INTEGRATION OF SCHEDULING AND CONTROLLER DESIGN FOR A MULTIPRODUCT CSTR

Transcription

1 ITEGRATIO OF SCHEDULIG AD COTROLLER DESIG FOR A MULTIPRODUCT CSTR Yunfei Chu, Fengqi You orhwesern Universiy 245 Sheridan Road, Evanson, IL 628 Absrac Though scheduling and conrol are radiionally considered separaely due o he differen operaional levels and ime scales, research on he inegraion of he wo problems becomes aracive recenly. I has been demonsraed ha he inegraion can resul in a collaboraive operaion wih higher economic profis, which is crucial for he fierce compeiions in curren process indusries. In his paper, we presen a novel inegraion approach, which simulaneously deermines he PI conroller design and producion scheduling decisions. The inegraed problem is solved o updae he conroller parameers in each ime slo and he conrol variable is hen deermined by he feedbac of he process measuremen according o he updaed conroller parameers. This feaure is differen from he previous inegraion sraegies which deermine he conrol variable direcly from he soluion of he inegraed problem. The advanage of he proposed sraegy is ha he inegraed problem can be solved in a large ime scale while he conroller wors in real ime. The wo layer framewor faciliaes he online implemenaion of he inegraion of scheduling and conrol. The proposed approach is illusraed hrough an example of a muliproduc CSTR. Keywords Scheduling and conrol, PI conroller, Decomposiion algorihm, Cyclic producion, Muliproduc CSTR Inroducion Manufacures in he curren process indusry confron rapid mare changes, sringen environmenal policies, and aggressive rival compeiions. In order o overcome he dwindling profi margins, i becomes increasingly imporan o opimize he global objecives by inegraing he informaion and he decision-maing among differen operaional levels (Grossmann 24; 25). Though scheduling and conrol are ransiionally considered separaely, a more economical operaion can be achieved by aing hem ino accoun simulaneously (Flores- Tlacuahuac and Grossmann 26; Harjunosi e al. 29; Munoz e al. 2). The process scheduling problem comprises he organizaion of human and echnological resources, he deerminaion of producion sequence and producion imes. The process conrol problem aims o regulae he process variables and ensure he dynamic rajecories racs he ones se o mee he operaional crieria. In a muli-produc manufacuring line, he ransiion period beween wo producs is deermined by he conrol sysem. Differen designs of conrol sysems will resul in differen values of process variables in he ransiion period, e.g. he ransiion maerial consumpion and he ransiion ime. These variables may significanly influence he scheduling decisions. On he oher hand, o design a conrol sysem, he ransiion sequence and he se poins are required o be deermined from he scheduling problem. Besides, he economic objecive funcion for used o design he conrol sysem design should be derived from he scheduling problem. The coupling of he

2 scheduling problem and he conrol problem necessiaes heir inegraion. The inegraed problem of scheduling and conrol is frequenly formulaed as a mixed ineger dynamic opimizaion (MIDO) (Baron e al. 998; Allgor and Baron 999). One soluion approach is o ransform he MIDO problem ino a mixed ineger nonlinear programming (MILP) by using he collocaion mehod (Cuhrell and Biegler 987; Kameswaran and Biegler 26). In his approach, he sae rajecories and he inpu rajecories are discreized simulaneously and he differenial equaions describing he dynamic behavior of he process are ransformed ino a large se of nonlinear algebraic equaions. This approach has been applied o he cyclic producion in a single CSTR (Flores-Tlacuahuac and Grossmann 26) and following exensions (Terrazas-Moreno e al. 27; Flores-Tlacuahuac and Grossmann 2). An alernaive o solving he MIDO problem is o use decomposiion mehod (ysrom e al. 25; Praa e al. 28). The scheduling problem is modeled as he maser problem of MILP while he conrol problem is formulaed as he primal problem of dynamic opimizaion (DO). The MIDO problem is solved by ieraions beween