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Hierarchical Optimal Operation of Continuous-Batch Processes
Preprints, 5th IFAC Conference on Nonlinear Model Predictive Control 5th IFAC Conference on Nonlinear Model Predictive Preprints, Preprints, 5th IFAC Conference on Nonlinear Model Predictive September 17-20, 2015. Seville, SpainAvailable online at www.sciencedirect.com Control Control September 17-20, 2015. Seville, Spain September 17-20, 2015. Seville, Spain
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Hierarchical Optimal Operation of Continuous-Batch Processes Hierarchical Hierarchical Optimal Optimal Operation Operation of of Continuous-Batch Continuous-Batch Processes Processes C. de Prada*. S. P. Cristea*, R. Mazaeda* C. C. de de Prada*. Prada*. S. S. P. P. Cristea*, Cristea*, R. R. Mazaeda* Mazaeda* *Dpt. Of Systems Engineering and Automatic Control, University of Valladolid, 47011, Valladolid *Dpt. Control, of Spain (Tel: and +34Automatic 983 423162; e-mail:University [email protected]). *Dpt. Of Of Systems Systems Engineering Engineering and Automatic Control, University of Valladolid, Valladolid, 47011, 47011, Valladolid Valladolid Spain (Tel: +34 983 423162; e-mail: [email protected]). Spain (Tel: +34 983 423162; e-mail: [email protected]). Abstract: This paper presents a methodology for the optimal operation of mixed continuous-batch Abstract: This paper presents methodology the optimal of continuous-batch processes a hierarchical of the local dynamic optimizers perform the Abstract: based This on paper presents aa decomposition methodology for for the problem: optimal operation operation of mixed mixed continuous-batch processes based on a hierarchical decomposition of the problem: local dynamic optimizers economic optimization of the batch units or continuous plants, while the upper layer deals with the processes based on a hierarchical decomposition of the problem: local dynamic optimizers perform perform the economic optimization of the batch units or continuous plants, while the upper layer deals scheduling of the batch units and the coordination of the flows and properties of sub-processes usingthe economic optimization of the batch units or continuous plants, while the upper layer deals with with thea scheduling of the batch units and the coordination of the flows and properties of sub-processes using novel simplified model built with predictions from the lower level. The approach is illustrated with a case scheduling of the batch units and the coordination of the flows and properties of sub-processes using aa novel simplified model built predictions from lower level. approach is with study the sugar industry corresponding thethe joint operation of the evaporation and crystallization novel from simplified model built with with predictions to from the lower level. The The approach is illustrated illustrated with aa case case study from the sugar industry corresponding to the joint operation of the evaporation and crystallization sections operated to maximize production withtominimum consumption. study from the sugar industry corresponding the joint energy operation of the evaporation and crystallization sections operated to production with energy consumption. sections operated to maximize maximize production with minimum minimum energy consumption. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Process optimization, Hierarchical control, Continuous-batch systems, NMPC, Sugar industry Keywords: Process optimization, Hierarchical control, Continuous-batch systems, NMPC, Sugar industry Keywords: Process optimization, Hierarchical control, Continuous-batch systems, NMPC, Sugar industry 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION Process optimization is being considered more and more as a Process optimization is considered more more key component in order to achieve an efficient operation Process optimization is being being considered more and and more as asinaa key component in order to achieve an efficient operation process plants. There are many ways in which optimization key component in order to achieve an efficient operation in in process plants. There are ways in can be used in the management of a optimization factory. For process plants. There are many many and wayscontrol in which which optimization can the and of factory. For instance, the in Execution System offers can be be used used inManufacturing the management management and control control of aa(MES) factory. For instance, the Execution System offers many examples of its application in maintenance systems, instance, the Manufacturing Manufacturing Execution System (MES) (MES) offers many of maintenance systems, instrumentation etc. Inin paper we will focus many examples examples management, of its its application application inthis maintenance systems, instrumentation management, etc. In this paper we will our attention onmanagement, topics related management instrumentation etc.toIn production this paper we will focus focus our on production management and, in particular, on the related optimalto about the values our attention attention on topics topics related todecisions production management and, in on optimal the of key variables that affect decisions the globalabout operation of a and,some in particular, particular, on the the optimal decisions about the values values of some key variables that affect the global operation process plant, considering aims such as maximizing of some key variables that affect the global operation of of aa process plant, considering aims such as maximizing production, minimizing production costs, maximizing energy process plant, considering aims such as maximizing production, minimizing efficiency so on. production production,and minimizing production costs, costs, maximizing maximizing energy energy efficiency efficiency and and so so on. on. Decisions about the production and operation of a process Decisions about production of plant are organized a set of layers, as in Decisions about the the hierarchically production and andin operation operation of aa process process plant are organized hierarchically in a set of layers, in Fig.1. are Darby et al. (2011). This is ainsimplified plant organized hierarchically a set of schematic layers, as asnot in Fig.1. Darby et al. (2011). This is a simplified schematic not covering other important features, but represents the main Fig.1. Darby et al. (2011). This is a simplified schematic not covering but the elements for theimportant purpose features, of the paper. Basic control is in covering other other important features, but represents represents the main main elements for the purpose of the paper. Basic control charge of keeping safety and stability of the plant under elements for the purpose of the paper. Basic control is is in in charge of keeping safety and stability of the plant under control, implementing the control room operators’ or upper charge of keeping safety and stability of the plant under control, implementing the room operators’ or layers The Model Predictive (MPC) layer control,decisions. implementing the control control roomControl operators’ or upper upper layers decisions. The Model Predictive Control (MPC) targets improving control by considering the interactions, layers decisions. The Model Predictive Control (MPC) layer layer targets improving control considering the disturbances and operation constraints associated to process targets improving control by by considering the interactions, interactions, disturbances operation constraints to units or smalland plants. Finally, the Realassociated Time Optimization disturbances and operation constraints associated to process process units or small plants. Finally, the Real Time Optimization (RTO) layer aims at computing the operation points of the units or small plants. Finally, the Real Time Optimization (RTO) layer aims at computing the operation points process units that optimizes production within its admissible (RTO) layer aims at computing the operation points of of the the process units that optimizes production within its admissible operating range and according to a certain criterion. A RTO process units that optimizes production within its admissible operating range to criterion. RTO system normally usesaccording large models covering a wholeA operating range and and according to aa certain certain criterion. A plant, RTO system normally uses large models covering a whole plant, but the variables characterizing the optimal operation points system normally uses large models covering a whole plant, but the the operation points are only a subset characterizing of the hundreds thousands of variables but the variables variables characterizing theoroptimal optimal operation points are only a subset of the hundreds or thousands of variables that only are present room. variables, if not are a subsetinofthethecontrol hundreds or These thousands of variables that the room. not computed by thein layer because of lackvariables, of modelsif that are are present present inRTO the control control room. These These variables, if and not computed by the RTO layer because of lack of models and computed by the RTO layer because of lack of models and
adequate computing tools, are normally decided by the plant adequate are decided the managers according tools, to experience or heuristics. quite adequate computing computing tools, are normally normally decided by byBut the plant plant managers according to experience or heuristics. But often, due to the complexity of the problem, lack of managers according to experience or heuristics. But quite quite often, due the complexity of the of information, slow etc. the about lack the key often, due to to thedynamics, complexity of decisions the problem, problem, lack of information, slow dynamics, etc. the decisions about the key variables mainly avoidetc. creating bottlenecks, information, slowtarget dynamics, the decisions aboutviolating the key variables target avoid bottlenecks, violating constraints or risking safecreating operation of the plant, not variables mainly mainly target the avoid creating bottlenecks, violating constraints or risking the safe operation of the plant, being as close as possible to the optimum operating point. constraints or risking the safe operation of the plant, not not being close as to point. Then, these optimal decisions areoptimum passed asoperating targets to the being as as close as possible possible to the the optimum operating point. Then, these optimal decisions are passed as targets to lower levels, MPC/control layer, that implements them in the Then, these optimal decisions are passed as targets to the lower MPC/control layer, them in process, but the overall functioning far from the optimum lower levels, levels, MPC/control layer, that thatisimplements implements them in the the process, but the overall functioning is far from the optimum one. Even if an RTO system is in operation, quite often the process, but the overall functioning is far from the optimum one. if in quite often the difference thesystem steady is state models in the RTO layer, one. Even Even between if an an RTO RTO system is in operation, operation, quite often the difference between the steady state models in the RTO layer, which facilitates solution the inassociated NLP difference between the steady stateof models the RTO layer, which the solution of associated NLP optimization problem, the linear in the MPC which facilitates facilitates the and solution of the theones associated NLP optimization problem, and the linear ones in the MPC controllers creates inconsistencies that lead to suboptimal optimization problem, and the linear ones in the MPC controllers performance. controllers creates creates inconsistencies inconsistencies that that lead lead to to suboptimal suboptimal performance. performance.
