The Computational Complexity Analysis of a MINLP-Based Chemical Process Control Design

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1 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : Abstract The Computational Complexity Analysis of a MINLP-Based Chemical Process Control Desin E. Ekawati, S. Yuliar Center for Instrumentation Technoloy and Automation Institut Teknoloi Bandun Bandun, Indonesia Received date : Mei 00 Accepted date : 9 Mei 00 It is essential that a chemical process plant is built based on a desin that interates the structural, operational and economic considerations. However, accomplishin such oal for a small section of a process system demands sinificant computational efforts. This project investiates the computational tractability of a mathematical optimization alorithm dedicated for such process control desin, the Dynamic Operability Framework. The taret output is a worst case computational cost of utilizin the framework to solve a process control desin problem in comparison with a standard Mixed Inteer n-linear Prorammin (MINLP problem. The cost is calculated based on the numbers of desin variables and constraints. This cost helps process desiners to make an informed decision of utilizin and improvin the framework as a tool for process desin. Introduction and Objectives One common oal in industrial chemical process desins is to achieve a specified product quality in feasible and profitable manners, despite the presence of feed fluctuations and other disturbances. One approach to accomplish such desin is by formulatin the simultaneous process and controller structure optimization as a mathematical prorammin problem, then subsequently utilizin a suitable computational alorithm to solve the problem. The Dynamic Operability Framework is an optimization alorithm dedicated for such purpose. The framework formulates the process control desin as a dynamic Mixed Inteer nlinear Prorammin (MINLP problem. The information processed by the framework consists of an objective (typically profit or operational costs, a set of Differential Alebraic Equations representin the process dynamics, a set of binary equations representin possible process control structures and interaction rules, a set of allowable ranes for operational conditions and a set of disturbance ranes. This information is processed throuh two layers of computation. The outer layer optimization selects the structure and parameters of the process units and controllers. The inner level tests the controllability and feasibility of the process dynamics subject to a known rane of disturbances over a iven period of time. The framework iterates between these levels to enerate the optimum solution. The size of the problem is typically lare. The computational complexities, i.e. requirement of CPU time, as well as Random Access Memory (RAM, may row with the number of optimization variables. These aspects should be considered seriously in handlin industrial cases that typically involves hundreds of dynamic variables. 55

2 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : In order to address such problem, this research investiates the computational tractability and optimality of the Dynamic Operability Framework. The results is the computational cost of a process desin, based on the information supplied to the framework, such as the number of process parameters, dynamic variables, operational rules and constraints. The cost analysis is the basis to determine the computational requirement and processin time to produce an optimum process desin throuh the framework. These will be especially useful to estimate the cost of a process desin, the cost of further modification to the framework, as well as the benefit in distributin of the computational tasks in a number of parallel computers. Process Synthesis Problem Due to inherent nonlinearities, chemical processes present varyin dynamic characteristics over different operatin conditions and different structures. Consequently, a certain dynamic and economic performances can only be achieved by optimizin the process and controller structures, in addition to optimizin the controller parameters. At this point, the difficulty arises due to stron interaction between both aspects, as these cannot be isolated and optimized separately. This interaction between structure and parameter in process desin, in pursuin the economic and control objectives, is best addressed within process synthesis problems. The synthesis problem is to select the optimum process structure, as well as its desin parameters. It involves riorous analysis of the existence and interconnection of unit operations, as well as the sizes and parameters of the components. The former clearly implies makin discrete decisions, while the latter implies makin a choice from amon a continuous space. The analysis ultimately determines the quality of process controllability and profitability in presence of disturbances and uncertainties. Over the last three decades, process synthesis problems have been investiated usin the heuristic, the physical insihts, and the mathematical prorammin approaches. The physical insihts yield in many essential desin procedures and uidelines. Nevertheless, the developments of automated synthesis procedures are mostly based either on the knowlede based heuristic, or on mathematical prorammin approach. The latter approach translates the process synthesis problem into the process control parameters and structures specification problem to satisfy the controllability and profitability objectives. These objectives are the consequence of two distinctive decisions. The first decision is the operational parameters, for instance, the size and operatin conditions of the unit operations. These are represented by continuous values. The second decision is structure; i.e. the existence of a unit, a stream flow rate, or an interconnection between process units. These may involve yes-no, if-then and other loic decisions, as well as selection of distinct operational ranes. Hence, these values are discrete. The consideration of these decisions ives rise to Mixed Inteer nlinear Prorammin (MINLP problems. 3 The Dynamic Operability Framework The Dynamic Operability Framework (DOF is a mathematical prorammin alorithm developed to select the best confiuration from a iven set of possib le process and controller structures (a superstructure. The oal is to find the process structure and parameters that produces the optimum value of process objective (i.e. profit or taret quality and keeps all dynamic responses within the desired output space (completely 56

