP), and Pipeline. 1. Introduction The oil industry has a strong influence upon the economic market. Research in this area may provide profitable solutions and also avoid environmental damages. The oil distribution problem is within this context, and pipelines provide an efficient way to convey products (Kennedy, 1993). However, the operational decision-making in pipeline systems is still based on experience, and no general framework has been established for determining the short-term scheduling of operational activities in pipeline systems. The approach to address (short-term) scheduling problems is manifold, but the struggle to Author to whom correspondence should be addressed:
[email protected]
1028 model and solve such problems within a reasonable computational amount has challenged the development of new optimization approaches. In the front line of such approaches, Operational Research (OR) and Constraint Programming (CP) optimization techniques are merging. More specifically, Mixed Integer Linear Programming (MILP) and Constraint Logic Programming (CLP) are at the confluence of OR and CP fields. The integration of CLP/MILP has been recognized as an emerging discipline for achieving the best that both fields can contribute to solve scheduling problems (Hooker, 2000). Following this tendency, this paper develops an optimization structure based on CLP and MILP techniques (with their well-known complementary strengths). This structure is used to aid the scheduling of activities in a real-world pipeline scenario.
2. Problem Description The considered problem involves the short-term scheduling of activities in a specific pipeline, which connects a harbor to an inland refinery. The pipeline is 93.5 km in length, and it connects a refinery tank farm to a harbor tank farm, conveying different types of commodities (gasoline, diesel, kerosene, alcohol, liquefied petroleum gas, etc). Products can be pumped either from refinery to harbor or from harbor to refinery. The pipe operates uninterruptedly, and there is no physical separation between successive products as they are pumped. Consequently, there is a contamination area between miscible products: the interface. Some interfaces are operationally not recommended, and a plug (small volume of product) can be used to avoid specific interfaces, but plug inclusions increase the operational cost. The scheduling process must take into account issues concerning product availability, tankage constraints, pumping sequencing, flow rate determination, usage of plugs, and operational requirements. The task is to specify the pipeline operation during a limited scheduling horizon (H), providing low cost operational procedures, and, at the same time, satisfying a set of operational requirements. An in-depth problem description can be found in Magat~o (2003). 3.
Methodology
An optimization structure to address this pipeline-scheduling problem was proposed by Magat~o et al. (2004). This structure, which is illustrated in Figure 1, is based on an MILP main model (Main Model), one auxiliary MILP model (Tank Bound), a time computation procedure (Auxiliary Routine), and a database (Data Base), which gathers the input data and the information provided by the other optimization blocks. The Tank Bound task involves the appropriate selection of some resources (tanks) for a given activity (the pumping of demanded products). Its main inputs are demand requirements, product availability, and tankage constraints. As an output, it specifies the tanks to be used in operational procedures. The Auxiliary Routine takes into account the available scheduling horizon, the product flow rate, and demand requirements. It specifies temporal constraints, which must be respected by the Main Model. The Main Model, which is based on MILP with uniform time discretization, determines the product pumping sequence and it establishes the initial and the final time of each pumping activity. The final scheduling is attained by first solving the Tank Bound and the Auxiliary Routine, and, at last, the Main Model. The Main Model must respect the
1029 parameters previously determined by the Auxiliary Routine. In this paper, we also use this optimization structure, but with one fundamental difference: the Main Model is b a s e d o n a c o m b i n e d CLP-MILP approach. In the former approach, the Main Model was just based on an MILP formulation, and, depending on the problem instance, it can demand a computational effort from minutes to even hours. The Tank Bound and the Auxiliary Routine, which demand few seconds of running, are essentially the same models of Magatfio et al. (2004). For simplicity, these models are not herein discussed, and they are considered to provide input parameters to the new CLP-MILP Main Model. In order to straightforward formulate the CLP-MILP model, a set of high-level modeling structures was created (details are given in Magatfio, 2003). Thus, the model builder just needs to establish a "high-level" CLP-MILP modeling statement, and, afterwards, CLP and MILP equivalent expressions could be automatically derived. In a modeling standpoint, the MILP vocabulary, which is just based on inequalities, is poorer than the CLP vocabulary (Williams and Wilson, 1998), and the high-level structures provided and insight to overcome the modeling difference between MILP and CLP techniques. Figure 2 illustrates a high-level "if and only if' proposition that is expressed according to CLP and MILP approaches +. In this proposition, a binary variable c~. has to be set to one if and only if E/a~/xy < bk, where J is the set of variables (/E J), K is the set of constraints (kEK), akj's are constraints coefficients on continuous variables Xj'S, b~'s are requirements. Moreover, Lk's and U~'s are, respectively, lower and upper bounds, such that Lk <-Eiak~j- bk < U~., and c is a positive small tolerance value. The CLP-MILP model is composed by both CLP and MILP formulations, which are iteratively invoked. Space restrictions preclude that the detailed mathematical formulation be presented (Magatgo, 2003). Basically, MILP is used to establish a continuous time scheduling model that enhances the traditional CLP search mechanisms (constraint propagation and domain reduction;) by providing (linear programming) relaxations to the CLP model during the search procedure ("upper/lower bounds"). Therefore, the scheme used for integrating CLP and MILP approaches is double modeling: each constraint is formulated as part of a constraint programming model and as part of a mixed integer model. The two models are linked and pass information to each other (Hooker, 2000). The double modeling scheme is implemented in the commercial tool ILOG OPL Studio 3.6.1 (ILOG, 2002). MAIN M O D E L L, i
• ank Bound ~
formulation Data ~ Base
Main Model
Auxiliary ~ / / ' Routine V
Figure 1. Optimization structure
I"
I
~- formulation
:"",a~JExample I "" ~., of high-level proposition:L~,..} k;.................................... Ic'Lpformulation" MIEPformulation:
Figure 2. CLP-MILP framework
+The CLP and MILP formulations are done by, respectively, one equivalence (+-~)and two inequalities. *For a detailed cxp|anation about CLP scarch mechanisms and a comparison bctwccn the main characteristics of CLP and MILP techniques see, for instance, Hooker (2000).
1030
4. Results This section initially considers an example involving the pumping of four products from the harbor to the refinery followed by other four pumped from the refinery to the harbor The illustrative example covers since the minimum scheduling horizon (H mi'= 114 hours), which is determined by the Auxiliary Routine, up to 144 hours. The considered case study represents a typical operational scenario of the real-world pipeline modeled. Table 1 provides information about the optimization structure simulation on a Pentium IV, 2.4 GHz, 1 Gbyte RAM. In Table 1, H is the scheduling horizon, MILP, CLP, and CLP-MILP labels refer to, respectively, a pure MILP, a pure CLP, and a combined CLP-MILP approach. Therefore, a numerical comparison amongst three different Main Model versions is presented. For each scheduling horizon (H), the optimization structure is run, and a specific operational cost is attained (see Table 1, column "Cost"). The Auxiliary Routine and the Tank Bound simulation data are neglected. These structures required a computational time lower than one second, for all illustrative instances (114 _
1031 Table 1. Main Model illustrative instances- MILP, CLP, and CLP-MILPjbrmulations H
Number
of Variables
Number
of Constraints
OptimaliU'
proved
in (s)
Nodes
Exploited
Cost
(h)
MILP
('LP
(q.P-MiI~P
MII~P
CLP
('I.P-MII.P
MILP
(?LP
(/I.P-MILP
MII.P
(/LP
('LP-MILP
114
361
273
417
1132
571
16211
15
0.8
5.0
9451_
57 5
575
174
115 116 117
425 425 425 425 425 425
305 305 3115 3O5 3(15 305
48 48 48
1324 1324 1324
631 631 631
IS60 1860 1,',;60
17 19 22
69 35;I 417
7.3 7.4 16~
11573 13492 15708
15 6 10' 62710 ~ 871) 10 ~
614 714 1655
175 176 177
48 48 48
1324 I ~24 1324
631 631 631
1861~ 1860 1860
23 34 42
629 1075 6312
16 16 _~_~
1502~ 23091 28509
1.25 I 0" 1.911 1()" 13.7 1(/'
1642 1783 3596
174 I 71 168
425 425 425
305 305 3O5
1324 1324 1324
631 631 031
1860
1860 1860
41 46 75
37 36 44
32258 491) 79
4073 3976 51146
164 160 156
425 425 425
3O5 115 3O5
1324 1324
631 631
1324
631
! N60 186(I 1860
85 61 64
39 ~8 49
56613 37645 44967
4509 448t)
126
48! 481 481 481 481 4Sl
5911
153 150 147
127 12,~
425 425
3O5 3O5
481
631 63
1860
1860
91 78
46 40
612t)2 48898
5419 4685
144 141
425 425 425
31t5 305 ]()'~
481 48! 481 481
1324 1324
129
!324 I ~24 1324
63 63 63
1860
45
25
! 860
411
25
25499 50686
2749 2879
138 139
1860
?6
26
22185
2988
140
425 425 425
~t15
481
~05 3t15
4,~1 481
1324 1324 1324
63 63 63
186() i 860 I S60
6(1 57 50
25 26 25
31741) 1651 27005
2888 2900 2783
139 140 141
425 425 425
305 305 3~15
481 481 481
1324 1324 1324
63 63 63
1860 I S60 I N60
41 84 64
26 26 26
47142 48055 348611
2878 2883 2844
142
136 137 138 139 141t
425 489 489
3115 ~7 ~3 V
1724 1516 t5t6
63 69 69
22 25
8N
26
489 489 489 489
33 337 337
1516
69
I5 !6
69
28 28
2547 2835 2873 3067 31)91
144 145 146 147 148
69 64
170 113 73 57
36620 27746 43455 62824 61t221
1516 1516
I N60 1110 !~() 21~10 2 i O0 100 i O0
73 52
141 142 143 144
481 545 545 545 545 545 5-15
29
32219
3116
149
77
29202
8445
150
I I ,~ I 19
1211 121
122 123 124 125
130 131 132
133 134 135
~,~,-l
25145
(~)
143
144
The data presented in Table 1 represent a realistic pipeline-scheduling scenario, which demanded running times of seconds to few minutes, at least for MILP and CLP-MILP approaches. In order to further investigate the computational effort trend presented by the Main Model, some hypothetical problem instances were also tested. Such instances do not necessarily represent typical operational scenarios. In fact, the main goal was to test MILP, CLP, and CLP-MILP approaches in theoretically more time-consuming problem instances. In such instances, basically, a fixed volume of each product is demanded; the number of demanded products is progressively increased from 8 to 12; and, the scheduling horizon varied from the minimum scheduling horizon (H '''j'') up to Ig"*"+ 6 hours (see details in Magatfio, 2003). Figure 4 summarizes the results of such experiment: it presents the average computational time of MILP, CLP and CLP-MILP models as a function of the total number of demanded products ("problem instance"). i
r
.
i [
( ' o x / v x . Schedtdin,,,,, H o r i z o n
i
.
.
.
.
.
.
A v e r a g e C o m l m l a l i o n a l Time
4o] 3~)
?'1
211
152 150 ] I~
1(1
144
I
(1 I IJ I It, l IN 1211 1
Figure 3. Cost
I
velwu,s"
4
scheduling horizon
Figure 4. Average computational timejbr MILP, CLP and CLP-MILP models
7
1032 Figure 4 indicates that, in average, the CLP-MILP model tends to solve this specific problem faster than the MILP and the CLP models as the total number of demanded products (instance) was increased. Obviously, one might argue that even the CLP-MILP model has an exponential-time computational behavior, but, for certain, the combined model could "go a step further" than both, the MILP and the CLP models.
5. Conclusions The economically important problem of product distribution through a real-world pipeline is addressed. The task is to predict the pipeline operation during a limited scheduling horizon, providing low cost operational procedures, and attending a series of operational requirements. The scheduling of operational activities has to take into account product availability, tankage constraints, pumping sequencing, flow rate determination, and a variety of operational procedures. The computational expense is concerned and an optimization structure is proposed (Figure 1). The core of such structure (Main Model) is based on a novel mathematical approach that combines Constraint Logic Programming (CLP) and Mixed Integer Linear Programming (MILP) techniques. The scheme used for integrating CLP and MILP is double modeling (Hooker, 2000), and a set of high-level modeling structures is used in order to straightforward formulate the CLP-MILP model. The large-scale model is implemented and solved in a commercial tool. Computational results have indicated that the combined CLP-MILP model performs better than both: the pure MILP model and the pure CLP model (Figure 4). Therefore, the integration of CLP and MILP approaches provided an effective alternative to deal with this pipeline-scheduling problem. Currently pipeline operation is still based on experience, but the developed model can aid the decision-making process. Furthermore, the optimization structure has indicated that economic improvements are feasible (e.g. Figure 3), and it has demonstrated the importance of model formulation to help solve problems with improved efficiency.
References Dantzig, G.B., 1963, Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, USA. Hooker, J.N., 2000, Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction. Wiley-Interscience Series in Discrete Mathematics and Optimization, New York, USA. ILOG, 2002, ILOG OPL Studio 3.6.1 - Language Manual. ILOG Corporation, France. Kennedy, J.L., 1993, Oil and Gas Pipeline Fundamentals. PennWell Publishing Company, Oklahoma, USA. Magat~o, L., 2003, Integrating Mixed Integer Linear Programming and Constraint Logic Programming. Technical Report, CEFET-PR/CPGEI, December 2003, 119 pages. Magat~o, L., Arruda, L.V.R. and F. Neves-Jr., 2004, A mixed integer programming approach for scheduling commodities in a pipeline. Computers & Chemical Engineering, 28 (1-2): 171-185. Williams, H.P. and J.M. Wilson, 1998, Connections between integer linear programming and constraint logic programming. INFORMS Journal on Computing, 10 (3): 261-264.
Acknowledgements The authors acknowledge financial support from ANP and FINEP (PRH-ANP / MCT PRH10 CEFET-PR).
European Symposium on Computer Aided Process Engineering- 15 L. Puigjaner and A. Espufia (Editors) © 2005 Elsevier B.V. All rights reserved.
1033
Scheduling of a pipeless multi-product batch plant using mixed-integer programming combined with heuristics Sebastian Panek ~, Sebastian Engell ~, Cathrin Lessner 2 University of Dortmund Process Control Lab (BCI-AST), University of Dortmund, Dortmund 44221, Germany 2 Axxom Software AG Axxom Software AG, Munich, Germany
Abstract The work presented here deals with short-term scheduling in the chemical industry. It proposes an alternative model to STN and RTN in the form of a MILP formulation which describes the plant at a lower level of detail. The aim is to reduce the complexity of the model and thus to increase the solution efficiency. The main simplification compared to STN/RTN is the that product stocks and mass balance constraints are ignored because discrete fixed-size batches are assumed. The proposed MILP formulation is used for the scheduling of a real-life example provided by a customer of Axxom, a lacquer production in a pipeless plant. Three types of recipes that involve cycles, rigid timing constraints between individual operations, and parallel allocations of stationary and mobile units are considered. Several dozens of production orders of different lacquer types in various colors with hundreds of operations have to be scheduled. A special solution procedure combines the solution of the MILP problem in two separate steps with three different heuristics which further reduce the model complexity. Several problem instances of the lacquer production are solved to demonstrate the successful application.
Keywords: supply chain management, scheduling, planning, multiproduct batch processing, pipeless plant
1. Introduction Research efforts in the past l0 years in the area of batch plant scheduling (batch sizing, resource allocation, sequencing) have led to powerful general modelling frameworks as STN (Kondili et al., 1993; Shah et al., 1993) and RTN (Schilling et al., 1996). They enable the user to express a large variety of constraints that might arise in the processing industry, such as capacity constraints on units and products, timing constraints between operations, sequence-dependent changeover procedures, various storage policies, complex production recipes etc. Within both frameworks, the representation of time turned out to be crucial for the performance of the solvers of the mathematical models. This led to continuous time STN/RTN representations with uniform and non-uniform
1034 grids (e.g. Ierapetritou et. al., 1998). Many recent publications focused on this problem by proposing improved modelling and solution approaches. For solver performance, it is usually beneficial to reduce the complexity of the model in terms of the number of equations and variables. This is of[en done by refining and adapting the constraints to exploit special properties of the plant and the production scheme. A discussion of various representations of time can be found in (Floudas et. al., 2004). We refer to two recent continuous-time modelling approaches here. The first one is slotbased and suitable for network-based production schemes involving mass balances and product flows (Pinto et. al. 1994). The second one is based on order-oriented sequential processes without time slots (Manne, 1960; Cerda et. al. 1997). The work presented here is strongly related to the latter approach. It aims at overcoming the difficulties mentioned above by an order-based modelling scheme. An alternative to STN and RTN in the form of a different MILP formulation which describes the plant at a lower level of detail is proposed. The aim is to simplify the representation of time and thus to increase the solution efficiency. An explicit representation is often avoided by establishing a minimal set of event points without predefined order. Sequencing decisions are encoded using dedicated binary variables which correspond to typical binary disjunctions arising in all sequencing problems. This leads to an efficient model formulation in which the resource assignment and the sequencing are decided in parallel. The main simplification compared to STN/RTN is that no product stocks and mass balance constraints are considered. Discrete fixed-size batches are assumed. This work also suggests a two step solution procedure. It combines the solution of the MILP problem in two separate steps with three different heuristics which further reduce the model complexity. The remainder of this paper is structured as follows: after stating the problem the modelling approach is described in detail. In the following section the solution procedure is discussed and the results of numerical experiments are given.
2. Problem Statement The proposed MILP formulation is used for the scheduling of a real-life example provided by a customer of Axxom, a lacquer production in a pipeless plant. The plant facilities consist of 5 mobile mixing vessels and 9 stationary processing units. The stationary units are: pre-dispersion line, main dispersion line, special pre-dispersion unit, dose distributor, two identical dose spinners, two identical filling stations and a laboratory for quality checks. Two mixing vessels have capacities of 19,000 litres, the other three vessels can hold 20,000 lives. The plant topology with traffic on limited routes, possible collisions and different distances is not considered here. Three basic recipes for lacquers, each with 6-8 operations are given. The recipes involve cycles, rigid timhlg constraints between individual operations, and parallel allocations of stationary and mobile units. Three different types of constraints involve either 1) starting or 2) ending or 3) ending and starting dates of individual operations within the recipes. Each of these constraints establishes a link between two operations and forces them to either start or to end within a time window defined by the constraint. The cycles in the production comprise two steps: dosing and quality checks. The termination of a cycle depends on the result of the last quality check. During the processing steps, no
1035 material flows from and to the vessels are considered such that the amount of product in the batch remains constant. Raw materials and the storage for end products are unlimited and available at any time. The market demand is represented by 29 production orders for different products and with irregular release and due dates spread over several months. The objective is to minimize the overall production cost which consists of penalties for missing the due dates. The recipe for a standard metallic lacquer is depicted in Fig. 1. The main processing steps are: assignment of a mobile mixing vessel, dosing, quality check, correction, second quality check and filling. The other recipes have a similar structure but involve two additional pre-processing steps. The mixing vessel holds the product during all steps and is thus allocated in parallel. Dosing
Quality check
Correction Mixin• v
e
Qualitycheck
Filling
~
Figure 1. The recipe graph of metallic [acqlter
3. Modelling A p p r o a c h 3.1 Simplifications and Assumptions Some simplifications with respect to the original problem are made. As the demanded product amounts and vessel capacities are quite similar and do not vary much, we assume equal mixing vessels with sufficient capacities to hold one batch of each product. Another simplification is related to the cycles. The results of the quality checks in the laboratory are a-priori unknown. Since the following steps depend on them, the overall number of processing steps is also unknown. We avoid stochastic modelling and assume that the first quality check always fails and the second one always succeeds. Thus, the loop in the recipe is always taken twice and we get 29 batches of different lacquer types in various colours with 202 operations in total that have to be scheduled. Since the lab is an unlimited resource quality checks may take place at any time arbitrarily often, but require a fixed amount of time. This allows us to remove all quality checks and the lab from the model. Appropriate timing constraints are established to make sure that the necessary minimal amount of time passes between the dosing, correction and filling operations. This reduces the total number of operations to 144 and the number of stationary units to 8.
3.2 The MILP Formulation The key idea of the formulation discussed here was originally proposed by (Manne, 1960) and developed further in (Cerda et. al. 1997). The main contribution is the consideration of additional constraints, simplifications and more complex recipes. The approach uses continuous event points for the starting and the ending dates of the operations instead of time grids and slots. For each pair of operations o and o' that can be processed on the same unit m, a binary variable is introduced to ensure mutual exclusion in resource utilization. It is set to 1 if o precedes o' and set to 0 otherwise. Since resource assignment is also subject of optimization, we define that this binary
1036 variable is 0 if either o or o' are assigned to an alternative unit (and no collision can occur). The resource assignment is modelled by additional binary variables. Starting and ending dates of operations are expressed by real variables. If an operation can be processed on alternative units, further copies of these variables are introduced for each possible unit. 3.2.1 Sets, P a r a m e t e r s a n d C o n s t a n t s
The following sets are used: jobs J , operations O, operations of job j : Oj c_ O, j E J , units (machines) M , alternative units for operation o: M o c_ M ,
operations to be
executed on unit m: 0 m c_ O , operations of job j that can be processed on unit m: Ojm c_ O , j ~ J , m c m .
Release and due dates of jobs: r e l j , d u e j , recipes of jobs: r e c j , j ~ J . Minimum and maximum operation durations on machine m " durom,duro+m,m ~ M , o c 0 m . Three types of timing constraints between starting and ending dates of operations are defined: ss (starting-starting constraints), es (ending-starting)and ee (ending-ending). Every constraint defines a time window [too,m,t+o,m] which limits the difference of the corresponding event points (e.g. the end of the dispersion and the beginning of the dosing). The time windows for the three types of constraints are denoted by: SS--
~SS +
.CS--
~CS +
CC--
.CC+
t
R;o,o E 0 m . The order of subsequent operations of a job is established using e s - c o n s t r a i n t s . A safe time horizon for Big-M constraints is HeN. too,m,too,m,too,m,too,m,too,m,too,m,C
3.2.2 Variables
Machine allocation variables: aom ~ {0,1},m ~ M , o E 0 m . Variables representing starting and ending dates of operations: Som,eom precedence relations of operations:
Poo'm E
E N,m E M,o E 0 m .
{0,1},mc M , o , o '
Variables
to encode
c 0m .
3.2.3 Equations
Every operation must be processed on one machine and started after the release date of
the job" Vo c O" Z
a°m = 1, V j C J , V o c O j " E
I
s°m >- r e l j ,
I
meMo
meMo
Starting and ending dates of operation o on machine m are set to 0 if another machine is assigned to o: V m ~ M , Vo ~ 0 m :Som <_H . a o m ,%m <- H ' a o m • Minimum and maximum operation durations: V m ~ M , Vo c 0 m "Sore + durom . aom
<_eom,Som + duro+m . aom >_ eom .
Timing constraints between operations: v j ~ J , V o , o' e Oj :
Z mEMo Z
m~Mo
Som + too'm <--
Z
So'm ~
meMo' "es+ > Corn + Ioo'm - Z
m~Mo
Z
Som + loo'm --
m~Mo s°' m
Z
m~Mo
m ~ meMo'
ee-
Z m~Mo
Z
%m
+es Z loo'm --
meMo Co'm,
Z
tee+ -> Z Corn + -oo'm
m~Mo
So'm
meMo' e° 'm •
m~a4o,
If o and o' are processed on m and o precedes o' then o' must not precede o: V m • M , V o , o ' • 0 m : Poo'm + Po'om <- aom,Poo'm + Po'om < ao'm,Poo'm + Po'om >- aom +ao'm - 1 .
