Plant-wide hierarchical optimal control of a crystallization process

9th International Symposium on Advanced Control of Chemical Processes 9th International Symposium on Advanced Chemical Processes June 7-10, 2015. Whistler, British Columbia,Control Canadaof 9th International Symposium on Control of Processes 9th International Symposium on Advanced Advanced Control of Chemical Chemical Processes Available online at www.sciencedirect.com June 7-10, British Columbia, Canada 9th International Symposium on Advanced June 7-10, 2015. 2015. Whistler, Whistler, British Columbia,Control Canadaof Chemical Processes June 7-10, 2015. Whistler, British Columbia, Canada June 7-10, 2015. Whistler, British Columbia, Canada

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IFAC-PapersOnLine 48-8 (2015) 1210–1215

Plant-wide hierarchical optimal control of a crystallization process Plant-wide Plant-wide hierarchical hierarchical optimal optimal control control of of aa crystallization crystallization process process Plant-wide hierarchical optimal control of a process Rogelio Mazaeda*, Smaranda P. Cristea*, crystallization César de Prada*

Rogelio Mazaeda*, Smaranda P. Cristea*, César de Prada* Rogelio Rogelio Mazaeda*, Mazaeda*, Smaranda Smaranda P. P. Cristea*, Cristea*, César César de de Prada* Prada* Rogelio Mazaeda*, Smaranda P. Cristea*, César de Prada* *Dpt.of Systems Engineering and Automatic Control, University of Valladolid, Spain (e-mail: rogelio@ cta.uva.es) *Dpt.of Systems Engineering Engineering andsmaranda@ Automatic Control, Control, University of Valladolid, Valladolid, Spain (e-mail: (e-mail: rogelio@ cta.uva.es) *Dpt.of and University of Spain (e-mail: autom.uva.es, e-mail: prada@ autom.uva.es) *Dpt.of Systems Systems Engineering and Automatic Automatic Control, University of Valladolid, Spain (e-mail: rogelio@ rogelio@ cta.uva.es) cta.uva.es) (e-mail: autom.uva.es, e-mail: prada@ autom.uva.es) *Dpt.of Systems Engineering andsmaranda@ Automatic Control, University of Valladolid, Spain (e-mail: rogelio@ cta.uva.es) (e-mail: smaranda@ autom.uva.es, e-mail: prada@ autom.uva.es) (e-mail: smaranda@ autom.uva.es, e-mail: prada@ autom.uva.es) (e-mail: smaranda@ autom.uva.es, e-mail: prada@ autom.uva.es) Abstract: This paper deals with the dynamic real time optimization of a benchmark model that represents Abstract: paper deals with the dynamic real time optimization of a benchmark model that represents Abstract: This paper deals the time optimization of model that a genuine This problem the crystallization section sugar factories. A most relevant of Abstract: This paperfound dealsatwith with the dynamic dynamic real real time of optimization of aa benchmark benchmark modelcharacteristic that represents represents aaAbstract: genuine problem found at the crystallization section of sugar factories. A most relevant characteristic of This paper deals with the dynamic real time optimization of a benchmark model that represents genuine problem found at the crystallization section of sugar factories. A most relevant characteristic of the control problem is given by its hybrid nature, mixing continuous control and the scheduling of the a genuine problem found at the crystallization section of sugar factories. A most relevant characteristic of controlproblem problem is given its hybrid nature, mixing continuous control the scheduling of the athe genuine atThe theby crystallization section of sugar factories. A mostand relevant characteristic of the is by its mixing continuous control the of crystallizer batch found units. optimal control task is approached non-centralized the control control problem problem is given given byplant-wide its hybrid hybrid nature, nature, mixing continuous control and and in the ascheduling scheduling of the the crystallizer batch units. The plant-wide optimal control task is approached in a non-centralized the control problem is given by its hybrid nature, mixing continuous control and the scheduling of the crystallizer batch batch units. The plant-wide plant-wide optimal control oftask task is ofapproached approached inforaaeach non-centralized hierarchical way with two-layers. The loweroptimal layer consists a setis MPCs, one in crystallizer, crystallizer units. The control non-centralized hierarchical way two-layers. The lower layer consists of aa set MPCs, one crystallizer, crystallizer batch units. Thelocal plant-wide control an approached infor aeach non-centralized hierarchical wayofwith with two-layers. The loweroptimal layerminimize consists oftask setis of of MPCs, onethe foreconomic each crystallizer, and is in charge taking the decisions that index related with behavior hierarchical way with two-layers. The lower layer consists of a set of MPCs, one for each crystallizer, and is charge of taking the local decisions that index related with hierarchical way two-layers. The lowerlayer layerminimize consists ofstate a setofof MPCs, onethe foreconomic each and is in in unit. charge ofwith taking the local decisions that minimize an index related with the economic behavior of each The higher level coordinator drives thean the whole plant near crystallizer, thebehavior overall and is in charge of taking the local decisions that minimize an index related with the economic behavior of each The level coordinator layer drives the the whole plant near thebehavior overall and is in unit. charge of higher taking local decisions that minimize anstate index related with the economic of each The level layer the state of the plant near economic optimum, while the respecting the constraints imposed the of existence of shared of each unit. unit. The higher higher level coordinator coordinator layer drives drives theby state of the whole whole plant resources. near the the overall overall economic optimum, while respecting the constraints imposed the of existence of shared of each unit. The higher level coordinator layer drives theby state the whole plant resources. near the overall economic optimum, while respecting constraints imposed by existence of shared resources. economic optimum, while respecting the imposed by the the integrated existence shared resources. Keywords: Predictive control, processthe control, hierarchical control, scheduling and reserved. continuous © 2015, IFAC (International Federation of constraints Automatic Control) Hosting by Elsevierof Ltd. All rights economic optimum, while respecting the constraints imposed by the integrated existence of shared resources. Keywords: Predictive control, process control, hierarchical control, scheduling and continuous Keywords: Predictive control, process control, hierarchical control, integrated scheduling and continuous plant control, hybrid optimal control. Keywords: Predictive control, process control, hierarchical control, integrated scheduling and continuous plant control, hybrid optimal Keywords: Predictive control,control. process control, hierarchical control, integrated scheduling and continuous plant plant control, control, hybrid hybrid optimal optimal control. control. plant control, hybrid optimal control. very simple continuously updated linear model of the 1. INTRODUCTION very simpleunits. continuously updated updated linear model model of the the very simple subordinate very simple continuously continuously updated linear linear model of of the 1. INTRODUCTION 1. INTRODUCTION subordinate units. 