3 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : feasible. The selection is made subject to a set of expected disturbances, operational constraints and operational loics. The eneral formulation of the Dynamic Operability Framework to solve the mentioned desin problem is iven below. minφ( z,θ z s.t. h ( z,θ ( z,θ (t,x(t,x (t,u(t, w(t,p, t z Z { z : z θ (t, x(t, x (t,u(t,w(t,p, t 0 (t, x(t,x (t,u(t,w(t,p, t 0 w(t w(t { x(t,u(t, w(t} x i j w ss ss θ ss x(t t EDS { θ : θ o w(t t ss l o ss z z } θ θ tt jd( y 0 max AOS θ ( w(t t t t ( 0 DOS w f θ r OCI DOS(w z { z, y, x,u, w } N N k k k u l u } i E j I jd j y {0,} w W { w : w u ss r t {t u(t t : t o o t t f l w w } u } ( Here, z is the aumented vector of continuous and binary desin variables z and y, and the initial conditions of state, manipulated, and output variables x ss, uss and wss respectively. The vector p contains the process parameters, and e contains the sampled disturbance profiles within the Expected Disturbance Space (EDS. The process model is represented by a set of differential alebraic equations hi. The time averae of the objective function profile is optimized within an optimization window. The set j is the feasible desin values for both plant and controller. The set jd is the subset of j, containin the loic propositions to solve the mixed-inteer problem related to the process control structure. The Desired Output Space (DOS defines the feasible dynamics of the output and measured variables in the aumented matrix w. The Achievable Output Space due to disturbances (AOS is the set of output dynamics due to e. The ratio between the size of DOS AOS and the size of DOS is the controllability index, r-oci. The external disturbances as well as process uncertainties are characterized as a set of step functions with uniformly distributed manitudes in EDS. Based on the above description, it is clear that DOF is a semi-infinite dynamic Mixed Inteer n-linear Prorammin ( SIDMINLP problem. The problem is solved in an iterative, twolayer dynamic optimization alorithm shown in Fiure. The outer layer is a combined steady state and dynamic multi-period MINLP problem, that considers the reulatory process dynamics due to the critical disturbance and uncertainty combinations k. At the initial outer level, k only consists of the nominal disturbances and uncertainties N. The optimum solution z * found in the outer layer is sent to the inner layer, which performs the feasibility test for the framework. This layer evaluates the process feasibility at z * and extracts the associated k from the convex hull of AOS. The iteration terminates when the outer layer converes and the inner layer does not introduce any additional k, and this typically happens in the second iteration. The outer layer consists of MINLP problem. The discrete variables represent superstructure that leads to Inteer Prorammin (IP Problem. The process model and operational constraints on each structure introduce nlinear Prorammin (NLP sub problems. 57

4 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : Branch and Bound (BB alorithm has been modified to solve these combined problems effectively. The search starts by solvin the NLP relaxation of the oriinal problem, then usin the solution as the bound of the problem. If the solutions of the discrete variables are all equal to the values defined at the discrete set, then the optimum solution is reached and the search is stopped. Otherwise, the search fixes all discrete variables to their respective closest discrete values. Any purely discrete equation within the MINLP formulation is enforced to eliminate infeasible nodes. For example, the constraint y+y eliminates the node [0,0]. The feasible nodes are considered as NLP sub problems. If one sub problem converes and the solution is better than the existin bound, this becomes the new bound, otherwise the node is eliminated. All previously evaluated binary combinations are listed to avoid repeated search. The inherent nonlinearities of process models and the solution of process dynamics in each NLP sub problem sinificantly contribute to the computational cost. This rows exponentially with the number of desin variables as described in Problem Backround. 3. The Computational Analysis of DOF The MINLP nature of the DOF alorithm requires the solution of one Inteer Problem (IP on distributin layer and one NLP at each nodes of the IP. These have been known as NP-Hard problems in. However, it should be noted that NP-Hardness is a worst case condition. There are features that make the DOF alorithm viable, such as [4]:. The r-oci test in the first inner layer uses the convex-hull of the process dynamics to determine all critically affectin disturbances k. The whole set of k is attended on the second outer layer, and typically no new disturbances detected by the subsequent inner layer. Therefore the alorithm is typically completed in two iterations.. The IP in the distributin layer is solved by the modified BB alorithm. The eneral BB complexity usually determined by the number of discrete variables. In the worst case of case of all 0/ decisions, all nodes have to be checked, hence the complexity is O(n. In DOF, the modified BB uses the pure loic proposition constraints to sinificantly eliminate infeasible nodes from the tree and reducin the computational complexity accordinly. For example, if only one of two equipments y and y is allowed to operate, then the associated loic propositions is y+y =. Then the possible nodes in enumeration tree are {[0,],[,0]}, instead of all four possible binary combination of [y,y]. The use of this proposition reduces 50% of computation efforts. 3. At each feasible node in the distributin layer, one dynamic NLP sub problem is solved. Once first converin sub problem with an objective value is found, this becomes the upper bound for the next sub problem. If the new sub problem shows any sin of inferiority, i.e. whether not converin or converin to poorer, then the node is fathomed and the alorithm resumes the search at the other nodes 58