Operations on the same machine must not overlap: V m • M , Vo, o' • 0 m :
1037 %m - So'm < H(I - Poo'm) + H(2 - aom eo, m -Som < H . Poo'm + H(2 - aom
-
- ao,tn).eom
-So,
m >_
a o , m ) . e o , m - S o r e >_
- H "Poo'm
-
H(2 - aom
-H(I - Poo'm) - H(2 - aom
-
ao,m)
- ao, m )
3.2.4 The o b j e c t i v e f i m c t i o n
The objective function penalizes the accumulated tardiness of the final operations of all jobs. These are the mixing vessel operations MV/ which finish later than the filling operations because of the cleaning of the vessels" min ~ - Z max { d u e / - eMv/m, 0}
o
.jcJ
4. S o l u t i o n P r o c e d u r e
The commercial package GAMS/Cplex was used. Parameter studies of various Cplex parameters yielded that dpriind = 1 clearly increased the solution performance. This setting was used for the procedure described below, the other parameters were set to their default values. The heuristics used here are similar to those in (Cerda et. al., 1997). • H1 - Non-overtaking of non-overlapping jobs: if d u e / < relj, then Vm e M, Vo e O/m,O' c 0i'm : eom < So'm H2 - Non-overtaking of jobs with equal recipes: if reci = recj,/~ d u e / < due/, then Vme M, Vo e O/m,O' e O7,m : eom <_So, m
•
•
H3 - Earliest due date: if d u e / < due/, then Vm c M, Vo c Olm,O' c 0i,,7 :eom < So, m .
Note that H1-H3 preserve enough degrees of freedom to keep the problem non-trivial. i-rl and H2 do not exclude possibly optimal schedules. The 2-step solution procedure is: 1. Apply heuristics H3 to the problem by fixing the corresponding p-variables. Solve the problem and save the integer solution which represents a valid schedule. 2. Relax the variables fixed in step 1. Apply H1 and H2 by fixing the corresponding p variables. Solve the problem using the solution from step 1 as initial solution. The solution obtained in step 1 is always a valid solution in step 2 and therefore can be used as an initial solution. This is because p-variables fixed to 1 in HI and H2 are also fixed in H3. Being the most restrictive heuristics H3 fixes more p variables to 1 than H1 and H2 together. Both steps were implemented within the same GAMS model. 5. O p t i m i z a t i o n
Results
The model and the solution procedure were implemented in the GAMS 21.3 language and solved with Cplex 9.0 on a 2.4 GHz Athlon machine with 1 GB of memory. Both steps of the solution procedure were limited to 20 min. computation time and a optimality gap of 5%. In order to investigate the scalability, several problem instances ranging from 10 to 29 jobs were solved. The job table was sorted according to due dates and for each instance, the first jobs from the table were taken. The results are shown in Table 1. The first column gives the problem size and the next 5 columns show the number of variables and equations of the problem as already reduced by Cplex. The final two columns show the solution times of both steps in seconds. All problem instances could be solved to optimality (accumulated tardiness and lower bound equal to 0) with reasonable computational effort (less than 700 seconds). While the number of real variables does not change between steps I and 2, the number of discrete variables increases considerably when the heuristics are switched from H3 to H I+H2. The first step of the solution procedure was always sufficient to compute optimal solutions. No
1038 better solutions could be derived within the second step. The time spent in step 2 was needed to determine that the optimal solution had been found in step 1. Hence, the EDD heuristics (H3) led to optimal solutions in all test cases. Table 1. Results of both stepsl and 2 in the scalability experiments. #Jobs
#gqus.l+2
#Vars. 1
#Vars. 2
#disc. Varsl
#disc. Vars2
Time l
Time 2
10 14 16 18 20 22 29
9807 19187 25005 31553 39007 47273 81903
1070 1901 2406 2942 3559 4234 6971
1376 2585 3320 4016 4886 5800 9194
762 1457 1898 2378 2923 3526 6043
1068 2141 2812 3452 4250 5092 8266
5.33 20.12 37.87 57.07 126.99 243.71 693.78
0.33 0.77 1.21 1.64 2.00 2.61 5.32
6. Summary, Conclusions and Remarks This work demonstrates a successful application of a MILP formulation for short and mid-term scheduling problems to a real-life example from the chemical industry. An example problem with more than 80,000 equations and 9,000 variables could be solved to optimality in less than 700 seconds. Although less powerful (in terms of modelling strength) than the widely used STN/RTN approach, our model is sufficiently expressive to model sequencing constraints, resource assignment, various timing constraints and complex recipes with parallel operations. Sequence-dependant changeover procedures and more complex objective functions, involving storage costs and makespan, can also be accommodated in the proposed formulation. Unlike STN/RTN, the batch sizes must be planned in a separate pre-processing step. The solution efficiency was improved by introducing a two-step solution procedure involving heuristics. The authors gratefully acknowledge financial support from the EU AMETIST project (IST-2001-35304). References
Cerda. J., G. Henning and I. Grossmann, 1997, A Mixed-Integer Programming Model fbr ShortTerm Scheduling of Single-Stage Multiproduct Batch Plants with Parallel Lines, Ind. and Eng. Chem. Res. 36 (1997), 1695-1707. Floudas, C. and X. Lin, 2004, Continuous-time versus discrete-time approaches for scheduling of chemical processes: a review, Comp. and Chem. Eng. 28 (2004), 2109-2129. Ierapetritou, M. and C. Floudas, 1998, Effective continuous-time formulation ~br short-term scheduling. 1. multipurpose batch processes, Ind. Eng. Chem. Res. 37 (1998), 4341-4359. Kondili, E., C. Pantelides and R. Sargent, 1993, A general algorithm for short-term scheduling of batch operations - i. milp formulation, Comp. and Chem. Eng. 17 (1993), 211-227. Manne, A. S., 1960, On the job-shop scheduling problem, Op. Res. 8, Issue 2 (1960), 219-223. Pinto, J. M. and I. E. Grossmann, 1994, Optimal cyclic scheduling of multistage continuous multiproduct plants, Comp. and Chem. Eng. 18 (1994)~ 797-816. Schilling, G. and C. Pantelides, 1996, A simple continuous-time process scheduling formulation and a novel solution algorithm, Comp. and Chem. Eng. 20 (1996), 1221-1226. Shah, N., C. C. Pantelides and R. Sargent, 1993, A general algorithm for short-term scheduling of batch operations- ii. computational issues, Comp. and Chem. Eng. 17 (1993), no. 2, 229-244.
European Symposiumon ComputerAided Process Engineering- 15 L. Pui~janerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1039
On the State-Task Network" Time Representations Christos T. Maravelias* Department of Chemical & Biological Engineering, University of Wisconsin-Madison 1415 Engineering Drive, Madison, WI 53706, USA
Abstract To model and solve complex Supply Chain problems we study the relationship between the discrete- and continuous-time State-Task Network (STN) representations. We show that the first is a special case of the second. We also propose a new mixed-time representation where the time grid is fixed but processing times are allowed to be variable and span an unknown number of time periods. The proposed scheme accounts linearly for holding and backlog costs, can be extended to handle continuous processes, accounts for due dates at no additional computational cost, and handles variable processing times; i.e. it combines the modelling advantages of both discrete- and continuous-time STN models. Finally, we present strong cutting planes that are used to enhance the solution of the proposed MIP model.
Keywords: Scheduling, State Task Network, Time Representations 1. Introduction The purpose of the paper is to present a time representation for STN models that can be readily used for Supply Chain (SC) optimization. Issues usually neglected in standalone scheduling models become important when scheduling is considered within SC: • Smooth demand patters at the customer-facing nodes are transformed into high demand peaks at the production facilities, due to the various inventory policies applied at the intermediate nodes (warehouses, distribution centres) to account for demand uncertainty. • Demand cannot always be satisfied and backlog costs are introduced to account for unmet demand and/or late deliveries. • Shipments are usually made at fixed time intervals which means that scheduling models have to handle multiple intermediate due dates at low computational cost. • Features such as seasonality of demand, low volume products with large minimum batch sizes, scheduled maintenance and shut-downs require long planning horizons. Thus, the lot-sizing problem (holding vs. changeover costs) becomes very important. To optimize the SC of process industries, therefore, current planning and scheduling methods should be amended to simultaneously account for lot-sizing, holding, backlog, changeover and shipment costs and multiple due dates, while solution methods should be enhanced to effectively solve problems over long time-horizons. The State-Task Network (STN) formulation (Kondili et al., 1993) and the equivalent Resource Task
[email protected]
1040 Network (RTN) formulation (Pantelides, 1994) are two powerful modelling tools that have been refined by several researchers (Schilling and Pantelides, 1996; Ierapetritou and Floudas, 1999; Mockus and Reklaitis, 1999; Castro et al., 2001; Maravelias and Grossamnn, 2003; Sundaramoorthy and Karimi, 2005) and extensively used for planning and scheduling problems. Nevertheless, continuous-time STN models do not account linearly for holding and backlog costs and are computationally expensive for the solution of problems with due dates, while discrete-time STN models are computationally expensive for operations with variable processing times. In this paper, we rigorously study the relationship between the two representations, and explore whether we can circumvent their limitations and combine their advantages.
2. Discrete-time Models: a Special Case of Continuous-time Models A compact form of the continuous-time model (M1) of Maravelias and Grossmann (2003) in Table 1, and the discrete-time model (M2) of Shah et al. (1993) in Table 2 are used as the basis of our analysis. Indices i, j, s, n and r are used to denote a task, unit, state, time point (or period) and shared resource, respectively. Table 1. C o n t i n u o u s - t i m e S T N m o d e l (M1) o f M a r a v e l i a s a n d G r o s s m a n n (2003).
Time Grid TO=0, Tu = H
T.+~_>T. Vn
(1) (2)
Assignment Constraints Z ( ~ W s ~ . , - Z W f . , ) <_1 Vj, Vn
(3)
icl(j) n'<_n
R,.o : R
Wf,,
Vi, gn
BSin_1 + Bpm_, = Bp,,, + Bf,,
B,
Wf,, < B f .
gi, Vn
S~. = S....., + Z p , s B f . + Zp,sBs,. < C. i~O(s)
Vi, Vn
Tf. >7",, + D i , , - H ( 1 - W s i . ) Vi, Vn Activation of binary variable WJj,, * Tf,,,_,
(5) (6) (7)
Vs.Vn
(9) (10) Vr, Vn (11)
i
Tf. < T,, + D~I, + H(1-Ws,,,)
Batch-size and Mass Balance Constraints BiMIN rr r vvsm < B&,, < B~MAXWsm Vi, Vn MAX
, - ~ R f , , . , +~_Rs,,.,
n
MIN
Vi, Vr, Vn
Calculation of duration Di, and finish time T f , D m =aiW&,, + fl~Bs,,, Vi, Vn (12)
(4)
Vi
Vi, Vr, Vn
RZ,.,1 = 7",,.WZ, + 6,,.sBf, ,
n'
ZWsi. = ZWf. n
Utility Constraints Rsi,.,, =Z~,.Ws~,,-6ir.Bsi.
Tf,,,_, > T._, - H(1-Wf,,)
(8)
Vi, Vn
(13) (14) (15) (16)
• Different constraints hold for zero-wait storage
i~l(s)
Table 2. D i s c r e t e - t i m e S T N m o d e l (M2) o f S h a h et al. (1993). T,, = n ( % ) =
nAt
(17)
Vn
n'<_n
2
2ve~,,,,<-1 W,v,
(18)
i d ( j ) n'>n-r+l
Bi~tlN Wsi,, <<-Bsi,, <- B i~ 4X~ si,' Ssn = S ..... 1 +
Vi, Vn
(19)
2 P',Bsi .... , - 2 pi,,.Bs~,, < C~ leO(s)
Vs, gn
(20)
id(s)
n'<_n arn = Z
Z(YirWSin,-[-(~iraSin,)~ i n'>_n-r
a f VIAX
VF, Vpl
(21)
In model (M1), binary Wsin (Wfn) is equal to one if task i starts at (finishes at or before) time point n E{O, 1, ..N}. The batch size of task i that starts at, is being processed at, and finishes at or before time point n is denoted by Bsi,, Bpi, and Bf~, respectively. The amount of state s at time point n is denoted by Ss, and the amount of resource r
1041 consumed by various tasks at time point n is denoted by R,.,,. The processing time of task i is denoted by Di,,. The start time of task i is always equal to time point T,, and thus time matching constraints are used only for the finish time, Tf~, of task i. The same variables Wsm, Bs~,,, S,n and R,.,, are used in model (M2) for uniformity. Model (M2) is obtained from model (M1) if A) the time grid is fixed, and B) the processing times of all tasks are constant multiples of the interval At=H/N of the fixed time grid. The implications of restrictions A and B are shown in Table 3.
Table 3. Restrictions applied to (M1). Restriction Implication Fixed time points Constant processing times Binary Wf,, function of Wsi,, Variable B/5,,function of Bsi,,
Constraint
7",, = nat Vn
(17)
Oi, , = riWsi, ,
(22)
Vi, Vn
Wf,, = Wsi...... Vi, Vn > r, (23) B~,, = Bs,..... Vi, Vn > r; (24) ~,'_
Variable B/in function of Bsi,,
Bp,, = ~.Bs,,,. Vi, Vn (25) n'2 tl- r i + 1
An outline of the reductions that can be applied to model (M1) follows: 1. If the time grid is fixed, equations (1)-(2) are replaced by eq. (17). 2. If we replace Wf,, from eq. (23) into eq. (3), we obtain eq. (18) of(M2). If no tasks that finish after H are allowed to start, eq. (4) is trivially satisfied due to eq. (23). 3. Eq. (5) is the same as eq. (19). If we replace Wf,, from eq. (23) into eq. (6), we obtain eq. (5) for n_>r,; i.e. eq. (6) is removed, if we plug eq. (24) and (25) into eq. (7) we obtain a constraint that is trivially satisfied, and hence dropped. By replacing Bfn from eq. (24) into eq. (8) we obtain eq. (20) of model (M2). Constraints (5)-(8) of (M1), therefore, are equivalent to constraints (19)-(20) of (M2). 4. If we replace Rsi,.,, from eq. (9) and Rf,.n from eq. (10) (which is a function of Wsin and Bsi,, due to equations (23) and (24)) into eq. (11) we obtain eq. (21) of (M2). 5. Constant processing times imply that eq. (12) is replaced by eq. (22). Constraints (13) and (14) are used to enforce the condition t h a t / f Wsi,,=l then Tf,,=T&,,+D~n, otherwise unconstrained, and cannot be simplified further. 6. Constraints (15) and (16) are used to enforce the following condition: if a task that started before point n finishes between T,,_l and T~, then Wf~=l. Variable Wf~, however, is uniquely defined by eq. (23), and constraints (15) and (16) are trivially satisfied, and therefore removed. The reduced model (MI*) consists of equations (17)-(21) of model (M2) and equations (13)-(14) and (22)-(25). Note that variables Wf~, D~, Bfn, Bp~ and Tf~ do not appear in equations (17)-(21), and that for any feasible solution of model (M2) (i.e. equations (17)-(21)), we can find values of variables Wf,,, Di,,, Bf~, Bpi,, and Tf,, that satisfy constraints (13)-(14) and (22)-(25). Moreover, a solution of (MI*) corresponds to a unique solution of (M2). In other words, the feasible regions of model (M2) and the reduced model (M1 *) in the space of variables Wsi~, B&,,, &,, and R,.n (i.e. when variables Wf,,, Din, Bf,,, Bp~,, and Tf,, are projected out) are exactly the same. The discrete-time model (M2), therefore, is equivalent to the reduced model (MI*), which means that model (M2) is a special case of the continuous-time model (M1) when restrictions A and B are applied.
1042
3. A New Time Representation In the previous paragraph, we showed that the assumption of the fixed-time grid is not necessarily coupled with the assumption of constant processing times. This allows us to develop a new time representation, where the time grid is fixed but processing times can be variable. Tasks are forced to start exactly at a time point but are allowed to span an unknown number of time intervals and finish anywhere within a time interval. The proposed MIP model (M3) consists of equations (3)-(17) and has a number of modelling advantages: • Compared to (M 1), it accounts linearly for holding and backlog costs. • Compared to (M 1), it accounts for due dates at no additional computational cost. • Compared to discrete-time models, it does not require the introduction of additional binary variables for the modelling of tasks with variable processing times. • It can be readily extended to account for continuous processes, with very good computational results (not presented in this paper due to space limitations). The only modelling drawback of model (M3) is that it cannot handle zero-wait storage policy between two batch processes. Zero-wait policy, however, is usually applied between two continuous processes, and this case is effectively handled.
4. Computational Improvements Model (M3) appears to "carry the computational curses" of both discrete- and continuous-time representations (i.e. large number of time intervals and poor LPrelaxation gaps). Nevertheless, the fact that the time points are fixed allows us to apply several computational improvements that greatly enhance the solution of model (M3). Note that similar improvements cannot always be applied to general continuous-time models. The most important enhancements are: 1. Tightening of big-M parameters: The inequality Tfn>T~ is always valid, which means that eq. (16) can be removed. We can also use H=O in eq. (14). Moreover, we can use H = D ~ X - A t in eq. (13) and in eq. (15), where D f tax (=~i+fliBiMAX) is the maximum processing time of task i. 2. The minimum Ni MIx and maximum N Y x number of time intervals a task i can span is determined by N/~N=/-Di~N/At ]and NyAX=[-D y X / A t ~, where DiMIN (=a4+fl~B~aylN) is the minimum duration of task i. This allows us to generate the valid inequalities of Table 4 that tighten the formulation. Additional valid inequalities can be developed and used in a branch-and-cut algorithm.
Table 4. Valid inequalities added to model (M3). O< N j~t~
O<_Nf f ~v
Wsi,, < ~ Wf,,+o Vi, Vn
(26)
Wf,, < ~ Ws,,_o Vi, Vn
0>_Ni ~°'~
(27)
O> N)~Lvi
n '<_n
~-'Ws,,. < 1 Vj, Vn --
(28)
ZWf"' < ZWsi"" Vi, Vn n'<_n
(29)
n"<_n - N ~ ~ *
i e I ( j ) n ' > n - Ni ~aN
Finally, the balance constraints of Sundaramoorthy and Karimi (2005) can be used for the calculation of the remaining processing time on an equipment unit, instead of big-M constraints ( 13)-(16).
1043
5. E x a m p l e and C o m p u t a t i o n a l Results Model (M3) is used for the scheduling of the STN in Figure 1 (modified from Kondili et al., 1993; data given in Table 5). The scheduling horizon is 10 hours and there are three orders for 20 kg of P 1 each (with intermediate due dates at t = 4, 6 and 8 hr). The objective is to meet the orders on time and maximize the revenues from additional sales. ) P1 ($10/kg) 40% ,I
-D
4O%
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o% IntBC
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1 B
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I
'
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i e°°'°
($10/kg)
C Figure 1. State-Task Network.
Table 5. Processing data (BAn"V/B~4vin kg, a in hr, fl in hr/kg).
Unit HT RI RII DC
Task ~ B 'vnx B~4.v 50 100 25 50 40 80 100 200
H
a 1 .
fi 0 .
a . 2 2 .
R1 fi . 0 0 .
R2
a . .5 .5 .
R3
fl .
.
.04 .025 . .
a . .25 .25 .
fl .02 .0125 .5
.01
To model intermediate due dates using a continuous-time model we have to introduce additional binary variables. In model (M3), however, we only introduce a (demand) parameter in eq. (8). The example is solved using both the continuous-time model (M1) with N=8, 9 and 10, and model (M3) with At=0.5, 0.33 and 0.25 hr (solution statistics given in Table 6). Model (M1) cannot be solved to optimality within 600 CPU sec, and the best solution has an objective value of $1,797.3 (found with N=8). However, a solution with an objective value of $1,805.4 is obtained using model (M3) with At= 0.25hr (N = 40) in 542.9 CPU sec. Model (M2), in other words, yields a better solution in less time than an "accurate" continuous-time model. Moreover, good solutions are also obtained with At = 0.5 ($1,720) and At = 0.33 ($1,763.3) in less than two minutes. The Gantt chart of the optimal solution of (M3) with At = 0.25 hr is shown in Figure 2. Table 6. Computational results of models (M1) and (M3) (At in hours).
(M1)
(M3)
N=8 N=9 N=IO At=0.5 At=0.33 At=0.25 LP-rel. 2,498.9 2,539.0 2,561.4 2,081.4 2,054.7 2,095.9 Objective 1,797.3 ~ 1,783.61 1,788.4 ~ 1,720.0 1,763.3 1,805.3 CPU-sec 2 600 600 600 43.1 76.3 542.9 Nodes 114,651 71,106 65,620 2,368 1,534 5,106 Optimality gap (%) 2.03 8.71 12.3 0.5 0.5 0.5 1 Best solution found after 600 CPU sec. 2 Using GAMS 21.3/CPLEX 9.0 on a Pentium M at 1.7 GHz; solved to 0.5% optimality gap.
1044 H RI RII
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Figure 2. Gantt chart of optimal solution of (M3) with At=0.25 hr.
The scheduling of batch process remains a very hard problem and better methods are needed, especially for the solution of problems over long scheduling horizons. Nevertheless, model (M3) can be used for the solution of batch scheduling problems with intermediate release and due times, for which continuous-time models cannot be solved to optimality within reasonable time. Furthermore, it can be used for the solution of problems where holding and backlog costs are important. Finally, the proposed representation can be extended to account for continuous processes, with very good computational results. More details can be found in Maravelias (2005).
6. Conclusions In this paper we formally show that discrete-time STN models are a special case of continuous-time models. A new mixed time representation (fixed grid, variable processing times) is proposed. Computational enhancements for the solution of the proposed representation are also presented.
References Castro, P.; Barbosa-Povoa, A. P. F. D.; Matos, H. An Improved RTN Continuous-Time Formulation for the Short-term Scheduling of Multipurpose Batch Plants. Ind. Eng. Chem. Res. 2001, 40, 2059-2068. Ierapetritou, M. G.; Floudas, C. A. Effective Continuous-Time Formulation for Short-Term Scheduling. 1. Multipurpose Batch Processes. Ind. Eng. Chem. Res. 1998, 37, 4341-4359. Kondili, E.; Pantelides, C. C.; Sargent, R. A General Algorithm for Short-Term Scheduling of Batch Operations - I. MILP Formulation. Comput. Chem. Eng. 1993, 17, 211-227. Maravelias, C. T.; Grossmann, I. E. A New General Continuous-Time State Task Network Formulation for the Short-Term Scheduling of Multipurpose Batch Plants. Ind. Eng. Chem. Res., 2003, 42(13), 3056-3074. Maravelias, C. T. A Mixed Time Representation for State Task Network Models. Submitted for Publication (2005). Mockus, L.; Reklaitis, G.V. Continuous Time Representation Approach to Batch and Continuous Process Scheduling. 1. MINLP Formulation. Ind. Eng. Chem. Res. 1999, 38, 197-203. Pantelides, C. C. Unified Frameworks for the Optimal Process Planning and Scheduling. Proceedings on the Second Conference on Foundations of Computer Aided Operations. 1994, 253-274. Shah, N.; E.; Pantelides, C. C.; Sargent, R. A General Algorithm for Short-Term Scheduling of Batch Operations- II. Computational Issues. Comput. Chem. Eng. 1993, 17, 229-244. Schilling, G.; Pantelides, C. C. A Simple Continuous-Time Process Scheduling Formulation and a Novel Solution Algorithm. Comput. Chem. Eng. 1996, 20, S1221-1226. Sundaramoorthy, A.; Karimi, I.A. A Simpler Better Slot-based Continuous-time Formulation for Short-term Scheduling in Multipurpose Batch Plants. Chem. Eng. Sci., In Press, 2005.