1. INTRODUCTION very simple continuously updated linear model of the subordinate units. A typical beet sugar factory can be viewed as the series subordinate units. 1. INTRODUCTION The paper isunits. organized as follows: in section 2 a description A typical beet sugar factory can be viewed as the series subordinate A typical beet sugar factory can be viewed as the series connection of two main departments. The first one is in Thethepaper paper is organized organized asisfollows: follows: insection section3 22the a description A typical beet sugar factory can be viewed as the series of The is benchmark modelas given; inin plant-wide The paper is organized as follows: in section section 2 aa description description connection of two departments. The first one is in A typical beet sugarmain factory can be viewed as the series connection of main departments. The first one in charge of extracting sucrose juice of the benchmark model is given; in section 3 the plant-wide connection of two two and mainpre-processing departments. the Thebeet first one is is in control The paper is organized as follows: in section 2 a description of the benchmark model given; in section 33 the the plant-wide task is posed as a is mathematical optimization problem of the benchmark model is given; in section plant-wide charge of extracting and pre-processing the beet sucrose juice connection of two main departments. The first one is in charge of and the sucrose and mainly of continuous operations. The control task is posed as a mathematical optimization problem chargeconsists of extracting extracting and pre-processing pre-processing the beet beet sucrose juice juice of the benchmark model is given; in section 3 the plant-wide control is posed as aa mathematical optimization problem and thetask proposed non-centralized control strategy is control task is posed as mathematical optimization problem and consists mainly of continuous operations. The charge of extracting and pre-processing the beet sucrose juice and mainly continuous operations. The downstream section, on theof hand, receives the incoming and the proposed non-centralized control strategy is and consists consists mainly ofother continuous operations. The and control task is posed as a mathematical optimization problem the proposed non-centralized control strategy is discussed, while, finally, in section 4, implementation details and the proposed non-centralized control strategy is downstream section, on the other hand, receives the incoming and consists mainly of continuous operations. The downstream section, onthe the crystallization other hand, hand, receives receives the incoming and juice to proceed withon of thethe commercial discussed, while, finally, in section 4, implementation details downstream section, the other incoming the and proposed non-centralized control strategy is discussed, while, finally, in section 4, implementation details are given simulation results are discussed. discussed, while, finally, in section 4, implementation details juice to proceed with the crystallization of the commercial downstream section, on the other hand, receives the incoming juice to to proceed proceed with the iscrystallization crystallization of the the commercial commercial product, a processwith which carried out, discontinuously, in are given and simulation are discussed. juice the of discussed, while, finally, results in section 4, implementation details are given and simulation results are discussed. given and simulation are discussed. product, aa process which carried in juice to proceed with the is of the commercial product, process which iscrystallization carried out, out, discontinuously, discontinuously, in are 2. PROCESS AND results BENCHMARK DISCUSSION semi-batch crystallizers. product, a process which is carried out, discontinuously, in are given and simulation results are discussed. 2. PROCESS AND BENCHMARK semi-batch crystallizers. product, a process which is carried out, discontinuously, in 2. PROCESS PROCESS AND AND BENCHMARK BENCHMARK DISCUSSION DISCUSSION semi-batch crystallizers. crystallizers. 2. DISCUSSION semi-batch (s),AND hereBENCHMARK defined as the excess concentration A particularly interesting plant-wide control problem is found Super-saturation 2. PROCESS DISCUSSION semi-batch crystallizers. Super-saturation (s), here defined as the excess A particularly interesting plant-wide control problem is found Super-saturation (s),solubility here defined defined as substance the excess excessinconcentration concentration A particularly particularly interesting plant-wide control problem is found found Super-saturation of sucrose over the of that water at the precisely at the interfaceplant-wide between these two main sections, (s), here as the concentration A interesting control problem is over solubility of substance in water the precisely the interface between these two main Super-saturation (s), here defined asdriving the excess A particularly interesting plant-wide control problem is found of of sucrose sucrose overofthe the solubility of that substance inconcentration water at the precisely atmultiple the interface between these two main sections, sections, current values temperature, is that the force of the at sugar where the at interchanges of mass and energy between of sucrose over the solubility of that substance in water at the precisely at the interface between these two main sections, current values of temperature, is the driving force of the sugar where the multiple interchanges of mass and energy between of sucrose over the solubility of that substance in water at the precisely at the interface between these two main sections, current values values of of temperature, is the the driving driving force in of industrial the sugar sugar where the multiple multiple interchanges ofneed masstoand and energy between crystallization process. The solubility of sucrose the continuous and interchanges discrete unitsof be energy coordinated. A current temperature, is force of the where the mass between crystallization process. The solubility of sucrose in industrial the continuous and discrete units need to be coordinated. A current values of