5 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : Start Define h i (z,y,x,dx/,w,u,p,t, z,y,x,dx/,w,u,p,t j (z,y,x,dx/,w,u,p,t, and DOS Initialise z 0, y 0, x 0 (0, w 0 (0, u 0 (0,s=, k = N y= y s+ (search new subproblem Feasible y s (subproblem? all s considered? Yes Yes Evaluate z, x(t, w(t, u(t, k for optimum superstructure infeasible Distributin Layer Tune z, x(t, w(t, u(t Yes r-oci=? Yes Converin? Minimum? Yes Outer layer Identify k Evaluate z*, y*, x*(t, w*(t, u*(t with EDS r-oci=? Inner layer Yes Finish Fiure Alorithm of the dynamic operability framework 59

6 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : Computational Measures To relate the computational costs of DOF problem, the followin measures are employed: The size of the optimization problem specified by the number of continuous desin variables nz, binary variables ny, and DAE describin the system i, operational constraints j and loic proposition jd, and samples of n. The contributions of these parameters to DOF complexity in relation to the alorithm depicted in Fiure are as follows:. The complexity of the NLP optimization problem in the outer level at the worst case is proportional to the number of continuous desin variables nz.. The complexity of the modified Branch and Bound alorithm in the outer level at the worst case is exponential to the number of binary variables ny, i.e. ny. However, the number of loic propositions can reduce the complexity, althouh at the worst case, only node is fathomed for each loic proposition. Hence at the worst case the complexity is ny- jd. 3. At the inner level, the complexity is proportional to the number of samples of (n since all are required to determine the convex hull of the process dynamics. Incorporatin all samples of is essential to DOF, in order to anticipate the non - convexities of the problem. The critically affectin disturbances k found in this inner layer in the worst case is also proportional to n. If the first inner level declares all as k, then the second outer layer attends the whole set of. Consequently the second inner layer does not find a new k. Therefore, even in the worst case, the alorithm concludes in two iterations. 4. The complexity of solvin the process dynamic at every levels is proportional to the size of process model hi. 5. Since the complexity of NLP is NP-Hard, a worst case duration to solve the Sequential Quadratic Prorammin alorithm to solve one NLP node for one continuous desin variables, TNLP is used to represent the NLP complexity. Similar term TDAE is used to represent the worst case time required to solve one DAE. 6. Based on the above contributions, the total complexity of DOF (CI is as follows: ny T NLP n z ( - jd TDAE n i CI ( To obtain more specific information about the complexity of the optimization problem defined above, the reduction of DOF complexity due to the numbers of problem parameters is investiated for two case studies. 5 Case Studies Two case studies of process and control structure selection of chemical process superstructures presented in the followin sections. These cases a re:. Dual Continuous Stirred Tank Reactors (CSTR. Five Effect Evaporator 5. The Dual CSTR The objective of this case study is to select both process and controller structure to maximize the net profit of the two Continuous Stirred Tank Reactors (CSTR system ( [3]. The net profit is defined as the objective function as follows: 60