he MILP problem and he DO problem. A challenge in mos inegraion approaches is ha he resuling conrol sysem from solving he MIDO problem is open loop, because he value of he conrol variable or he process inpu is deermined by he opimizaion algorihm once and for all while he curren measuremen of he process variables are no aen ino accoun. Due o he ineviable uncerainies and disurbances in he real process, an open loop conrol is scarcely applicable in pracice. A new inegraion sraegy which generaes a closed-loop conrol sysem is proposed recenly (Zhuge and Ieraperiou, 2). The closed-loop conrol is realized by solving he inegraed problem beween he sampling duraion of he measuremen. This sraegy is similar o he one adoped in model predicive conrol (MPC). However, solving he MIDO in real ime is sill formidable for mos curren solvers. The challenge in he inegraion of scheduling and conrol arises from he differen scales of he wo problems in he ime domain. The conroller needs o wor in real ime o deal wih he disurbance in he process immediaely while he scheduling decisions have o be deermined in a much larger ime scale due o he complexiy of he opimizaion problem involving discree decision variables. Therefore, implemenaion of he inegraed problem in he large ime scale resuls in an open loop conrol problem while he implemenaion in real ime maes he opimizaion problem formidable. In his paper, we propose a novel inegraion approach o address hese challenges. The main difference beween he proposed sraegy from he exising ones is ha here is a closed loop conroller for he process and he inegraed problem is solved o deermine he parameers of he conroller insead of he direc value of he conrol variable. The value of he conrol variable is deermined by he conroller according o he measured process variables along wih he conroller parameers calculaed from he inegraed problem. This sraegy employs he feaure of differen ime scales for he scheduling and he conrol. The closed-loop conroller wors in real ime while is parameers are updaed by solving he inegraed problem during he producion period. To solve he formulaed MIDO, we propose a decomposiion mehod. The MIDO is decomposed ino a scheduling problem of MILP and a series of conrol problems of DO in he ransiion periods. The proposed approach is illusraed hrough an example of a muliproduc CSTR. For mulaion of scheduling and conrol problems This secion presens he formulaion of he scheduling problem and he conrol problem. The formulaion varies according o differen ypes of he producion recipe and differen dynamic behaviors of he process. A producion model which is widely sudied in he lieraure is he manufacure of muliple producs in CSTRs (ysrom e al. 25; Flores-Tlacuahuac and Grossmann 26; Mira e al. 2). Due o is populariy, he focus of his paper is placed on his ype of model, which is, specifically, he cyclic producion in a muliproduc CSTR (Pino and Grossmann 994; Flores- Tlacuahuac and Grossmann 26). In he producion cycle displayed in Figure, a number of producs are cyclically manufacured in a single CSTR. The cycle ime is pariioned ino slos and only one produc is produced in one slo. Each slo is composed of wo periods: he ransiion period and he producion period. The produc is produced only in he producion period when he CSTR runs in he seady sae. The ransiion period represens he changeover beween differen producs in which he oupu of he CSTR varies wih ime. Producion cycle Slo Slo 2 Slo Transiion period Oupu Producion period Time Figure : Cyclic producion of a muliproduc CSTR