temperature, is the driving force of the sugar where the multiple interchanges of mass and energy between crystallization process. The solubility solubilityand of sucrose sucrose in industrial the continuous continuous and(fig. discrete units has needbeen to be be coordinated. A crystallization syrups dependsprocess. on its concentration on the in temperature benchmark model 1), which used by the FP7, The of industrial the and discrete units need to coordinated. A itssolution. concentration and on the in temperature benchmark (fig. 1), which used by the FP7, crystallization The solubility of sucrose the continuous discrete units has needbeen to be coordinated. A syrups syrups depends on and on benchmark model (fig. has by the thedepends purity process. ofon The latter as industrial the mass EU Networkmodel of and Excellence “Highly-complex and Networked syrups depends ontheits its concentration concentration and defined on the the temperature temperature benchmark model (fig. 1), 1), which which has been been used used by the FP7, FP7, and the purity of the The latter defined as the mass EU Network of Excellence “Highly-complex and Networked syrups depends on itssolution. concentration andof onallthethe temperature benchmark model (fig. 1), which has been used by the2014), FP7, and and the purity of the solution. The latter defined as EU Network of Excellence “Highly-complex and Networked of sucrose with respect to the masses dissolved Control Systems (HYCON2)” (HYCON/WP5, the mass mass EU Network of Excellence “Highly-complex and Networked and the purity of the solution. The latter defined as the of sucrose with respect to the masses of all the dissolved Control Systems (HYCON2)” (HYCON/WP5, 2014), and the purity of the solution. The latter defined as the mass EU Network of Excellence “Highly-complex and Networked of sucrose with respect to the masses of all the dissolved Control Systems (HYCON2)” (HYCON/WP5, 2014), substances. For moderately supersaturated syrups, where s is captures the essential characteristics of the real plant. The Control Systems (HYCON2)” (HYCON/WP5, 2014), of sucrose with respect to the masses of all the dissolved substances. For moderately s is captures the essential characteristics of the real plant. The of sucrose with respect to supersaturated thezone, masses of syrups, all the where dissolved Control described Systems (HYCON2)” (HYCON/WP5, 2014), substances. For moderately supersaturated syrups, where captures the essential characteristics of the real plant. The inside the so called metastable the crystallization occurs problem in the sugar benchmark is important in its is captures the essential characteristics of the real plant. The substances. For moderately supersaturated syrups, where ss is inside the so called metastable zone, the crystallization occurs problem described in the sugar benchmark is important in its substances. For moderately supersaturated syrups, where s is captures the essential characteristics of the real plant. The inside the theas so the called metastable zone, the existing crystallization occurs problem described inbethe theconsidered sugar benchmark benchmark is important important inofits itsa inside basically growing of thezone, already crystals. For own rightdescribed but it canin as a specific examplein so called metastable the crystallization occurs problem sugar is as of already existing For own right but it considered aa specific example inside the so the called metastable thezone crystallization occurs problem described sugar benchmark isindustries, important inof basically as the growing of the the already existing crystals. For own right but situation it can caninbe bethe considered asprocess specific example ofitsaaa basically higher values of growing s, however, thezone, labile mightcrystals. be entered, more general occurring inas where basically as the growing of the already existing crystals. For own right but it can be considered as a specific example of higher values of s, however, the labile zone might be entered, more general situation occurring in process industries, where basically as the growing of the already existing crystals. For own right but it can be considered as a specific example of a higher values of s, s, however, the labile labile zone zone might be be entered, entered, more general situation occurring occurring in process industries,section where higher and this would imply the spontaneous appearance of new it is general not uncommon to have in some downstream values of however, the might more situation process industries, where and this would imply the spontaneous appearance of new it is not uncommon to have some downstream section higher values of s, however, the labile zone might be entered, more general situation occurring in process industries, where and this thisdirectly would out imply thesolution. spontaneous appearance appearance of of new new it is is not not uncommon uncommon to semi-batch have some some units, downstream section and crystals of the consisting of batch orto (e.g. reactors, would imply the spontaneous it have downstream section of the consisting of batch or reactors, and thisdirectly would out imply thesolution. spontaneous appearance of new it is not uncommon to semi-batch have some downstream section crystals directly out of the solution. consisting of or semi-batch units, (e.g. reactors, membrane whose needunits, to be (e.g. coordinated in crystals crystals directly out of the solution. consisting filters) of batch batch or operation semi-batch units, (e.g. reactors, The sugar crystallizers heresolution. described, use evaporation as the membrane filters) whose need tocoming be (e.g. coordinated in crystals directly out of the consisting of batch or operation semi-batch units, reactors, membrane filters) whose operation need in order to keep pace with the flowrate from the membrane filters) whose operation need to to be be coordinated coordinated in The sugar here described, The sugar crystallizers crystallizers here described, use use evaporation evaporation asinthe thea mechanism for creating super-saturation. They workas order to keep pace with the flowrate coming from the The sugar crystallizers here described, use evaporation as the membrane filters) whose operation need to be coordinated in order to keep upstream order to plant. keep pace pace with with the the flowrate flowrate coming coming from from the the seeded mechanism for creating super-saturation. They work The sugar crystallizers here described, use evaporation asin mechanism for creating creating super-saturation. Theyis work work intheaaa semi-batch fashion.super-saturation. The seed of crystals introduced upstream plant. mechanism for They in order to keep pace with the flowrate coming from the