Acknowledgements The author would like to thank Professor Ignacio Grossmann for stimulating discussions on the time representation of STN models.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjaner and A. Espufia (Editors) © 2005 Elsevier B.V. All rights reserved.
1045
Optimization of Biopharmaceutical Manufacturing with Scheduling Tools - Experiences from the Real World Charles A. Siletti a*, Demetri Petrides a and Alexandros Koulourisb aIntelligen, Inc. New Jersey, USA bIntelligen Europe Thermi-Thessaloniki, Greece
Abstract This paper presents industrial experience with a resource-constrained batch process scheduling program. The batch process representation is loosely based on the ISA $88 batch process standard. This representation allows the import of batch process information from other software, e.g. batch process simulators. The scheduling algorithm is a non-optimization approach that proceeds in two steps. First a bottleneck analysis is done to determine a lower bound on the process cycle time, and all the batches are scheduled accordingly. Second, if conflicts remain, they are resolved by applying progressively aggressive modifications to the schedule. This approach to scheduling was tested on several biotech processes. These processes consist of a sequence of batch steps performed with dedicated equipment. The scheduling challenges in biotech processes lie in the ancillary operations: media and buffer preparation, vessel and line cleaning, and chromatography column preparation. Such operations may use shared resources and may serve to couple process suites with otherwise dedicated equipment. These considerations are further complicated by variability in process durations. Three case studies, which are based on a process for the manufacture of monoclonal antibodies (MABs), illustrate the value of a constrained-resource scheduling tool for biotech processes. In the first case study, the scheduling tool shows that auxiliary cleaning equipment can limit batch production. A second case study shows how scheduling tools can calculate the size of a purified water system. A third case study illustrates how to use scheduling tools to mitigate the effects of process variability.
Keywords: scheduling,
process modelling, biotech, pharmaceutical manufacture
1. Introduction Biotechnology manufacturing capacity is currently in very high demand. In recent years the estimated capacity utilization has been 90% for both microbial and mammalian cell culture production (S. Fox et al., 2001). Some biotech firms have estimated potential revenue losses of well over $100 million due to lack of Author to whom correspondence should be addressed:
[email protected]
1046 manufacturing capacity (R. Rouhi, 2002). Thus there is substantial motivation to improve process efficiency in biotechnology manufacturing.
1.1 Bioprocessing Overview A typical biopharmaceutical process consists of two parts: an upstream process in which a living organism produces the product in a raw form and a downstream process in which the product is purified. Most biotech upstream processes employ either microbial fermentation or mammalian cell culture. From a scheduling viewpoint, biotechnology processes generally have the following features: • They are batch processes • Primary processing equipment is dedicated to a particular processing step • Wait-time between process steps is either zero or limited • From 20 to 30 buffers may be made for each batch and each has a limited life • There is some variability in processing times especially in the upstream processes • Equipment cleaning is common after most process steps and often requires auxiliary clean-in-place (CIP) skids, which are not dedicated Biopharmaceutical manufacture is regulated, and manufacturers need to prove, through studies or clinical trials, that a process change will not adversely affect the product. Manufacturers therefore tend to avoid any direct changes in the process itself.
1.2 Scheduling Challenges For most bioprocesses, scheduling the main process does not pose a significant challenge. Because the steps usually proceed with no wait, the timing is fixed when the batch start is fixed. Scheduling challenges arise in the support operations such as cleaning and buffer preparation. Such support operations may impose unexpected limits on the process. Common utilities, such as purified water, may impose scheduling limits because they impose limits on both peak and average resource consumption. Finally, planning for process variability and failures presents a significant challenge.
2. A Scheduling Tool for Bioprocessing Most bioprocess manufacturers employ spreadsheets for process scheduling because spreadsheets are inexpensive, readily available, easy to learn and highly flexible. For more complicated situations, however, spreadsheets have very clear drawbacks including poor visualization tools and poor maintainability. Furthermore, spreadsheet solutions are usually "owned" by an individual and may be difficult to transfer to another individual or to another site. The scheduling tool and approach described in this section maintains many of the advantages of the spreadsheet approach while minimizing the disadvantages. Pekny and Reklaitis (1998) describe a generic scheduling tool consisting of graphical user interface, a representation layer, and problem formulation and solution layers.
2.1 The Interface The interface should provide both an easy means of entering and maintaining scheduling information and appropriate displays of the scheduling outputs. The equipment occupancy chart is the most popular way to display scheduling information
1047 for bioprocesses. Equipment occupancy charts, as shown in Figure 5, display Ganttstyle time-bars for equipment.
2.2 The Representation Layer The representation consists of a model for the process and its associated resources and constraints. The instructions for making a single batch of a product constitute a recipe. The recipe structure, which is loosely based on the ISA SP88 standard, consists of unit procedures and operations. A unit procedure is a primary process step and is assigned a single unit of primary equipment. Operations are individual tasks within a unit procedure. All resources other than primary equipment, i.e. auxiliary equipment, labor, materials and utilities, are associated with operations. Operation durations may be fixed or rate and/or equipment dependent. Rate dependent durations are linearly dependent on the batch size. Scheduling precedence relationships exist among operations. Specifically, an operation may have any of the following timing dependencies: (1) the operation starts at the beginning of the batch, (2) the operation starts simultaneously with a reference operation, (3) the operation starts at the end of a reference operation or (4) operation finishes at the start of a reference operation. In addition to the relationships above, an operation may have a fixed or flexible shift time. A fixed shift indicates the time after (or before) which the dependency condition is fulfilled that the operation actually starts. A flexible shift indicates the maximum time that an operation may be delayed.
2.3 The Formulation and Solution Layers There is an abundance of proposed problem formulations and solution algorithms for scheduling problems. The goal of the scheduling methodology described below is to allow a user to generate and evaluate a schedule interactively. The user enters a plan with one or more campaigns each of which consists of a number of batches of a particular recipe and either the due date or start date. The system lays out a preliminary schedule using the cycle-time analysis and conflict resolution methodology. The user may edit the resulting schedule. The system schedules a campaign by scheduling batch starts according to the estimated cycle time. The cycle time for a recipe is the average time between consecutive batch starts. The minimum cycle time, Tcycle, is estimated by the following relation from Biegler et al. (1997). Tcycle = Max(Ti/Ni) j o t i = (1, M)
(1)
77 is the duration of unit procedure i, Ni is the number of alternative primary equipment units, and M is the number of unit procedures. This estimate does not account for shared primary equipment, auxiliary equipment, or down-time, so a conflict resolution scheme is employed to ensure that the resulting schedule is feasible. Conflicts are resolved by (1) selecting other equipment, (2) adding a flexible delay, or (3) delaying the start of a batch.
3. B i o p r o c e s s i n g Case Studies The case studies draw on a process for producing a monoclonal antibody product. The process, shown in Figure 1, consists of multiple upstream suites and a single
1048 downstream suite, which is detailed in Figure 2. The upstream process is limited by the production bioreactor, which has a processing time of 12 days. Multiple upstream suites allow for a batch every 48 hours, while the downstream suite has a minimum cycle time of 33 hours. The limiting equipment in the downstream process is the buffer hold vessel, DCS- 103. Upstream(multiplesuites) Tcycle= 48 h
Downstream(singlesuite) T c y c l e = 33 h
I I..... lum H Bio- ~ Prep. reactor
PrimaryR ...... y H naietPor H I..... hange H Hydrophobic k l Final I Viralinactivation Chromatography Chromatography Chromatography Filtration Figure 1. Monoclonal Antibody Process
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Figure2. MAB Downstream Details
3.1 Case Study 1 CIP Skids The objective is to schedule the downstream process to accommodate upstream improvements that would reduce the upstream cycle time to 36 hours. When the system is given a target cycle time of 36 hours, it reports that the process may not be scheduled. In fact target cycle times of 46, 36 and 35 hours aren't met. Target cycle times of 48, 45, and 34 hours, however, are achieved. The equipment occupancy chart in Figure 3 reveals the problem. CIP-SKID-1 is conflicting from batch to batch. The cleaning skid is used to clean the bioreactor harvest tank, V-101 and is required at the same time to clean the IEX elution buffer tank in the second batch, so the second batch is delayed at the expense of the cycle-time target.
1049 +~
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The key to resolving the problem lies in understanding where to add flexibility. While a delay in a step that affects the product would probably not be allowable in a biopharmaceutical process, a short delay in cleaning a vessel would be acceptable. In this example, a delay of up to 2 hours before cleaning the buffer tanks allows any cycletime target greater 33 hours.
3.2 Case Study 2 Estimating Water for Injection Requirements Purified water known as water for injection (WFI) is used extensively in bioprocesses both for the process and for cleaning vessels. A WFI system consists of a still, surge tank and circulation system. The still capacity surge vessel requirements are dependent on the production schedule. A plot of WFI consumption, shown in Figure 4, gives the basic design parameters. Under peak conditions a four-hour WFI turnover is chosen. The plot shows the instantaneous consumption (light red), the 4-hour average consumption rate (dark blue) and the 4-hour cumulative consumption (brown). The approximate still capacity can be set to peak average rate (9,000 L/h) and the vessel size to the peak 4-hour consumption (35,000 L).
~ i ................i
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3.3 Case Study 3, Scheduling for Uncertainty An important source of uncertainty in bioprocesses arises from variability in bioreactor durations combined with the long processing times. In the MAB process above, the completion time of an upstream batch may easily vary by a day.
1050 An analysis of the cycle time shows that a 24-hour delay in the upstream process will cause a conflict in the downstream schedule. The upstream cycle time is 48 hours, and the downstream cycle time is 33 hours. A 24 hour delay in the upstream process reduces the time between consecutive branches as follows: ( 4 8 h - 24h) < 33 h. As noted earlier, the cycle-time limiting procedure is a buffer hold step. Buffers are normally made up about a day in advance and held. See the DCS equipment in the first batch in Figure 5. In an emergency, buffer preparation could reasonably be delayed as long as the buffers are ready in time for the chromatography. The user interactively resets the start times for the buffer preparation steps in batch 3 and shifts the start of batch 2 by 24 hours as shown in Figure 5. "(:i!f::!~:~:£1 : i q "
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Figure 5. Schedule for a Delay in the Second Batch
4. Conclusion For most bioprocesses, tight constraints dictate much of the scheduling. Under such conditions interactive scheduling tools can deliver a considerable benefit even if they do not generate mathematically optimized schedules. References Biegler, L. T., I. E. Grossmann and A. W. Westerberg, 1997, Systematic Methods of Chemical Process Design, Prentice Hall, Upper Saddle River, New Jersey, 721. Fox, S., L. Sopchak and R. Khoury, 2001, A Time to Build Capacity, Contract Pharma, September. Pekny, J. and G. Reklaitis, 1998, Towards the Convergence of Theory and Practice: A Technology Guide for Scheduling~Planning Methodology, In Proceedings of Foundations of Computer-Aided Process Operations, J. Pekny and G. Blau, Eds., AIChE, 91. Rouhi, R., 2002, No Vacancy, Chemical and Engineering News, 80, 7, 84-85.
European Symposiumon ComputerAided Process Engineering- 15 L. Pui~ianerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1051
Advances in Robust Optimization Approaches for Scheduling under Uncertainty Stacy L Janak a and Christodoulos A. Floudas a* aDepartment of Chemical Engineering Princeton University Princeton, NJ 08544-5263
Abstract The problem of scheduling under uncertainty is addressed. We propose a novel robust optimization methodology, which when applied to Mixed-Integer Linear Programming (MILP) problems produces "robust" solutions that are, in a sense, immune against uncertainty. The robust optimization approach is applied to the scheduling under uncertainty problem. Based on a novel and effective continuous-time short-term scheduling model proposed by Floudas and coworkers (Ierapetritou and Floudas 1998a, 1998b; lerapetritou et al. 1999; Janak et al. 2004; Lin and Floudas 2001; Lin et al. 2002, 2003), three of the most common sources of uncertainty in scheduling problems can be addressed, namely processing times of tasks, market demands for products, and prices of products and raw materials. Computational results on a small example with uncertainty in the processing times of tasks are presented to demonstrate the effectiveness of the proposed approach.
Keywords: Process scheduling, uncertainty, robust optimization, MILP 1. I n t r o d u c t i o n The issue of robustness in scheduling under uncertainty has received relatively little attention, in spite of its importance and the fact that there has been a substantial amount of work to address the problem of design and operation of batch plants under uncertainty. Most of the existing work has followed the scenario-based framework, in which the uncertainty is modeled through the use of a number of scenarios, using either discrete probability distributions or the discretization of continuous probability distribution functions, and the expectation of a certain performance criterion, such as the expected profit, which is optimized with respect to the scheduling decision variables. Scenario-based approaches provide a straightforward way to implicitly incorporate uncertainty. However, they inevitably enlarge the size of the problem significantly as the number of scenarios increases exponentially with the number of uncertain parameters. This main drawback limits the application of these approaches to solve practical problems with a large number of uncertain parameters. A recent review of scheduling approaches, including uncertainty, can be found in Floudas and Lin (2004). Author to whom correspondence should be addressed:
[email protected]
1052 In this work, we propose a novel robust optimization approach to address the problem of scheduling under uncertainty. The underlying framework is based on a robust optimization methodology first introduced for Linear Programming (LP) problems by Ben-Tal and Nemirovski (2000) and extended in this work for Mixed-Integer Linear Programming (MILP) problems.
2. Problem Statement The scheduling problem of chemical processes is defined as follows. Given (i) production recipes (i.e., the processing times for each task at the suitable units, and the amount of the materials required for the production of each product), (ii) available equipment and the ranges of their capacities, (iii) material storage policy, (iv) production requirement, and (v) time horizon under consideration, determine (i) the optimal sequence of tasks taking place in each unit, (ii) the amount of material being processed at each time in each unit, (iii) the processing time of each task in each unit, so as to optimize a performance criterion, for example, to minimize the makespan or to maximize the overall profit. The most common sources of uncertainty in the aforementioned scheduling problem are (i) the processing times of tasks, (ii) the market demands for products, and (iii) the prices of products and/or raw materials. An uncertain parameter can be described using discrete or continuous distributions. In some cases, only limited knowledge about the distribution is available, for example, the uncertainty is bounded, or the uncertainty is symmetrically distributed in a certain range. In the best situation, the distribution function for the uncertain parameter is given, for instance, as a normal distribution with known mean and standard deviation. In this paper, we will discuss bounded uncertainty as well as uncertainty with a known distribution.
3. Robust Optimization for MILP Problems Consider the following generic mixed-integer linear programming (MILP) problem:
Min / Max crx + d r y x,y
s.t.
Ex + Fy = e Ax+By<_ p x L <x<x
y =
(1)
U
0,1}
Assume that the uncertainty arises from both the coefficients and the right-hand-side parameters of the inequality constraints, namely, aim, blk and Pl. We are concerned about the feasibility of the following inequality.
Zat~xm +Zbt~yk < p, m
(2)
k
Our objective here is to develop a robust optimization methodology to generate "reliable" solutions to the MILP program, which are immune against uncertainty. Two types of uncertainty are addressed, (i) bounded uncertainty and (ii) uncertainty with a known distribution.
1053 3.1. Bounded Uncertainty Suppose that the uncertain data range in the following intervals:
[a,m-a,,~ I<-Ela,,~ I, Ib,~.-b,~. I<_cIb,~. I, I~,-P, <-cIp, [
(3)
a~,,,, blk and Pl are the "true" values, al,,,, b/a and Pi are the nominal values, and e > 0 is a given (relative) uncertainty level. We call a solution (x,y) robust if: (i) (x,y) is feasible for the nominal problem, and (ii) whatever are the true values of the coefficients and parameters within the corresponding intervals, (x,y) must satisfy the /-th inequality constraint with an error of at most 6 max]l, p~], where 8 is a given infeasibility tolerance. Given an infeasibility tolerance 8, to generate robust solutions, the following so-called (~,8)-interval Robust Counterpart (IRC[~,8]) of the original uncertain MILP problem can be derived.
Min / Max
cTx + dry
X, y , It
s.t.
Ex + Fy - e Ax+By<_ p m
m ~M I
_< p~
x
L
k
kcK t
(4)
- c[p~ [+,5" max]l, ]p~ [], Vl
<x m
Vm
y~ - {0,1}, V k
where Ml and /£1 are the set of indices of the x and y variables, respectively, with uncertain coefficients in the /-th inequality constraint. The derivation of this formulation can be found in the full-length manuscript of Lin et al. (2004). Note the mathematical model given in (4) remains an MILP model and compared to the original deterministic MILP problem, the robust counterpart has a set of auxiliary variables Um and a set of additional constraints relating the variables Xm and u,,,. 3.2. Uncertainty with a Known Distribution Assume that in inequality constraint l, the true values of the uncertain parameters are obtained from their nominal values by random perturbations"
a~m - (1 + g~:l,, )aim,
blk - (1 + g~:,k )b,k,
Pl - (1 + ~ 4 z ) P l
(5)
where ~'/,,, ~'/a and ~:1are independent random variables and e > 0 is a given (relative) uncertainty level. In this situation, we call a solution (x,y) robust if: (i) (x,y) is feasible for the nominal problem, and (ii) for every l, the probability of the event
Z ~ImXm -+-Z b/kYk > Pl -3t-(~max]l,] P t ] m
is at most K, where 8 > 0 is a given
k
feasibility tolerance and K > 0 is a given reliability level. If the distributions of the random variables ~l,,,, ~lk and ~:t in the uncertain parameters are known, it is possible
1054 to obtain a more accurate estimation of the probability measures involved. Denote a new random variable ~ as the following:
-- Z
~'m l alm I Xm + ~ ~Zk I b/k l Y k - ~z I P, I
mcM l
(6)
k~K l
Assume that the distribution function of ~ is"
F;(A) - Pr{~ _ 2} - 1 - P r { ~ > 2} - 1-~c
(7)
where • is a given reliability level and the inverse function (quantile) can be represented as follows:
f~ -1 ( 1 - K:) - f(,Z,I aim
I Xm,I bzk I y~-,I p,
1)
(8)
Then, given an infeasibility tolerance, 6, and a reliability level, K, to generate robust solutions, the following so-called (e,6,~:)-Robust Counterpart (RC[~,8,K]) of the original uncertain MILP problem can be derived. The additional constraints in the RC problem:
~-'~atmXm + ~ btkyk +~f(A,I alm I xm,I btk l Yk,I p, I) ,,
k
(9)
__ p, + 6 max[1,l Pt I], v/ Several different distribution functions can be modeled this way including the uniform distribution, normal distribution, difference of normal distributions, and several discrete distributions such as Poisson or binomial (Janak et al., 2005). This robust optimization methodology can be applied to address the problem of scheduling under uncertainty, including three classes of problems: (i) uncertainty in processing times/rates of tasks, (ii) uncertainty in market demands for products, and (iii) uncertainty in market prices of products and raw materials. In this work, we will only consider uncertainty in the processing times/rates of tasks.
4. E x a m p l e P r o b l e m Consider the following example process that was first presented by Kondili et al. (1993) and has been widely studied in the literature. Two products can be produced from three feeds according to the state-task network as shown in Figure 1. The objective is to maximize the profit from sales of products manufactured in a time horizon of 12 h. The continuous-time formulation proposed by Floudas and coworkers (Ierapetritou and Floudas 1998a, 1998b; Ierapetritou et al. 1999; Janak et al. 2004; Lin and Floudas 2001; Lin et al. 2002, 2003) is used to solve this simple scheduling problem. The example is implemented with GAMS (Brooke et al., 1988) and solved using CPLEX 8.1 on a Linux 3.0 GHz workstation. The "nominal" solution is shown in Figure 2, which features intensive utilization of units U2 and U3 and an objective value (profit) of 3639. However, this solution can become completely infeasible when there is uncertainty in the processing times of the tasks. Consider the case where the uncertainty of the processing times is bounded and the (relative) uncertainty level, ~, is 15% and the infeasibility tolerance level, 8, is 10%. Then, by solving the IRC[~,8] problem, a "robust" schedule is obtained, as shown in Figure 3, which takes into account
1055 .Product 1
Heating ~
l
I
IntAB
4 o.4
( ~
Io.~ r , ,
Reaction 2 [ . . . . ~ ) ~,
Feed A
0.6
re E
I Feed B
0"8lReacti°n31
I
T0.5
0.1
_I
0
0.2
Product 2
Feed C
Figure 1. State-task networkjbr the example problem.
uncertainty in the processing times. Compared to the nominal solution, the robust solution exhibits very different scheduling strategies, such as task-unit assignments and task timings. The robust solution ensures that the robust schedule obtained is feasible with the specified uncertainty level and infeasibility tolerance. However, the resulting profit is reduced, from 3639 to 2887, which reflects the effect of uncertainty on overall production. A comparison of the model and solution statistics for the nominal and robust solutions can be found in Table 1. [30.00
U4 89_00 T2 50_00 T2
It? U2
52_00 T[ I
U1
80_00 T3 50_C~0 W3
I I 20_(210 T[ t
1
52_0 0 T1
."
I
80.00 T4
8C~_00 T2
I 5":u:u') T3
I
[
80_00 T3 50_0C~ T3
50_00 T4
[ I
4
5
6
8
7
T5
I 68.75 I T4 50_00 T4 I [ 6_00 T1 I
I
[
3
[ [8.75
T5
I
9
40:30 T3
10
11
12
Figure 2. Optimal solution with nominal processing times.
97_ 50
L[,4
I 80.0 (-i T2 50.00 T2
U3 U2
l_0 0_0 0
U1
T[
II II 3 6_0 0
II-- T[ t
0
8,0_(J(-i T3 50.130 T3
1
80.00 T3
il I I
50.00 T4
I~-
i 47.5 (-I T4 I I
72_ 1_9
T5
I
I 2219 T4
]
74.00 T2
I
T5
50.00 T4
I
I
80_ 00 T3 50.00 T3
II II
I I
i i
,c. "~
t
i
t
i
L
i
I
3
4
5
6
7
8
•~.
I
10
i
11
Figure 3. Robust solution with uncertainty level (~)of 15%, infeasibilit)' tolerance (0")of 10%.