Fig. 1. Hierarchical decision layers for process control and Fig. operation Fig. 1. 1. Hierarchical Hierarchical decision decision layers layers for for process process control control and and operation operation Alternatively, the RTO and MPC layers can be combined in a Alternatively, the and layers can economic MPC optimal dynamic operation problem asin Alternatively, theorRTO RTO and MPC MPC layers can be be combined combined ininaa economic MPC or optimal dynamic operation problem as economic MPC or optimal dynamic operation problem as in in
2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 294Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2015 responsibility IFAC 294Control. Copyright © 2015 IFAC 294 10.1016/j.ifacol.2015.11.298
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Engell et al. (2007), Gonzalez et al. (2001).This approach solves the inconsistency problem between layers and it is well stablished for continuous processes. Nevertheless, many plants include some parts where some process units operate in batch or semi-batch modes, which implies that scheduling of these units should be part of any optimization problem, complicating substantially the formulation and solution of the problem. Additionally, the complexity of the large-scale optimization problems that appears in these cases must be balanced with sensible computing times in order to allow for real-time implementation, which calls for some form of distributed computation.
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consumed by the continuous plant depends on the energy demands of the batch one. Hence, decisions on the optimal process functioning must be taken considering the joint operation of both plants.
Fig. 2. Schematic of continuous and batch plants operating in series with interchange of materials and energy.
Some contributions have been presented in the past to the problem of integrating scheduling and dynamic optimization Prata et al. (2008), Terrazas-Moreno et al. (2007) but few of them considers batch processes, Prada et al. (2009). Nevertheless, the subject has received increased attention due to the opportunities opened by the integration of different decision layers and the use of improved optimization methods and tools, Harjunkoski, et al. (2014), Nie et al. (2015).
2.2 Plant optimization Typical targets for this type of plants can be classified broadly into two groups: In the first one we assume that production flow is fixed, either in the continuous feed side or in the final products of the batch plant. The target is to operate the process minimizing energy or costs, while respecting quality specifications as well as other safety or process constraints. Minimization of energy use or costs, or some Resource Efficiency Indicators (REIs) such as specific energy consumption or yield, are natural candidates to be used as cost function.
This paper intend to contribute to the field studying the special plant wide optimization problems that results when continuous plants must operate together with batch ones in a synchronized way interchanging materials and energy and considering some of the large-scale optimization difficulties that must be faced. The paper proposes a hierarchical/decentralized architecture that facilitates the implementation of the RTO, taking into account what can be considered more adequate from the point of view of the control room. The use of the proposed architecture is illustrated with an example taken from the sugar industry: the joint operation of the sections of evaporation and crystallization.
In the second group of targets, the aim is to maximize production making the best use of the available resources (energy or others) and respecting, as before, process and quality constraints. Concerning the continuous plant, the characteristics of the material and energy streams connected to the batch plant, as well as the ones of the energy and materials feeds, define its operation point and the interaction with the other plant. Hence, those variables associated with these streams (such as steam pressure or concentration of a product), and belonging to the degrees of freedom of the process, must be included in the decision variables u c of the problem, characterized by a set of equality and inequality equations describing the dynamic model h c and constraints g c affecting this plant as in (1):
The paper is organized as follows: after the introduction, section 2 deals with the formulation of the problem and section 3 presents the hierarchical/ distributed architecture. Then, section 4 shows the case study of the sugar factory and section 5 provides results corresponding to several operational modes. The paper ends with some references. 2. HYBRID PLANT OPTIMIZATION
h c ( x c , x c , u c ) = 0 g c (x c , u c ) ≤ 0
2.1 The process
(1)
Regarding the batch plant, there are two problems that must be considered: the first one is the local optimal control of every batch unit j, with decision variables ubj, and the second one is the scheduling of the operation of the whole set of units taking into account the limited shared resources coming from the continuous plant and other possible constraints. So, in the operation of the batch plant, besides individual dynamic models h bj and constraints g bj for the batch units, with local decision variables u bj , , as in (2), we must consider the global material balances and decision variables like starting times tij of these units. Notice that the operation of every batch unit, in terms of batch duration or mass and energy demands per cycle, may depend on the quality of the
The type of processes considered are represented schematically on Fig.2, where we can see a continuous plant that processes a flow of raw products using a certain amount of energy, followed by another plant whose process units operate in batch mode, with materials and energy coming from the continuous plant. Notice that both plants are interacting: It is obvious that the batch plant depends in its operation on the quantity and quality of the energy and materials it receives from the continuous plant, but, conversely, the inflow of raw products of the continuous plant is limited by the capacity of the batch one according to the schedule of the units, and the energy 295
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corresponding supply received from the continuous plant. Also, the batch plant throughput, which depends heavily on the scheduling of the batch units, may act as the bottleneck of the whole process and must be matched to the continuous plant production as in (3), where f and r describe the balance of material and energy at the interface as well as the associated constraints.
h bj ( x bj , x bj , u bj ) = 0 g bj ( x bj , u bj ) ≤ 0
j = 1,..., B
the overall plant dynamics and the interactions among the process units, so it should be built using simplified models including only the elements that are essential for the coordinated operation of the sub-processes. If the problem is formulated within the framework of economic MPC, then, it will include predictions of the future behaviour of some key variables, which can be provided by the optimizers of the lower layer, besides mass and energy balances governing the interaction of the continuous and batch units and the dependence of the batch operation with the properties of the continuous streams. Changes in these ones being incorporated by mean of sensitivities computed in the lower level. A model for the coordinator layer built in this way avoids the heavy internal mathematical model of the process units, while retaining the main elements for proper decision making, being a good balance between adequate process representation and easiness of computation. The upper layer model is used in a dynamic optimization problem with economic aim to generate the batch units scheduling and targets and constraints for the interacting streams and optimizers of the lower layer, solving an optimization problem that includes all constraints required for the global safe operation of the batch and continuous units.
(2)
f c ( x c , u c ) = ∑ f bj ( x bj , u bj ) j
rc ( x c , u c ) ≤ ∑ rbj ( x bj , u bj )
(3)
j
On any case, any formulation of an optimization problem must consider the dynamic and discrete nature of scheduling, and the interactions with the continuous plant, making the problem a complex one. The fact that the whole process must be considered simultaneously calls for centralized formulations which lead to MINLP problems, but its mixed continuous-batch nature imposes a heavy computational load that does not favour a solution of the problem to be used in real-time. In addition, physical implementation of the solutions must consider the human factor, that is, the personnel dealing with the system and its maintenance, which calls for advanced applications close to the process units, and as simple as possible. In the control room of a large process factory, different operators are in charge of different sections of the plant and centralized optimization applications does not provide the required flexibility for adaptation to the distributed responsibilities. Another consideration refers to expandability and facility of adaptation to the upsets or different number of units in operation that occur from time to time. All these factors lead to consider decentralized architectures able to provide similar performance than a centralized one, but having better implementation characteristics.
The idea is summarized in Fig.3, with the upper layer in charge of computing the most favourable operating conditions of the sub-processes compatible with the joint operation of the whole system and a lower layer of Economic MPCs (EMPC) performing optimal control of every continuous or batch sub-process. Notice that the main task of the coordination layer is not mainly finding optimal set points for the lower layer according to a global economic optimum, as in a typical RTO-MPC setting, but to establish global feasible ranges that facilitate the optimal operation of the subprocesses, which, in turn, create new conditions and material and energy demands that the upper layer must satisfy generating another schedule or constraints and targets. The interaction of both layers allows for recursive improvements until convergence to a plant-wide optimum is eventually reached.
3. HIERARCHICAL / DECENTRALIZED ARCHITECTURE FOR OPTIMIZATION The approach proposed in this paper considers the continuous plant and the batch units as the natural single pieces to build an optimization strategy due to the fact that local models can be more easily defined for each of them and also considering that the corresponding optimal operation of each process unit can be likely computed and implemented following the existing structure of the control room. Nevertheless, it is clear that the solution of the individual optimization problems will be neither adequate nor globally optimal unless the interactions among them, the global behaviour and the scheduling of the batch units are taken into account explicitly. This requires a coordination layer dealing with them, which must be placed hierarchically on top of the process units optimizations.
Fig. 3. Hierarchical and distributed architecture for global optimization of the hybrid system. Notice that one important characteristic that this architecture resides in the fact that it allows parallelism in the computation of the heavier tasks that correspond to the distributed local optimizers, facilitating real-time optimization. At the same time, the modular architecture facilitates both, the activation or deactivation of units if there are not in service, and an easy expandability to new units.
This layer cannot repeat the large model of the centralized approach, but should contain the key element that describe 296
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4. SUGAR FACTORY CASE STUDY
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the juice from the previous evaporator. The vapours generated in the second effect being used also to heat the crystallization section. The evaporators present a decreasing profile of pressures and temperatures, and increasing concentrations, with the pressure in the juice chamber of the last effect being maintained with the help of a barometric condenser for the vapours.
In order to illustrate and test the previous ideas, a case study taken from the sugar industry is presented. It corresponds to a benchmark proposed within the FP7, EU Network of Excellence “Highly-complex and Networked Control Systems (HYCON2)”, and described in detail in Mazaeda et al, (2014). The benchmark is fully accessible in HYCON/WP5 (2014) where additional information is provided as well as the full process simulator. It presents the problem of the joint operation of the evaporation and crystallization sections of a sugar factory and responds to the schematic of Fig.4. Previous contributions can be seen in Hernandez et al. (2014), where a centralized approach is presented, as well as in Mazaeda et al.(2015), that uses an approach similar to the one in this paper, but limited to the batch section..