upstream plant. upstream plant. semi-batch The crystals is introduced mechanism for creating super-saturation. They in a seeded semi-batch fashion. The seed seed of ofprocess, crystals is work introduced at the beginning offashion. the crystallization and this initial This article describes the application of a plant-wide non- seeded seeded semi-batch fashion. The seed of crystals is introduced upstream plant. at the beginning of the crystallization and this initial This article describes the application of aa plant-wide nonseeded semi-batch fashion. The seed ofprocess, crystals isobjective introduced at the beginning of the crystallization process, and this initial This article describes the application of plant-wide nonpopulation is made to grow along the cycle. The is centralized hierarchical optimizing control strategy to the This article describes the application of a plant-wide non- at the beginning of the crystallization process, and this initial population is made to grow along the cycle. The is centralized hierarchical optimizing control strategy to the at the beginning of the crystallization process, andobjective this initial This article describes the application of a plant-wide nonpopulation is made to grow along the cycle. The objective centralized hierarchical optimizing control strategy to the to avoid both dissolution and spontaneous nucleation, by problem presented by the sugar crystallization benchmark. is centralized hierarchical optimizing control strategy to the population is made to grow along the cycle. The objective is to avoid both dissolution and spontaneous nucleation, by problem presented by the sugar crystallization benchmark. population is made to grow along the cycle. The objective is centralized hierarchical optimizing control strategy to the to both and nucleation, by problem presented the sugar the super-saturation inside the meta-stable The novelty of the by article on one hand, benchmark. on the fact always to avoid avoidkeeping both dissolution dissolution and spontaneous spontaneous nucleation, by problem presented by the resides, sugar crystallization crystallization benchmark. always keeping the super-saturation inside the meta-stable The novelty of the article resides, on one hand, on the fact to avoid both dissolution and spontaneous nucleation, by problem presented by the sugar crystallization benchmark. always inside the meta-stable The of article one hand, on the If this the goalsuper-saturation is achieved, then size that the hybrid of the on plant always keeping keeping the super-saturation insidethe the crystal meta-stable The novelty novelty of the thenature article resides, resides, on one and hand,its on specific the fact fact zone. If this goal is achieved, then crystal that the of plant its always insidethe the meta-stable The ofrequires thenature article resides, one and hand,between on specific the fact zone. keeping If (CSD) this the goal is resulting achieved, then the crystal size that novelty the hybrid hybrid nature of the the plant and its specific distribution of super-saturation the population would havesize the characteristics a very tighton integration the zone. zone. If this goal is achieved, then the crystal size that the hybrid nature of the plant and its specific of population have the characteristics requires aa very tight integration the zone. If (CSD) thischaracteristics: goal is resulting achieved, thenwidth thewould crystal size that the decisions, hybrid nature of plant and ofbetween itsthespecific distribution (CSD) of the the resulting population would haveright the characteristics requires very tightscheduling integration between the distribution desired quality a narrow and the discrete related to the the batch distribution (CSD) of the resulting population would have the characteristics requires a very tight integration between the desired quality characteristics: a narrow width and the right discrete decisions, related to the scheduling of the batch distribution (CSD) of the resulting population would have the characteristics requires a very tight integration between the desiredsize. quality characteristics: narrow widthisand and the right right discrete decisions, related to the the decisions schedulingtaken of the the batch mean A typical beet sugar crystallizer operated by crystallizers, with the continuous for batch their desired quality characteristics: aa narrow width the discrete decisions, related to scheduling of mean size. A typical beet sugar crystallizer is operated by crystallizers, with the continuous decisions taken for their desired quality characteristics: a narrow width and the right discrete decisions, related to the scheduling of the batch mean size.a configurable A typical typical beet beetrecipe sugarconsisting crystallizerof is isthe operated by crystallizers, with the the continuous decisions taken fora their their enforcing following optimal intra-run operation. On decisions the other taken hand,for new mean size. A sugar crystallizer operated by crystallizers, with continuous enforcing aa configurable consisting of following optimal intra-run operation. On the other hand, new mean size. Ainitial typicalcharge beetrecipe sugar crystallizer isthe operated by crystallizers, with the continuous decisions taken foraaof their enforcing configurable recipe consisting of the following optimal intra-run operation. On the other hand, new main steps: of syrup, concentration, seed, coordination scheme is used in the upper level the optimal intra-run operation. On the other hand, a new enforcing a configurable recipe consisting of the following main steps: initial charge of syrup, concentration, seed, coordination scheme is used in the upper level the enforcing a configurable recipe consisting of All the the following optimal intra-run operation. On the other hand, aof new main steps: initial charge of syrup, concentration, seed, coordination scheme is used in the upper level of the crystal grow, tightening and mass discharge. steps, controller hierarchy, that involves the on the fly creation of a coordination scheme is used in the upper level of the main steps: initial charge of syrup, concentration, seed, grow, initial tightening and ofmass discharge. All the steps, controller hierarchy, the fly creation main steps: charge syrup, concentration, seed, coordination schemethat is involves used inthe theon upper level of oftheaa crystal controller crystal grow, grow, tightening tightening and and mass mass discharge. discharge. All All the the steps, steps, controller hierarchy, hierarchy, that that involves involves the the on on the the fly fly creation creation of of a crystal controller hierarchy, that involves the on the fly creation of a crystal grow, tightening and mass discharge. All the steps,