7 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : Φ N 0(Q f C fn Q f C fn 0.3(Q f Q f 0.0Cool 0.Cool 0.(Q f Q f (3 The dynamic variables are the product compositions C, C and temperatures T, T. The feasible operatin conditions are defined by the temperatures T, T, the amount of heat transfer between coolant and reactor Cool, Cool and the final product composition Cp. The coolant flow rates mc and mc are manipulated variables that may control either T or Cool and T or Cool, respectively. The problem is to decide the existence of the flow rates Qf, Qf, Q3 and Q4. At least one of Qf or Qf, and one of Q3 or Q4 should exist, which leads to series, parallel, or combination process confiuration. These decisions are facilitated by assinin four additional binary variables yp, yp, yp3, and yp4 to the respective flow rates. Furthermore, the effect of reactor volumes V and V to both process and controller structures is evaluated. Therefore, the desin variables are Qf, Qf, Q3, Q4,,, V, V. yc, yc, yp, yp, yp3, and yp4 as well as the initial conditions of Tss, Tss, Css, Css, Coolss, and Coolss. This superstructure is assessed in a closed-loop condition with a Proportional Interal (PI controller structure. The optimum parameters values are optimized further usin the scalin factors and. The binary assinments related to controller and process structure are iven in Table and Table respectively. Splitter Feed Q f Q f mc CSTR FC TT CoolT T Cool C, T Mixer Q m Q 4 Splitter mc CSTR Q 3 FC TT CoolT T Cool C, T C p Q m Mixer Fiure Dual CSTR problem 6

8 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : The superstructure model is as follows: dc Ec / RT V koe CV Qf(Cf C e y T di c ss T e dt V Ec / RT Dhk oe CV Qf(Tf T Cool e y Cool c ss e Cool di Uamccp Cool (Tc T U m cp a c m c m css K c e I K c e I Q 3 Qf Qf e yc Tss T e di Q m Q Q 3 f y Cool c ss e e Cool di (4 C m (Q T m f C Q Q (Q m f f C m f T Q Q f T dc Ec / RT V koe CV Qm(C m C f m c m css K K K c c int e e e I I I int dt V Ec / RT Dhkoe CV Qm(Tm T Cool Uamccp Cool (Tc T U m cp a c Q m Q m Q 4 (Q4C Q Cp Q m m C 6

9 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : Table Control parameters and loical assinments Control loop Binary Proportional Reset i Control pair Assinment ain Ki times I T - m c y c = Cool - m c. y c = T - m c y c = Cool - mc. yc = int T - mc yc = idx = controller parameter index Table Process structure and binary assinments Binary assinments Decision Binary assinments Decision yp = 0 Qf off yp = Qf on yp = 0 Qf off yp = Qf on y p3 = 0 Q 3 off y p3 = Q 3 on y p4 = 0 Q 4 off y p4 = Q 4 on The associated constraints for these desin variables are as follows: : T : Cool 30 : C : 0 : 0 : V 4.5 : V 4.5 : y f f p 0.05y 0.05y 0.05y 0.05y y p p p p3 p : T : Cool f f : : : V 6 : V 6 : y 350 f 3 4 p3 Q 0.8y 0.8y 0.8y 0.8y y 0 f p4 0.8 p p p3 p4 (5 All process dynamics are simulated over 00 seconds time horizon with -second intervals, and the outputs are Cool, Cool, mc, mc, and N; in addition to the state variables T, T, C, C, I, I, I, and I. The constraints 7-4 relates the binary decisions to the flow-rate existence. These relationships uarantee the amounts of flow-rate if yp=, and none otherwise. At least one of Qf or Qf and one of Q3 or Q4 should exist, and these are 63

10 Condenser Heater J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : represented by 3-4, which are evaluated in the inteer feasibility test. The combination of the constraints 3-4 and the binary variables ive rise to a 36 nodes enumeration tree at the dynamic MINLP problem. 5. The Five-Effect Evaporator The five-effect evaporator superstructure is shown in Fiure 3. The evaporator consists of one fallin film, three counter-current forced circulation and one super-concentrator units. The controlled variables are the liquor levels hp-5, the product density P5 and the product temperature TP5. The candidates of manipulated variables are the product flow rates QP-5, the live steam MS3-5, and the vaporization rate in the final stae MV5. The enery transfers involve the live steam MS3-5, the vapor MV-5, the condensates MC-5, and the recycle streams MHF-4. The disturbances are the fluctuations of the feed flow rate QF, the feed temperature TF and the coolin water temperature TCW, which have been sampled and held every 30 minutes. The interaction between these variables ives rise of 3 Differential Alebraic Equations for each tank that develop the process model. V 4 Live steam (S Coolin Water (CW V Flash Tank LC HD V3 V 34 HD 3 Flash LC Tank 3 DC S 3 HD 4 S S 4 5 HD Flash 5 LC Tank 4 V 5 Flash Tank 5 LC DC Hot Well (HW F V Flash Tank I LC HF P C P 4 I I 3 3 I 4 C HF P C 345 C 3 HF 3 HF 4 P 4 C 5 P 5 Fiure 3 Five effect evaporator The superstructure facilitates the distribution of live steam to the last three staes, the mixin of vapor from stae 3-4, and the alternative matchin between measurements and manipulated variables. The process structure is associated with the binary decision variables for the distribution of live steams, yp3-5. The variables determine the operation of three, four or five-effect modes to achieve the taret density. At least one of the staes 3 or 4 has to be operated; and if both are, the vapor from both staes is mixed. 64