3 The scheduling problem aims o maximize he producion profi by deermining he sequence in which he producs are manufacured as well as he producion rae and he duraion of each period in a ime slo. Since he ransiion period is also dependen on he conrol sysem of he CSTR, i can achieve a beer overall performance by inegraing conrol problem ino he scheduling problem han solving he wo problems separaely because radeoffs can be made for conflicing objecives. The deailed formulaions of he scheduling problem and he conrol problem are presened as follows. Scheduling of cyclic producion Objecive Funcion max ϕ = ϕ ϕ2 ϕ3 () where φ is he produc revenue rae, φ 2 is he invenory cos rae, and φ 3 is he raw maerial cos rae. They are defined as p i c i= ϕ = CWi (2) T s ϕ2 = Ci ( GT i c Wi) Θi (3) T 2 c i= Tc c r ϕ 3 = C F d T (4) Since he cyclic producion is assumed o be carried ou repeaedly, each erm in he objecive funcion is calculaed as he averaged value in a producion cycle. The cos of conrol in he ransiion periods is no formulaed explicily in he objecive funcion, bu considered implicily in he cos of raw maerials, which depends on he conrol sysem and radeoffs oher coss. The objecive funcion can include some penaly erms for he difference beween he process variables and he se poins (Flores-Tlacuahuac and Grossmann 26). Though i is widely used in conrol sysem design, he objecive funcion is difficul o be qualified economically and incorporaed in he scheduling problem (Zhuge and Ieraperiou, 2). Thus, we do no include i in he proposed model. Produc Assignmen Consrains ξi =, i (5) = ξi =, i (6) i= ξ j = βij =, j, (7) i= ξ j βij = ξi, i, (8) j= The binary variable ξ i denoes if he produc i is assigned o he slo. The number of slos is assumed o be equal o he number of producs, denoed by. Consrain (5) indicaes ha each produc can be manufacured only once wihin a producion cycle while consrain (6) implies ha only one produc is manufacured in each slo. The binary variable β ij denoes if produc i is preceded by produc j in slo. The sequence variable β ij is relaed o he assignmen variable ξ i according o consrains (7) and (8). Since he producs are manufacured cyclically, he produc manufacured in he firs slo of a producion cycle follows he one in he las slo, which is refleced by he equaliy of β i ij = ξ = j in consrain (7). As suggesed in Wolsey (997), he sequence variables can be replaced by coninuous variables beween and, insead of binary variables. This significanly reduces he number of discree variables and improves he compuaional efficiency. Demand Consrains Wi Tc Di, i (9) W = GΘ, i () i i i The variable W i denoes he oal amoun of he produc i produced in each producion cycle and is averaged value needs o mee he demand rae D i, which is assumed o be consan. The producion amoun is deermined by he muliplicaion of producion rae G i and producion ime Θ i, according o he equaion (). Timing Relaion e s p = + +, () s = (2) e s, = + (3) e T (4) c p = i, i= max i i, i, βijτij, i= j= (5) ξ (6) = (7) Θ i = i, i (8) = The slo sars from ime s o ime e. The saring ime of he firs slo is se as zero while he end ime of he las slo is bounded by he cyclic ime T c from above. Each slo consiss of he ransiion period and he producion period. The duraions of he wo periods are

4 denoed by p and respecively. The duraion of he producion period is relaed o he producion ime of he produc i in he slo in equaion (5) and he equaion (6) ses he upper bound for he producion ime. The duraion of he ransiion period is dependen on he ransiion ime from he produc j o he produc i, denoed by τ ij, which is deermined by he subsequen conrol problem. The equaion (8) calculaes he oal producion ime of he produc i, which is used o calculae he produced amoun W i in he equaion () and he objecive funcion φ 2 in he equaion (3). Conrol Sysem in Transiion Period The produc is manufacured only in he producion period when he CSTR says in he seady sae. However, he ransiion beween differen producs is dependen on he dynamic behavior of he CSTR. Informaion in he ransiion period, e.g. he ransiion ime and he flow rae of he raw maerial, is required in he scheduling problem. To rerieve he informaion, he conrol sysem design for he CSTR is aen ino accoun. Suppose he dynamic behavior of he CSTR is described by a se of ordinary differenial equaions (, ) = (, ) x = f x u y g x u (9) where x() conains he sae variables, y() denoes he process oupu, and u() is he process inpu (he conrol variable). The goal of conrol sysem design is o deermine he value of u() such ha he oupu y() has he desired properies. An objecive funcion can be inroduced o quanify