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except mass discharge, are carried out while keeping vacuum condition in the chamber, so that mass boiling would occur at low temperatures, preventing caramelization. The energy for heating the mass is provided by a heat exchanger incorporated in the design of the unit. The exchanger’s primary receives saturated steam from the common source. Pst

Sugar Crystallization Benchmark Heating steam Evaporated steam

Syrup to process

Wst_local_1

Crys. I

Wst_local_2

Crys. II

Wst_local_3

Crys. III

Feed liquor buffer tank

Fig.1. Sugar crystallization benchmark The pressures of the heating steam at exchanger’s primary and of the evaporated steam inside the crystallization chamber are both enforced by their respective PI type controllers. The set-points of these controllers are provided by the sequencing program implementing the recipe. In addition to pressure sensors, the unit is also equipped with online measurements of the level attained by the mass in the chamber and for its concentration. This last variable takes into account the presence of the dissolved substances (sucrose plus impurities) and of the sugar crystals suspended in the solution or mother liquor which appear after the seeding step. The first three steps of the recipe are in charge of loading the initial amount of feeding syrup to approximately half its capacity, to proceed next with its concentration for achieving a super-saturation capable of hosting the initial seed.

Level (%)

Pan level (%)

Syrup conc.=76%

Pan level (%) Syrup conc.=71%

Syrup purity=95% Syrup conc.=73%

Syrup purity=95%

Syrup purity=91%

(b)

(a)

Time (s)

evolution of the batch (corresponding to a different level in the chamber) a specific set-point for the concentration is given, chosen in such a way as to approximately discount the expected increase in the mass of crystals and the decrease in solution purity, so that the resulting conditions translate into a sucrose super-saturation in the meta-stable zone. When the level reaches its maximum value, the slurry is concentrated further and, finally, is discharged. The benchmark that has been put forward (fig. 1) consists of three identical crystallizers that are fed from the common source represented by a buffer tank of the syrup to process. This vessel seeks to accommodate the continuous upstream syrup processing units with the discontinuous crystallizers. The latter have also to compete for the needed heating steam. The basic problem to solve is that of proposing a schedule for the crystallizer’s cycles, so that the buffer tank is not overflowed nor drained. This has to be done while trying to solve, at the same time, the difficulties imposed by the limited availability of the heating steam. So, in this context, it is illustrative to understand the impact that the properties of the incoming syrup, disturbances from the benchmark’s point of view, have on the duration of the crystallizer cycle. Figure (2) shows that an increase of the concentration and of the purity of the syrup leads both to a reduction of the cycle duration, provided that the evaporation rate is not changed. But the cycle duration, and thus the throughput of the crystallizers, can also be influenced by modifying the evaporation rate imposed in the units. This can be achieved, for example, by adjusting the set-points to the heating steam pressure regulatory loops. This is here accomplished by multiplying the nominal profile used by the sequencing program of the crystallizer by a factor pF centered at one. The effect of this factor is shown in fig. (3.a). It is important to notice that this acceleration comes at a price: since the crystallization kinetics has its own rhythm, which depends also on the purity of the mother liquor, an attempt to rush-up the strike beyond a certain limit would imply an increase in super-saturation (fig. 3.b), since the aggregate surface of the mass of crystal will not grow at the rate needed to keep up with the increasing presence of sucrose. Heating pressure loop SP profile

Time (s)

Fig.2. Impact on batch length of syrup (a) conc. (b) purity

PF=1.0 PF=0.8

PF=1.2 PF=1.0

s

The crystal-growing step is the lengthier and most important of the cycle. The goal is keep super-saturation in the metastable zone by manipulating the amount of syrup fed to the chamber. This syrup input seeks to compensate the dissolved sucrose that migrates to the surface of the growing crystals and the water that is evaporated by boiling. Super-saturation control is indirectly achieved by stipulating the concentration of the mass to enforce at each point of the evolution of the cycle. Super-saturation and the suspension concentration are obviously correlated but the relation is not so straightforward. Both depend on the mother liquor concentration but the slurry concentration also takes into account the presence of the mass of crystals. Super-saturation, on its turn, depends on temperature and on purity. In the industrial crystallizer here taken as reference, the total mass concentration is used nonetheless for conducting the process. At each point in the

Pressure (bar)

PF=1.2

s

Wst

1211

Sucrose super-saturation

(a)

PF=1.2 PF=1.0 PF=0.8

PF=0.8

(b)

Time (s)

Fig.3. Impact of pF on a) batch length and b) super-saturation The first-principles model of the benchmark is described in Mazaeda et al, (2014) and the references therein. It includes the dynamic models of each crystallizer, and of the buffer tank. It also incorporates a very simple model of the saturated

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steam source. The latter represents the limited availability of this resource by means of a linear static law relating an increase in the consumed steam flowrate with a decrease of the provided pressure. The model of the crystallizers and of the tank includes the needed mass and energy balances. Mass balances to the syrup include the balances to its three components: water and dissolved sucrose and impurities. In the crystallizer, after seeding, there is also the need to follow the evolution of the sugar crystals mass. The crystallizer model requires balances to the steam evaporated in the chamber and for the heating steam in the primary of the heat exchanger. The objective of the model is to serve as a test case for evaluating plant-wide control strategies. The emphasis has been mainly put in capturing the relation of the batch crystallizers with the common resources represented by the syrup tank and the steam source. Then, special care has been taken, for example, to reflect the non-linear and timevarying behavior of the equivalent global heat transfer coefficient of the heat exchanger. This coefficient, and thus the steam consumed by the crystallizer, diminishes along the strike as the level of the mass in the chamber and its viscosity increases, affecting mass circulation. The heating steam profile also determines the shape of the syrup input profile, due to the presence of the suspension concentration regulatory loop. So it has a similar diminishing pattern, parallel to of the one exhibited by the evaporation rate. The growing of the crystals is customarily described in the population balance framework. Here however, for the sake of simplicity, a lumped model describing the aggregate mass of sugar grains has been used, as in Lauret et al, (1998). 3. PLANT-WIDE OPTIMAL CONTROL PROBLEM The sugar benchmark problem exhibits a pattern that can be found in process industries with some frequency. The general situation is the following: there is a downstream sub-plant, consisting of a group of batch or semi-batch parallel units (the sugar crystallizers), whose combined throughput needs to match the flowrate (syrup) arriving from the upstream factory. Since the match has to be found between continuous and discrete units, it should be understood in some average sense, typically by requiring for the level of an intermediate buffer vessel to be kept between safe limits. In addition to depend of a common stock to process, it is not rare for the discrete units to compete for some limited shared auxiliary resource (steam). In general, the duration of the cycle of the semi-batch units, and thus their throughput, would depend on some properties of the stream to process (concentration and purity of the syrup). In some cases, there exists a margin for impacting the duration of the cycle by manipulating some control inputs (the heating pressure set-points factors pF). So, from the point of view of control, and turning again our attention to the specific sugar crystallization case, the throughput of the parallel crystallizers should be regulated to match the flowrate of incoming syrup, or equivalently to avoid the overflow or draining of the buffer tank, by choosing appropriately the short-time scheduling of the semi-batch units. The scheduling task can be eased by shortening the duration of the individual cycle at the expense of increasing the consumption of steam.