The basic control system is shown in Fig.4 and includes level controllers in the evaporators, fresh juice flow control, supply steam to the first effect pressure control and a controller of the brix at the output syrup, that operates manipulating the vapours flow to the barometric condenser. Notice that the syrup flow cannot be manipulated independently if these controllers are in operation, as it is imposed by the balance of dry matter. In this way, the pressure set point of the supply steam and the brix set point of the syrup are the two key variables that define the evaporation operation, as reflected in Fig.5. Brix, purity and flow of the fresh juice can be considered as the main disturbances, with the inflow being sometimes also a manipulated variable according to the mode of operation.
4.1 Process plant and control system description The picture depicts the evaporation section on the left and the crystallization section on the right with while Fig.5 shows a block diagram of them with the main streams involved.
Regarding the crystallization section, it incorporates a buffer tank for supplying syrup to three semi-batch crystallizers that can operate in parallel following a certain recipe to produce sugar crystals. The operation in the crystallizers, also called tachas or vacuum pans, starts with the load of an initial amount of syrup, followed by its concentration by boiling using the vapours from the evaporation section as heat source, with the purpose of achieving the required supersaturation. The crystallizers work under vacuum to facilitate evaporation and prevent caramelization. Then, in the following step, the seed of tiny crystals takes place, which constitutes the population that will grow in size in a controlled manner in the next step: the growing of the grain. In this stage, while saccharose moves to the sugar crystals, oversaturation in the solution is maintained by a mixture of evaporation and addition of new syrup. When the crystals reach the required size and the crystallizer is full, the strike ends and the resulting massecuite is discharged. Fig.6 depicts a typical time evolution of the level in a crystallizer where, after the initial load, it grows slowly up to the end of the strike.
Fig. 4. Evaporation and crystallization sections with basic control loops.
Fig. 5. Block diagram of the two sections with the main streams involved and global decision variables marked as yellow points.
Syrup brix 75%
The purpose of the evaporation section is to increase the concentration of a sugar juice fed to it up to a certain value in the syrup at the output of the section. This concentration is normally measured in brix degrees (% in weight). The operation is carried out in a set of three evaporators working in a multiple effect configuration with the first one receiving a flow F of fresh juice in the juice chamber and a flow F v of saturated steam at a certain pressure P s in the heating chamber. The juice boils and, as the water is evaporated, its sugar concentration increases, while the vapours generated are used to heat the following evaporator, which is fed with
Syrup brix 71%
Fig.6. Time evolution of the level of a crystallizer for two different values of the syrup brix Fig. 6 shows the level profile for two values of the brix of the feed, showing that decreasing brix enlarges the cycle time. 297
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Also, one could see that the amount of steam consumed from the evaporation section increases if the syrup brix of the feed decreases, showing the strong interaction between the quality of the syrup from the evaporation and the crystallizers operation. Purity also plays an important role in the crystallizer operation but, in order not to complicate too much the process description and concentrate the focus in the hierarchical optimization, it will not be considered explicitly here as, in a certain way, it parallels the brix effect.
4.2 Process Optimization Two different modes of operation have been selected: In the first one, the factory is processing a certain amount of beets, so that the flow of fresh juice at the evaporators input can be considered fixed (but not necessarily constant) by the previous processes and the aim is to globally operate the joint continuous-batch process (that is, select the values of the five key variables and the scheduling of the batch units) with minimum energy (steam) consumption and avoiding bottlenecks, in spite of possible disturbances, e.g. changes in the fresh juice brix or flow. Quality of the sugar production is assured by the local controllers implementing the recipe of the crystallizers, but different cycle duration or steam or juice consumption are expected depending on the operation of the evaporation and local steam pressure SP in the crystallizers, which, in turn, will affect the energy efficiency and plant throughput, creating possible bottlenecks.
In the same way, decreasing the heating vapour pressure will enlarge the cycle time. This can be seen in Fig.7, where the upper graph displays profiles of pressure over a full cycle and the lower one the corresponding level time evolution. Additionally, decreasing pressure decreases supersaturation and improves crystal size distribution.
PFactor=1.2
In the second mode of operation, the aim is to increase the flow of fresh juice as much as possible, keeping also the previous aims of minimizing energy consumption. In this case, the juice feed flow becomes both, a target and a new decision variable.
PFactor=1.0
PFactor=0.8
The solution follows the scheme of Fig.3, with a coordination layer and four local dynamic process optimizers in the lower layer: one for the evaporation section and one for each of the three sugar crystallizers. The economic MPC attached to the evaporation section, solves the dynamic optimization problem (4) every sampling time, minimizing specific steam consumption (or maximizing fresh juice flow according to the mode of operation), under the constraints imposed by the evaporation model h evap and constraints g evap affecting pressure, temperature and brix. In (4), x e refers to the state and u e to the decision variables. The main decision variable is the set point of the supply steam pressure (plus the flow of fresh juice in the corresponding mode of operation), while the set point of the syrup brix is imposed by the coordination layer because it is a key variable governing the interaction between both sections. In the same way, predictions of the steam demands from the crystallizers are computed in the upper coordination layer and passed to the evaporation local NMPC controller as a boundary condition.
PFactor=0.8 PFactor=1.2
PFactor=1.0
Fig.7 Crystallizer heating pressure profile (upper) and level profile for three different pressure values. According to the crystallizer recipe, the pressure follows a certain profile, but it can be modified as in Fig.7. This can be seen as applying a factor P factor to a nominal profile. This factor can be considered as the main knob the operators dispose to modify the vacuum operation, as most of the other variables are fixed by the local sequencing control according to its recipe. Similarly, we will consider syrup brix of the feed as the main disturbance.
min ∫ Fsteam dt / ∫ Fdt Ps
A local control system implements the sequence of steps included in the recipe, with local controllers of level, brix of the massecuite, heating steam pressure and vacuum. We will assume that the local control system performs its duties adequately, so that it is able to guarantee the right crystal growing at the end of the strike, at least for an acceptable range of operating conditions. This leaves the set point of the heating chamber pressure controller as the main knob for the operators to influence the crystallizer operation, as mentioned above. Notice that the aim of operating with low steam consumption does not conflict with the local one of maintaining crystal quality. In addition, the starting times of the crystallizers define the batch plant operation, leaving a total of six degrees of freedom per cycle in this plant as reflected in Fig.5.
h evap ( x e , x e , u e ) = 0
(4)
g evap ( x e , u e ) ≤ 0 The dynamic model of the evaporation section is a DAE system composed by mass and energy balances, as well as by thermo-dynamical equilibriums and flow equations involving 157 equations and variables, including 13 states. Notice that, after solving (4), the future profile of the syrup flow to the crystallization section can be computed easily and passed later to the coordination layer. The constraints refer mainly to the admissible ranges for pressure and brix in the evaporators. The model has being implemented in the EcosimPro simulation environment.
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In the same way, every crystallizer j has an economic NMPC attached to it, formulated as (5):
min Pfactor
layer EMPCs predictions, as the ones from the crystallizers syrup and steam consumptions displayed in Fig.9. As mentioned before, similar profiles are obtained for the syrup flow from the evaporation. The grey model is built on the fly capturing the key elements that influence the global dynamics and involving the most important variables at this level, being a good compromise between process complexity and on-line computational load.
∫ Fvapjdt
h bj ( x bj , x bj , u bj ) = 0 g bj ( x bj , u bj ) ≤ 0
299
(5)
Tbatchj ≤ Tbatch _ max Here the aim is to perform the crystallization cycle consuming as less steam from evaporation as possible, which also improves crystal size distribution, being the main decision variable P factorj , a factor applied to the steam pressure profile that can be varied within a range. Besides the dynamic model h bj , the constraints include among others the super-saturation of the syrup in the crystallizer, which must be maintained in a range in order to avoid the creation of spontaneous new crystals or crystal dissolutions. In addition, another constraint is highlighted in (5): the maximum cycle time allowed T batch_max , which is fixed by the coordination layer, as well as the starting time of the cycle T loadj . The model is composed by mass and energy balances plus equilibriums and crystal growing equations, totalling 227 variables for each crystallizer, including 30 states. The model includes discontinuities corresponding to the different stages, but the dynamic optimization problem (5) can be solved with current gradient-based optimization software using a sequential approach, as in Fig.8, due to the fact that the number and type of discontinuities is kept constant by the control system, Galan et al. (1999). At every iteration of the NLP optimizer a set of the decision variables are proposed to the dynamic model simulation, evaluating de cost function and constraints as required by the NLP. A similar architecture is used to solve problem (4)
Fig.9 Typical profiles of syrup and steam flows to the crystallizers Referring to the simplified schematic of Fig.10, that capture the main elements related to global process operation, total demand of steam to the evaporation section can be computed from the individual profiles of steam consumption and the schedule of the crystallizers. This demand, needed to solve the evaporation model, will be passed to the evaporation NMPC after each execution of the coordination layer. In the same way, material balances around the syrup storage tank can be calculated using the profiles of the individual crystallizer’s demands of syrup, ordered over time according to the schedule, and the syrup outflow predictions provided by the evaporation. Feasible operation in the use of these shared resources is required and can be achieved imposing appropriate conditions on the level of the storage tank and pressure and continuity on the steam demand.