he performance of an inpu profile and he opimal inpu can be calculaed by minimizing his objecive funcion * arg max ϕc sp (, ), u = y y u u (2) u () where φ c is he objecive funcion. In an oupu feedbac conrol sysem, u() is dependen on he conrol error which is he difference beween he process oupu y() and he se poin y sp. The dependence of he oupu y() on he inpu u() is deermined by he sysem of equaions (9). I should be noed ha he conrol sysem is designed on behalf of he scheduling problem in he upper level. So he objecive funcion should be derived from he scheduling problem. Besides he objecive funcion, he design procedure also requires oher informaion from he scheduling problem, e.g. he producion sequence and he se poin of he oupu. Since he conrol problem is coupled wih he scheduling problem, he wo problems can be solved simulaneously. Inegraion of scheduling and conrol resuls in an MIDO problem. A common soluion is o discreize he differenial equaions by he mehod of simulaneous collocaion (Flores-Tlacuahuac and Grossmann 26; ie and Biegler 2). The ime domain is firs discreized by a se of grid,,,. Then, in each inerval of poins { } [ ], +, =,,, he soluion o he differenial equaion is approximaed by a n-h order polynomial and he values of he sae variables a he grid poins are expressed by + i i i= n x = x + h bf (2) where h = + - is he inerval lengh, b i is he coefficien which is dependen on he ype of polynomial used o approximae he soluion, and f i is he righ hand side of he differenial equaion evaluaed a he roos of he polynomial (collocaion poins). Applying he collocaion mehod, he differenial equaions (9) are ransformed ino a se of nonlinear algebraic equaions (2) and he MIDO problem is reformulaed as an MILP problem. Inegraion of PI Conroller Design wih Scheduling The previous secion presens he framewor of formulaing he scheduling and conrol problems. However, he sraegy for inegraing he wo problems is no unique. This secion sars from invesigaing some exising sraegies presened in he lieraure and hen proposes a new sraegy which circumvens difficulies in he exising sraegies. The resuled inegraion model is sill an MIDO. Insead of he collocaion mehod which solves he scheduling problem and he discreized conrol problem simulaneously, we propose a decomposiion mehod, which separaes he dynamic opimizaion in he conrol problem from he MILP in he scheduling problem. The decomposiion maes i possible o use differen solvers for each problem and also provides an insigh ino he radeoff beween he wo problems. The inegraed problem is finally solved by ieraions beween he dynamic opimizaion and he MILP. Sraegy for inegraing scheduling and conrol The firs sraegy for he inegraion is o simulaneously solve he wo problems ogeher (Flores- Tlacuahuac and Grossmann 26) and hen apply he calculaed inpu o conrol he dynamic sysem in he ransiion period. The main deficiency of his inegraion sraegy is ha he conrol sysem is open-loop. Since he

5 uncerainies and disurbances are ineviable in a conrol sysem, an open loop conroller is scarcely applied in pracice. Wihou he feedbac, he conrol variable can only follow he specified value no maer wha he oupu is. To improve he deficiency, a closed-loop sraegy is proposed (Zhuge and Ieraperiou, 2). Insead of solving he inegraed problem once, he problem is solved repeaedly beween wo sampling poins of he process oupu. The real ime informaion of he oupu can be aen ino accoun o calculae he new inpu value. This sraegy is similar o he one used in model predicive conrol (MPC) and i can significanly improve he performance of he conrol sysem since he uncerainies and disurbances are deal wih by he feedbac. However, he main drawbac of his sraegy is he expensive compuaional effors since he MIDO problem has o be solved on line. To cope wih he difficuly in he exising sraegy, a new sraegy is proposed in Figure 2. process variables schedule conroller conroller parameers u() process y() beween he scheduling algorihm and he conroller can be done in a large ime scale. For example, he conrol sysem can provide informaion of he curren