We have both, the scheduling of the crystallizers and the three individual pF acceleration/delaying factors, for specifying the average combined throughput of the parallel units. This gives flexibility but it is redundant. In cases like this, it is reasonable to choose the existing degrees of freedom optimally. In plant-wide problems, a profit related performance criterion is usually taken: prices of the products minus costs of raw materials and auxiliary resources. Typically, the value of products far outweighs the costs, so the optimal solution is equivalent to maximize production, a value determined by the bottleneck of the whole factory. But in the case of partial downstream plants, however, since the attainable production rate is determined by the upstream plant, a reasonable optimal strategy will be one that minimizes the costs of ancillary resources (steam in the benchmark considered) as highlighted in (Skogestadt, 2004). t =t

min J = min ∫t = 0end Wst dt P , Load u

(1)

f overall ( x , x, u , p, t ) = 0

(2)

Lmin ≤ LBuf (t ) ≤ Lmax

(3)

smin,i (t ) ≤ si (t ) ≤ smax,i (t )

(4)

P f min,i ≤ Pf i (t ) ≤ Pf max,i

(5)

fi

i

Load i ∈ {0,1} i ∈ {1,2,3}

(6)

In view of the above, the dynamic plan-wide optimal control for the sugar benchmark is stated as shown in equations (1)(6). The cost to minimize is related to the consumption of heating steam mass flowrate (Wst) along the entire exercise considered. This steam flow can be put in relation with the cost of the required energy; but there are, also other benefits in working at lower values. For example, a lower steam flowrate means that the crystallizer is conducted at a slower pace, and at correspondingly lower super-saturations, resulting in a more uniform CSD. The decision variables are the discrete variables (6) on when to start each cycle (Loadi) and the continuous accelerating/delaying factors (pFi) limited to lie in some range (5). In (3), foverall represents the dynamical equations modelling the entire plant. The level of the buffer tank should remain between the specified limits [Lmin,Lmax]. At this point, some considerations related to the scope of the optimal control for the sugar crystallization benchmark are in order. Notice that a problem such as (1)-(6) would, in general, require the optimal intra-run control of each of the semi-batch crystallizers. This task is, by itself, quite challenging, see for example, (Mesbah et al, 2012). The optimal control of the crystallizer should, in general, strive for maximizing the yield, guaranteeing some target CSD characteristics related with the quality of the product while minimizing the cost of processing the batch. Batch crystallization processes are complex, highly non-linear and uncertain. This last characteristic is aggravated in the industrial context due to the lack of enough sensors. The access to on-line information on the CSD is typically absent, especially in food processing industries. Important variables such as, for example, the super-saturation need to be inferred

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from others. The use of observers, as part of the optimal control strategy, in this type of setting, is a necessity. In the context of the sugar crystallization benchmark, however, the interest is in the global aspect of the problem. In conformity with this, and to avoid obscuring the plant-wide strategy with complex details which are very specific of the crystallization example, it is assumed that the original recipebased control is good enough. The factors (pFi), are, however, available for adjusting the duration of the cycles. Since an undue acceleration could cause the increase of supersaturation (s) in a way that would compromise quality, the controller manipulating the pFi commands, should assume the responsibility of guaranteeing that s remains always on safe limits, well within the meta-stable zone, as expressed by the path restrictions (4). Notice that we are considering, for the sake of simplicity, that the super-saturation is directly available. 3.1 Hierarchical plant-wide optimal control solution Plant-wide problems, such as the one represented by our case study, have been traditionally approached in a hierarchical fashion. A well-known strategy is the Real Time Optimization scheme (RTO), which assumes the existence of at least two layers combining an economic optimization with a static model in the upper layer and Model Predictive Controller (MPC) type of controller in the lower one (Engell, 2007). The RTO approach has the drawback of having to keep two different models of the same process. An alternative that is rapidly gaining interest is the use of economic MPC (eMPC). This is a single layer approach where the receding horizon scheme is directly applied to the optimization of the economic index. Economic MPC has been the subject of recent academic interest, concerning, among other issues, the topic of guaranteeing stability (Ellis, et al, 2014). For the plant-wide case, however, the use of a single, monolithic e-MPC controller approach might work fine for small plants; but for real-sized problems it could rapidly become computationally challenged. In (Hernández, et al, 2014) an essentially equivalent variant of the sugar crystallization optimal control problem has been satisfactorily solved using an original two-level strategy. The authors propose to solve the complete continuous dynamic problem optimally, choosing the pFi factors that minimize a cost related index for a fixed given scheduling in the lower layer using the complete dynamical model of the plant. Then, the upper layer uses a static model for refining the current scheduling by guarantying the global mass-balances. The problem has been solved off-line for the entire fixed horizon of several cycles into the future. For on-line application, the described strategy would probably require a receding horizon type of scheme (e-MPC type controller) in the lower layer as means of incorporating disturbances. Note that also in this case, the dealing with large plants remains a problem.