Fig.10 Simplified schematic of the hybrid evaporationcrystallization process
Fig.8 Sequential approach for solving optimization problems of the lower layer
the
Nevertheless, it is important to notice that, in the optimization problem to be solved in the coordination layer, the profiles cannot be considered constant because they may change as a result of the future decisions taken on the schedule or on the syrup brix SP of the evaporation. These changes are considered by constructing a model for the profiles that incorporates the sensitivities of these flows to the variables passed from the upper layer: maximum cycle time of crystallizers T bmax and syrup brix b as in (6).
dynamic
After solving (5), the profiles of the steam and syrup demands of every crystallizer are computed and passed to the upper coordination layer. 4.3 Coordination layer
Fvapj = Fvap0 j +
Problems (4) and (5) cannot provide sensible and feasible solutions unless global balances are satisfied and constraints respected, which is the main task of the upper layer. For this purpose, a surrogate model is built at this level mixing physical balances that must always be satisfied with the profiles of the interacting variables provided by the lower
∂Fvapj ∂Tb max
Fsyrupj = Fvyru0 j + Fe = Fe 0 +
299
(Tb max − Tb max 0 ) +
∂Fsyruj ∂Tb max
∂Fe (b − b 0 ) ∂b
∂Fvapj
(Tb max − Tb max 0 ) + j = 1...3
∂b
(b − b 0 )
∂Fsyrupj ∂b
(b − b 0 ) (6)
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The sensitivities can be computed locally by the EMPCs of the lower layer and passed to the coordination one. In (6) the sub index 0 refers to the profile obtained from the corresponding EMPC and F e stands for the syrup flow from evaporation. The full demand of each steam or syrup stream is built by adding these profiles shifted in time according to the schedule of the crystallizers.
syrup flow to the syrup tank as an additional decision variable that is also maximized in the cost function. This value is passed as an upper limit to the evaporation EMPC that also includes fresh juice flow as a new decision variable and as a target to maximize. Prediction Horizon (NP complete cycles for each pan) Control Horizon (Nc cycles for each pan, Ej: Nc=2)
Regarding the schedule of the batch units, in this case study the problem can be formulated as selecting their successive i starting times over the prediction horizon T loadji for every tacha j, so that the operation constraints are satisfied in the best possible way. In this way, as T loadji are continuous variables, the use of mixed integer codes is avoided. Obviously, the schedule of the crystallizers is part of the coordination problem, where other decision variables are involved. In particular, the syrup brix SP of the evaporation is used as a decision variable, as mentioned before (assuming no steady state error in the corresponding PID controller). As the problem may have multiple solutions, the one that provides the best conditions for the sub-processes operation is sought, that is, the one that minimizes the total steam demand to the evaporation, which corresponds with maximum possible length cycle in the crystallizers and maximum time interval between cycles of a crystallizer. Maximum cycle time is incorporated as another decision variable, while both, the interval between cycles, are considered as well as targets to maximize. The problem is formulated in continuous time domain with a prediction and control horizons covering several cycles, as the ones displayed in Fig 11, where the levels in the units have been used to identify the different cycles of the three crystallizers:
min
Tloadj , brix , Tbatch _ max
∑∫F
vapj
TLoad1_1
TLoad1_2
Load1_1
LL
Load1_2
... TLoad2_1
TLoad2_2
Load2_1
LLL
.
Load2_2
... TLoad3_1 Load3_1
TLoad3_2 Load3_2
Fig.11 A schedule of the crystallizers showing the control and prediction horizons comprising an integer number of cycles. 5. SIMULATION RESULTS The approach has been tested in a process simulator implemented in the modelling environment EcosimPro E.A.I (2015) for different operating conditions. In this section, some results corresponding to an exercise in which we try to maximize the flow of juice processed, while minimizing steam consumption, are provided for one day of operation.
dt − ∑ Dt ji − Tbatch _ max
s.t. extended profiles (6) 0 ≤ Tloadji ≤ Tloadji+1 j = 1,2,3
...
L
i = 1,2,..., N u
Tmin ≤ Tloadj( i +1) − Tloadji − Tbatch _ max = Dt ji ≤ Tmax TB min ≤ Tbatch _ max ≤ TB max
(7)
L min ≤ L buffer ≤ L max D min ≤ Tbatch _ max − Tbatch _ max 0 ≤ D max d min ≤ b − b 0 ≤ d max
dm buffer / dt = Fe − ∑ Fsyrupj
dm buffer b / dt = Fe b e − ∑ Fsyrupjb Here, the constraint in the buffer tank level L buffer avoids possible bottlenecks, while the ones on the changes of T batch_max and b are used to include some type of trust region, to guarantee the validity of (6). The last two differential equations reflect mass balances in the buffer tank. The coordination layer generates the brix set point of syrup and steam demand for the evaporation, starting times of the crystallizers and the maximum allowed cycle times that are passed to the EMPCs.
Fig.12 Time evolution of the fresh juice to evaporation
If the global target is to maximize the fresh juice processed, while still minimizing steam use, then the optimization problem of the coordination layer is changed to include the
Fig.13. Schedule of the crystallizers represented with the pan levels. Every crystallizer in a different colour. 300
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Fig.12 shows the time evolution of the flow of fresh juice to the evaporation, clearly reaching a maximum, while Fig.13 displays the optimal schedule of the crystallizers by means of the levels in the pans. Fig.14 gives the syrup level in the storage tank, which must be maintained between 20 and 90%, and Fig. 15-16 the supply fresh steam to the evaporation, and pressure in the crystallizers header, showing that plant-wide optimum operation is feasible.
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approach has been illustrated with a case study corresponding to the joint operation of the evaporation and crystallization sections of a sugar factory problem. ACKNOWLEDGEMENT The authors wish to thank the Spanish MINECO for its support through project DPI2012-37859. REFERENCES Darby M. L., Nikolaou M., Jones J., Nicholson D. (2011). RTO: An overview and assessment of current practice, Journal of Process Control.Vol. 21, 874–884. de Prada, C.; Grossmann, I. E.; Sarabia, D.; Cristea, S. (2009) A strategy for predictive control of a mixed continuous batch process. Journal of Process Control, 19, 123–137. Empresarios Agrupados Int.EcosimPro, last relese (2015) http://www.ecosimpro.com/products/ecosimpro/ Engell, S. (2007). Feedback control for optimal process operation. Journal of Process Control, vol. 17, 203-219. Galán S., Feehery W.F., Barton P.I., (1999) Parametric sensitivity functions for hybrid discrete/continuous systems, Applied Numerical Mathematics,31, 17–47. Gonzalez Ana I., Zamarreño J.M., de Prada C. (2001) Nonlinear model predictive control in a batch fermentator with state estimation, European Control Conference, Porto, Portugal, ISBN: 972-752-047-2 Hernández R., Simora L., Paulen R., Wegerhoff S., Mazaeda R., Engell S., de Prada C. (2014). Optimal Allocation of Shared Resources in an Integrated Sugar Production Plant. Computer Aided Process Engineering series, vol. 33, pp. 637-642. Harjunkoski, I.; Maravelias, C. T.; Bongers, P.; Castro, P. M.; Engell, S.; Grossmann, I. E.; Hooker, J.; Mndez, C.; Sand, G.; Wassick, J. (2014) Scope for industrial applications of production scheduling models and solution methods. Computers & Chemical Engineering, 62, 161 – 193. HYCON/WP5 (2014). Sugar show case. available fro download at: hycon.isa.cie.uva.es/home. Mazaeda, R., Acebes, L.F., Rodríguez, A., Engell, S., Prada, C. (2014). Sugar Crystallization Benchmark. Computer Aided Process Eng. series, vol. 33, pp. 613-618. Mazaeda R., Cristea S. P., de Prada C. (2015) Plant-wide hierarchical optimal control of a crystallization process, Int. Symposium on advanced Controlof Chemical Processes, ADCHEM2015, IFAC, Wishtler, British Columbia, Canada Nie Y., Biegler L.T., Villa C.M., Wassick J.M. (2015) Discrete Time Formulation for the Integration of Scheduling and Dynamic Optimization, Ind. Eng. Chem. Res., 54 (16), pp 4303–4315 Prata, A.; Oldenburg, J.; Kroll, A.; Marquardt, W. (2008) Integrated scheduling and dynamic optimization of grade transitions for a continuous polymerization reactor. Computers &Chemical Engineering, 32, 463–476. Terrazas-Moreno, S.; Flores-Tlacuahuac, A.; Grossmann, I. E. (2007) Simultaneous cyclic scheduling and optimal control of polymerization reactors. AIChE Journal, 53, 2301–2315.
Fig.14 Buffer tank level to be kept between 20 and 90%.