process variables a he end of he ransiion period and he inegraed problem can be solved based on he curren informaion during he producion period. The new calculaed conroller parameers are hen sen bac o he conroller a he beginning of he nex ransiion period. Since he conroller exiss all he ime, he conrol sysem is in he closed loop even if here is no communicaion beween he upper level scheduling and he lower level conroller. The proposed inegraion sraegy can be applied o any parameerized conroller and he mos widely used one is he radiional PID conroller. The srucure of he PID conroller is shown in Figure 3. The conroller is composed of hree erms which are he proporional erm, he inegral erm, and he derivaive erm of he conrol error. The inpu of he process sysem is he sum of he hree erms. Since he derivaive erm will amplify he high frequency noise due o he differeniaion operaion, i is ofen omied in pracice and a PI conroller is usually implemened. K P e() e() = y sp - y() K I e() + u() Figure 2: Sraegy for inegraing scheduling and conrol The main difference beween he proposed sraegy from he exising ones is ha here is a closed loop conroller for he process and he inegraed problem is solved o deermine he parameers of he conroller insead of he direc inpu value. The inpu value is deermined by he conroller according o he measured process variables along wih he conroller parameers deermined in he inegraed problem. The advanage of his sraegy includes: () Fiing he pracical implemenaion srucure: Due o differen ime scales, scheduling and conrol are usually implemened a differen levels. The real ime conroller is implemened a he lower level in a PLC while he scheduling algorihm is implemened a he higher level in a PC. The exising sraegies for inegraing scheduling and conrol brea he common srucure and a new sysem is required o build for he implemenaion. However, he proposed sraegy can sill wor in he curren srucure. Wha needs o do is o ransfer informaion abou conroller parameers and process variables beween he wo levels and his is easy o achieve by modern hierarchical conrol sysems. (2) Balancing he conrol sysem performance wih he compuaional effors: Since he inegraed problem only deermines he conroller parameers, here is no need o solve he problem in real ime. Communicaion K D de()/d Figure 3: Srucure of PID conroller The differenial equaion of a PI conroller is = = sp x e e e y y u = Ke P + Kx I e (22) The conroller has wo parameers: he proporional gain K P and he inegral gain K I. Though many mehods exis o une he wo parameers, none of hem ae ino accoun he objecive funcion of he scheduling problem. Thus, we propose o deermine hese wo parameers by solving he inegraed problem of scheduling and conrol. Algorihm for he inegraed problem Simulaneous discreizaion presened in he previous secion is an alernaive o solving he inegraion sraegy proposed above. However, afer discreizaion a large scale of nonlinear algebraic equaions are derived. Suppose he number of grid poins o discreize he ime domain is, and c collocaion poins are seleced in each ime segmen beween wo adjunc grid poins. There are oally c equaions and variables derived by he collocaion mehod. Moreover, his number is only for one

6 sae variable in one ransiion period. Considering he number of ransiion periods and he number of saes x, he oal number of equaions and variables generaed by he discreizaion mehod is x c. Furhermore, he collocaion mehod is inrinsically an approximaion mehod by polynomial and i is nonrivial o deermine he grid poins and he collocaion. In he inegraed scheduling and conrol problem, discreizaion poins are probably required o be deermined for each ransiion period. For a producion cycle manufacuring producs, he number of possible ransiion periods is - and he case-by-case invesigaion of each one is difficul. Las, in he dynamic opimizaion problem for he proposed sraegy, he conroller parameers are calculaed insead of he ime funcion of he inpu. There is no need o discreize he decision variables. Therefore, anoher alernaive is adoped and a decomposiion algorihm is proposed. To