11 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : Table 3 Control loops and parameters Decision Output In- put Kc (hr te Decision Out- put Input Kc (hr te - h P-4 Q P-4 5 always active y C4 P5 Q P PI control yc hp-4 QP-4 I control yc5 P4 MS34-30 y C h P5 Q P5 y C6 P5 M S5 00 yc hp5 MV5 0.5 yc7 P5 MV5-00 y C h P5 Q P5 5 P control y C8 T P5 M S5-0. yc3 hp5 MV5 0 yc9 TP5 MV The controller structure is represented by the binary decision variables yc-9, which activate the control-loops properties of a multi-loop Proportional Interal (PI feedback control scheme as summarized in 65

12 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : Table 3. The ains Kc-9 and the reset times -9 are pre-desined for stability. The binary variables are constrained such that each manipulated and controlled variable can only be used once within the control scheme, and only activated if the associated process structure is active. For instance, hp5-qp5 can only be activated if stae 5 is active. The parameters of active control loops can be tuned further with the scalin variables 0-9. The process optimization problem is associated with the maximization of the profit subject to the feasible process dynamics, as well as the loics of interconnection between the process and controllers. 6 Results and Discussions The CSTR problems parameters include 4 continuous desin variables, 6 binary desin variables, differential alebraic equations, constraints and loic propositions. Meanwhile, the Evaporator problems parameters include 40 continuous desin variables, 9 binary desin variables, 60 differential alebraic equations, constraints and loic propositions. The worst case durations of operation are measured as the maximum duration for both cases in a Pentium IV Personal Computer,.4 GHz, 5 MB RAM with 0.05% confidence. Based on these measurements, TNLP and TDAE are 0.5 seconds and 0. seconds respectively. Based on equation (, the complexity indices for these cases, in comparison to the duration of experimental optimization results are: Table 4 Comparison between computational index and experimental duration Case CI (in seconds Dual CSTR Evaporator Experimental duration (in seconds Compared to the empirical results, the theoretical worst-case computational indices are twofold of the experimental durations. These show that the MINLP problems are solved efficiently in comparison to the worst case complexity. In these cases, the loic propositions provide the most contribution to fathom the structural nodes of the modified Branch and Bound alorithm. At cases, the proportionality and the ap between CI and the experimental duration ives a promisin uses as the measure of computational efforts of process control desin based on mathematical prorammin approach and further study of assessin the efficiency of optimization alorithms. 7 Conclusion The worst case complexity of Dynamic Operability Framework has been formulated based on the size of the optimization problem specified by the number of continuous desin variables nz, binary variables ny, and DAE describin the system i, operational constraints j and loic proposition jd, and samples of n. The resultin complexity indices are relatively proportional to the experimental duration and the aps provide a promisin use as the quantification of the efficiency of the alorithm. 8 References [] Blondel, V. D., Tsitsiklis, J. N., Automatica, 999(35: [] Blondel, V. D., Tsitsiklis, J. N., Automatica, 000(36:

13 J.Oto.Ktrl.Inst (J.Auto.Ctrl.Inst Vol (, 00 ISSN : [3] Ekawati, E. and Bahri, P. A. Journal of Process Control, 003(3-8: [4] Ekawati E. and Bahri, P.A. CHEMECA 004, 004. [5] Floudas, A., Mixed Inteer nlinear Prorammin, 995. [6] Goyal, V., Ierapetritou, M. G., Computers and Chemical Enineerin, 004 [7] Nishida, N., Stephanopoulos, G. and Westerber, A. W., AIChE Journal, 98(7: [8] Papadimitriou, C. and Steilitz, K., Combinatorial Optimization; Alorithms and Complexity, 98 [9] Till, J., Enell, S., and Panek, S., Stursber, O, IFAC-Conference Analysis and Desin of Hybrid Systems, 003 [0] Wan, L., Journal of Systems Science and Complexity, 00(4- [] Wolff, E. A., Skoestad, S., Hovd, M. and Mathisen, K. W., IFAC workshop on Interactions between Process Desin and Process Control, 99 67