Fig.15. Supply steam flow to the evaporation
Fig.16. Steam pressure in the collector to the crystallizers 6. CONCLUSIONS A novel approach for the optimal operation of a class of combined continuous-batch processes has been presented with the purpose of offering a solution that, avoiding the bottlenecks created by traditional heuristic management, could be implemented with a sensible computational load. Its main points are the hierarchical decentralized architecture and the simplified grey model of the coordination layer. The architecture provides feasible targets and operating ranges to the lower level as well as flexibility for adding or removing process units, updating the upper layer model, even on-line, or changing the optimization targets and fits into the control room organization. The distribution of tasks and the simplified upper model, besides the parallelization of the computation, offers competitive computation times without the complexity of other distributed architectures. The 301
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Hierarchical Optimal Operation of Continuous-Batch Processes Hierarchical Hierarchical Optimal Optimal Operation Operation of of Continuous-Batch Continuous-Batch Processes Processes C. de Prada*. S. P. Cristea*, R. Mazaeda* C. C. de de Prada*. Prada*. S. S. P. P. Cristea*, Cristea*, R. R. Mazaeda* Mazaeda* *Dpt. Of Systems Engineering and Automatic Control, University of Valladolid, 47011, Valladolid *Dpt. Control, of Spain (Tel: and +34Automatic 983 423162; e-mail:University [email protected]). *Dpt. Of Of Systems Systems Engineering Engineering and Automatic Control, University of Valladolid, Valladolid, 47011, 47011, Valladolid Valladolid Spain (Tel: +34 983 423162; e-mail: [email protected]). Spain (Tel: +34 983 423162; e-mail: [email protected]). Abstract: This paper presents a methodology for the optimal operation of mixed continuous-batch Abstract: This paper presents methodology the optimal of continuous-batch processes a hierarchical of the local dynamic optimizers perform the Abstract: based This on paper presents aa decomposition methodology for for the problem: optimal operation operation of mixed mixed continuous-batch processes based on a hierarchical decomposition of the problem: local dynamic optimizers economic optimization of the batch units or continuous plants, while the upper layer deals with the processes based on a hierarchical decomposition of the problem: local dynamic optimizers perform perform the economic optimization of the batch units or continuous plants, while the upper layer deals scheduling of the batch units and the coordination of the flows and properties of sub-processes usingthe economic optimization of the batch units or continuous plants, while the upper layer deals with with thea scheduling of the batch units and the coordination of the flows and properties of sub-processes using novel simplified model built with predictions from the lower level. The approach is illustrated with a case scheduling of the batch units and the coordination of the flows and properties of sub-processes using aa novel simplified model built predictions from lower level. approach is with study the sugar industry corresponding thethe joint operation of the evaporation and crystallization novel from simplified model built with with predictions to from the lower level. The The approach is illustrated illustrated with aa case case study from the sugar industry corresponding to the joint operation of the evaporation and crystallization sections operated to maximize production withtominimum consumption. study from the sugar industry corresponding the joint energy operation of the evaporation and crystallization sections operated to production with energy consumption. sections operated to maximize maximize production with minimum minimum energy consumption. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Process optimization, Hierarchical control, Continuous-batch systems, NMPC, Sugar industry Keywords: Process optimization, Hierarchical control, Continuous-batch systems, NMPC, Sugar industry Keywords: Process optimization, Hierarchical control, Continuous-batch systems, NMPC, Sugar industry 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION Process optimization is being considered more and more as a Process optimization is considered more more key component in order to achieve an efficient operation Process optimization is being being considered more and and more as asinaa key component in order to achieve an efficient operation process plants. There are many ways in which optimization key component in order to achieve an efficient operation in in process plants. There are ways in can be used in the management of a optimization factory. For process plants. There are many many and wayscontrol in which which optimization can the and of factory. For instance, the in Execution System offers can be be used used inManufacturing the management management and control control of aa(MES) factory. For instance, the Execution System offers many examples of its application in maintenance systems, instance, the Manufacturing Manufacturing Execution System (MES) (MES) offers many of maintenance systems, instrumentation etc. Inin paper we will focus many examples examples management, of its its application application inthis maintenance systems, instrumentation management, etc. In this paper we will our attention onmanagement, topics related management instrumentation etc.toIn production this paper we will focus focus our on production management and, in particular, on the related optimalto about the values our attention attention on topics topics related todecisions production management and, in on optimal the of key variables that affect decisions the globalabout operation of a and,some in particular, particular, on the the optimal decisions about the values values of some key variables that affect the global operation process plant, considering aims such as maximizing of some key variables that affect the global operation of of aa process plant, considering aims such as maximizing production, minimizing production costs, maximizing energy process plant, considering aims such as maximizing production, minimizing efficiency so on. production production,and minimizing production costs, costs, maximizing maximizing energy energy efficiency efficiency and and so so on. on. Decisions about the production and operation of a process Decisions about production of plant are organized a set of layers, as in Decisions about the the hierarchically production and andin operation operation of aa process process plant are organized hierarchically in a set of layers, in Fig.1. are Darby et al. (2011). This is ainsimplified plant organized hierarchically a set of schematic layers, as asnot in Fig.1. Darby et al. (2011). This is a simplified schematic not covering other important features, but represents the main Fig.1. Darby et al. (2011). This is a simplified schematic not covering but the elements for theimportant purpose features, of the paper. Basic control is in covering other other important features, but represents represents the main main elements for the purpose of the paper. Basic control charge of keeping safety and stability of the plant under elements for the purpose of the paper. Basic control is is in in charge of keeping safety and stability of the plant under control, implementing the control room operators’ or upper charge of keeping safety and stability of the plant under control, implementing the room operators’ or layers The Model Predictive (MPC) layer control,decisions. implementing the control control roomControl operators’ or upper upper layers decisions. The Model Predictive Control (MPC) targets improving control by considering the interactions, layers decisions. The Model Predictive Control (MPC) layer layer targets improving control considering the disturbances and operation constraints associated to process targets improving control by by considering the interactions, interactions, disturbances operation constraints to units or smalland plants. Finally, the Realassociated Time Optimization disturbances and operation constraints associated to process process units or small plants. Finally, the Real Time Optimization (RTO) layer aims at computing the operation points of the units or small plants. Finally, the Real Time Optimization (RTO) layer aims at computing the operation points process units that optimizes production within its admissible (RTO) layer aims at computing the operation points of of the the process units that optimizes production within its admissible operating range and according to a certain criterion. A RTO process units that optimizes production within its admissible operating range to criterion. RTO system normally usesaccording large models covering a wholeA operating range and and according to aa certain certain criterion. A plant, RTO system normally uses large models covering a whole plant, but the variables characterizing the optimal operation points system normally uses large models covering a whole plant, but the the operation points are only a subset characterizing of the hundreds thousands of variables but the variables variables characterizing theoroptimal optimal operation points are only a subset of the hundreds or thousands of variables that only are present room. variables, if not are a subsetinofthethecontrol hundreds or These thousands of variables that the room. not computed by thein layer because of lackvariables, of modelsif that are are present present inRTO the control control room. These These variables, if and not computed by the RTO layer because of lack of models and computed by the RTO layer because of lack of models and
adequate computing tools, are normally decided by the plant adequate are decided the managers according tools, to experience or heuristics. quite adequate computing computing tools, are normally normally decided by byBut the plant plant managers according to experience or heuristics. But often, due to the complexity of the problem, lack of managers according to experience or heuristics. But quite quite often, due the complexity of the of information, slow etc. the about lack the key often, due to to thedynamics, complexity of decisions the problem, problem, lack of information, slow dynamics, etc. the decisions about the key variables mainly avoidetc. creating bottlenecks, information, slowtarget dynamics, the decisions aboutviolating the key variables target avoid bottlenecks, violating constraints or risking safecreating operation of the plant, not variables mainly mainly target the avoid creating bottlenecks, violating constraints or risking the safe operation of the plant, being as close as possible to the optimum operating point. constraints or risking the safe operation of the plant, not not being close as to point. Then, these optimal decisions areoptimum passed asoperating targets to the being as as close as possible possible to the the optimum operating point. Then, these optimal decisions are passed as targets to lower levels, MPC/control layer, that implements them in the Then, these optimal decisions are passed as targets to the lower MPC/control layer, them in process, but the overall functioning far from the optimum lower levels, levels, MPC/control layer, that thatisimplements implements them in the the process, but the overall functioning is far from the optimum one. Even if an RTO system is in operation, quite often the process, but the overall functioning is far from the optimum one. if in quite often the difference thesystem steady is state models in the RTO layer, one. Even Even between if an an RTO RTO system is in operation, operation, quite often the difference between the steady state models in the RTO layer, which facilitates solution the inassociated NLP difference between the steady stateof models the RTO layer, which the solution of associated NLP optimization problem, the linear in the MPC which facilitates facilitates the and solution of the theones associated NLP optimization problem, and the linear ones in the MPC controllers creates inconsistencies that lead to suboptimal optimization problem, and the linear ones in the MPC controllers performance. controllers creates creates inconsistencies inconsistencies that that lead lead to to suboptimal suboptimal performance. performance.
Fig. 1. Hierarchical decision layers for process control and Fig. operation Fig. 1. 1. Hierarchical Hierarchical decision decision layers layers for for process process control control and and operation operation Alternatively, the RTO and MPC layers can be combined in a Alternatively, the and layers can economic MPC optimal dynamic operation problem asin Alternatively, theorRTO RTO and MPC MPC layers can be be combined combined ininaa economic MPC or optimal dynamic operation problem as economic MPC or optimal dynamic operation problem as in in
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Engell et al. (2007), Gonzalez et al. (2001).This approach solves the inconsistency problem between layers and it is well stablished for continuous processes. Nevertheless, many plants include some parts where some process units operate in batch or semi-batch modes, which implies that scheduling of these units should be part of any optimization problem, complicating substantially the formulation and solution of the problem. Additionally, the complexity of the large-scale optimization problems that appears in these cases must be balanced with sensible computing times in order to allow for real-time implementation, which calls for some form of distributed computation.