decompose he inegraed problem, i is firs required o analyze how he conrol problem enering he scheduling problem. The analysis can also provide an insigh ino he radeoff beween he wo problems. The conrol sysem only affecs he variables in he ransiion periods, specifically he ransiion ime and he ransiion flow rae of he maerial, defined as F ( ). The ransiion ime is highly coupled wih oher variables in he scheduling problem while he ransiion flow rae only affecs he objecive funcion of he raw maerial cos. To separae he ransiion flow rae from oher variables, he objecive funcion () can be expressed as ϕ ( zf,,, F ) ( z) ( z) zf,,, F = ϕ ϕ ϕ 2 3 ( ) (23) where z is defined o include all decision variables excep F,, F. For he shor noaion, define { F },, F F = (24) By separaing he ransiion flow raes, he objecive funcion can be maximized by wo seps, since max ϕ max max ϕ (25) { } z, { F } z F By expanding he expression of φ, he righ side in he equivalence (25) urns ou o be ( { }) max ϕ ( z) ϕ ( z) min ϕ z, F z 2 { F } 3 or equivalenly (26) * ( ϕ( z) ϕ2( z) ϕ3( z { F })) * = ϕ max, z { } F ( { }) s.. F arg min 3 z, F { } (27) By subsiuing he expression of φ 3, he inner minimizaion problem is C min { F } F r T c = = min F d, for each p + p F d + F d (28) where he ransiion flow rae and he producion flow rae are defined as,,,, F = F + (29) s F = F + (3) p s p The equivalence of he wo problems in he expression (28) is due o he fac ha in he inner minimizaion of φ 3 F * are all regarded as { } he variables excep parameers. Therefore, he inegraed problem can be solved by ieraions beween he scheduling problem and a series of conrol problems (Figure 4). scheduling problem conrol problems * max ( ϕ( z) ϕ2( z) ϕ3( z, { F })) z F * z F * z min F d... K, K min F d K, K P I Figure 4: Decomposiion of scheduling problem and conrol problem The scheduling problem is formulaed in he previous secion, which can be summarized as max ϕ Eq. ( 4) s.. Assignmen consrains Eq. (5 8) Demand requiremens Eq. (9 ) Timing relaion Eq. ( 8) P I (3) The closed-loop conrol problem including he process dynamic sysem and he conroller is formulaed as s.. I = min x (32) * F KP, KI F

7 , = = = (, ) = sp max x = f x F x e e x F F y g x F e y y F = Ke P + Kx I e (33) min e e e max, (34) min F F F (35) The informaion which he conrol problem requires is he value of he ransiion ime and he ransiion sequence j i. Having he informaion, each conrol problem is a dynamic opimizaion and can be solved independenly. The dynamic opimizaion of he conrol sysem aims o minimize he inegral ransiion flow rae. To evaluae he inegral, a new variable of x F () is inroduced, which denoes he inegral of he ransiion flow rae from o. Consequenly, a new differenial equaion of x F () is insered. Besides his equaion, he se of he differenial equaions (33) is derived by combining he sysem equaions (9) and he equaions describing he PI conroller (22). The conrol variable u() is subsiued by he feed flow rae F ( ). Under he PI conroller, he process oupu will reach he se poin afer he ransiion period. The end of he ransiion period is indicaed by he fac ha he process oupu y() consanly says in a bound around he se poin of y sp, or equivalenly he conrol error e() falls ino he inerval bounded by e min and e max in he inequaliies (34). The bounds of he conrol error are frequenly se as ±2% of he se poin. The dynamic model is simulaed beyond he ransiion ime while objecive funcion of he conrol problem is only calculaed up o he ransiion ime. The las wo inequaliies (35) are se on he conrol variable, which is usually bounded due o he limied power of he pracical insrumen (sauraion of he acuaor). Case Sudy To illusrae he presened inegraion sraegy of scheduling and conrol, he cyclic producion in a muliproduc CSTR is sudied. The recipe and daa of he model are given in Flores-Tlacuahuac and Grossmann (26). The following reacion 3 R P, r = C (36) R 3 R aes place in an isohermal CSTR for manufacuring five producs, A, B, C, D, and E. The dynamic model of he CSTR is described by ( ) dcr F = Cin CR + rr (37) d V where C in is