5. Conclusions In this work, we propose a new approach to address the scheduling under uncertainty problem based on a robust optimization methodology, which when applied to MixedInteger (MILP) problems produces "robust" solutions which are in a sense immune
1056 Table 1. Model and solution statistics for the example problem.
Profit CPU Time (s) Binary Variables Continuous Variables Constraints
Nominal Solution 3638.75 0.40 64 346 489
Robust Solution 2887.19 10.10 64 346 713
against uncertainties in both the coefficients in the objective function, the left-hand-side parameters and the right-hand-side parameters of the inequality constraints. A unique feature of the proposed approach is that it can address many uncertain parameters. The approach can be applied to address the problem of production scheduling with uncertain processing times, market demands, and/or prices of products and raw materials. Our computational results show that this approach provides an effective way to address scheduling problems under uncertainty, producing reliable schedules and generating helpful insights on the tradeoffs between conflicting objectives. Furthermore, due to its efficient transformation, the approach is capable of solving real-world problems with a large number of uncertain parameters (Lin et al., 2004). References Ben-Tel, A. and A. Nemirovski, 2000, Robust solutions of Linear Programming problems contaminated with uncertain data, Math. Program. 88, 411. Brooke A., D. Kendrick and A. Meeraus, 1988, GAMS: A User's Guide, San Francisco, CA. Floudas, C.A. and X. Lin, 2004, Continuous-Time versus Discrete-Time Approaches for Scheduling of Chemical Processes: A Review, Comp. Chem. Engng. 28, 2109. Ierapetritou, M.G. and C.A. Floudas, 1998a, Effective Continuous-Time Formulation for ShortTerm Scheduling: 1. Multipurpose Batch Processes, Ind. Eng. Chem. 37, 4341. Ierapetritou, M.G. and C.A. Floudas, 1998b, Effective Continuous-Time Formulation for ShortTerm Scheduling: 2. Continuous and Semi-continuous Processes, Ind. Eng. Chem. 37, 4360. Ierapetritou, M.G., T.S. Hene and C.A. Floudas, 1999, Effective Continuous-Time Formulation for Short-Term Scheduling: 2. Multiple Intermediate Due Dates, Ind. Eng. Chem. 38, 3446. Janak, S.L., X. Lin and C.A. Floudas, 2004, Enhanced Continuous-Time Unit-Specific EventBased Formulation for Short-Term Scheduling of Multipurpose Batch Processes: Resource Constraints and Mixed Storage Policies, Ind. Eng. Chem. Res. 43, 2516. Janak, S.L., X. Lin and C.A. Floudas, 2005, A New Robust Optimization Approach for Scheduling under Uncertainty: II. Uncertainty with Known Distribution, submitted for publication. Kondili, E., C.C. Pantelides and R.W.H. Sargent, 1993, A General Algorithm for Short-Term Scheduling of Batch Operations - I. MILP Formulation, Comp. Chem. Engng. 17, 211. Lin, X., E.D. Chajakis and C.A. Floudas, 2003, Scheduling of Tanker Lightering via a Novel Continuous-Time Optimization Framework, Ind. Eng. Chem. Res. 28, 2109. Lin, X. and C.A. Floudas, 2001, Design, Synthesis and Scheduling of Multipurpose Batch Plants via an Effective Continuous-Time Formulation, Comp. Chem. Engng. 25,665. Lin, X., C.A. Floudas, S. Modi and N.M. Juhasz, 2002, Continuous-Time Optimization Approach for Medium-Range Production Scheduling of a Multiproduct Batch Plant, Ind. Eng. Chem. Res. 41, 3884. Lin, X, S.L. Janak and C.A. Floudas, 2004, A New Robust Optimization Approach for Scheduling under Uncertainty: I. Bounded Uncertainty, Comp. Chem. Engng. 28, 1069.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1057
Proactive approach to address robust batch process scheduling under short-term uncertainties A. Bonfill, A. Espufia and L. Puigjaner Universitat Polit6cnica de Catalunya, Chemical Engineering Department, ETSEIB, Diagonal 647, E - 08028 Barcelona, Spain e-mails: [anna.bonfill, antonio.espuna, luis.puigjaner]@upc.edu
Abstract A contribution is made in the area of proactive scheduling with the aim to properly define the scheduling problem explicitly incorporating the effects of short-term uncertainties. The idea is to identify a robust initial schedule with the flexibility to react to unexpected events with minimum effects. The problem is modelled using a stochastic optimization approach where not only a set of anticipated scenarios can be considered, but also the capability to react to events once they occur. A stochastic genetic algorithm is developed to efficiently identify robust schedules with minimum expectance for the wait times and idle times that commonly arise in the operation of batch processes with variable operation times and machine breakdowns. The application of the proposed modelling framework to different batch processes shows the flexibility of the identified initial schedule and highlights the importance of exploiting the information of the uncertainty at the decision stage.
Keywords: Proactive schedule, rescheduling, robustness, uncertainty. 1. Introduction Process variations and incomplete information are inherent characteristics of any process system, and flexibility to respond quickly and effectively to the dynamic and uncertain environment has become an essential feature for effective scheduling. Research in scheduling under uncertainty has mostly been focused either on reactive scheduling algorithms, implemented according to the actual situation of the plant once the uncertainty is realized or unexpected events occur, or on proactive scheduling approaches, which tend to generate schedules that are in some sense robust or insensitive to a priori anticipated uncertainties. The execution of optimal schedules based on nominal parameter values and the implementation of rescheduling strategies to face disruptions could result cumbersome and might lead to inefficient or costly reconfigurations as well as to plant nervousness. On the other hand, a robust schedule exhibits an optimum expected performance, but it is not likely to be the optimum one for the actual scenario that will finally occur. Both methodologies have usually been implemented independently, and relatively little attention has been given to the consideration of short-term uncertainties proactively (O'donovan et al., 1999; Kim and Diwekar, 2002; Jensen, 2003).
1058 The incorporation of rescheduling aspects at the time of scheduling is proposed in this work. The identification of a robust initial schedule with the flexibility to react to unexpected events with minimum effects is pursued by explicitly addressing the major effects of variable processing times and equipment breakdowns in short-term scheduling of batch processes. These effects can be characterized by their main consequences in terms of task scheduling. On one hand, if a breakdown occurs and/or the actual processing time of a task is longer than the scheduled one, the time spent by batches waiting for the availability of the next unit increases. This might lead to unexpected delays, and eventually result in quality problems for sensitive or unstable materials that force the rejection of batches with the consequent increase of operational costs. On the other hand, if processing times are shorter than the scheduled ones, idle times appear and subsequent equipment under-utilization occurs (Figure 1). The approaches proposed so far that recognize the importance of considering the uncertainty into the decision level do not explicitly address not even analyze these critical situations that can arise during the execution of the schedule. However, the knowledge of this uncertainty can be usefully incorporated at the time of scheduling to reduce the gap between theory and practice, thus reducing reschedule requirements and improving the overall plant performance avoiding the occurrence of the full force of a perturbation. It is highly desirable to balance the trade offs between robustness, rescheduling and performance, and develop an initial schedule able to absorb anticipated disruptions, thus minimizing their effect on planned activities while maintaining acceptable plant performance. Predicted schedule
°, {ii!iii{i{
o~
jii{gi~iiiig~iigii
u2
~:~ii~ii"ii~i~ii~~
................... ..............the scheduled ones "";...................... WAIT TIME (wt) []
--, wt .................. u l i:~i~ii~i~'~i~ii~{~ iiiii~i~ ...........
I
wt
o'it/ca/for unstab/e materia& ...........Shorter times than ....................... the scheduled ones
U2
~,<~,~D~.--it
......
~::~J~J~i::i<~!~J} ...........
poor p/ant uti/izat/on
Figure 1. Effects of processing times variability.
2. Problem statement The scheduling problem addressed comprises a multistage batch plant with a set of orders to be fulfilled, the set of processing stages required by each order, a set of units where they can be processed, the operations required by each stage, and the processing times of each operation, represented by probability distributions. This information has been modelled using the Process-Stage-Operation hierarchical approach defined by the standard ISA-$88 (ANSI/ISA, 1995), which provides a standard terminology and hierarchical structured models for batch processes.
1059
Due to the uncertainty in the operation times a detailed schedule is not pursued, but only the minimum information to be released to the batch process control is established. This information is related to the production sequence, the assignment of units to stages and the initial (expected) processing time of each process. The effects that may arise due to the uncertainty are explicitly managed by minimizing a weighted combination of the expected makespan and the expected wait times resulting from the execution of the schedule under a set of anticipated scenarios. The following assumptions are made: • From the schedule, the control level (ISA $88) requires only information related to the sequence, the assignment of units to stages and the processes start times. Then production proceeds according to the control recipe. • The Non-Intermediate Storage policy (NIS) between stages is assumed. • Within each stage, all the operations must be executed without interruption. When at the end of a stage (or before a transfer operation) the next unit is not available, a wait time has to be introduced. On the other hand, if a unit is available before the time it is required by the next stage, an idle time appears. Uncertainty associated with operation times can be represented indistinctly by discrete or continuous probability distributions. The scenario-based representation of the uncertainty is then adopted by sampling over all the probability space to approximate the expectation of the objective function. Other sources of short-term uncertainties can be easily incorporated within the proposed framework at the expense of a larger number of scenarios required to represent significantly all the uncertain space. 3. M o d e l l i n g The integrated framework for planning and scheduling of batch chemical plants developed by Cant6n (2003) is used in this work for the modelling of the system. A set of tools, integrating both heuristic rules and optimization algorithms, are available to establish the number of batches to be performed, the sequence, and the assignment of production stages to specific units. Based on the characteristics of the problem and the specified objective function, different strategies combining these tools can be used. I
[ Genetic
Algorithm
OPTIMIZER ~ . ~
MODELER
Makespans
Wait times S
/ /
/
New
L
!
I Generation ["
Individuals OF/Fitness
TOPs
MODEL ~
•
I I
~
Mks/Wts I
"' ~
Y
' evaluation l
................. ~ .................... Uncertain ~ - -
Figure 2. Stochastic optimization framework.
Selection Crossover
Term~nabon
_ ~ e crlterlon r ~ F
~
........C.!~°ve.r....P.r.°!~ a.b.i.!!t~...............................................................................................................
EEMk+ Wt]
:i:~: ,~
S a m p l i n g loop
Mutation probability Crossover method Mutationmethod
1% overlapping
Schedule min
Initial batch start times Production sequence Assignment
E[Makespan + Wait time] \
Populationsize I Numberof generations
~ .
be£
Parameters
~ • ~-Sampling I / I ........ -
~ Schedulewith , I rain E[Mk+Wt] k
Parameters (scenario)
Figure 3. stochGA solution procedure.
1060 A stochastic genetic algorithm (stochGA) has been designed and implemented within this scheduling framework. The GAlib C++ library (Wall, 1996) has been used with customised genome classes. Each individual of the population identifies an initial schedule, and is encoded using a mixed representation involving a real-valued string for the initial batch times, a permutation representation for the sequencing decisions, and a string of integer values for the assignment. For the reproduction process, suitable operators have been implemented in each part of the solution vector. The developed stochGA involves two recursive loops (Figure 2), and the algorithm proceeds as represented in Figure 3. There is an outer optimization loop controlled by the genetic algorithm, which directs the search to the identification of the initial times of each batch, the sequence and the assignment decisions that minimize a weighted combination of expected makespan and wait times, thus introducing robustness features as pointed out before. This outer procedure incorporates an inner sampling loop within which a set of probable scenarios are anticipated by sampling over the probability space to evaluate the probabilistic fitness for each individual. Specifically, the expected performance of each individual (schedule) is evaluated by computing for each scenario the wait time and makespan values that would occur when implementing its sequence, assignment and initial times under the assumed rescheduling policy; that is, the capability to react to events once they occur is considered when evaluating the individuals. The number of scenarios required to obtain a given accuracy for the actual mean and standard deviation of the performance measure is assessed.
4. Results The proposed methodology has been applied to a case study based on a process plant involving 3 production stages and 8 operations. A scheme of the process is shown in Figure 4. Uncertainty in the operations of loading, heating and discharging has been introduced and characterized with a uniform distribution function. Two different products can be produced and five orders have been considered for scheduling.
IPr°~andkUrel
3f-~~ ~
~FprOcedure3~ Reactor k 1// Pr°cedure "l Reactor2/
....
Transfer
k
J
i/ ii iiiiit heating dischargeI clean
Procedure1 load I hold Transfer ) time
Figure 4. Case study.
The scheduling problem was solved using the proposed stochastic modelling and assuming a right-shifting rescheduling strategy. For comparison purposes, the deterministic problem considering only one scenario with nominal operation times was next solved. The production sequence, the assignment and the processes start times thus obtained were fixed, and the makespan and wait time values that would arise after the occurrence of each particular scenario were computed. The results obtained are summarized in Table 1. Figures 5 and 6 depict the schedules that would be executed in the plant for the nominal scenario and for one of the randomly-generated scenarios, respectively.
1061 Table 1. Production sequence, initial batch times, expected makespan and wait time, and makespan and wait times in the nominal scenario (Mk ........ Wt......,), .for the deterministic and stochastic optimized schedules.
Batch
Deterministic Product
Tin
Stochastic Product
Tin
1
A
0.0
A
0.0
2 3 4 5
B B A A
5.6 l l.2 19.4 25.3
B A B A
5.8 13.8 19.8 25.6
Mk .....i~al Wt .....inal
39.0
39.7
0.0
0.0
From these results it can be observed that the deterministic modelling overestimates the system performance. Although the makespan and wait time values of the deterministic schedule are optimal in the nominal scenario, when the schedule is used to face the uncertainty the expected makespan and wait time value raises about 4%. The schedule identified with the stochastic approach shows a better expected performance over the anticipated scenarios of processing times. Despite the simplicity of the analyzed case study, and the relatively small variability associated with the uncertain operations, it is important to notice the consequences of neglecting the known uncertainty and the quick loose of optimality when implementing a deterministic schedule. Concerning the rescheduling features, different policies can be followed when the uncertainty is revealed or unexpected events occur besides avoiding changes once the execution of a schedule has already started: • resolution of the new scheduling problem from scratch, • right-shifting, generating the new schedule from the initial one just delaying the operations affected by the event, more sophisticated rescheduling methods.
Reactor1
'
.
.....
JI'
]
.
.
.
.
i
.
.
'J °
Reactor2 . . . . .
Tankl:]::]tliti
. . . . . . . . . . . .
, !] lt:i
(a) Makespan" 39; wait time: 0
(b) Makespan: 39.7; wait time" 0
Figure 5. Schedules that would be executed ![the nominal scenario.finally occurred according to (a) the deterministic and (b) the stochastic optimization approaches.
Particularly, it can be considered that once a breakdown or an unreasonable wait time is detected, tasks can be just right-shifted or reassigned to alternative units, batches can be
1062 immediately rejected and new ones ordered, thus avoiding unnecessary wait times. In such a case, this knowledge related to the rescheduling policy should be incorporated proactively at the time of scheduling to improve flexibility and plant performance. Reactor1
+ JBmi~ I~1__~+,+++o+,+++ ++°+++++~+,+l~++ .........+++~ t +~ +li
Pu+ ................... _j ...........+ ...............++ ...................... + +..............+ +................................ ~+ .
(a) Makespan: 39.7; wait time: 1
.
.
.
.
.
..
(b) Makespan: 39.7; wait time: 0
Figure 6. Schedules that would be executed in a particular scenario according to (a) the deterministic and (b) the stochastic optimization approaches.
5. C o n c l u s i o n s A stochastic modelling and optimization approach is proposed in this work to address the processing times uncertainty arising in scheduling of batch processes. A robust initial schedule is identified which shows reduced expected wait times and acceptable line occupation, thus reducing eventual quality problems or unexpected delays. The applicability of the proposed framework highlights the importance of exploiting the information of the uncertainty at the decision stage by incorporating not only anticipated scenarios but also suitable reactions to improve the flexibility and the final quality of the schedule's overall performance.
References Cant6n J., 2003, Integrated support system for planning and scheduling of batch chemical plants, PhD Thesis, Universitat Polit6cnica de Catalunya, Espafia. (http://www.tdx.cesca.es/TDX0707103-125200/index_cs.html) International soc. for measurement and control, ANSI/ISA-S88.01-1995, Batch Control, Part I: Models and Terminology. Jensen, M.T., 2003. Generating Robust and Flexible Job Shop Schedules using Genetic Algorithms. IEEE Transactions on Evolutionary Computation, 7, 275 - 288. Kim. K., Diwekar, U.M., 2002 Efficient Combinatorial Optimization under Uncertainty. 1. Algorithmic Development. Ind. Eng. Chem. Res., 41, 1276- 1284. ) O'donovan, R., Uzsoy, R., McKay, K.N., 1999. Predictable scheduling of a single machine with breakdowns and sensitive jobs. Int. J. Prod. Res., 37, 4217 - 4233. Wall, M., 1996. Galib: A C++ library of genetic algorithm components. (http://lancet.mit.edu/ga) Acknowledgements Financial support received from the spanish Ministerio de Educaci6n, Cultura y Deporte (FPU research grant to A. B.) and Ministerio de Ciencia y Tecnologia (project DPI2002-00856); from the Generalitat de Catalunya (project I0353); and from the European Community (contracts GIRD-CT-2000-00318 and MRTN-CT-2004-512233) is fully appreciated.
European Symposiumon Computer Aided Process Engineering- 15 L. Puigjaner and A. Espufia (Editors) ,9 2005 Elsevier B.V. All rights reserved.
1063
A Rigorous MINLP for the Simultaneous Scheduling and Operation of Multiproduct Pipeline Systems R. Rejowski Jr. a, J. M. Pinto a'b* ~Department of Chemical Engineering, University of Sio Paulo Av. Prof. Luciano Gualberto, t.3,380 - Silo Paulo, SP, Brazil .05508-900 bChemical and Biological Sciences and Engineering, Polytechnic University 6 Metrotech C e n t e r - Brooklyn, NY, USA 11201
Abstract The problem addressed in this work is based on a system that is composed by a refinery that transfers multiple products through a single pipeline connected to several depots. The depots must supply local consumer markets with large amounts of oil derivatives. The objective of this work is to develop a rigorous formulation for pipeline scheduling. This novel formulation for pipeline scheduling uses a continuous time representation and also considers a rigorous hydraulic approach for pipeline operations. The MINLP formulation is compared to a hydraulic linear formulation that parameterizes flow and yield rates. The results show that the MINLP presents better solutions and computational performance.
Keywords: pipeline, scheduling, hydraulic, continuous time representation.
1. Introduction Pipelines transfer large amounts of different petroleum types and their products at a lower cost than any other transportation mode. Pipeline scheduling is a challenging operation due to their multiproduct nature and to the large amounts of energy required. Westerlund et al. (1994) solve the optimal pump configuration that consists of a set of series and parallel pumps. Van den Heever and Grossmann (2003)show a hydrogen supply chain network where optimal flow rates are obtained from a detailed hydraulic representation. Neiro and Pinto (2004) and Rejowski Jr. and Pinto (2004) present mixed-integer models for planning and scheduling of pipeline operations, respectively. The objective of this work is to develop a hydraulic formulation for pipeline scheduling. The formulation is based on a continuous time representation that handles variable flow rates in the pipeline.
2. Problem Description A refinery must transfer P petroleum products to D depots connected to a single pipeline. The depots have to satisfy product demands determined by local consumer
Author to whom correspondence should be addressed:
[email protected],
[email protected]
1064 markets. Furthermore, several booster station configurations may be utilized for pumping the products along the time horizon (Figure 1).
[--]0=1
~ p=p Him]
1;:;::::;::::::i::: : i: : ::::i :: ::::1
~-~¢~..¢~.
.~/"
.......
-~'kt.. 7"..: k = 1 !::_:.. I k=0 I
..:.
.;.:..............................................,.Q~,. "-.,
'~..,,]c=2
............... 577515 T;77TC777/T7;5TT77;7>>!.CUUT,,,,75;.7/#7..~=,~"
I I
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....................................................................... ~< .................. i...;...;..,.;..~.,.i..A':R'~'/~--~', .................. ,/ .........i ! ~ , ~ " i ~ A . ; "
/
! 5-/
i
Rermery
I~{"~~ ~L'JF~ '(beginning /~k=0
....................................................................................................................................................................................................... Q# [ m 3 / h ]
of the' operation)
Figure 1 - Distribution pipeline system The refinery stores its products in dedicated tanks and these load their derivatives at predetermined amounts and initial and end time instants along the time horizon. Whenever the pipeline receives any product from the refinery, the same amount leaves it and feeds the depots at the same time in order to satisfy volume and flow rate constraints. The depots have to control their inventory levels and fulfill product demands. Due to demand variations, the pipeline operation is in most cases intermittent. Figure 1 also shows the system and the pump operating curves. The hydraulic behavior depends on the sequencing of products and their allocation inside the pipeline, the flow rate variations, the topographical profile of the pipeline and diameter variations. The pump curves operating depend on the pump types, adjustable speeds, and represent the energy that is provided by the booster stations to transfer the products. Moreover, the pipeline operating point is given by the intersection of these two curves.
3. Optimization Model The MINLP formulation is based on the previously MILP developed by Rejowski Jr. and Pinto (2004), which was however based on a discrete time formulation. The objective function is shown by equation (1).
CERp.VRp,k +
Min C = ~ k=l P
p=l D
C E D p , d . V D p , j , k .A k p=l d=l
K
P
P
D
K
+2 Z Z CP~.OPWd.k.Ak +Z 2 Z Z CONTACTp,p,.TYpa,p,,k p = l d = l k=l
p = l p'= d = l k=l
(1)
1065 The terms in brackets represent the inventory costs at the refinery and at the depots that take into account the variable time interval duration (A~). The next term accounts for the pumping costs, that are composed by the overall power consumption by the booster stations (OPWa.~.) and the time interval that the pipeline remains in operation (&). The last term corresponds to costs associated to product interface formation (Rejowski Jr. and Pinto, 2004).
3.1 Temporal and refinery constraints The refinery sends the products to its tanks in lots (i) with predefined amounts and initial and end instants, as shown in Figure 2. The temporal constraints must satisfy the operational time horizon (/-/), according to equation (2) and their initial (Ti~) and end instants (Tf ) along the transfer operation, as in equations (3) and (4), respectively. The mass balances at the refinery are given by (5) and their volumes are bounded in (6). ~:i T'- 1
R e r m e ~ w P r odu c t i o n L o t s
[
i = 1 R P I , I,~
:
7)
T~/Tf~: slta11.,,'end t h n e foi~ loading operation for product lot i composed by p r o d u c t p at t h n e / e l K ~ : Set of t i m e h l t e r v a l s that receive a product lot f r o m t h e
1
f
2
rermel.w p r o d u c t i o n
.#eL,
/~O: 1 ~ t t i m e h l t e r v a l b~fore production lot )'~ , : 1~1[ t i m e h l t e r v a l olr production lot i
I#1
--
i
I
" Tim e ~
Figure 2 - Production lots and continuous time representation K
~-'A k - H
(2)
k:l
77i
Vi
(3)
/"k - T~
Vi
(4)
Vp, d:l,k
(5)
Vp, k
(6)
~" A k
-
k _
i
VR,~-VRZERO,+~_, ~_, (RP,.:.~,xA~-XR ~,xUj) i I k'~k
VRMIN ~r < VR ~. < VRMAX k
3.2 Pipeline scheduling and depot constraints The idea of this approach is to assign logical variables XVp.l,k ~: to each (product, depot, pack, time). The pipeline operation can be expressed by disjunction (7) (Rejowski Jr. and Pinto, 2004) that is composed by two additional terms that relate the flow rate variations and the time interval durations. This linear disjunction can be transformed into mixed-integer constraints. Boolean variable Yd,k controls the pipeline segment operation and is expressed as logical 0-1 variable XSa.~ in mixed-integer form.