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consumed by the continuous plant depends on the energy demands of the batch one. Hence, decisions on the optimal process functioning must be taken considering the joint operation of both plants.
Fig. 2. Schematic of continuous and batch plants operating in series with interchange of materials and energy.
Some contributions have been presented in the past to the problem of integrating scheduling and dynamic optimization Prata et al. (2008), Terrazas-Moreno et al. (2007) but few of them considers batch processes, Prada et al. (2009). Nevertheless, the subject has received increased attention due to the opportunities opened by the integration of different decision layers and the use of improved optimization methods and tools, Harjunkoski, et al. (2014), Nie et al. (2015).
2.2 Plant optimization Typical targets for this type of plants can be classified broadly into two groups: In the first one we assume that production flow is fixed, either in the continuous feed side or in the final products of the batch plant. The target is to operate the process minimizing energy or costs, while respecting quality specifications as well as other safety or process constraints. Minimization of energy use or costs, or some Resource Efficiency Indicators (REIs) such as specific energy consumption or yield, are natural candidates to be used as cost function.
This paper intend to contribute to the field studying the special plant wide optimization problems that results when continuous plants must operate together with batch ones in a synchronized way interchanging materials and energy and considering some of the large-scale optimization difficulties that must be faced. The paper proposes a hierarchical/decentralized architecture that facilitates the implementation of the RTO, taking into account what can be considered more adequate from the point of view of the control room. The use of the proposed architecture is illustrated with an example taken from the sugar industry: the joint operation of the sections of evaporation and crystallization.
In the second group of targets, the aim is to maximize production making the best use of the available resources (energy or others) and respecting, as before, process and quality constraints. Concerning the continuous plant, the characteristics of the material and energy streams connected to the batch plant, as well as the ones of the energy and materials feeds, define its operation point and the interaction with the other plant. Hence, those variables associated with these streams (such as steam pressure or concentration of a product), and belonging to the degrees of freedom of the process, must be included in the decision variables u c of the problem, characterized by a set of equality and inequality equations describing the dynamic model h c and constraints g c affecting this plant as in (1):
The paper is organized as follows: after the introduction, section 2 deals with the formulation of the problem and section 3 presents the hierarchical/ distributed architecture. Then, section 4 shows the case study of the sugar factory and section 5 provides results corresponding to several operational modes. The paper ends with some references. 2. HYBRID PLANT OPTIMIZATION
h c ( x c , x c , u c ) = 0 g c (x c , u c ) ≤ 0
2.1 The process
(1)
Regarding the batch plant, there are two problems that must be considered: the first one is the local optimal control of every batch unit j, with decision variables ubj, and the second one is the scheduling of the operation of the whole set of units taking into account the limited shared resources coming from the continuous plant and other possible constraints. So, in the operation of the batch plant, besides individual dynamic models h bj and constraints g bj for the batch units, with local decision variables u bj , , as in (2), we must consider the global material balances and decision variables like starting times tij of these units. Notice that the operation of every batch unit, in terms of batch duration or mass and energy demands per cycle, may depend on the quality of the
The type of processes considered are represented schematically on Fig.2, where we can see a continuous plant that processes a flow of raw products using a certain amount of energy, followed by another plant whose process units operate in batch mode, with materials and energy coming from the continuous plant. Notice that both plants are interacting: It is obvious that the batch plant depends in its operation on the quantity and quality of the energy and materials it receives from the continuous plant, but, conversely, the inflow of raw products of the continuous plant is limited by the capacity of the batch one according to the schedule of the units, and the energy 295
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corresponding supply received from the continuous plant. Also, the batch plant throughput, which depends heavily on the scheduling of the batch units, may act as the bottleneck of the whole process and must be matched to the continuous plant production as in (3), where f and r describe the balance of material and energy at the interface as well as the associated constraints.
h bj ( x bj , x bj , u bj ) = 0 g bj ( x bj , u bj ) ≤ 0
j = 1,..., B
the overall plant dynamics and the interactions among the process units, so it should be built using simplified models including only the elements that are essential for the coordinated operation of the sub-processes. If the problem is formulated within the framework of economic MPC, then, it will include predictions of the future behaviour of some key variables, which can be provided by the optimizers of the lower layer, besides mass and energy balances governing the interaction of the continuous and batch units and the dependence of the batch operation with the properties of the continuous streams. Changes in these ones being incorporated by mean of sensitivities computed in the lower level. A model for the coordinator layer built in this way avoids the heavy internal mathematical model of the process units, while retaining the main elements for proper decision making, being a good balance between adequate process representation and easiness of computation. The upper layer model is used in a dynamic optimization problem with economic aim to generate the batch units scheduling and targets and constraints for the interacting streams and optimizers of the lower layer, solving an optimization problem that includes all constraints required for the global safe operation of the batch and continuous units.
(2)
f c ( x c , u c ) = ∑ f bj ( x bj , u bj ) j
rc ( x c , u c ) ≤ ∑ rbj ( x bj , u bj )
(3)
j
On any case, any formulation of an optimization problem must consider the dynamic and discrete nature of scheduling, and the interactions with the continuous plant, making the problem a complex one. The fact that the whole process must be considered simultaneously calls for centralized formulations which lead to MINLP problems, but its mixed continuous-batch nature imposes a heavy computational load that does not favour a solution of the problem to be used in real-time. In addition, physical implementation of the solutions must consider the human factor, that is, the personnel dealing with the system and its maintenance, which calls for advanced applications close to the process units, and as simple as possible. In the control room of a large process factory, different operators are in charge of different sections of the plant and centralized optimization applications does not provide the required flexibility for adaptation to the distributed responsibilities. Another consideration refers to expandability and facility of adaptation to the upsets or different number of units in operation that occur from time to time. All these factors lead to consider decentralized architectures able to provide similar performance than a centralized one, but having better implementation characteristics.
The idea is summarized in Fig.3, with the upper layer in charge of computing the most favourable operating conditions of the sub-processes compatible with the joint operation of the whole system and a lower layer of Economic MPCs (EMPC) performing optimal control of every continuous or batch sub-process. Notice that the main task of the coordination layer is not mainly finding optimal set points for the lower layer according to a global economic optimum, as in a typical RTO-MPC setting, but to establish global feasible ranges that facilitate the optimal operation of the subprocesses, which, in turn, create new conditions and material and energy demands that the upper layer must satisfy generating another schedule or constraints and targets. The interaction of both layers allows for recursive improvements until convergence to a plant-wide optimum is eventually reached.
3. HIERARCHICAL / DECENTRALIZED ARCHITECTURE FOR OPTIMIZATION The approach proposed in this paper considers the continuous plant and the batch units as the natural single pieces to build an optimization strategy due to the fact that local models can be more easily defined for each of them and also considering that the corresponding optimal operation of each process unit can be likely computed and implemented following the existing structure of the control room. Nevertheless, it is clear that the solution of the individual optimization problems will be neither adequate nor globally optimal unless the interactions among them, the global behaviour and the scheduling of the batch units are taken into account explicitly. This requires a coordination layer dealing with them, which must be placed hierarchically on top of the process units optimizations.
Fig. 3. Hierarchical and distributed architecture for global optimization of the hybrid system. Notice that one important characteristic that this architecture resides in the fact that it allows parallelism in the computation of the heavier tasks that correspond to the distributed local optimizers, facilitating real-time optimization. At the same time, the modular architecture facilitates both, the activation or deactivation of units if there are not in service, and an easy expandability to new units.
This layer cannot repeat the large model of the centralized approach, but should contain the key element that describe 296
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4. SUGAR FACTORY CASE STUDY
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the juice from the previous evaporator. The vapours generated in the second effect being used also to heat the crystallization section. The evaporators present a decreasing profile of pressures and temperatures, and increasing concentrations, with the pressure in the juice chamber of the last effect being maintained with the help of a barometric condenser for the vapours.
In order to illustrate and test the previous ideas, a case study taken from the sugar industry is presented. It corresponds to a benchmark proposed within the FP7, EU Network of Excellence “Highly-complex and Networked Control Systems (HYCON2)”, and described in detail in Mazaeda et al, (2014). The benchmark is fully accessible in HYCON/WP5 (2014) where additional information is provided as well as the full process simulator. It presents the problem of the joint operation of the evaporation and crystallization sections of a sugar factory and responds to the schematic of Fig.4. Previous contributions can be seen in Hernandez et al. (2014), where a centralized approach is presented, as well as in Mazaeda et al.(2015), that uses an approach similar to the one in this paper, but limited to the batch section..