he concenraion in he feed flow and C R () is he concenraion in he ouflow. The flow rae F() is he conrol variable. The design and ineic parameers are C in = mol/l, V = 5 L, = 2 L 2 /(mol 2 h). Oher daa for he process are lised in Table. Table. Process daa for he case sudy Type F C R G Demand Price Invenory (L/hour) (mol/l) (g/h) (g/h) ($/g) cos ($/g) A B C D E The objecive is o maximize he produc profi per hour. Decision variables include he producion sequence, he producion ime and he ransiion ime in each slo, and he conroller parameers in each ransiion period. The MIDO of he inegraed problem is formulaed and solved by he decomposiion mehod. Table 2 liss he scheduling resuls obained from he opimal soluion o he inegraed problem. Table 2. Opimal scheduling resul Slo Produc Producion Transiion Produced ime (h) ime (h) amoun (g/h) A E D C B The opimal producion sequence is A E D C B. The produced amoun of each produc mees he demand. The cyclic ime is 93.4 hour and he objecive funcion is ϕ = = $/h. Table 3. Conroller parameers Transiion K P K I A E E D D C C B B A The PI conroller parameers in each ransiion period are deermined from he inegraed problem and lised in Table 3. I is seen ha he conroller parameers are se differenly for differen ransiion periods. This is an indicaor ha he conrol problem is dependen on he scheduling resuls. The dynamic profiles of he process

8 inpu and he oupu for he closed-loop conrol sysem in each ransiion period are displayed in Figure 5. The ransiion period sars a and ends a he ime mared by he verical dash line. Due o he inegral erm in he PI conroller, he conrol error will ulimaely end o zero and he process oupu can reach he se poin in he seady sae. To deermine he ransiion ime or he ime he sysem eners ino he seady sae, wo horizonal dash lines around he seady sae are ploed. They are se as, respecively, ±2% of he seady sae value. Afer he ransiion ime, he process oupu says beween he wo dash lines, indicaing he end of he ransiion period. concenraion concenraion concenraion concenraion concenraion A E 2 3 ime.6 E D ime.4 D C ime.4 C B ime.2 B A ime flow rae flow rae flow rae flow rae flow rae A E 2 3 ime 3 E D ime D C ime 5 4 C B ime 5 4 B A ime Figure 5. Opimal dynamic profiles of he concenraion in he ouflow (he process oupu in he lef panel) and he feed flow rae (he conrol variable in he righ panel) for each ransiion period To compare he scheduling resul, he producion sequence is generaed by a rule of humb, i.e. A B C D E. The ransiion is made gradually from he produc wih he lowes concenraion o he produc wih he highes concenraion The scheduling resuls of he producion sequence are calculaed and lised in Table 4. Table 4. Scheduling resul of rule-of-humb sequence Slo Produc Producion Transiion Produced ime (h) ime (h) amoun (g/h) A B C D E The cyclic ime is 97.6 hour and he objecive funcion is ϕ = =647.5 $/h. The profi is much less han he one produced by he opimal soluion. Whenever a disurbance occurs in he process, he disurbance can be aenuaed by he PI conroller which wors in real ime. So here is no need o re-solve he whole inegraed problem immediaely. The policy adoped in he case sudy is ha he inegraed problem is re-solved a he beginning of each producion period and he scheduling resuls as well as he conroller parameers are reurned o he conroller before he sar of he subsequen ransiion period. This means he inegraed problem is only solved once in each producion period and he requiremen on he compuaional ime is ha he inegraed problem can be solved wihin each producion period, which is ofen large enough o do so. To demonsrae his on-line rescheduling policy, a numerical experimen is conduced. The opimal soluion o he inegraed problem is applied and a disurbance is inroduced in he ransiion period of A E. The dynamic profiles of he process oupu and inpu are displayed in Figure 6. concenraion A E ime flow rae A E ime Figure 6. Dynamic profiles under he disurbance. The disurbance occurs a he ime of.5 hour and i is finally eliminaed by he PI conroller. The sysem eners ino he seady sae a he ime of.26 hour which is larger han ha wihou he disurbance. Though he ransiion ime in his period is enlarged by he disurbance, he process can sill successfully ransfer o he producion period. This is a significan difference from he open loop conrol where a sligh disurbance can cause a failure in he producion ransiion. This is also differen from he real ime rescheduling sraegy since he conroller parameers are sill he ones calculaed from