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one for each batch crystallizer, in the bottom layer and then a higher level coordinator, also implemented as an e-MPC, in charge of the scheduling of the batch units and of the tuning the local problems to drive the plant’s state near the optimal operation. It seems reasonable to state the local problem as solving the trade-off between the throughput of the unit (which could be increased by reducing the duration of batch assuming an approximately constant sugar yield) and the consumption of heating steam (Wst_local). This kind of compromise is usually solved using the prices of the product and costs of auxiliary resources. In this case, however, this approach would not work for fixed prices: nothing is gained by rushing unnecessarily the batch because the objective is, for the parallel crystallizers, to match the incoming syrup. On the contrary more steam would be wasted without obtaining any global benefits. There are also restrictions on the availability of syrup and of steam that can only be assessed form the plant-wide perspective. What is needed is some kind of artificial knob for each local optimal problem, adjusted by the global coordinator in a way that leads the solutions of the subordinate problems to approach the plant-wide optimum of minimizing the total steam consumption. In the case at hand the, the role of this adjusting knob is played by the parameter TBatch_max, the maximum allowed duration of the batch.

min J single = min Pf

TBatch

pF

(7)

∫ Wst _ loc dt

0

Subject to:

TBatch < TBatch _ max

(8)

f single ( x , x, u, p, t ) = 0

(9)

smin (t ) ≤ s(t ) ≤ smax (t )

(10)

p F min ≤ p F (t ) ≤ p F max

(11)

Thus, each local e-MPC should seek to minimize its consumption of heating steam (Wst_loc) (7) by manipulating its corresponding pF variable between the stablished limits (11), provided that it complies with requirement (8) for the actual duration of the batch (TBatch) to be below a certain maximum value (TBatch_max). The optimization must be carried out subject to the dynamic equations of the unit’s model (9) and it should comply with the path restrictions on supersaturation (10). The coordinator’s task is also approached in an economic optimal control setup as the minimization of the total consumption of steam in the prediction horizon (h), as in (12)-(17).

min

TBatch_maxi ,TLoad _ i

J cost_all =

t =h min Wst dt TBatch_maxi , ∆Tini _ i t = 0

∫

(12)

Subject to:

In order to cope with this scale problem, a non-centralized strategy is followed in this paper. It uses a hierarchical architecture that has three independent e-MPC controllers, 1214

Lmin ≤ LBuf (t ) ≤ Lmax dmbuffer dt

= wsyrup _ in − wsyrup _ out

f st _ src ( Pst ,Wst ) = 0

(13) (14) (15)

IFAC ADCHEM 2015 1214 June 7-10, 2015. Whistler, BC, Canada Rogelio Mazaeda et al. / IFAC-PapersOnLine 48-8 (2015) 1210–1215

wsyrpi = wsyrp0 i +

∂wsyrpi ∂Tbatch _ mxi

(Tbatch _ max

∂wst _ locali

wst _ loci = wst _ loc0 i +

∂Tbatch _ max i

i

− Tbatch _ mx0 i

(Tbatch _ mx

)

(16)

)

− Tbatch _ mx0 i (17)

i

At the coordinator level, the decision variables are the mentioned tuning parameters for each of the three crystallizers (TBatch_max_i) and the time of occurrence of the starting commands for each batch (TLoad_i). Notice that for this coordinator-subordinate non-centralized solution to work, the global and local problems should be consistent (Findesein, 1982), which is the case at hand: compare (12) with (7).

expressing a linear approximation of the predicted common resource requests of the subordinate units (Wsyrpi,Wst_loci) as a linear function of the adjusting parameters (fig. 4). Notice that in (12), the original mixed-integer non-linear program (1)-(6), defined in terms of the discrete decisions (Loadi), has been reformulated into a completed continuous one using instead the starting time of each batch (TLoadi). This kind of reformulation is well posed (Prada et al, 2008) when the number of switches to apply is known beforehand. This is exactly the case, since the prediction horizon (h) used by the e-MPC of the coordinator, would be chosen, at each sampling interval, so as to encompass a future fixed number of complete cycles for each subordinate crystallizer (fig. 5). Prediction Horizon (NP complete cycles for each pan) Control Horizon (Nc cycles for each pan, Ej: Nc=2)

The optimal problem posed in the coordination layer should enforce the restriction of the buffer tank level (13). It should also include, as restrictions, the dynamic model of the same vessel (14) and the static model (fst_src) expressing the real source steam relation between the common pressure (Pst) and the total mass flowrate of steam drawn by the three crystallizers (15). Steam source

Pst

Wst_local1

IBuff

Crys. I

Crys. II

Wst_local_2 Wsyrp2

Wsyrp1

e-NMPC I

e-NMPC

TBatch_mx1 Ioad1

∂Wst _ loc1

,

Ioad1_2

... TIoad2_2

TIoad2_1 Ioad2_1

Ioad2_2

... TIoad3_1

TIoad3_2 Ioad3_2

Fig. 5. Prediction horizon used in coordinator. Wst_local_3

4. IMPLEMENTATION AND RESULTS

Crys. III

Wsyrp3

e-NMPC II

...

TIoad1_2

Ioad3_1

Wsyrup_out

Coordinator

LL

LLL

Evaporated steam

Buffer tank

TIoad1_1 Ioad1_1

Hierachical Control

Wst Syrup to Process (Wsyrup_in)

L

e-NMPC III

∂Wsyrp_ 1

∂TBatch_ mx1 ∂TBatch_ mx1 Wst_loc0_1 Wsyrp0_1

Fig. 4. Hierarchical plant-wide optimizing control. But the models of the tank and of the steam source are not enough. The coordinator has to have information on the future profile of the demands of syrup and steam for each of the subordinate crystallizers and on how these requirements on the common resources would change with respect to the duration of the individual batches. This information could be obtained, of course, if the coordinator had access to the complete model of the factory but that is what we strive to avoid for guaranteeing the computational scalability of the proposed solution. The strategy proposed in this paper consists in having the subordinate e-MPCs to send to the coordinator, at each sampling interval, updated information on the predicted profiles of the demands of the common resources (Wsyrp0i, Wst_loc0i) (obtained as part of the MPC solution), along with information on the sensitivity of those profiles to changes in the individual crystallizer tuning parameter (Tbatch_max_i). The sensitivities are calculated by finite differences: once the current solution is obtained, a simulation with a slightly changed value for the batch duration is carried out. This information should be incorporated as restrictions (16)-(17): a very simple model