The basic control system is shown in Fig.4 and includes level controllers in the evaporators, fresh juice flow control, supply steam to the first effect pressure control and a controller of the brix at the output syrup, that operates manipulating the vapours flow to the barometric condenser. Notice that the syrup flow cannot be manipulated independently if these controllers are in operation, as it is imposed by the balance of dry matter. In this way, the pressure set point of the supply steam and the brix set point of the syrup are the two key variables that define the evaporation operation, as reflected in Fig.5. Brix, purity and flow of the fresh juice can be considered as the main disturbances, with the inflow being sometimes also a manipulated variable according to the mode of operation.
4.1 Process plant and control system description The picture depicts the evaporation section on the left and the crystallization section on the right with while Fig.5 shows a block diagram of them with the main streams involved.
Regarding the crystallization section, it incorporates a buffer tank for supplying syrup to three semi-batch crystallizers that can operate in parallel following a certain recipe to produce sugar crystals. The operation in the crystallizers, also called tachas or vacuum pans, starts with the load of an initial amount of syrup, followed by its concentration by boiling using the vapours from the evaporation section as heat source, with the purpose of achieving the required supersaturation. The crystallizers work under vacuum to facilitate evaporation and prevent caramelization. Then, in the following step, the seed of tiny crystals takes place, which constitutes the population that will grow in size in a controlled manner in the next step: the growing of the grain. In this stage, while saccharose moves to the sugar crystals, oversaturation in the solution is maintained by a mixture of evaporation and addition of new syrup. When the crystals reach the required size and the crystallizer is full, the strike ends and the resulting massecuite is discharged. Fig.6 depicts a typical time evolution of the level in a crystallizer where, after the initial load, it grows slowly up to the end of the strike.
Fig. 4. Evaporation and crystallization sections with basic control loops.
Fig. 5. Block diagram of the two sections with the main streams involved and global decision variables marked as yellow points.
Syrup brix 75%
The purpose of the evaporation section is to increase the concentration of a sugar juice fed to it up to a certain value in the syrup at the output of the section. This concentration is normally measured in brix degrees (% in weight). The operation is carried out in a set of three evaporators working in a multiple effect configuration with the first one receiving a flow F of fresh juice in the juice chamber and a flow F v of saturated steam at a certain pressure P s in the heating chamber. The juice boils and, as the water is evaporated, its sugar concentration increases, while the vapours generated are used to heat the following evaporator, which is fed with
Syrup brix 71%
Fig.6. Time evolution of the level of a crystallizer for two different values of the syrup brix Fig. 6 shows the level profile for two values of the brix of the feed, showing that decreasing brix enlarges the cycle time. 297
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Also, one could see that the amount of steam consumed from the evaporation section increases if the syrup brix of the feed decreases, showing the strong interaction between the quality of the syrup from the evaporation and the crystallizers operation. Purity also plays an important role in the crystallizer operation but, in order not to complicate too much the process description and concentrate the focus in the hierarchical optimization, it will not be considered explicitly here as, in a certain way, it parallels the brix effect.
4.2 Process Optimization Two different modes of operation have been selected: In the first one, the factory is processing a certain amount of beets, so that the flow of fresh juice at the evaporators input can be considered fixed (but not necessarily constant) by the previous processes and the aim is to globally operate the joint continuous-batch process (that is, select the values of the five key variables and the scheduling of the batch units) with minimum energy (steam) consumption and avoiding bottlenecks, in spite of possible disturbances, e.g. changes in the fresh juice brix or flow. Quality of the sugar production is assured by the local controllers implementing the recipe of the crystallizers, but different cycle duration or steam or juice consumption are expected depending on the operation of the evaporation and local steam pressure SP in the crystallizers, which, in turn, will affect the energy efficiency and plant throughput, creating possible bottlenecks.
In the same way, decreasing the heating vapour pressure will enlarge the cycle time. This can be seen in Fig.7, where the upper graph displays profiles of pressure over a full cycle and the lower one the corresponding level time evolution. Additionally, decreasing pressure decreases supersaturation and improves crystal size distribution.
PFactor=1.2
In the second mode of operation, the aim is to increase the flow of fresh juice as much as possible, keeping also the previous aims of minimizing energy consumption. In this case, the juice feed flow becomes both, a target and a new decision variable.
PFactor=1.0
PFactor=0.8
The solution follows the scheme of Fig.3, with a coordination layer and four local dynamic process optimizers in the lower layer: one for the evaporation section and one for each of the three sugar crystallizers. The economic MPC attached to the evaporation section, solves the dynamic optimization problem (4) every sampling time, minimizing specific steam consumption (or maximizing fresh juice flow according to the mode of operation), under the constraints imposed by the evaporation model h evap and constraints g evap affecting pressure, temperature and brix. In (4), x e refers to the state and u e to the decision variables. The main decision variable is the set point of the supply steam pressure (plus the flow of fresh juice in the corresponding mode of operation), while the set point of the syrup brix is imposed by the coordination layer because it is a key variable governing the interaction between both sections. In the same way, predictions of the steam demands from the crystallizers are computed in the upper coordination layer and passed to the evaporation local NMPC controller as a boundary condition.
PFactor=0.8 PFactor=1.2
PFactor=1.0
Fig.7 Crystallizer heating pressure profile (upper) and level profile for three different pressure values. According to the crystallizer recipe, the pressure follows a certain profile, but it can be modified as in Fig.7. This can be seen as applying a factor P factor to a nominal profile. This factor can be considered as the main knob the operators dispose to modify the vacuum operation, as most of the other variables are fixed by the local sequencing control according to its recipe. Similarly, we will consider syrup brix of the feed as the main disturbance.
min ∫ Fsteam dt / ∫ Fdt Ps
A local control system implements the sequence of steps included in the recipe, with local controllers of level, brix of the massecuite, heating steam pressure and vacuum. We will assume that the local control system performs its duties adequately, so that it is able to guarantee the right crystal growing at the end of the strike, at least for an acceptable range of operating conditions. This leaves the set point of the heating chamber pressure controller as the main knob for the operators to influence the crystallizer operation, as mentioned above. Notice that the aim of operating with low steam consumption does not conflict with the local one of maintaining crystal quality. In addition, the starting times of the crystallizers define the batch plant operation, leaving a total of six degrees of freedom per cycle in this plant as reflected in Fig.5.
h evap ( x e , x e , u e ) = 0
(4)
g evap ( x e , u e ) ≤ 0 The dynamic model of the evaporation section is a DAE system composed by mass and energy balances, as well as by thermo-dynamical equilibriums and flow equations involving 157 equations and variables, including 13 states. Notice that, after solving (4), the future profile of the syrup flow to the crystallization section can be computed easily and passed later to the coordination layer. The constraints refer mainly to the admissible ranges for pressure and brix in the evaporators. The model has being implemented in the EcosimPro simulation environment.
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In the same way, every crystallizer j has an economic NMPC attached to it, formulated as (5):
min Pfactor
layer EMPCs predictions, as the ones from the crystallizers syrup and steam consumptions displayed in Fig.9. As mentioned before, similar profiles are obtained for the syrup flow from the evaporation. The grey model is built on the fly capturing the key elements that influence the global dynamics and involving the most important variables at this level, being a good compromise between process complexity and on-line computational load.
∫ Fvapjdt
h bj ( x bj , x bj , u bj ) = 0 g bj ( x bj , u bj ) ≤ 0
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(5)
Tbatchj ≤ Tbatch _ max Here the aim is to perform the crystallization cycle consuming as less steam from evaporation as possible, which also improves crystal size distribution, being the main decision variable P factorj , a factor applied to the steam pressure profile that can be varied within a range. Besides the dynamic model h bj , the constraints include among others the super-saturation of the syrup in the crystallizer, which must be maintained in a range in order to avoid the creation of spontaneous new crystals or crystal dissolutions. In addition, another constraint is highlighted in (5): the maximum cycle time allowed T batch_max , which is fixed by the coordination layer, as well as the starting time of the cycle T loadj . The model is composed by mass and energy balances plus equilibriums and crystal growing equations, totalling 227 variables for each crystallizer, including 30 states. The model includes discontinuities corresponding to the different stages, but the dynamic optimization problem (5) can be solved with current gradient-based optimization software using a sequential approach, as in Fig.8, due to the fact that the number and type of discontinuities is kept constant by the control system, Galan et al. (1999). At every iteration of the NLP optimizer a set of the decision variables are proposed to the dynamic model simulation, evaluating de cost function and constraints as required by the NLP. A similar architecture is used to solve problem (4)
Fig.9 Typical profiles of syrup and steam flows to the crystallizers Referring to the simplified schematic of Fig.10, that capture the main elements related to global process operation, total demand of steam to the evaporation section can be computed from the individual profiles of steam consumption and the schedule of the crystallizers. This demand, needed to solve the evaporation model, will be passed to the evaporation NMPC after each execution of the coordination layer. In the same way, material balances around the syrup storage tank can be calculated using the profiles of the individual crystallizer’s demands of syrup, ordered over time according to the schedule, and the syrup outflow predictions provided by the evaporation. Feasible operation in the use of these shared resources is required and can be achieved imposing appropriate conditions on the level of the storage tank and pressure and continuity on the steam demand.