9 he iniial inegraed problem and he inegraed problem is no solved a his momen. Table 5. Scheduling resul under disurbances Slo Produc Producion Transiion Produced ime (h) ime (h) amoun (g/h) A E D C B The inegraed problem is re-solved wih he updaed informaion from he conroller from he beginning of he producion period in which he produc E is manufacured. The re-scheduling resuls are lised in Table 5. I is seen ha he producion sequence is no affeced by he disurbance. However he producion ime and he ransiion ime are changed. The produc profi is changed o ϕ = = $/h, which is only a slighly lower han he opimal resul wihou he disurbance. Conclusion A new sraegy for inegraing scheduling and conrol is presened for a muli-produc CSTR. A PI conroller is se o conrol he dynamic behavior of he process in he ransiion period beween wo producs. The conroller parameers are opimized simulaneously wih he produc scheduling problem. The conroller wors in real ime while he inegraed problem can be solved in a large ime scale, e.g. wihin each producion period. The presened inegraion sraegy overcomes he difficulies in previous sraegies in he lieraure. A radeoff is made beween he performance of he closed loop conrol sysem and he compuaional effors for solving he inegraed problem. The presened sraegy is illusraed by a case sudy of a CSTR producing five producs. omenclaure Decision Variables e() conrol error e max upper bound on conrol error e min lower bound on conrol error F maerial flow rae (g/h) p F ( ) flow rae in producion period in slo F ( ) flow rae in ransiion period in slo F max upper bound on ransiion flow rae F min lower bound on ransiion flow rae G i producion rae of producion i (g/h) K P proporional gain of PID conroller K I inegral gain of PID conroller derivaive gain of PID conroller K D s e T c u() W i x() x e () x F () y() y sp z sar ime of slo (h) end ime of slo (h) cycle ime (h) conrol variable/process inpu produced amoun of produc i (g) sae variable in process dynamic sysem inegral of conrol error inegral of ransiion flow rae process oupu se poin of he oupu all decision variables excep he ransiion flow rae Gree Leer β ij - variable o denoe if produc i is preceded by produc j in slo φ objecive funcion ($/h) τ ij ransiion ime from produc j o produc i (h) i processing ime of produc in slo (h) max upper bound on processing ime (h) p processing ime in slo (h) ransiion ime in slo (h) Θ i oal processing ime of produc i (h) ξ i - variable o denoe if produc i is assigned o slo Parameers p C i price of produc i ($/g) invenory cos of produc i [$/(g h)] s C i r C D i References raw maerial cos ($/g) demand rae of produc i (g/h) number of producs/slos Allgor, R. J., Baron, P. I. (999). Mixed-ineger dynamic opimizaion I: problem formulaion. Compuers & Chemical Engineering 23 (4-5): Baron, P. I., Allgor, R. J., Feehery, W. F., Galan, S. (998). Dynamic opimizaion in a disconinuous world. Indusrial & Engineering Chemisry Research 37 (3): Cuhrell, J. E., Biegler, L. T. (987). On he opimizaion of differenial-algebraic process sysems. AIChE Jounal 33 (8): Flores-Tlacuahuac, A., Grossmann I. E. (26). Simulaneous cyclic scheduling and conrol of a muliproduc CSTR. Indusrial & Engineering Chemisry Research 45 (2): Flores-Tlacuahuac, A., Grossmann, I. E. (2). Simulaneous cyclic scheduling and conrol of ubular reacors: Single producion lines. Indusrial & Engineering Chemisry Research 49 (22): Grossmann, I. E. (24). Challenges in he new millennium: produc discovery and design, enerprise and supply chain opimizaion, global life cycle assessmen. Compuers & Chemical Engineering 29 ():

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