1066 Y
t
,
k
d
,
k
d = XTp,d,k X Vp,l,k
gp,l = 1
XVj,,. k = X V p,l-l,k-1 J
gp, l = 2 ..... L,
XVt;~,,,k = XV~'/,,,k_,
V p , l = 2 ..... L d
XDp,d.k + XTp,,,+,,k= XVj.,~, ,k-1
~7fP
XDv.j. k + XTp.j+,. k = 0
Vp
Vp,~-
A STOP LO < A k < n cp -STOP
A, = U d / ( v k . A ) L O < Vk < _V k -_
xv;',,, = xv;',,,,
UP
_v k = 0
vk
Vd> 1, k=2,...,K
(7)
If YJ,k is true, the first pack of the segment takes the feed variable values (XTp.d,k or d = 1). The products in the pipeline are displaced from pack 1-1 to pack 1. The product stored at pack Ld can be sent to depot d (XDp,d,k) o r to the next segment (XTp.J+l.a). The velocity limits (vk) must be respected when the pipeline operates and its time interval (&) is equal to the ratio between the pack size (Uj) and the flow rate used at time k. If the segment remains idle, Yj,a is false and all packs keep their contents. Interfaces are detected when the first two packs from the segments store different products. Moreover, any pipeline segment can only be stopped if it does not contain any interfaces. The depot constraints are similar to the ones of the refinery and demands must be met at the end of the operation. Finally, Rejowski Jr. and Pinto (2004) propose integer cuts that relate demands and the initial inventory at the depots and in the segments. The temporal variables are disaggregated into two parcels. The first one (Ale) regards
XRp,k for
the pipeline operation, whereas the second ( A ,2) considers the time that the pipeline remains idle. Equations (9) and (10) state that the time that the pipeline operates at interval k must be equal to the ratio between the pipeline volume pack (Ud) and its operational flow rate
(v,.A).
Equation (11) imposes zero values to A~k whenever the
pipeline remains idle. The product velocities must take positive values when the pipeline is operative, according to (12). A~ = A~1 + A~2
vk
(8)
iX'k" (v,.A) = U~.X'S~
Vd, k
(9)
Vd, k
(10)
Vd, k
(11)
Vd, k
(12)
AL°sToP'~I - X S a
uP k] < - A 2h -< A s r o p ' I 1 -
vkL°.XSd, ~ -< vk -< vUe'xsjk
,k
X S a ,k
3.3 Flow rate and hydraulic constraints The hydraulic model is described by disjunction (13). If Yj,~ is true, the friction factor
(fd,/,k) follows equation (13a), where logical 0-1 variable XVpdt,k is multiplied by an exponential that depends on the product physical properties. The friction losses for each pack I take into account the friction factor, the pipeline internal diameter ( D i n ) , the pack extension ( L E X T I ) and the pipeline flow rate, according to (13b). The yield rates (r/k) used by the booster stations are represented by an mt-h order polynomial that depends on
1067
the velocity. The booster stations have several stages (N) that can be activated during the operation from binary variable ( NS,;~k ) and their power consumption ( PW,~/~ ) must satisfy lower and upper bounds, according to (13d). Finally, the energy balance over the pipeline is given by (13e) and the overall power consumption from (1) is given in (13f). If the pipeline remains idle, the friction factor and losses, the power consumed by the booster stations and the yield rates equal to zero, as shown by (13g) to (13k).
J]LZ.k = ,~__,XV:/.Lk.(a,,'vk ' ~' )
VI < Lj
(13a)
_
Y/,k
P
lw/;,.,., = LEX7],.,.f,.z , .,,~ /2.Di,,
,,,--cm.(.,)'"
+c,,,, (.,)
.... '+...+c .... .(,,) ..... +...+co ,,3c)
PW]°.NSj,k <_PWII~, < P L,I
(13b)
V! <_ L,
'"~
VI < L, (13g)
/w/]L,./< = 0
VI_< Lj (13h)
11/` = 0
(13d)
~.,g.p,,.(zj~-zj~ ,)+ ~,h.t'/]L,.~, vkA = l-1
PWJ~,,,
(13e)
Vd, k
(13i)
W,.~ = 0
L,/
[ 0
./],.,.~ = 0
Vn
OP Wj,k = 0
(13j)
( 13k)
_
t~1
OPW~/'k = £PWJ/' ~
(13f)
"'~
4. Examples This section addresses two examples that are composed by a depot. In both examples the proposed MINLP (M1) is compared with an MILP proposed by Rejowski Jr. and Pinto (2003) (M2). In M2, the pipeline operates with fixed flow and yield rates and uses a discrete time representation. Data for this example are given in table 1. GAMS/CPLEX/DICOPT++ (Brooke et al., 2000) was used to implement and solve the models. Examples E1 and E2 consider a 100 and 150 hours time horizon, respectively. Table 1 - Data for Example Folwmlation
Number (?['Time Intervals (K) E1 (H=I00 hours)
MINLP (MI) MILP (Rejowski Jr. and Pinto, 2003) (M2)
Production Profile Product Gasoline (1) Diesel Oil (2) LPG (3) Jet fuel (4)
14 20 (5 hours each) 50 first hours 50 last hours
(p= 1, p=2) (1)=3,p=4) CERp[$/m3.h] CED ¢,.j[$/m'.h] RPp,i,z [xl0-= m3/h] 0.010 0.009 0.005 0.007
0.0240 0.0250 0.0240 0.0250
Product
VRZERO~,[xl0 -2 m 3]
2.5 2.5 1.5 1.5 VDZERO¢,.j[xlO-=m ~]
Gasoline (1) Diesel Oil (2) LPG (3) Jet fuel (4)
1,000 1,050 400 700
150 200 200 200
E2 (H=150 hours) 15 30 (5 hours each) 100 first hours 50 last hours
(p= 1, p=2) (p=3, p=4) CONTACTp4,,[xl0 -2 $]
(1,2)/(1,3)/(1,4) 100/150/120 (2,1)/(2,3)/(2,4) 100/150/120 (3,1)/(3,2)/(3,4) 150/X/X (4,1)/(4,2)/(4,3) 120/X/X XVZEROt,.~u DEMp,d [xl0-= m 3] El/E2 0 /=1,2,3,4,5,6,7,8 0 0
170/150 100/100 10/110 50/50
Table 2 shows the objective function values as well as the computational performance for both examples and models.
1068 Table 2 - Results for the proposed example Example - Model E 1.M 1 E 1M2 Relaxed Solution (C) 26,042.2 8,372.3 Solution Found (C) 26,060.2 38,247.5 Gap (%) 0.000 CPU Time [s] 197.6 3,544.7 Number of Equations 2,286 5,296 Number of Variables 1,537 3,217
E2.M 1 34,927.2 45,865.1 532.9 2,448 1,647
E2.M2 10,326.01 103,756.8 0.8100 54,900.0 7,936 4,827
The MILP presented by Rejowski Jr. and Pinto (2003) (M2) has several limitations. Firstly, it requires a large number of time intervals to synchronise the loading and unloading tasks. Consequently, a much larger model is created that increases CPU times. Furthermore, model M1 demands fewer time intervals for the operation and allows variations in the flow and yield rates within the time horizon. Another important aspect of formulation MDH besides its fixed time interval duration is that it can also provide flow rates that result in low yield rates for the booster stations during the horizon due to the tasks synchronisation at the refinery, as shown in Figure 2. Model E 1.M 1 found a solution that is 46.8% higher than model El.M2, whereas model E2.M1 found a solution with total costs 126% higher than model E2.M2. These factors result in formulations with fewer time intervals and with better objective function values for this challenging problem. Figure 3 shows the yield and flow rates for examples E2.M1 and E2.M1. Note that in E2.M1 the pipeline remains idle in the first 50 hours, whereas in E2.M2 it does not operate at the end of the operation. ~-MI •,I,- y l , l[I ~ l e
E2-M? ~, 11o4o r:.,le ~-6
6.'r
6.T
i66
,.';4.
"r66
,,66 •
6
J66
~';
. . . . . .
-- ,k,~
J66
6
Figure 3 - Yield and flow rates for E2.M 1 and E2.M2
5. Conclusions This work addresses the simultaneous multiproduct pipeline scheduling and hydraulic operation. The resulting MINLP showed better results than a previous MILP.
References Brooke, A., Kendrick, D., Meeraus, A. A., 2000, The Scientific Press. Redwood City, USA. Neiro, S., Pinto, J. M., 2004, Comp. & Chem. Eng., 28, 871. Rejowski Jr.,R., Pinto, J.M., 2003, AIChE Annual National Meeting [446y], San Francisco, USA. Rejowski Jr., R. & Pinto, J.M., 2004, Comp. & Chem. Eng., 28, 1511. Van den Heever, S.A.; Grossmann, I.E., 2003, Comp. & Chem. Eng., 27, 1813. Westerlund, T., Petterson, F., Grossmann, I.E., 1994, Comp. & Chem. Eng., 18, 845.
Acknowledgements The authors would like to thank FAPESP (01-10944/9) and CAPES (BEX1312/03-0).
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) ~i 2005 Elsevier B.V. All rights reserved.
1069
Multicommodity transportation and supply problem with stepwise constant cost function Zoltan Lelkes a*, Endre Rev a, Tivadar Farkas a, Zsolt Fonyo a, Tibor Kovacs b and Ian Jones b aBudapest Univ. of Technology and Economics, Department of Chemical Engineering H- 1521, Budapest, Hungary bSABMiller Europe P.O. Box 53, H-1553, Budapest, Hungary
Abstract A new model has been developed for multicommodity transportation and supply chain problems including stepwise constant costs. The model is expressed as an MILP problem. The modelling equations are presented here. The new model has been tested on multicommodity problem of SABMiller Europe, and compared to other methods from the literature. A feasibility checking method has been developed for large scale MILP problems having binary variables in the objetive function only. This feasibility checking is made by solving a special relaxed LP problem; and the most probable physical reason is pointed out by the feasibility check results in case of infeasibility.
Keywords: supply chain, transportation, multicommodity, MILP 1. Introduction The transportation problem of a commodity deals with a set of source sites, and a set of destinations. The task is to satisfy all the demands of the destinations and utilize all the supply of source sites with minimum transportation cost. This is the basis of much more complicated problem classes, like supply chain allocation and distribution. There is no standardized model for supply chain allocation problems accepted in the literature. According to Vidal and Goetschalckx (1997) the problem involves the determination of the number, location, capacity, and type of manufacturing plants and warehouses to use; the set of suppliers to select; the transportation channels to use; the amount of raw materials and products to produce and ship among suppliers, plants, warehouses, and customers; and the amount of raw materials, intermediate products, and finished goods to hold at various locations in inventory. Here we consider a much simpler problem class where several commodities are produced in several plant sites with capacity constraints, and distributed to several destination sites according to demands and transportation constraints. We deal with the special case where fix cost contribution to the objective can be expressed as a stepwise
*Author to whom correspondence should be addressed:
[email protected]
1070 constant function. The stepwise cost function is a consequence of working with shifts. In case of the usual constraints, the task can be expressed as a mixed-integer linear programming (MILP) problem, where integer variables are applied for deciding on opening the plants, and on operating the manufacturing lines inside the plants, and on the number of necessary shifts in each plant. The logic or binary decisions are usually dealt with some special techniques, like Big-M or Convex Hull. The models found in the literature usually do not consider stepwise constant functions, although Ttirkay and Grossmann (1996) deal with stepwise nonlinear (i.e. discontinuous) cost functions. Neither do they utilize the tendency of costs to monotonously increase (more shifts are more expensive). Initialization is also difficult with the literature models. We have developed a new model that utilizes the main characteristics of the presented problem class, but not too specialized that would prevent it from being useful in similar cases.
2. Problem description Given a set of plant sites (simply called plants), customer sites (customers), and products. Customers have specified demand for products, which has to be supplied. Products are produced in plants, and packed in the finite number of packaging lines of the plants. Plants have the capabilities to produce certain types of products only. The processing and packaging capacity of the products in the plants are given. Figure 1 illustrates the problem for three plants, three packaging lines in each plant, and for ten customers. Processing plants
Packaging lines
Customer sites
Figure 1. Problem description The target is to minimize the sum of distribution costs, production costs, and packaging costs. The distribution costs are proportional to the amount of transported products. Production and packaging costs consist of variable cost, which is proportional to the amount of the products, and of fix costs. Fix costs are given by stepwise constant functions, as is shown in Figure 2. Intervals for defining the production fix costs can be given e.g. according to the utilization of the processing plant, and in each interval a constant fix cost can be specified. Similar cost functions can be given for packaging
1071 lines, according e.g. to the working shifts. The steps in the production fix costs are normally monotonously increasing with exception of a single point at zero capacity that may be higher that zero and express the cost of closing a plant.
Q
bc3
bc2
o
bco
x
I
p
I
)
1
o ~
bc~
I I I I I
i I I !
bao
bal
I I I I I I I I
I I I I I I I I I
I ! ! I I I I I ! ! I I
I ! I I I ! I ! ! I ! I
ba2
ba3
Amount of product
Figure 2. Shape of thefix cost.fimction The main difficulty is solving a large scale problem with discontinuous objective function. Discontinuity is handled with logic or binary variables; and the computational labour and time increase exponentially with the number of such variables.
3. Model formulation Our intention was possibly applying the best relaxation to the discontinuous equations and variables. Because the problem is of a large scale one, assignment of good initial values are also important. Since we deal with a large set of constraints, finding feasible initial values is not a trivial task. Finally, we have to assign as tight bounds to the variables as possible. We work as follows. The logical decisions on operating plants and shifts are transformed to algebraic equations, in MILP form. The fix cost (C) of the plant depends on the amount of product (A). The range of variable A is divided into a number of intervals i ( i ~ I ) . The borders of these intervals are given by the upper bound of interval i, bai. The constant fix cost of the plant in interval i is given by parameter bci. Binary variables )'i are assigned to the intervals. If the value of the independent variable is in or above interval i then yi=l, else y~=0. Thus, if the value of the independent variable is in the kth interval ( k ~ I ) then y;~= 1, and yi>z.=0. This can be expressed by Equation 1, where variable Da~ is the length of interval i. Equation 2 expresses monotinicity, i.e. the cost cannot be smaller than the cost of the kth interval. Equation 3 specifies that the intervals have to be considered one after the other.
icI
1072 (2)
C>-Z(yi'Dci) ici
i~{ili~I,i>O} i~{ili~I,i-O}
Yi <- Yiq
Yi --0
(3) (4)
where Dai and Dci are parameters specified by Equations 5-9.
Da i -ba i -bai_ 1
i ~ {i l i ~ I, i > 1}
(5)
D a i - ba i
i ~ {i [ i ~ I , i - 1}
(6)
Oa i -
0
i ~ {i l i ~ I , i -
0}
(7)
DC i -
bc i -bc,_
i ~ {i l i ~ I , i > 0}
(8)
i ~ {i l i ~ I , i - 0}
(9)
Oc i
1
-- b c i
Tight bounds to all the non-decision variables are computed, based on the problem formulation, from bounds given to the decision variables, see Chapter 5. Note also that Equations 1-2 need not be given as equality constraints because of the monotonity of the cost function, and because the cost is minimized during the optimization.
4. Example problems Results of test runs on a middle scale and a large scale problems, given by SABMiller Europe, with charactesistics summarized in Table 1, are presented here.
Table 1. Characteristicsof the exampleproblems Problem Small Large
Processing plants 3 25
Packaging lines Products in a plant 3 13 5 100
Customer sites 67 250
Steps of cost function of plants lines 6 4 5 6
Table 2. Comparison of models Problem
Small Large
Model Multi-M Ttirkay Convex Hull New model New model
Number of equations 4,035 3,927 3,981 3,852 671,626
Number of Number of variables binary v. 5,632 66 5,632 66 5,764 66 5,632 66 1,281,201 875
Number of Solution time iterations (CPU sec) 3,384 1,046 2,350 750 1,121 453 543 312 437,364 19,357
The middle scale problem is used to compare the efficiency of some usual model formulations applied to the given problem type. The problem was solved by GAMS (Brooke et al., 1998) using CPLEX as MILP solver on a PC Pentium 4 CPU 2.4 GHz.
1073 The results are collected in Table 2. The same optimum was found in all the cases. The number of iterations and the solution time in CPU sec are shown in the last two columns. The Ttirkay model is a forward development of the Big-M technique. The Convex Hull technique applies tighter bound / better relaxation than either Big-M or Ttirkay, that is why the solution properties improve according to this sequence. Our new methodology utilizes the monotonicity property; that must be the main reason of the improvement. The lower row in Table 2 demonstrates that large scale problems become solvable with our suggested model formulation. The problem was solved using the same solver on the same machine as above. The solution was found with 1,33% relative gap between the best integer and the best relaxed solution.
5. Feasibility check and solution methodology Checking feasibility may involve examining all the binary combinations in general case. Our special formulation, however, applies binary variables in the terms of the cost function only; and a relaxed LP problem (RLP) can be generated by excluding those terms from the cost function. Any (LP) problem, see below, can be extended (and called LPV) by introducing vn (negative perturbation) and vp (positive perturbation) variable arrays: m
min min
w-
z - cx Ax
- b = 0
x > O,b > O, x e R "
,-:1
(LP)
vp-
m
~-'vp,. + ~ v n , . vn + Ax
,.:1 - b = 0
(LPV)
x > O,b > O, vp > O, v n > _ O xeR",vpeRm,vneR
m
where m is the number of equations. LPV always has a solution; LP has a solution if, and only if the optimum of LPV is w:0; if the optimum of LPV is w=0, vp*=O, vn*=O, and x*, then x* is a feasible solution of LP. If w:/:O, vp*:/:O, and/or vn*:/:O then RPV is infeasible. Which element(s) of the array v=[vn, vp] is(are) nonzero tells us which constraint(s) is(are) voilated. If there were not minimum capacity utilizations specified in the original problem then the solution of RLP would be always a feasible solution of the original problem, as well. But such minimum utilizations are specified, and binary variables related to the existence of plants cannot be excluded, involving a rather difficult problem. Instead, we check the feasibility of the most probable binary combination only; this is the case that all the plants included in the model work with some capacity. The final program is illustarted in Figure 3. The problem data are collected in MS Acces, and transformed into GAMS readable format using mdb2gms.exe (Kalvelagen, 2004). The GAMS model has three main parts. (1) First the feasibility of the problem is checked using LPV. If w:/:0 solution is found then the program stops, and reports the value of the nonzero perturbation variables. (2) RLP is solved in the other case, and provides with proper initial values for the variables. (3) Finally, the original MILP, formulated according to the new modelling equations, is solved. The results of the GAMS run is transformed into MS Access format using GDXViewer (Kalvelagen,
1074 2004). The result data can be read in MS Access, or it is transformed into graphical form by MS MapPoint. This latter form is illustrated in Figure 4. with a theoretical example including 4 processing plants and 24 customers. Circles are assigned to the customers; their size visualise the total demand of the customer, whereas circle sectors represent what parts of the demand are satisfied from different sources.
i• .........
. . . . . . . . .
Data in mdb2gms.I MS Access I ~
-"
I
J'] Solve LPV~
I
es
i •
2[Initialization
I. . . .
___
l[
Solvethe
I
I .
i
i
i
• GDX Viewe~
i
i .
i
,
~
.
i
.
i
.
i
.
i
.
i
=
~
.
i
,
i
w
~
MS Access
+
q
MS MapPoint Figure 3. Algorithm of the final program
.!!!:¸~: :i:i ..e.
::::::::::::::::::::::::::: ::::::::::P:::::::::::[
Figure 3. Graph visualization of the results of a theoretical example
6. Conclusions and recommendations The new model works well for the studied problem with objective function including terms with stepwise constant cost functions. Test on middle case problems resulted in better computation properties than Big-M or Convex Hull, and large scale problems can also be solved with it. The relaxed formulation (RLP) in its transformed form LPV, together with the elaborated GAMS program, can be successfully applied to check the feasibility of the problem prior to trying MILP solution. When RLP is infeasible, the results of LPV provides useful information on the possible reasons of the infeasibility. References
Brook, A., D. D. Kendrick and A. Meeraus, 1992, GAMS. A User's Guide. boyd & fraser, USA Kalvelagen, E., 2004, http://www.gams.com/-erwin/interface/wtools.zip Tt~rkay, M. and I.E. Grossmann, 1996, Ind. Eng. Chem. Res. 35, 2611-2623. Vidal, C.J., and M. Goetschalckx, 1997, Eur. J. Op. Res. 98, 1-18. Acknowledgements
This study was partially supported by OTKA T037191 and OTKA F046282.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1075
Design and Planning of Supply Chains with Reverse Flows Maria Isabel Gomes Salema a, Ana Paula Barbosa Pdvoa bY, Augusto Q. Novais c aCMA, FCT, UNL, Monte de Caparica, 2825-114 Caparica, Portugal bCEG-IST, Av. Rovisco Pais, 1049-101 Lisboa, Portugal CDMS, INETI, Est. do Pago do Lumiar, 1649-038 Lisboa, Portugal
Abstract A multi-product model for the design of global supply chains with reverse flows is proposed. Two levels of decisions are considered, one strategic and one tactical. The first is modelled through a macro perspective of time where the determination of the network structure and flows is accomplished. At tactical level, a micro perspective of time is considered, where production planning and inventory management are addressed in detail. A mixed integer linear programming formulation is developed which is solved with standard Branch and Bound techniques. The model accuracy and suitability are studied using a case study.
Keywords: Supply Chain design, Optimization, Reverse Logistics, Planning.
1. Introduction In modern society, used products constitute a major challenge. Governments are facing overflowed landfills, while creating legislation to shift product management responsibility towards the producers. Used/disposed products are now placed in a different perspective, as company managers perceive new business opportunities whereby these products should be returned to factories for remanufacturing/recycling. Consequently, the supply chain must be extended and no longer terminate at the end customers, but account also for the products return. Only recently the scientific community started looking into this problem. Thus the reverse logistics problem appears as an emerging field where only a limited number of models have been, so far, presented in the literature. These are essentially case study dependent and mostly consider the reverse flow on its own and not as an integral part of the supply chain. As the most generalized published models, one should refer to: Fleischmann et al. (2001), where forward and reverse flows of a given product are integrated, with no limiting capacities in the involved facilities and flows; Jayaraman et al. (2003), who proposed a MILP model for the reverse distribution problem, without taking into account the forward flow; Fandel and Stammen (2004), that proposed a MILP general model for extended strategic supply chain management, based on a twostep time structure, but where no testing of adequacy to any example/case was explored; and finally Salema et al. (2004) who developed a capacitated multi-product design
Author to whom correspondence should be addressed,
[email protected]
1076 network model where forward and reverse flows were considered, the flows differ not only in terms of structure but also in the number of products involved. Within these few works, one important area of research not yet explored, is the simultaneous design and planning of such structures (Goetschalckx et al., 2002). In the present paper, we look into this issue and propose an optimal multi-product supply chain design model where both forward and return flows are integrated considering simultaneously the design and planning of such structures. A Mixed Integer Linear Programming formulation is developed where two different perspectives are employed for the treatment of time: a macro perspective (strategic), for the determination of the network structure and flows, and a micro, more detailed perspective (tactical), for the production planning and inventory management. An illustrative case-study is solved showing the model applicability.