Fig.10 Simplified schematic of the hybrid evaporationcrystallization process
Fig.8 Sequential approach for solving optimization problems of the lower layer
the
Nevertheless, it is important to notice that, in the optimization problem to be solved in the coordination layer, the profiles cannot be considered constant because they may change as a result of the future decisions taken on the schedule or on the syrup brix SP of the evaporation. These changes are considered by constructing a model for the profiles that incorporates the sensitivities of these flows to the variables passed from the upper layer: maximum cycle time of crystallizers T bmax and syrup brix b as in (6).
dynamic
After solving (5), the profiles of the steam and syrup demands of every crystallizer are computed and passed to the upper coordination layer. 4.3 Coordination layer
Fvapj = Fvap0 j +
Problems (4) and (5) cannot provide sensible and feasible solutions unless global balances are satisfied and constraints respected, which is the main task of the upper layer. For this purpose, a surrogate model is built at this level mixing physical balances that must always be satisfied with the profiles of the interacting variables provided by the lower
∂Fvapj ∂Tb max
Fsyrupj = Fvyru0 j + Fe = Fe 0 +
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(Tb max − Tb max 0 ) +
∂Fsyruj ∂Tb max
∂Fe (b − b 0 ) ∂b
∂Fvapj
(Tb max − Tb max 0 ) + j = 1...3
∂b
(b − b 0 )
∂Fsyrupj ∂b
(b − b 0 ) (6)
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The sensitivities can be computed locally by the EMPCs of the lower layer and passed to the coordination one. In (6) the sub index 0 refers to the profile obtained from the corresponding EMPC and F e stands for the syrup flow from evaporation. The full demand of each steam or syrup stream is built by adding these profiles shifted in time according to the schedule of the crystallizers.
syrup flow to the syrup tank as an additional decision variable that is also maximized in the cost function. This value is passed as an upper limit to the evaporation EMPC that also includes fresh juice flow as a new decision variable and as a target to maximize. Prediction Horizon (NP complete cycles for each pan) Control Horizon (Nc cycles for each pan, Ej: Nc=2)
Regarding the schedule of the batch units, in this case study the problem can be formulated as selecting their successive i starting times over the prediction horizon T loadji for every tacha j, so that the operation constraints are satisfied in the best possible way. In this way, as T loadji are continuous variables, the use of mixed integer codes is avoided. Obviously, the schedule of the crystallizers is part of the coordination problem, where other decision variables are involved. In particular, the syrup brix SP of the evaporation is used as a decision variable, as mentioned before (assuming no steady state error in the corresponding PID controller). As the problem may have multiple solutions, the one that provides the best conditions for the sub-processes operation is sought, that is, the one that minimizes the total steam demand to the evaporation, which corresponds with maximum possible length cycle in the crystallizers and maximum time interval between cycles of a crystallizer. Maximum cycle time is incorporated as another decision variable, while both, the interval between cycles, are considered as well as targets to maximize. The problem is formulated in continuous time domain with a prediction and control horizons covering several cycles, as the ones displayed in Fig 11, where the levels in the units have been used to identify the different cycles of the three crystallizers:
min
Tloadj , brix , Tbatch _ max
∑∫F
vapj
TLoad1_1
TLoad1_2
Load1_1
LL
Load1_2
... TLoad2_1
TLoad2_2
Load2_1
LLL
.
Load2_2
... TLoad3_1 Load3_1
TLoad3_2 Load3_2
Fig.11 A schedule of the crystallizers showing the control and prediction horizons comprising an integer number of cycles. 5. SIMULATION RESULTS The approach has been tested in a process simulator implemented in the modelling environment EcosimPro E.A.I (2015) for different operating conditions. In this section, some results corresponding to an exercise in which we try to maximize the flow of juice processed, while minimizing steam consumption, are provided for one day of operation.
dt − ∑ Dt ji − Tbatch _ max
s.t. extended profiles (6) 0 ≤ Tloadji ≤ Tloadji+1 j = 1,2,3
...
L
i = 1,2,..., N u
Tmin ≤ Tloadj( i +1) − Tloadji − Tbatch _ max = Dt ji ≤ Tmax TB min ≤ Tbatch _ max ≤ TB max
(7)
L min ≤ L buffer ≤ L max D min ≤ Tbatch _ max − Tbatch _ max 0 ≤ D max d min ≤ b − b 0 ≤ d max
dm buffer / dt = Fe − ∑ Fsyrupj
dm buffer b / dt = Fe b e − ∑ Fsyrupjb Here, the constraint in the buffer tank level L buffer avoids possible bottlenecks, while the ones on the changes of T batch_max and b are used to include some type of trust region, to guarantee the validity of (6). The last two differential equations reflect mass balances in the buffer tank. The coordination layer generates the brix set point of syrup and steam demand for the evaporation, starting times of the crystallizers and the maximum allowed cycle times that are passed to the EMPCs.
Fig.12 Time evolution of the fresh juice to evaporation
If the global target is to maximize the fresh juice processed, while still minimizing steam use, then the optimization problem of the coordination layer is changed to include the
Fig.13. Schedule of the crystallizers represented with the pan levels. Every crystallizer in a different colour. 300
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Fig.12 shows the time evolution of the flow of fresh juice to the evaporation, clearly reaching a maximum, while Fig.13 displays the optimal schedule of the crystallizers by means of the levels in the pans. Fig.14 gives the syrup level in the storage tank, which must be maintained between 20 and 90%, and Fig. 15-16 the supply fresh steam to the evaporation, and pressure in the crystallizers header, showing that plant-wide optimum operation is feasible.
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approach has been illustrated with a case study corresponding to the joint operation of the evaporation and crystallization sections of a sugar factory problem. ACKNOWLEDGEMENT The authors wish to thank the Spanish MINECO for its support through project DPI2012-37859. REFERENCES Darby M. L., Nikolaou M., Jones J., Nicholson D. (2011). RTO: An overview and assessment of current practice, Journal of Process Control.Vol. 21, 874–884. de Prada, C.; Grossmann, I. E.; Sarabia, D.; Cristea, S. (2009) A strategy for predictive control of a mixed continuous batch process. Journal of Process Control, 19, 123–137. Empresarios Agrupados Int.EcosimPro, last relese (2015) http://www.ecosimpro.com/products/ecosimpro/ Engell, S. (2007). Feedback control for optimal process operation. Journal of Process Control, vol. 17, 203-219. Galán S., Feehery W.F., Barton P.I., (1999) Parametric sensitivity functions for hybrid discrete/continuous systems, Applied Numerical Mathematics,31, 17–47. Gonzalez Ana I., Zamarreño J.M., de Prada C. (2001) Nonlinear model predictive control in a batch fermentator with state estimation, European Control Conference, Porto, Portugal, ISBN: 972-752-047-2 Hernández R., Simora L., Paulen R., Wegerhoff S., Mazaeda R., Engell S., de Prada C. (2014). Optimal Allocation of Shared Resources in an Integrated Sugar Production Plant. Computer Aided Process Engineering series, vol. 33, pp. 637-642. Harjunkoski, I.; Maravelias, C. T.; Bongers, P.; Castro, P. M.; Engell, S.; Grossmann, I. E.; Hooker, J.; Mndez, C.; Sand, G.; Wassick, J. (2014) Scope for industrial applications of production scheduling models and solution methods. Computers & Chemical Engineering, 62, 161 – 193. HYCON/WP5 (2014). Sugar show case. available fro download at: hycon.isa.cie.uva.es/home. Mazaeda, R., Acebes, L.F., Rodríguez, A., Engell, S., Prada, C. (2014). Sugar Crystallization Benchmark. Computer Aided Process Eng. series, vol. 33, pp. 613-618. Mazaeda R., Cristea S. P., de Prada C. (2015) Plant-wide hierarchical optimal control of a crystallization process, Int. Symposium on advanced Controlof Chemical Processes, ADCHEM2015, IFAC, Wishtler, British Columbia, Canada Nie Y., Biegler L.T., Villa C.M., Wassick J.M. (2015) Discrete Time Formulation for the Integration of Scheduling and Dynamic Optimization, Ind. Eng. Chem. Res., 54 (16), pp 4303–4315 Prata, A.; Oldenburg, J.; Kroll, A.; Marquardt, W. (2008) Integrated scheduling and dynamic optimization of grade transitions for a continuous polymerization reactor. Computers &Chemical Engineering, 32, 463–476. Terrazas-Moreno, S.; Flores-Tlacuahuac, A.; Grossmann, I. E. (2007) Simultaneous cyclic scheduling and optimal control of polymerization reactors. AIChE Journal, 53, 2301–2315.
Fig.14 Buffer tank level to be kept between 20 and 90%.
Fig.15. Supply steam flow to the evaporation
Fig.16. Steam pressure in the collector to the crystallizers 6. CONCLUSIONS A novel approach for the optimal operation of a class of combined continuous-batch processes has been presented with the purpose of offering a solution that, avoiding the bottlenecks created by traditional heuristic management, could be implemented with a sensible computational load. Its main points are the hierarchical decentralized architecture and the simplified grey model of the coordination layer. The architecture provides feasible targets and operating ranges to the lower level as well as flexibility for adding or removing process units, updating the upper layer model, even on-line, or changing the optimization targets and fits into the control room organization. The distribution of tasks and the simplified upper model, besides the parallelization of the computation, offers competitive computation times without the complexity of other distributed architectures. The 301