The plant-wide control strategy described in the previous section has been tested in the benchmark simulator. The local MPCs use control vector parametrization and a sequential approach in order to solve the optimization stated in (7-11). The numerical mathematical programming algorithm used is of the SQP type as developed in SNOPT. Since the process to be controlled is of the semi-batch type, the very simple intra-run optimization has been performed using a prediction shrinking horizon that reaches the end on the current cycle. In order to keep the problem simple, the control horizon has been taken as equal to one. The interval for performing the optimization has been chosen to be of 30 minutes, but data has been gathered with intervals of 300 s. For this simulation test, the model of the crystallizers are considered as perfect and it is assumed that the controllers has full access to the state of their respective unit. Of course, as justified previously, a real implementation would need to consider more sophisticated intra-run optimization strategies and the use of observers. The coordinator MPC, solving problem (12-17), is also executed every 30 minutes and is also implemented in sequential fashion. The prediction horizon is variable since it is stretched for one entire cycle into the future for each of the subordinate crystallizers (see Fig. 5). The control horizon is of one sampling interval and an evolutive optimization algorithm was used at this level. The first optimal control simulation has been carried out along 48 hours. The disturbances to the process are represented by the incoming syrup that should be processed on time. The concentration and purity are considered to be fixed at typical values of 76% and 95% respectively, while

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the syrup mass flow-rate undergoes a important abrupt change, going from 14.5 kg/s to 18.5 kg/s at the hour 24 (86400s). 100

Crystallizers levels (%) Pan I Pan II

80

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experiments has been carried out using a single computer for running three subordinate controllers and coordinator. With the use of a distributed computer platform, now possible due to the non-centralized approach, the situation would be even more favourable.

Pan III

60

Total steam pressure (bar)

2.2

40

2.0

20

1.8

(a) 1.6 90

Buffer Tank Ievel (%)

1.4

80

1.2

70

1.0 0

60

50 0

50 000

Time (s)

100 000

150 000

PF decision variables

1.1

1.0

0.9

0.8

0.7 50 000

Time (s)

100 000

150 000

In fig. 6.a, the profile of the level inside each batch crystallizer is shown, representing the attained schedule. The evolution of the level of the buffer tank is depicted in fig. 6.b, making evident that the task of processing all the syrup has been accomplished along the exercise. Figure 7, shows the pFi for each crystallizer Notice that in the second half of the exercise, for higher values of the syrup input to the plant, the coordinator issues the starting commands at a higher frequency while the lower level controllers decide to use a greater amount of heating steam, to keep pace with the higher flowrate of syrup into the plant. Maximum allowed batch duraction (s)

Tbatch_mx_sup Tbatch_mx Tbtach_mx_inf

8500

7500

6500 0

50 000

Time (s)

100 000

150 000

REFERENCES

Fig. 7. Local units PF factors.

9500

100 000

A novel hierarchical plant-wide economic optimization scheme has been developed. It has been applied to a sugar crystallization process. Each batch crystallizer is operated by its own e-NMPC seeking to optimize an index related to the local economic performance of the unit, while the upper layer takes care of the scheduling and the management of common resources while minimizing the total consumption of steam.

(b)

1.2

0

Time (s)

6. CONCLUSIONS

Fig. 6. (a) Level of batch units. (b) Level of buffer tank. 1.3

50 000

Fig. 9. Heating steam pressure.

(b)

150 000

Fig. 8. Coordinator tuning parameter Tbatch_max1. Figure 8 shows the Tbatch_max coordinating parameter for the case of the first pan, along with the dynamically calculated band that limits the size of its excursions to keep the linear approximation valid. Notice that as the conditions are more demanding to the end of the exercise, the coordinator requests from the subordinates a shorter batch to increase throughput at the cost of using more steam. The fig. 9 shows the evolution of the pressure in the common steam source. Finally, it is of interest to mention that the total amount of processing time for all the controllers is always much less that the considered sampling interval of 30 min. Simulation

Ellis, M., Durand, H., Christofides, P.D. (2014). A tutorial review of economic model predictive control methods. Journal of Process Control, vol. 24, pp. 1156-1178. Engell, S. (2007). Feedback control for optimal process operation. Journal of Process Control, vol. 17, pp. 203219. Findesein, W. (1982). Decentralized and hierarchical control under consistency or disagreement of interests. Automatica, vol. 18, pp. 647-664. 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. HYCON/WP5 (2014). Sugar show case. Online available: hycon.isa.cie.uva.es/home. Lauret, P., Boyer, H., Gatina, J.C. (2000). Hybrid model of sugar boiling process. Control Engineering Practice, vol. 8, pp. 299-310. 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. Mesba, A., Huesman, A.E.M., Kramer, H.J.M., Nagy, Z.K., Van den Hof, P.M.J. (2011). Real-time control of semiindustrial fed-batch evaporative crystallizer using different direct optimization strategies. AIChE Journal, vol. 57, America Inst. of Chem. Eng., pp. 1557-1569. Prada, C. de, Sarabia, D., Cristea, S.P. and Mazaeda, R. (2008). Plant-wide control of a hybrid process. Adaptive Control and Signal Processing, vol 22, pp. 124-143. Skogestadt, S. (2004). Control structure design for complete chemical plants. Computers and Chemical Engineering, vol 28, pp. 219-234.

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