2. Problem Definition Figure 1 shows a schematic representation of ~ l~~ a supply chain network with reverse flows. A ~ ~ ~ two echelons forward distribution network is considered where factories are linked to ~l"llili"iiFactory iiliii:;i~i'"i"i"ii;'"i:i~' i"~ customers through warehouses. No direct
connection, factory-costumer, is allowed. The
~
Customer
~ - - ' [ : i ~ DisassemblyCentre
same applies for the reverse flow where a two Figure 1: Distribution network with echelons structure is also present, customers reverse flows. being linked to factories through disassembly centres. Again, customers cannot send their products directly to factories, since products must pass through disassembly centres. Forward and returned products might be treated as independent since we consider that original products may loose their identity after use (e.g. paper recycling paper products after use are simply classified as paper). However, if required, it is also possible to track the same product in both flows. Furthermore, a disposal option is considered within the structure in study. At the disassembly centres if collected products are found to be unsuitable for remanufacturing, a disposal option is made available. Using these structural options the model considers two levels of decisions at different time scales. A "macro time" scale, where the supply network is designed, and a "micro time" scale, where planning activities are set (e.g. macro production and/or storage planning). These time scales can be years/months, years/trimester, month/days or whichever combination suits the problem. The chosen facilities will remain unchanged throughout the time horizon while the throughput will undergo changes. Flows are not necessarily instantaneous (customers will not have their demand satisfied in the same time period when products are manufactured) and thus travelling times might be considered. A travelling time is modelled as the number of micro time periods that a product takes to flow from its origin to its destination. If all travelling times were to be set to zero, a multi-period location/allocation model would be obtained. Finally, a profit function is assumed for the objective function where revenues and transfer plus selling prices are considered. The former are defined whenever there are products flowing between facilities (from factories to warehouses or from disassembly
1077 centres to factories) and the latter whenever products are sent to customers by warehouses or collected by disassembly centres. In terms of costs different terms are identified: investment costs (whenever a facility is chosen), transportation costs, production costs, storage costs and penalty costs (for non-satisfied demand or return). In short, the proposed model can be stated as follows.
Given" • the investment costs • the amount of returned product that will be added to the new products • travelling time between each pair of network agents • the minimum disposal fraction and for each macro period and product: • customers' demand and return values, • the unit penalty costs for non satisfied demand and return, and in addition, for each micro period: • the unit transportation cost between each pair of network agents, • the maximum and minimum flow capacities, • the factory production unit costs, • the facilities unit storage costs, • the maximum and minimum production capacities, • the maximum storage capacities, • the initial stock levels, • the transfer prices between facilities, • customers' purchase and selling prices. Determine, the network structure, the production levels and storage levels, the flow amounts, and the non-satisfied demand and return volumes. So as to, maximize the global supply chain profit. 3. I b e r i a n
Case
3.1 Case description This example was created based on a company that operates in the Iberian Peninsula. This company needs to determine the network design for a supply chain that will involve two forward products (F1 and F2) and one single returned product (R1). At the strategic level customers are grouped into 16 clusters, where each cluster is named after the city it represents. Customers' clusters, from now on designated simply as customers, are respectively located in Barcelona, Bilbao, Braga, Coimbra, la Corufia, Granada, Lisbon, Madrid, Malaga, Oviedo, Porto, Santander, Saragossa, Seville, Valencia and Valladolid. Six of these cities are possible sites to locate warehouses and/or disassembly centres (Barcelona, Bilbao, Lisboa, Madrid, Porto and Sevilla). For the factories there are only two possible locations: Lisbon and Madrid. In terms of time, a macro period is defined over ten years and a micro period over twelve months per year: macro period - "year" and micro period = "month". Since the model considers a horizon of ten years, some data have to be estimated. These include the demand and return volumes as well as variations in costs over the years. These
1078 estimates were based on some assumptions: transportations costs are proportional to the distance between each city and after the first year an inflation rate of 3% (or some other convenient value) is applied to these and all other costs; if flows link cities in different countries, a tax is applied to the corresponding transportation cost namely, 6% from Portugal to Spain and 3% from Spain to Portugal; in the first year, customers' demand is made equal to a fraction of the city inhabitants (a value between 0.8 and 0.9) while in the remaining years a variation factor (ranging from 0.98 to 1.08) is considered, allowing for an increase or decrease in the demand value; in each year, customers' returns is set as a 0.8 fraction of the corresponding demand. The problem also assumes zero initial stock levels; for product recycling F1 incorporates 0.6 of product R1 and product F2 incorporates the remaining 0.4 of product R1; the disposal fraction is set to zero; minimum and maximum capacities are defined for production (0.8"106 and 1.0"106 , respectively); no limit is imposed on flows; travelling time is set to nil, which seems a reasonable assumption given the chosen time scale (years/month) and the particular geographical area under study. 3.2 Results
The resulting MILP model was solved by GMAS/CPLEX (built 21.1), in a Pentium 4, 3,40 GHz. The model is characterised by 46 337 variables (14 binary) and 5 703 constraints and took about 392 CPU seconds to solve (0% optimality gap). The optimal value found for the objective function is 96xl 09 currency units and the optimal network is characterised by a single factory location (Madrid). /
//
.o................
i--
Figure 2a: Forward networks
Figure 2b: Reverse networks
In the forward flow, the Madrid factory serves four warehouses (Bilbao, Lisboa, Madrid and Porto). The associated connections are depicted in Figure 2a: solid lines link the factory to warehouses and dotted lines connect warehouses to customers. One can see that Lisboa and Bilbao have just one customer and that Madrid acts has a geographical centre. Every warehouse supplies the customer located in the same city, while there is one single customer which is served by a warehouse outside the country (Coruna is served by Porto). The number of connections between the two countries is therefore small, which is a result of the taxation imposed for flows between countries. Finally, all customers had their demand fully supplied. In terms of the reverse network (Figure 2b), all six different locations were chosen for the disassembly centres (Barcelona, Bilbao, Lisboa, Madrid, Porto and Seville). As in
1079 the forward network, every disassembly centre serves the customer located in the same city. Concerning the tactical level of decision three different analyses can be made, respectively for production, storage and distribution. As the model produces a large wealth of information, only some examples will be presented.
Figure 3. Facto~ production plan.
In terms of production, Madrid factory operates continually at the minimum level. To meet demand fluctuations, this factory has the products returned by the disassembly centres. To illustrate the optimal production plan, the first, fourth and tenth years are depicted in Figure 3. In terms of storage, the optimal solution leads to a zero stock policy. This is a result of the negative impact that storage has on the objective function (i.e. inventory is costly). Finally in terms of distribution, four examples are given. Each one refers to a different network level. il ir :
• 11 !i!
i:i
i
i:i
.... Iliiill i!l il Figure 4. Distribution.17ows between Madrid and Lisboa.
Figure 5. Demand served to Braga's customer.
Figure 4 shows in detail the flows between the factory and the warehouse located in Lisboa. The relaxation of the capacity constraints resulted in very dissimilar flow patterns. In Figure 5, Braga's customer supplying plan is depicted. The difference between amounts supplied has to do with the simultaneity of supplies received by this customer, as noted in the third and sixth years. On the contrary, in the ninth year there are three separate visits. In the remaining years, every product is supplied in one single separate delivery. With regard to return volumes, Figure 6 shows the total return of product R1 between customers and disassembly centres. The dissimilarity among values is a consequence of
1080 the customers return volumes. Lastly, the return of Lisboa disassembly centre is depicted in figure 7. One can see that each year returns are sent to remanufacture (Madrid factory, labelled Mad) as well as to the disposal option (f0). However, the number of times these activities are performed, varies between once and twice a year. ~:~.~
i
-
.......................
o:o ~ : ~ -i...................................................... I
:~ ° ° ° ' ~
~
::~.~;~E~
.)~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~°~ ~ ' ~
...............................
..................
I
. . . . . . . . . . . . .
......... '........ I I .... . . . . . . . . .
:~
Figure 6." Total return for the first year.
...... II i .?..l...... ...I...!..........
....... i...... iI!
¢~~ , ~
...................................................... m .......................................................
Figure 7." Lisboa disassembly centre return
flOWS.
4. Conclusions In this paper, a multi-period, multi-product design model for a supply chain network with reverse flows is proposed. Two time levels are modelled allowing the establishment of two different levels of decisions: a strategic level defining the network design of the supply chain and a tactical level defining the production, storage, distribution and returning planning. The obtained MILP formulation is fairly flexible since several products can be considered in both networks; different time units can be correlated (years/semesters, years/months, months/days...); limits can be imposed on flows, production and storage capacities; and different travelling times are allowed. The mathematical formulation which supports this model, while it is likely to increase significantly in complexity with the problem dimension, still appears as an important tool to help the decision making process at the strategic and tactical levels of the supply chain management decisions. In order to overcome the computational burden of such a formulation, different solution techniques are now being explored to speed up resolution. Further research is also being undertaken with a view to both strengthen the model formulation and to treat production planning with greater detail, with the introduction of bills of materials.
References Fandel G. and M. Stammen, 2004, Int.J.P.E. 89: 293-308. Fleischmann M., P. Beullens, J.M. Bloemhof-Ruwaard and L.N. Van Wassenhove, 2001. POM 10: 156-173. Goetschalckx M., C.J. Vidal and K. Dogan, 2002. EJOR 143: 1-18. Jayaraman V., R.A. Patterson and E. Rolland, 2003. EJOR 150: 128-149. Salema MI, AP Barb6sa-P6voa and AQ Novais, 2004. POM (submitted).
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjaner and A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1081
Heterogeneous Batch Distillation Processes: Real System Optimisation Pommier S6bastien a, Massebeuf Silvbre a, Gerbaud Vincent a, Baudouin Olivier b , Baudet Philippe b and Joulia Xavier a* aLaboratoire de G6nie Chimique, 118 route de Narbonne, F-31077 Toulouse Cedex 04, France bproSim SA, Stratbge OffShore, Bfitiment A - BP 2738 F-31312 Lab6ge Cedex, France
Abstract in this paper, optimisation of batch distillation processes is considered. It deals with real systems with rigorous simulation of the processes through the resolution full MESH differential algebraic equations. A specific software architecture is developed, lying on the ProSim BatchColumn simulator and on both SQP and GA numerical algorithms. The efficiency of the proposed optimisation tool is illustrated by a case study of heterogeneous batch solvent recovery by distillation. Significant economical gains are optained along with improved process conditions. For such a multiobjective complex problems, GA is preferred to SQP that is able to improve specific GA solutions.
Keywords: Optimisation, Batch Distillation, Heterogeneous Azeotrope
1. Introduction Solvent recovery is a major issue in the pharmaceutical and specialty chemical industries. In that purpose, batch distillation is a separation process of choice. For azeotropic or close-boiling mixtures, the addition of an entrainer, partially or totally miscible with one of the initial binary mixture components, is viable and its choice is the first key issue of azeotropic batch distillation. A whole set of entrainer selection rules has been published for both homogeneous and heterogeneous azeotropic distillation for the separation of azeotropic binary mixtures or close boiling components (Rodriguez Donis, 2001a and 2001b). These rules also hint at a feasible sequence of batch needed to perform the separation together with the initial feed stream location in the ternary diagram. But the optimisation of the batch sequences is a second key issue and this contribution validates a framework for the optimisation of complex distillation.
2. Problem definition The goal of batch sequences optimisation is to minimise an overall economical criterion while respecting constraints such as purity, .... It can be considered as a nonlinear constraint optimisation problem. The classical formulation involves an objective function ~ , equality constraints and inequality constraints (gi and hj respectively):
Author/s to whom correspondence should be addressed :
[email protected]
1082 Min
f (x)
gi(x) = 0
i = 1,..., ng
hj(x) < 0
j = 1,...,n h
(1)
2.1. Objective function The objective f u n c t i o n f i s the summation of six cost functions ci: Table 1. Economical cost functions taken into account in the objective function 6
cost f = Z ci object function 1 cl c: c3 c4 c5
expression
immobilisation energy load entrainer column treatment
c~ = c: = c3 = c4 =
used variable
al.t + bl a:. Q a3.L a4.E
c5 = as.R + b5
t = total separation duration Q = total required energy L = global column load E = entrainer amount added initially R = residual column load
lit
C6
tanks treatments c6 = Z a~'Tk + b6k Tk = final load of each of the nrtanks (including still) k=l
2.2. Constraints The constraints of the NLP problem are defined with respect to target purity and/or quantity specifications at the end of the distillation process. Each constraint hj is expressed as follows: (2)
h j : x k - xik obj
where x/k and X~obj are the effective and target fraction of component i in tank k.
2.3. Action variables Action variables are chosen among all the available running specifications of the batch process, that is a collection of successive tasks and the initial load of entrainer (Table 2). Table 2. Available action variables
Available action variable (* for each task i) Entrainer load Boiling duty * Subcooling temperature *
Task duration * Reflux ratio of light phase * Reflux ratio of heavy phase *
3. Problem resolution 3.1. Overall resolution software architecture The proposed optimisation methodology lies on a rigorous simulation of the considered batch processes. Most of the variables values required to evaluate the objective function and the constraints are calculated through this process simulation. From a defined column configuration and defined initial settings, a full MESH (Material balance,
1083 Equilibrium, Summation of molar fraction, Heat Balance) set of differential algebraic equation is solved using the ProSim BatchColumn software (ProSim SA, France). Main results from the batch simulations are mass and composition in each distillate tank and in the boiler, as well as the total heating and cooling duties. The economical optimisation criterion and the constraints values is evaluated from these results. These evaluations stand for the heart of the resolution software architecture, as shown in
Figure 1. Optimisation algorithms offer strategies to change the values of the action variables in order to solve the constraint minimisation problem. Simulation
Batch Column simu1ator ~~,,,,~~.~
Lsettings [
Action
i
Manager ~,q variables J criteria ~ evaluation ~ Action variab1es • Simulation '7 ~ ~ j ~ Objectivefunction value
results
simulation
1 Optimisation pack
Constraints value
objective function and constraints evaluation
optimisation
Figure 1. Optimisationsoftware architecture 3.2. Optimisation methods Two optimisation techniques are used: a SQP-based deterministic method, and a home made Genetic Algorithm as a stochastic one. The SQP algorithm is the donlp2 tool, available at www.netlib.org (Spellucci, 1998). It incorporates the exact ll-merit function and a special BFGS quasi-Newton approximation to the Hessian. The optimisation problem is strictly equation (1). The genetic algorithm is real-coded. In order to use such an unconstrained optimisation technique, the constraints are introduced into the objective function by penalty terms. The optimisation problem aims then at finding the minimum of the following it; function:
{O with
P, (x) -
80
if gi (x) -- O tf g; (x) ~ 0
{O and
Qi(x)-
oo
if hi (x) <-O (fh~(x)>O
and Pi ° and Qi ° are weighting factors proportional the inverse of the squared tolerances on the constraints.
4. Separation of pyridine from water using toluene as entrainer 4.1 Problem settings We study the separation of the minimum temperature homoazeotropic binary mixture w a t e r - pyridine. According the Rodriguez-Donis et al. (2002) and Skouras (2004), the separation is possible using a heterogeneous entrainer. Toluene is added to the mixture, forming a minimum temperature homoazeotrope with pyridine and a minimum
1084 temperature heteroazeotrope with water. Three distillation regions exist with the w a t e r toluene heteroazeotrope being the unstable node in each region and the stable node being the pure vertexes. The distillation boundaries are strongly curved and tangent to the vapour line at the heteroazeotrope, like any residue curve in the VLLE region. During the heterogeneous batch rectification process, removal of the aqueous phase in the decanter is expected and reflux of either the non-aqueous phase or a combination of both decanter phases is possible. In this work, the whole non-aqueous decanter phase is refluxed. This operating mode is called Mode B by Skouras (2004) who described elegantly the heterogeneous batch distillation process issue and feasibility in complement to Rodriguez-Donis et al. (2002). According to Skouras (2004), the initial charge composition must be above the line p y r i d i n e - aqueous phase to make the process feasible. The batch distillation boundary has no impact on mode B process feasibility (but does on mode A, see Skouras (2004)). The residue curve/distillation boundaries have no impact on feasibility despite their curvature. Table 3. VLL and LL Thermodynamic parameters (liquid." NRTL," gas." ideal gas)
Parameter value (cal/mol)
Aijo
Ajio
otijo
Water- Toluene Water- Pyridine Toluene - Pyridine
3809.1 1779.18 264.64
2776.3 416.162 -60.34
Aijv
0.2 0.6932 0.2992
AjiT
21.182 0 0
GtijT
-7.3179 0 0
0 0 0
F o r N R T L : gij-gjj=Aijo+AijT.(T-273.15); gji-gi~=Aj~o+AjiT.(T-273.15); ~ij=Gtij0+GtijT.(T-273.15)
.....
298K LLE enveloppe
toluene [sn] (383.8K)
VLLEResidueenVeloppecurves
/#, ,W~I!,.,, \
.................................. Residue curve boundary . . . . .
......
_ j~, ~ Still path
Batch distillation boundary Vapour line Az tol-pyr [sa] (383.3K)
' / ',
o o o o Distillate path
',
/ \ '\ .~" ....................... 7\............................ ~!~ A
#
'~
/
'~z~/~,:j
/;:\,,
/%
'\\
/",X /\,,
/./,<,,/\/ ',,X/., /',, /',, , / ,, ° ........................... )...,
/ ",, \
... XD,R~
..............-/~
Az tol-wat [un] (357K)
i
pyridine [sn] 1388.3K)
/
Az wat-pyr [sa] (368K)
Figure 2. Water/Pyridine/Toluene ternary diagram
\
water [sn] (373.1K)
1085 The column has 16 plates (incl. decanter), initial charge of 100 moles with molar fractions [water: 0.4; toluene: 0.1; pyridine: 0.5] is placed in the still. Decanter and plate holdup equals 5 and 1 moles respectively. All plate efficiency are set to unity. Decanter is subcooled to 25°C. Column pressure is 1 atm. and pressure drop is 0.05 atm. The process consists of three tasks: filling, infinite reflux and distillation (water phase removal). Thermodynamic model parameters are given in table 3. These parameters are used to compute VLLE in the column and 25°C LLE in the decanter. The optimisation problem aims at minimising overall costs and satisfying two molar composition inequality constraints: • hl(Xwater)> 0.992 in the distillate tank at the process end. • h2(xpyridine)> 0.95 in the still at the process end. Costs c~, c2, c4, and c5 (table 1) are used with arbitrary cost factors not detailed here. Five action variables are defined: entrainer load (FE); heat duty (Qb/ROc) and task duration (t/Roe) for the infinite reflux task; heat duty (Qb/Dist) and task duration (t/Dist) for the distillate removal task. Tolerances in the constraints are set to 0.001. For the GA, initial population equals 100 to ensure efficient sampling of the five dimensional space; stop criteria equals 0.1. For the SQP, centered gradients are used. Influence of costs factors and optimiser parameters are not considered in this presentation. 4.2 Results and discussion Results of the optimisation are given in table 4. The initial point does satisfy the constraints on purpose (with respect to tolerances for hl(Xwater)). If not, results not shown here indicate that the initial criteria is very high because constraint h2(xpyr~d~ne) is not satisfied and its Qi ° factor is proportional to 1/(0.0012). Only the GA finds a solution while the SQP does not. Table 4. Optimisation results
initial
SQP
GA
SQP after GA 0.806
FE (kg)
2
1.954
0.866
Q j R oc (kcal/hr)
3000
2998
1212
1209
t/R oc (hr)
0.2
0.154
0.043
0.026
QJDist (kcal/hr) t/Dist (hr)
3000 0.6
3000 0.533
1437 0.647
1433 0.617
Criteria f
155.468
13 7.182
82.090
77. 882
Criteria ft~
183.453
NA
82.090
NA
-4.24E-04
- 7.10E-05
8.9E-04
1.4E-03
h2(Xpyridine)
5.38E-03
9. 70E-03
9.1E-03
-5. OE-04
Water purity
0.992
0.992
0.993
0.993
Pyridine purity
0.955
0.960
0.959
0.949
Water recovery
100%
100%
95%
93%
Pyridine recovoy
82%
82%
85%
87%
Gain
0%
12%
47%
50%
For results in table 4, the SQP improvement of initial conditions is slight. GA ends after 10 population generations but the after the first generation, the criteria has already
1086 improved to f=fp=90.03 with a mean criteria f=128.3 and fp= 1.0 106 indicating a wide sampling of the solution space. An important reduction of the heat duty is observer along with a reduction of the entrainer load. SQP improvement of the GA solution is acceptable. Of course, results are dependant of the cost factors and problem setting. Indeed, for the separation considered, the entrainer load can be much lower to ensure a feasible separation (see skouras, 2004). When done with a cost of entrainer c4 25 times greater than other costs, the optimiser logically finds a best entrainer load equal to 0.294 kg, but heating required is greater. Besides, for a given purity, Qb/Dist and t/Dist are linked: heating more implies less time to obtain as much water. On this particular problem and on all problems of batch distillation process optimisation we have performed, the SQP does not perform well when the initial point is infeasible, whereas the GA is always able to find a suitable solution to the problem. In fact, SQP should be used on a feasible solution like the final result of a GA run. This indicates that the GA should always be ran first, unless a specific tuning on a feasible solution is sought. Simulation of the optimal solution of the SQP optimisation done after the GA (last column) leads to the composition profiles reported in figure 2 for the still and the distillate composition. As can be seen, the distillate at the end of the infinite reflux period lies on the vapour line and rapidly shifts to the LLE envelope at 298K. The still path moves towards the pyridine vertex. Those paths are in agreement with the path already published (Rodriguez-Donis et al., 2002; Skouras, 2004)
5. Conclusion An optimisation framework coupling stochastic GA and deterministic SQP approaches has been devised and suited for batch distillation processes, homogeneous or heterogeneous, single batch or batch sequence. In the present contribution, validation is done on a double task single rectifier heterogeneous batch process for the separation of water- pyridine with toluene. The five action variable problem is solved. The use of GA followed by an SQP is the recommended choice. As careful weighting of the action variables shows, such a powerful optimisation tool should be used by users well acquainted with the process expected behaviour.
References Rodriguez Donis I., Gerbaud V. and X. Joulia, 2001a, Ind. Eng. Chem. Res., 40, 2729-2741. Rodriguez Donis I., Gerbaud V. and X. Joulia, 2001b, Ind. Eng. Chem. Res., 40, 4935-4950. Rodriguez-Donis I., Gerbaud V. and X. Joulia, 2002, AIChE Journal, 48 (6), 1168-1178. Skouras S., 2004, PhD Thesis available at http://www.nt.ntnu.no/users/skoge/publications/. Spellucci P., 1998, Math. Prog., 82, 413-448.
Acknowledgements This work was performed in collaboration with ProSim SA (Toulouse, France), and supported by the french environment and energy agency (ADEME).
European Symposiumon ComputerAided Process Engineering- 15 L. Pui~ianerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1087
Modelling and Optimisation of Distributed-Parameter Batch and Semi-batch Reactor Systems Xiaoping Zheng, Robin Smith, and Constantinos Theodoropoulos* School of Chemical Engineering and Analytical Science University of Manchester, Manchester, M60 1QD, UK
Abstract Macro-mixing effects in batch and semi-batch reactors are investigated by constructing 3-dimensional models using a network of zones (NoZ) discretisation. System dynamics including volume changes due to continuous feeding are successfully predicted. Detailed flow fields are calculated from phenomenological correlations which include parameters such as reactor size and configuration, impeller type and speed and fluid physical properties. The Proper Orthogonal Decomposition method is subsequently applied to extract reduced models from the large-scale NoZ-based ones that can be used for computationally efficient design, optimisation and optimal control studies.
Keywords: network of zones, non-ideal mixing, proper orthogonal decomposition
1. Introduction Batch and semi-batch reactors are increasingly employed in the industrial production of fine and specialty chemicals, pharmaceuticals, polymers and crystals due to their economic efficiency, including low capital investment and typically high yields, and their versatility to operate with a range of reactants and products. Nevertheless, due to the complex system dynamics involving turbulent flows and mixing phenomena, optimal and safe reactor scale-up and operation are challenging tasks, increasing considerably the uncertainties in reactor design and set-up (Zaldivar et al, 1996, Brucato et al. 2000). Modelling approaches that have been proposed to predict and optimise batch reactor performance range form simple compartment-based models (David et al., 1992, Cui et al., 1996) which use only a few parameters and can yield good agreement with experimental results, but can not give detailed insight for detailed design, to more complex models involving detailed CFD-based flow field simulations (e.g. Brucato et al., 2000, Bakker et al., 2001). Turbulent CFD models can give detailed flow information, but they are computationally intensive, commonly used k-~ models introduce inaccuracies by assuming isotropic turbulence and more advanced models (e.g. LES) require parameter tuning through experimental validation (Vrabel et al., 2000). In this work we have adopted the network-of-zones (NoZ) approach (Nienow et al. 1992, Rahimi and Mann, 2001, Hristov and Mann, 2002), where the reactor is finely discretised in a number of cells. The NoZ model can successfully predict the effects of
Author to whom correspondence should be addressed:
[email protected]
1088 macro-mixing, reactor design and operating conditions. It can be integrated with CFDbased flow calculations (Brucato et al 2000) as well as with models describing micromixing and drop or bubble dispersion and movement phenomena. Here, we calculate detailed flow fields in a computationally efficient way by using complex correlations involving a large range of design parameters (Platzer and Noll, 1988) and we have developed a new network structure to deal with reaction volume change effects.
2. N e t w o r k of zones m o d e l Existing NoZ models assume that the reaction volume in every zone does not change during a batch cycle. However, in reality there are changes due to reaction, feeding or mass transport. In this work an improved model is proposed where the network of zones is constructed based on the volume of the reactor. Fig. 1 depicts the structure of this new model, which involves 3 types of zones: a. Empty zones above the fluid surface; b. halfempty zones containing the fluid surface and c. Fluid-occupied zones below the surface level. If the reaction volume changes during the batch process, the surface of the fluid will move up into the empty zones or down towards the fluid-covered zones leaving more empty zones at the top. Accordingly, the mathematical description of this model includes equations describing reaction volume changes. Consider the mass transfer between one zone (i, j, k) and its adjacent zones. Equations (1), (2) and (3) are three ordinary differential equations (ODEs) describing the rate of change of volume VO,j,~) , molar amounts, N~0,j,~),and concentrations C oJ,~) of each component l, in this zone.
~lf-empty zones Fluid-covered zones !ili~
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C i/.,-1,j,k)
- Vol _ f l o w _ in (. 1 , j , k ) X -
dt
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_in(i+l,j,k ) x C i l i + l , j , k ) - t -
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f l o w _ in(i,j.,,_l)x C(i,j,k_l) + Vol _ f l o w _ in(;,j,,+l)x C ( i _ l , j , k + l ) t~r
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V o l f l o w _ i n and Vol__/qow_out are the volumetric convective flow rates through the surfaces of the zone, calculated by integrating the flow field across each surface. Rct_rate and F d rate and M o l a r _ v o l : are the reaction rate, feed rate and component l molar volume, respectively. Each zone is assumed to be well-mixed, its volume is determined by the level of discretisation and remains fixed during the simulation. For the fluid-covered zones, reaction volume is the same as the zone volume, so the volume changes from equation (2) are used to compute the new level of the fluid surface. Beyond a volume change threshold the flow field is re-computed. Note that only equations (2) and (3) need to be solved being both expressed in terms of equation (1).
2.1 Flow field computation In several works employing NoZ models, the flow rate through surfaces of zones was calculated only based on impeller overall circulation convective capacity (Nienow et al., 1992; Rahimi and Mann, 2001; Hristov and Mann, 2002). This flow rate was then equally distributed through all the zones. There are several drawbacks in this method. It can not capture velocity variations at different regions inside the reactor vessel and does not fully exploit the benefits of the NoZ model. Also, it uses too few reactor configuration parameters, thus it cannot be used for reactor design and optimisation. Usually other parameters like tank size, impeller type, baffle number and size are of great interest in reactor engineering. Here, a sophisticated correlation system (Desouza and Pike, 1972; Platzer and Noll, 1988) is used which divides the flow field into three characteristic model parts: (1) rotational flow due to impeller rotation; (2) Circulating flow in the "bulk" cells; (3) distinct jet flow (near the impeller). Different correlations have been developed for these 3 parts. The correlations can give detailed 3-D flow field information, i.e. velocity vectors at every cell in the vessel. The parameters involved include reactor configuration (tank size, impeller type, size and position and baffle number and size), operating conditions (impeller speed, fluid depth and feed position) and fluid properties (density and viscosity). This model has been tested against CFDbased calculations giving results of comparable accuracy for a wide range of parameters (Zheng et al 2004). Furthermore it is much more computationally efficient.
3. Case study A system proposed by Nienow et al. (1992) in order to characterise imperfect macromixing and partial segregation in a stirred semi-batch reactor is considered. The predictions of our model were compared with experimental results and simulation results reported in the literature. The chemistry is based on a pair of parallel reactions
A
kl >S
A+B
~~ >R
where A (a diazonium salt) is initially charged in the reactor and B (a pyrazolone) is added continuously at a constant rate. R, a dyestuff is the product and S the unwanted
1090 by-product. The reaction kinetics, operating conditions and kinetic parameters are reported in (Nienow et al., 1992). The height and diameter D of the vessel are both 0.3m. Four 0.1D strip baffles were used with a Rushton turbine with diameter DI=D/3 and clearance C=D/3. A 3-D NoZ model was constructed using 20 zones in each direction (axial, radial and circumferential -8000 zones in total) resulting in a system of 48000 ODEs which were integrated in time using DASPK (Maly and Petzold, 1996). Fig.2 shows a comparison between results from the 3-D network (diamonds) experimental results (squares) and simulation results from the literature (triangles- Nienow et al 1992) for a range of impeller rotation speeds. Simulations assuming ideal mixing (circles) over-predict the yield. Our 3-D simulations agree very well with the experiments and are in better agreement than the literature results, which are, however, close since the volume change effects are small in this case. Further parametric studies have shown that better yield can be achieved by supplying both feeds continuously from the same feed position near the tip of the impeller. These results along with results from a second case study where volume change effects were more pronounced (Paul and Treybal 1972) are presented in a forthcoming publication (Zheng et al, 2004). 0.97
0
0
0
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0
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.....~i~.....!i
~
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1O0
150
200
250
300
350
Rotation Speed(RPM)
Figure 2. Comparison between yield predictionsfrom our 3-D model, and experimental and simulation results from the literature.
i~i~:!~:~~!i:~:i:!i? i~i,i:~,i~:~'~~i~:i
3 seconds
18 seconds
Figure 3. Concentration snapshots o f the product R on a vertical plane in the reactor vessel at t=3 and t = 18 s. The blue (red) colour denotes low (high) concentration.
1091 Fig. 3 shows concentration distribution profiles of species R at a vertical plane inside the reactor at t-3 and 18 s. The right side is the reactor centreline. The impeller rotation speed was 78 RPM. Blue (red) colour denotes low (high) concentration. The concentration at the top empty zones is zero. As it can be seen, areas of lower mixing intensity are the comers of the reactor, the impeller shaft and the circulating zones. As time progresses reactants in these parts eventually participate in the reactions and are converted to products or by-products.
3. Reduced model The NoZ model coupled with flow correlations typically results to systems containing (hundreds of) thousands of ODEs. The simulation of large-scale ODE-based systems is nowadays achievable in realistic CPU times with large yet reasonable memory requirements. Nevertheless, optimisation studies and optimal control design and implementation cannot be based on such large-scale systems since a huge number of function evaluations is required. In this work we have employed the Proper Orthogonal Decomposition method (POD) (Holmes et al., 1996) to extract accurate low-order models from the full-scale ones. In POD a small number of semi-empirical eigenfunctions are computed from a database of detailed full-scale simulations (or even experiments) that can capture the energy of the system i.e. can accurately describe the system in the parametric range of interest. The dynamic low-order model can then be obtained by a Galerkin projection of the governing equations onto these few basis functions. POD has been used successfully in a number of works (e.g. Rowley et al, 2004; Cizmas et al, 2003; Shvartsman et al 2000). Here the s c a l a r - v a l u e d method is employed (Rowley et al. 2004) computing POD modes for each variable (concentrations and reaction volume). We have constructed a simulation database for the case study presented above, by performing simulations using the NoZ model at 3 different rotation speeds: 39 RPM, 197 RPM and 302 RPM recording snapshots every 0.5s. It was found that 20 basis functions for each species (100 in total) and only 1 basis function for the volume were sufficient to capture 99.9 % of the energy of the system. A Galerkin projection of equations (1)-(3) onto these eigenfunctions produced a reduced model of only 101 ODEs that can accurately predict the system behaviour.
~:~:'i!!!ii!i!ii!ili!
i 3 seconds
.........
l
i 18 seconds
Figure 4.Concentration snapshots of the product R at t=3 & 18 s on a vertical plane in the reactor obtained from the reduced model. The blue (red) colour denotes low (high) concentration.
1092 In Fig. 4 concentration profiles obtained from the reduced model at the same conditions as the profiles showed in Fig. 3 are depicted. As it can be seen the agreement between the full-scale and the reduced model results is excellent both for the short term (3s) and for the longer term (18s) dynamics. It is worthwhile to note that the case simulated here (impeller speed 78 RPM) is not included in the simulation database. Results of this reduced model at other conditions also show the same agreement with results from the full model. It can be concluded that the reduced model can predict the system behaviour very well requiring much less computer memory and CPU time.
4. Conclusions We have constructed 3-D models of batch and semi-batch reactors using a network of zones discretisation. The computational domain is discretised in an appropriately large number of cells and local velocity distributions are computed by detailed flow correlations. Mass balances coupled with volumetric changes are then superimposed onto the computed flow resulting in large-scale ODE-based systems. The model can successfully predict the effects of non-ideal macro-mixing and includes a large number of important design and operating parameters than can be used for system scale-up, optimisation and control. The POD method was subsequently used to extract reduced computationally-amenable models from the full-scale ones that can be efficiently employed in parametric studies, model-based optimisation and optimal control.
References Bakker, A., A.H. Haidari and L.M. Oshinowo 2001, Chem. Eng. Prog., 97, 45. Brucato A., M. Ciofalo, F. Grisafi, R. Tocco 2000, Chem. Eng. Sci. 55,291. Cizmas, P.G., A. Palacios, T. O'Brien and M. Syamlal 2003, Chem. Engi. Sci. 58, 4417. Cui, Y.Q., R.G.J.M. van del Lans, H.J. Noorman and K. Luyben 1996, Chem. Eng. Res. Des. 74, 261. David, R., H. Muhr and J. Villermaux 1992, Chem. Eng. Sci. 47,2841. Desouza, A. and R.W. Pike 1972. Can. J. Chem. Eng. 50, 15. Holmes P., J.L. Lumley and G. Berkooz 1996, Turbulence, coherent structures, dynamical systems and symmetry, Cambridge University Press. Hristov, H.V. and R. Mann 2002, IChemE, 80, 872. Maly, T. and L.R. Petzold 1996, Appl. Numer. Math. 20, 57. Nienow, A.W., S.M. Drain, A.P. Boyes, R. Mann, A.M. E1-Hamouz, and K.J. Carpenter 1992. Chem. Eng. Sci. 47, 2825. Paul, E. L. and R.E. Treybal 1971 AIChE J. 17, 718. Platzer, B. and G. Noll 1988. Chem. Eng. Proc. 23, 13. Rahimi, M. and R. Mann 2001 Chem. Eng. Sci. 56, 763. Rowley, C.W., T. Colonius and R.M. Murray 2004 Physica D: Nonlin Phen. 189, 119. Shvartsman, S. Y., C. Theodoropoulos, R. Rico-Martinez, I.G. Kevrekidis, E.S. Titi and T.J. Mountziaris 2000 J. Proc. Control, 10, 177. Vrabel P, R.G.J.M van der Lans, K.Ch.A.M. Luyben, L. Boon and A.W. Nienow 2000, Chem. Eng. Sci. 55, 5881. Zaldivar, J. M., H. Hernfindez and C. Barcons 1996 Thermochimica Acta, 289, 267. Zheng X., R. Smith and C. Theodoropoulos, Manuscript in preparation.
European Symposium on Computer Aided Process Engineering - 15 L. Puigjaner and A. Espufia (Editors) © 2005 Elsevier B.V. All rights reserved.
1093
Optimal Start-up of Micro Power Generation Processes Paul I. Barton a*, Alexander Mitsos a, and Benoit Chachuat a a
Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue (room 66-464), Cambridge, MA 02139-4307, USA
Abstract Fuel cell based systems are a promising alternative to batteries in man-portable power generation applications. These micro power generation processes must operate fully autonomously and automatically without the intervention of operators. Operational considerations are indeed so important that they influence the optimal design, following the paradigm of interaction of design and operation. This paper presents a methodology for the simultaneous optimization of design and operation of such systems. We illustrate the methodology with two case studies, focusing on the start-up procedure. A small rechargeable battery is necessary to satisfy the power demand during the start-up while the device temperature is too low for power generation. The optimization problem is formulated as minimization of the mass of fuel and battery required to heat the device up to operating temperature.
Keywords: man-portable power; micro power generation; micro fuel-cell system; optimal start-up operation; dynamic optimization
1. Introduction The widespread use of portable electric and electronic devices increases the need for efficient man-portable power supplies (up to 50 W). Currently, batteries are the predominant technology in most applications, even though they have a large environmental impact, high cost and relatively low gravimetric and volumetric energy densities; furthermore, the upper limit on performance is now being reached. Out of the alternatives that are possible, we are focusing on micro scale power generation devices based on the electrochemical conversion of common fuels and chemicals, such as hydrocarbons or alcohols, in fuel cells. These process-product hybrids have the potential to yield much higher energy densities than state-of-the-art batteries, because the above mentioned fuels have very high energy contents and fuel cells can in principle achieve very high efficiencies. Since most power consuming devices are operated periodically and have rapidly changing power demands, the dynamics and automated operation of portable power production are very important and must be considered thoroughly. In this paper, the focus is on the optimal start-up of micro power generation processes. It is most likely that the devices will be coupled with a relatively small, rechargeable battery; its role is to ensure that the power demand is met when the fuel cell is unavailable or can only Author/s to whom correspondence should be addressed: pib@mi t. edu
1094 satisfy part of the demand, to provide the energy needed to heat the stack up to a temperature at which chemical and electrochemical reactions are fast enough, or to provide an electric spark for the initiation of a combustion reaction.
2. Methodology Our methodology relies on the assumption that intermediate fidelity models can approximate the performance of the devices and can be used for optimization purposes. Our models do not require the specification of a detailed geometry and rely mainly on first-principles, containing only a minimal number of modeling parameters. This is possible because the relative importance of physical phenomena at the micro scale makes one-dimensional spatial discretization sufficient. We assume that the molar flux in the gas channels of the fuel processing reactor, fuel cell and burners is convective in the flow direction (PFR), and axial diffusion can be neglected; on the other hand we assume that diffusion in the radial direction is fast enough to ensure a uniform profile in the cross-section. These assumptions have the consequence that micro-fabricated units such as reactors or fuel cells can be approximated by an idealized model using 1-D species balances, without the inclusion of the diffusion term. We neglect the pressure drop along the gas channel and assume an ideal gas. Back-of-the-envelope calculations based on the expected device dimensions using Hagen-Poiseuille's equation provide an estimated pressure drop in the order of a few kPa, i.e., a relative pressure drop of a few percent only. We note that this value is in good agreement with the measurements for a micro-fabricated reactor made by Arana (2003). As a consequence, no solution of momentum equations is necessary. We further assume that heat transfer is fast enough, so that the temperature throughout the device, or regions of the device, is near uniform. This is typically the case at the micro-scale for silicon based reactors. Finite element simulations were also performed, which confirm this assumption. It is important to note that considering a uniform temperature allows one to not specify a particular geometry for the unit operations and their arrangement in the stack. Otherwise, not only the generality of our study would be inherently affected, but problems would also be encountered as several micro devices and components of the proposed processes are not fully developed thus far. Due to material constraints and technological limitations the start-up time will be in the order of at least one minute, much longer than the residence time of gases in the process, which is in the order of ms. We therefore assume pseudo-steady-state concentration profiles along the various units at each time instant. This assumption allows us to solve the concentration profile at each time step using an integration along the spatial axis, similar to the the steadystate case (Chachuat et al., 2004) without requiring method of lines semi-discretization of the state variables; in some cases this assumption even allows the calculation of analytical solutions for the concentration profile. It should be noted that if one wanted to explicitly calculate the material stresses developed, a fully transient model would be necessary. The objective of the start-up problem is to bring the fuel cell to its nominal temperature while minimizing the total mass (battery and fuel) required for this heat-up and meeting the nominal power demand at all times. In the case studies we assume that the battery
1095 can also be used for heat-up of the device. Additional constraints can also be specified, such as a maximum rate of change for the temperature based on structural stability considerations, or requirements concerning the emission of toxic gases. Since different operating modes are described by different sets of equations (e.g., discharging and recharging of the battery), the start-up problem is formulated as a hybrid discrete/continuous dynamic optimization problem (Lee and Barton, 2002). This optimization problem is solved by using recent developments in numerical methods for dynamic optimization with hybrid systems embedded.
3. Case Studies 3.1. Case Study 1: Butane Based Process A very promising process for micro power generation is the partial oxidation of butane, with subsequent electro-chemical conversion of the generated syngas in a Solid Oxide Fuel Cell (SOFC) (Mitsos et al., 2004a); one of the main advantages of this process is that butane has a very high energy content, and partial oxidation is an exothermic reaction. Therefore, oxidation of the fuel cell effluents is sufficient to overcome the heat losses at steady-state operation. A conceptual flowsheet for the process is shown in Figure 1; the reactor, fuel cell and catalytic burner are assumed to be thermally coupled and operate at a common uniform temperature. The drawbacks of this process are that butane partial oxidation has not yet been demonstrated at the micro-scale and limited kinetic data are available; therefore the model presented should be considered preliminary and the results qualitative rather than quantitative. air
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We now present results obtained from optimization of the butane based process at a nominal power output of 1 W and a nominal operating temperature of 1000 K. Figure 2 illustrates the optimal profile, obtained by applying a piecewise constant approximation with 50 control segments of equal duration to solve the problem. The optimal start-up procedure duration was determined to be around 150 s. The number of time intervals has an insignificant influence on the start-up performance in terms of the objective function.
1096 2
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3.2. Case Study 2: Ammonia Based Process Ammonia is often considered as a potential energy source in fuel cell systems, e.g., Metkemeijer and Achard (1994), because ammonia decomposition produces hydrogen. A drawback of this process is that ammonia is corrosive and toxic and therefore tight constraints regarding the emission of residual ammonia need to be imposed. Also, ammonia decomposition is an endothermic reaction and therefore a heat source is required. While oxidation of part of the hydrogen produced could be used to provide the necessary heat, a more promising approach Mitsos et al. (2004b) is the use of a secondary fuel with a high energy density, such as butane. In Chachuat et al. (2004), we have considered optimal steady-state operation of the process shown in Figure 3 and we now extend this work to transient operation.
1097 air
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Figure 3. Conceptual lTowsheetfor ammonia based process.
The scope of this paper does not permit a detailed discussion of the start-up procedure for the ammonia based process. Instead we present how, for a fixed design, the fuel energy density (in Whr/Kg) changes as a function of the power output. This consideration is very important for the transient operation, since at start-up it is plausible to gradually increase the flow through the fuel cell. Furthermore, the power demand of power consuming devices is time varying and there is a trade-off between running a process away from its optimal operating point and consuming more energy from the auxiliary battery. In Figure 4 the design optimized for a nominal power demand of 10 W (Chachuat et al., 2004) is used and the energy density based on optimal steady-state operation is shown in comparison to the design optimized for the given power output, it should be noted that we do not consider a variation of the operating temperature, assuming that the design was done for the maximal allowable temperature, based on material constraints. The energy density is maximal at a power demand slightly higher than the one for which the system was designed (~ 10.1 W). For power demands lower than the design power output, the heat generation is small, and relatively more butane needs to be burned to compensate for the heat losses. For an increasing power output, the ammonia flowrate is increased and the fractional conversion in the reactor and fuel cell is decreased; the constraints on residual concentrations of ammonia and nitric oxide in the outlets are only met with a largely increased flow of oxygen, which results in higher requirements for heat and a higher butane flow rate. The flow rate for oxygen quickly becomes so large that the pressure drop through the system would become significant, making the process practically infeasible and also violating one of our main modeling assumptions. This case study shows that it is necessary to consider variations in the power demand during the design of the system, and a promising method is stochastic optimization.
1098
.........
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4. Conclusions A methodology for the simultaneous design and operation of fuel-cell based micro power generation devices was presented and demonstrated by considering the start-up procedure for two fuel-cell systems based on fuel processing in conjunction with a Solid Oxide Fuel Cell. The models used are of intermediate fidelity, allowing the optimization of design and operation without the specification of an exact geometry. The formulated methodology allows the solution of the formulated problems using dynamic optimization tools. The case studies show a strong interaction between design and operation. We are currently extending our methodology and case studies to the whole operating cycle as well as varying power demands. We anticipate that using stochastic optimization formulations will allow a significant improvement in process performance.
References Arana L.R., 2003. High-Temperature Microfluidic Systems for Thermally-Efficient Fuel Processing. Ph.D. thesis, Massachusetts Institute of Technology. Chachuat B., A. Mitsos and P. I. Barton, 2004. Optimal design and operation of micro power generation employing fuel cells. Submitted. Lee C. K. and P. I. Barton, 2002. Modeling, simulation, sensitivity analysis and optimization of hybrid systems. ACM Transactions on Modeling and Computer Simulation 12, 256. Metkemeijer R. and P. Achard, 1994. Ammonia as a feedstock for a hydrogen fuel cell; reformer and fuel cell behaviour. Journal of Power Sources 49, 271. Mitsos A., I. Palou-Rivera and P.I. Barton, 2004a. Alternatives for micropower generation processes. Industrial & Engineering Chemistry Research 43, 74. Mitsos A., M. M. Hencke and P. I. Barton, 2004b. Product engineering for man-portable power generation based on fuel cells. AIChE Journal (in press).
Acknowledgements This work was supported by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the Army Research Office under Grant DAAD 19-011-0566. We would like to acknowledge Klavs F. Jensen and the other members of the MIT ~tChemPower team for stimulating discussions.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1099
Performance monitoring of industrial controllers based on the predictability of controller behavior Rachid A Ghraizi a, Ernesto Martinez b, C6sar de Prada a, Francisco Cifuentes c, and Jos6 Luis Martinez c ~Dpt. of Systems Engineering and Automatic Control Faculty of Sciences, University of Valladolid, 47011, Spain bINGAR-Institute of Development and Design Avellaneda 3657, $3002 GJC - Santa Fe. Argentina CRepsol-YPF, Dpt. of Advanced Control Paseo de la Castellana 278, Madrid, Spain
Abstract This study focuses on performance assessment of industrial controllers. A methodology based on the concept of the predictability of controller errors is proposed for performance monitoring. The proposed approach is based on evaluating controller behaviour by analysing the time series of its error and to verify the existence predictable patterns beyond the control horizon in each one of the controlled variables of the process. To favour its implementation in a plant information system a performance index is proposed. For effectiveness of the monitoring algorithm, proper selection of some tuning parameters depending on the type of loop (temperature, level, pressure, etc.) is discussed. Examples using industrial data from a refinery are provided.
Keywords: Process control, controller performance, loop monitoring, performance benchmarking, fault diagnosis. 1. Introduction With the increasing complexity of control structures and the sheer number of controllers in modern process plants, the automation of performance monitoring tasks is a key issue to grasp the benefits of advanced control systems and real-time optimization (Thornhill, et al., 1999). In process plants there are thousands of control loops whose performance demands continuous supervision. Human personnel simply cannot have the budget of attention to handle this overwhelming task which renders many loops to remain open or providing a service much below the required standards. Abnormal operation of control loops can make a significant impact not only in the economy but also in the safety of the process. During the last decade several monitoring techniques have been developed. Desborough and Harris (1992, 1993) focusing on the comparison of the actual controller variance to ideal of a minimum variance controller. Thornhill, et al., (1999), proposed the prediction of the error to determine the performance of a SISO controller. Ghraizi, et al., (2004), suggest a practical index for performance monitoring of a control loop based
1100 on the analysis of the predictability of the error time series emphasizing proper selection of the control horizon using engineering judgment. The contribution of our work is based on the proposal of a procedure to obtain an index that allows the controller monitoring in closed loop and to evaluate its performance using predictions to detect the existence of predictable patterns in the time series of the error associated to each one of the controlled variables of the process. The method was applied to analyze off line some loops of PIDs in a petrochemical plant.
2. Monitoring methodology The performance-monitoring concept revolves around the idea of predictability of controller behavior beyond a chosen control horizon. Assuming the control horizon b has been chosen appropriately, the behavior of a perfectly working controller cannot be predicted beyond the interval of time during which any disturbance entering the loop up to a present time is supposed to be compensated (see Fig. 1 for details). On this ground, there may exist different alternatives to detect patterns of predictability in the time series associated to controller errors and manipulated variable changes. It is worth noting that as seen from time t, the controller error after time t+b of a properly working controller cannot be distinguished from a random walk stochastic process. Over the control horizon, the controller behavior is fully predictable since it corresponds to its own control policy built-in by design.
Figure 1: presentation of the prediction of the error.
It is worth discussing first the meaning of the control horizon b for a regulatory control task. Whatever the internal workings (PID, Predictive, etc.) of a controller, the value of b represents a sound engineering decision that takes into account among other things process dynamics, type of service and acceptable control energy. Let's denote by a scalar e(t) the controller error whereas ;(t) stands for the prediction of such error based on past values of the controller error, and possibly, control actions generated by the controller. The difference between the actual and predicted controller errors is the residue r(t) whose means and variance provide relevant information regarding the predictability of a controller behavior. The calculation of a performance index from a given data set demands some way of estimating future controller errors. The easiest way to do this is to propose a regression model of the following form: e(t + b) - a 0 + a l e ( t ) + a 2 e ( t - 1) + a 3 e ( t - 2) + ... + a m e ( t -
m + 1)
(1)
Where m is the model order and ai are the parameters to be fitted upon data using for example least-square regression. The Predictability Index (PI) is calculated to bear some similarity with the one proposed by Harris (1989) to measure the current
1101 performance regarding the best performance that can be achieved using a minimum variance controller, 2 PI - 1 cr r (2) mse
Where, cy2 is the residue variance and mse is the mean square error. Similar calculations can be used to define a measure of the predictability of controller outputs. For a given interval of time, if a controller does not exhibit a predictable behavior beyond the control horizon, c~ ~ mse gives rising to a near zero value of PI. As the controller behavior is more predictable rose increases relative to o-r2 , which in turn increases PI. For a controller exhibiting an easily predictable behavior (e.g., output saturation) o-j <<mse(t) and PI= 1. It is possible to define confidence intervals for sample estimations of the predictability index, which allows using control charts to detect excursions associated to loop malfunctions. The estimate to the confidence interval is carried out according to the following equation: r
T=11 a/Zn-1 --
~
~;
--
(3)
n
Where t~-a/Z,,-~ is the Student statistic, a is the level of confidence, n and or,. are, respectively, the size of the group of the data and the variance of the noise.
3. Parameter tuning It is necessary to provide some guidelines on how some parameters involved in the calculation of P1 should be selected. Parameter m, represents the order of the regression model. This parameter should have a value that is big enough to capture the characteristics of the time series of the error to reflect the predictable components in the model. As a rule of thumb, m should have a value slightly bigger than the loop settling time. Too high a value for m creates problems of overfitting and poor extrapolation capabilities in the model, which will affect the sample estimation of the PI index. Parameter n, is the size of the data sample and it should to take into account the trade off between index variance and data homogeneity. A very small size of the data set gives rise to big variance in the index distribution, while a too big data set is mixes heterogeneous data, which may mask a lot of important information. Since index calculation uses the error of controller and not the controlled variables, it is not necessary that the loop remains in the same set point, but it is important that the characteristics of the loop are the same throughout (Ghraizi, et al., 2003), such that, sensors, valves, control algorithms should not be altered by calibration or tuning. Regarding the sampling interval tm, is necessary to avoid an excessive or insufficient sampling. If the data are frequently sampled, the impulse response of the closed loop is not established inside the m samples. With low frequency sampling, the impulse response is only established inside a few samples and the important loop characteristics are not captured between the samples (Thornhill, et al., 1999; Stanfelj, et al, 1993). Parameter b represents the control horizon, which coincides with the prediction horizon for the time series model. It has been analysed by different authors like Harris, (1989), Desborough, and Harris, (1992), Harris, et al., (1996), Stanfelj, et al., (1993). In our work, we have observed that b should be equal to the loop settling time, independently
1102 of the type of the loop so that so it can reflect the necessary prediction characteristics in a control loop.
4. Industrial data analysis In order to test the index, several analysis were performed with a Toolbox implementing several monitoring functions. The length of the batches considered was n = 1000 samples of real plant data. The following graphs show the analysis of different loops taken from a petrochemical plant. The parameters were adjusted to the nature of the loops. In this way, we choose tm=5 sec., b = 15 (number of samples), m=30 for flow control loops. For level ones tm=60sec, b=30, m-30, (in this case n=720), for pressure tin-5 sec., b=5, m=30, and for temperature tm=60sec., b = 15, m-30. In the left part of all the graphs, we can observe the batches of data and their analyses, while in the right part we can see a zoom of a certain area of them to visualize some details. The upper graph corresponds to the controlled variable and its set point, while the values of the error and its predictions are in the graph in the middle. Finally, in the lower one, the values of the performance index P I are displayed. In figure 2 one can observe 17 batches of 1000 data of a flow loop which performance deteriorates at the 13 th batch, due to a perturbation that has affected the process. The performance index at the beginning doesn't have high values what indicates that the error has little predictability, as shown in the right hand side, but when the perturbation appears, the manipulated variable saturates and the P I almost is equal to one. Notice the bottom right hand side graph showing that the error can be predicted easily this time. Once the process returns to its normal state, the error, and the index takes a small value again. Data
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In figure 3 one can observe results from a level loop where a change in the behaviour takes place after t=1000 proximately. At the beginning, the PI has a small value but when the change takes place, PI increases reflecting the predictability of the error. By the contrary, the Harris index, shown in the bottom right hand side graph, does not performs equally well remaining in a low value which means bad tuning, not taking into account the special characteristics of an average level control.
1103
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Figure 4 displays data from a cascade loop in which a temperature output is following a changing set point very slowly with a significant steady error. In this case, the P I has high values all the time, and in the extended graph o f the right is seen that the error is completely predictable. In addition, the Harris index is consistent with this result. Data T e m p e r a t u r e
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Finally, graph 5 shows the data o f a pressure loop performing well. In this case, the error is not predictable and the values o f PI are always low. Also, the Harris index is high but shows a higher variability than the PI. Data pressure •
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1104 Figure 6 shows the main window of the developed Toolbox, which allows not only perform analyses of a control loop using the index IP, but also other analyses based on cross-correlation, power spectrum, impulse response, Harris index, etc., that can be used to confirm or perform a deeper analysis in order to detect the possible cause of the controller's bad operation. w~
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5. Conclusions This paper presents results showing a promising way of analysing the performance of industrial controllers using a time series of the error to detect the existence of predictable patterns. An index was computed to achieve this analysis evaluating the residuals between the controller's error and its prediction and some rules have been proposed to adjust the parameters of the method. Finally, it was applied to plant data showing a good behaviour.
References Desborough, L., Harris, T. 1993. Performance assessment measures for univariate feedforward/feedback control. Canadian Journal of Chemical Engineering, 71,605-616. Desborough, L., Harris T., 1992 Performance assessment measures for univariate feedback control, Can. J. Chem. Eng. 70, 1186-1197. Ghraizi R. A., Martinez E., de Prada C, 2004, Anfilisis del comportamiento de los controladores industriales, XI congreso latinoamericano de control automfitico Ciudad de La Habana, Harris T. J., (1989 Assessment of control loop performance, Can. J. Chem. Eng. 67, 856-861. Harris T.J., Boudreau F., MacGregor J.F., 1996 Performance assessment of multivariable feedback controllers, Automatica 32, 1505-1518. Stanfelj N., Marlin T.E., MacGregor J.F., 1993 Monitoring and diagnosing process control performance: the single loop case, Ind. Eng. Chem. Res. 32, 301-314. Thornhill N. F., Oettinger M., Fedenczuk P., 1999 Refinery-wide control loop performance assessment, Journal of Process Control 9 109-124.
European Symposiumon ComputerAided Process Engineering- 15 L. Puigjanerand A. Espufia(Editors) © 2005 Elsevier B.V. All rights reserved.
1105
A Systematic Approach to Plant-Wide Control Based On Thermodynamics Luis T. Antelo, Irene Otero-Muras, Julio R. Banga and Antonio A. Alonso* Process Engineering Group, Instituto de Investigaciones Marinas-CSIC C/Eduardo Cabello, 6 - 36208 Vigo, Spain
Abstract In this work, a systematic approach to plant-wide control design is proposed. The approach combines ingredients from process networks, thermodynamics and systems theory to derive robust decentralized controllers that will ensure complete plant stability. The proposed methodology will be illustrated on a reactor network exhibiting complex behaviour.
Keywords: Plant-wide Control, Process Networks, irreversible Thermodynamics, Inventory Control.
1. Introduction Over the years the area of plant-wide control has attracted the process engineering community as a challenging problem which drives continuing research efforts. A number of solutions to it were suggested, lying in between the following two extremes: a hierarchical decomposition of the original problem based on heuristic rules (Buckley, 1964; Luyben et al., 1997; Skogestad, 2002) and a mathematically oriented approach based on the solution of a given large scale mixed integer nonlinear programming dynamic optimization problem (see Biegler and Grossman, 2004, for an excellent review). Unfortunately, both lines of attack are hampered by a number of drawbacks which prevent their systematic application to general classes of process plants: the hierarchical approach usually leads to conflicting decisions only unravelled on a case by case basis and the mathematically one is limited by the high dimensionality of the problem and the restrictions imposed by the definition of the objective function. To overcome these issues, we combine previous results that link thermodynamics with passivity and Lyapunov theory (Farschman et al., 1998; Hangos et al., 1999; Alonso and Ydstie, 2001), and apply them to systematically design and/or to analyse stable decentralized control structures for process plants. In this way, the approach leads to hierarchical decentralized control structure which simultaneously ensures convergence of mass and energy inventories and guarantees robust stabilization of the intensive variables. The paper is structured as follows: In section 2, a formal representation of chemical process plants in terms of interconnected mass and energy networks is presented. The thermodynamic formalism and its consequences in designing decentralized controllers are presented in section 3. Finally, in Section 4 the approach is illustrated on a reaction network. *Author to whom correspondence should be addressed:
[email protected]
1106
2. The Underlying Structure of Process Networks A process network is defined by a number j = l ..... n of well mixed homogeneous material regions connected by material and energy fluxes we will refer to as nodes, plus an extra region j = 0 which represents the environment. To each node j in the network we associate a state vector zjs R ~+~of the form zj=(n/, .... n~,d) r, where n~ represents the mole numbers of component i, # is the internal energy and c stands for the total number of chemical species. Nodes and environment are connected by a set of n convective fluxes we refer to as f ~ It~+Cand p j ~ ) ~ R +for component and energy, respectively. Since energy is transported through material flows, we also have that pj(O) = 0. Note that the formalism as it stands can easily accommodate multiple connections between modes by defining ~b~ - £ a j k ~ b ~ , where ~. stands for either material or energy fluxes and %~ k=l
represents the fraction of ¢~ directed to the node k so that £ a i k = 1 (Hangos et al., k=l
1999).Dissipative transfer between nodes in the network is included through vectors cpk• R +de (with i=l .... c) a n d ~ E R +d'' which stand for mass and energy respectively. Network dynamics obey standard conservation principles for mole number and energy which, with some abuse of notation, can be formally stated as: h~ - N ¢ f ~ + N ¢CPk + v W
n~. , f k • ~+,. , k _ 1,.. ., c;
(1)
it - N o p + N~,~ + Q
u, p • R +"
(2)
where N¢ •N'×" denotes a convective column conservation matrix (Hangos et al., 1999).Ne ~ N,×Jc , N ~ R "×J'' are matrices which define network dissipative transfer interconnections, and the extra-terms v W and Q in Eqns (1)-(2) have been included to account for chemical reaction units and external heat sources, respectively. Alternative network dynamic representations can be derived from the fundamental network dynamics (1)-(2). In this way, the corresponding mass network representation can be easily obtained by defining transformations m - 2 crknk and ~b= 2 c r ~ f k k =1
with crk
k =1
being molecular weight of the k-specie. Inventory network representations will result from projecting Eqns (1)-(2) onto the linear operators P~ • R "-de×" and P~ • R È-d'×È satisfying P~ I N ~ v 1 - 0
and P~ I N ~ v I - 0, respectively, so that"
n"~I - N o . f ~
n ~l , f ~ • ~ +(,-de) ; k - 1,...,c;
(3)
i~ - Nopz
uz, pz ~ I~ +~"-~''~
(4)
3. Hierarchical Design of Decentralized Controllers 3.1. Thermodynamic foundations of Process Networks To start with, we consider that each node in the process network with volume vj is equipped with a continuous and twice differentiable scalar function Sj (zj, vj )" R +(c+2) ~ R with the following properties"
1107 1. S / ( z , v / )
is a first order homogeneous function in all their arguments so that
Sj - ~ z j +/~v/
(5)
2. S j (z/, v l)is strictly concave with respect to the vector zj. Function ~., as defined, coincides with classical entropy (Callen, 1985). From Property 1, it follows that entropy is an additive function so that for a process network its total entropy is of the form
S- ~S/(z/,v/).
Property 2 ensures that for
(z,v)
,/=1
constants, S has a maximum over the convex set: ,,
/~,
~,/
(6)
v .,
jl
.i--1
This, in fact, constitutes an evolution criterion as any thermodynamic system evolves in C so to maximize S. Finally, concavity also ensures the existence of a well defined oneto-one map between the vector of intensive properties /~j and the vector of densities
p~- v~lz/.
One main implication, of interest in process networks, is that
whenever ~b - 0 network states will evolve so to maximize S. This can be easily shown by noting that ~b - 0 :zz>~ - 0 for k = 1..... c and p(O) = 0 so that Eqns (3)-(4) become .,~
n~ -z~ I - 0 . Since material and energy inventories are constant, we are in the set C and the evolution criterion applies. It is worth mentioning that as the network evolves, entropy is being produced so by computing the time derivative of S along (1)-(2) we obtain: _..I
i
i
Since S is bounded from above by its maximum, we also have that a ( z ] ) - 0
(7*)" --/
,/=1
- MArg ax
C
S
'
the
equilibrium
StClte
for
of the network. The entropy balance
represented by Eqn (7) adopts the following form for open process networks:
dS
---P
+¢~ (,u/,~b)
(8)
where ~ (,u/ , 6) denotes the entropy flow through the network, which by definition becomes zero at the equilibrium state of the network, i . e . .
(~i, 6) - 0.
3.2. Decentralized control design In order to preserve and/or to enforce the stability of extensive as well as intensive plant states, we make use of previous results in system theory that links thermodynamics with passivity and Lyapunov theory (Hangos et al, 1999; Alonso and Ydstie, 2001). The underlying idea is to exploit the concavity of the entropy function to construct natural storage and Lyapunov candidates and to employ them to design stabilizing (usually high gain) controllers. However, as discussed previously, concavity can only be attained once the network states are in the set C defined in (6). This fact, motivates a hierarchical control design decomposition in which mass and energy inventory controllers are first designed to ensure that the network states will remain in the set C. In particular, the
1108 mass and energy inventory control layers consist of linear proportional controllers of the form:
~b1- ~b~+ c% (rnl -m~ )
pl - p~ + co,,(ul -u~ )
(9)
where c% and co are appropriate gain matrices constructed so that Nocom and N~co are negative definite. By applying (9) to the inventory network (3)-(4), it is straightforward to show that mt--4 m~ and u~ ~ u~ exponentially fast. Integral action could be also included in (9) without substantially altering the stability properties of the closed loop network (Farschman et al, 1998). From the thermodynamic properties discussed in Section 3.1, it also follows that since the network states are in the interior of the set C, its entropy will remain upper and lower bounded and consequently, from Eqn (8), we have that: T
Lim ~(P + ~ )dz - O
(10)
T--+~ 0
Unfortunately, condition (10) does not prevent the density states pj = v~lzj, and thus their intensive counterparts from exhibiting complex dynamic behaviour such as multiplicities. However, since the network remains in C, the network entropy is strictly concave and intensive variable control can be designed by, for instance, methods discussed in Alonso and Ydstie (2001) to ensure convergence of the network density states. This will be illustrated next on a representative example.
4. Case Study: A N o n - i s o t h e r m a l C S T R We illustrate the proposed hierarchical control design methodology on a jacketed nonisothermal CSTR in which an liquid phase irreversible exothermic reaction A -~ B takes place. A detailed description of the process model and parameters can be found in Alonso and Banga (1998).
4.1. Mass and energy inventory control Figure 1 depicts a schematic representation of the reactor and heat exchanger as a process network consisting of two nodes connected by a dissipative heat transfer term. The mass and energy inventory control approach discussed in Section 3.2 is diagrammatically represented in Figure 2. As shown in the Figure, mass inventory control results into a LC control loop which fix the residence time in the reactor. In order to implement the energy inventory control, an internal energy observer was designed to estimate the energy content in the reactor, and the estimate used as the measurement for the FC loop acting on coolant flow rate. As discussed previously, the proposed controllers guarantee inventories to converge to the desired set-point. However, the control structure does not ensure control of temperature and conversion in the reactor. Such behaviour motivates the thermodynamic analysis presented next.
4.2. Thermodynamic analysis of the reactor network. The inventory control structure allows us to re-write process network dynamics in terms of intensive variables, which for this case takes the form:
C
1109 where CA and T are composition and temperature, respectively, p is the reaction rate and 0,/7, and ), are dimensionless physical parameters related with residence time and heat transfer areas (see Alonso and Banga, 1998).
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Figure 1. Reactor process network
Figure 2. Proposed control structure
The entropy balance associated to this network takes the form given in (8) with entropy production and entropy flow being of the form: R
(12)
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d7 p
7"
dc
rdc (c::' - cA)o- g
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) is represented in Figure 3
along the set C defined by (6), which in our case corresponds with the manifold of constant mole number and energy. As it can be seen from the Figure, there are three points in C satisfying the steady-state entropy balance which correspond with an unstable and two stable stationary points. Note that the stability or instability can be easily established from the curvature of the systems entropy. 0.5:
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50 (s)
Figure 4. Concentration and Temperature evolution under intensive variable control
4.3. Intensive variables control In order to ensure stabilization of the intensive variables (reactor concentration and temperature) the inventory control configuration is completed with an intensive variable control loop acting on the coolant temperature To. Although different control structures could serve to that purpose, in this work a PI temperature controller and a non-linear exact linearizing control (Alonso and Banga, 1998) were tested, demonstrating for both
lllO cases state stabilization. The evolution of temperature and concentration under intensive variable control is presented in Figure 4. Finally, some control experiments were carried out in order to test the unstabilizing effect of input constraints on the closed loop dynamics. In this case, the thermodynamic analysis reveals new points in the region C satisfying the stationary entropy balance (see Figure 5) which corresponds with either stable, unstable or even limit cycles caused by total inventory fluctuations. A representation of one of such oscillatory regimes is depicted in Figure 6. a)
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'
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Figure 6.Evolution of the state and manipulated variable in the presence of input constraints
5. Conclusions In this contribution, a systematic plant-wide control design methodology has been presented. The approach, which combines tools and concepts from systems theory and thermodynamics, allows the design of decentralized control structures which ensure stabilization of both plant extensive and intensive variables. The proposed methodology has been illustrated on a reactor network exhibiting complex behaviour.
References Alonso, A.A. and J.R. Banga, 1998, Design of a class of stabilizing nonlinear state feedback controlled with bounded inputs. Industrial & Engineering Chemistry Research, 37, 131. Alonso, A.A. and B.E. Ydstie, 2001, Stabilization of distributed systems using irreversible thermodynamics. Automatica, 37, 1739. Biegler,L.T. and I.E. Grossmann, 2004, Retrospective on optimization. Computers & Chemical Engineering, 28, 1169. Buckley, P.S., 1964, Techniques of process control. John Wiley & Sons. Callen, H.B., 1985, Thermodynamics and an introduction to thermostatistics. John Wiley & Sons. Farschman, C.A., K.P. Viswanath and B.E. Ydstie, 1998, Process Systems and Inventory Control. AIChE Journal, 44, 1841. Hangos, K.M., A.A. Alonso, J.D. Perkins and B.E. Ydstie, 1999, Thermodynamic approach to the structural stability of process plants. AIChE Journal, 45,802. Luyben, M.L., B.D. Tyreus and W. Luyben, 1997, Plantwide control design procedure. AIChE Journal, 43, 3163. Skogestad, S., 2002, Plantwide control: Towards a systematic procedure. In CAPE 10 and ESCAPE 12. Elsevier, 57.
Acknowledgements The authors acknowledge the financial support received from the Spanish Government (MCyT Projects PPQ2001-3643) and Xunta de Galicia (PGIDIT02-PXlC40209PN).
European Symposiumon Computer Aided Process Engineering- 15 L. Puigjaner and A. Espufia(Editors) cC~2005 Elsevier B.V. All rights reserved.
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A Multiple Model, State Feedback Strategy for Robust Control of Non-linear Processes F. Y. Wang a* p. Bahri b, p. L. Lee c and I. T. Cameron
a
a Division of Chemical Engineering, The University of Queensland Brisbane, Qld 4072, Austral
