Format And Type Fonts CCHHEEMMIICCAALL EENNGGIINNEEEERRIINNGG TTRRAANNSSAACCTTIIOONNSS VOL. 39, 2014 A publication of The Italian Association of Chemical Engineering www.aidic.it/cet Guest Editors: Petar Sabev Varbanov, Jiří Jaromír Klemeš, Peng Yen Liew, Jun Yow Yong Copyright © 2014, AIDIC Servizi S.r.l., ISBN 978-88-95608-30-3; ISSN 2283-9216 DOI: 10.3303/CET1439025 Please cite this article as: Bakošová M., Oravec J., 2014, PDLF-based robust MPC of a heat exchanger network, Chemical Engineering Transactions, 39, 145-150 DOI:10.3303/CET1439025 145 PDLF-Based Robust MPC of a Heat Exchanger Network Monika Bakošová * , Juraj Oravec Slovak University of Technology in Bratislava, Faculty of Chemical and Food Technology, Institute of Information Engineering, Automation, and Mathematics, Radlinského 9, 812 37 Bratislava, Slovak Republic monika.bakosova@stuba.sk The paper is focused on the case study of the advanced control of the heat exchanger network (HEN). The HEN is used for cooling petroleum produced by distillation. The robust model predictive control (RMPC) strategy is implemented to find the optimal control actions taking into account the boundaries on the control inputs. RMPC approach is also able to design the controller managing the process uncertainties. The aim is to demonstrate that the HEN robust model predictive control can be improved and the energy efficiency can be optimized using the parameter-dependent Lyapunov functions (PDLF). The simulation results confirmed also the energy savings. 1. Introduction The heat exchangers (HEs) are often used in the chemical industry. Due to the fact that heat losses can rise up to 50 % (Čuček et al., 2013), there is the necessity to implement the advanced optimization-based control algorithms, e.g. model-based predictive control (MPC) (Bemporad and Morari, 1999). It was demonstrated in Pan et al. (2013) that the non-linear model of HEN can be found as the solution of an optimization problem. In the paper Walmsley et al. (2013) the optimization was utilized to determine optimal structure of a HEN. The HEN control using PID controllers was studied e.g. in Ipsakis et al. (2013). The uncertain HE control via H2 and H∞ approaches was studied in Vasičkaninová and Bakošová (2013). As the HENs belong to the key devices in the petroleum industry (González et. al, 2006) with high energy demands, it is important to find optimal control of the HEN. It was shown in Bakošová and Oravec (2013) that the robust MPC strategy decreased energy consumption during the HEN operation. The aim of this paper is to investigate further possibilities to increase the energy savings. In the presented case study the HEN was utilized to cool the petroleum. The hot petroleum was the product of distillation and the water was the cooling medium. Three robust MPC approaches were designed and the control performance of three counter-current shell-and-tube HEs in series was studied by simulations for two scenarios in each strategy, the worst and the best case scenarios. 2. Controlled heat exchangers Based on the previous research, the controlled process was adopted from Bakošová and Oravec (2013) and is briefly described next. The simple HEN is composed of three identical counter-current shell-and- tube HEs in series. The feed of the HEN to be cooled down is the petroleum as a product of distillation in a refinery. Petroleum flows in the inner tubes and the cooling water in shell of every heat exchanger. The tubes of the HEs are made from steel. The controlled variable is the temperature of the outlet stream of the petroleum from the 3rd HE. The control input is the volumetric flow rate of the inlet cold water into the 3rd HE. The objective is to decrease the outlet temperature of the petroleum to the reference value 45 °C and to minimise the energy demands measured by the total consumption of cold water. The technological parameters and control conditions are the same as in Bakošová and Oravec (2013) and are summarized in Table 1, where n is the number of HE’s tubes, l is the length of the HE, din,1 is the inner diameter of the tube, dout,1 is the outer diameter of the tube, din,2 is the inner diameter of the HE, Ah is the total heat transfer area, V is the volume, cp is the thermal capacity, ρ is the density, Tin is the inlet temperature, q is the volumetric flow rate. The subscripts 1 and 2 refer to the water and petroleum, respectively. The 146 Table 1: Technological parameters and reference values of HEs Parameter Unit Value Parameter Unit Value n 1 40 Tin,1 °C 20.0 l m 6 Tin,2 °C 180.0 din,1 m 19×10 -3 T1 (1),S °C 75.8 din,2 m 414×10 -3 T1 (2),S °C 48.0 dout,1 m 25×10 -3 T1 (3),S °C 30.8 Ah m 2 16.6 T2 (1),S °C 113.0 V1 m 3 91.2×10 -3 T2 (2),S °C 71.3 V2 m 3 716.5×10 -3 T2 (3),S °C 45.3 q2 S m 3 s -1 5.8×10 -3 T1 (1),0 °C 87.1 cp,1 J kg -1 K -1 4.186×10 3 T1 (2),0 °C 55.7 cp,2 J kg -1 K -1 2.140×10 3 T1 (3),0 °C 34.4 ρ1 kg m -3 980.0 T2 (1),0 °C 118.4 ρ2 kg m -3 810.0±16.2 T2 (2),0 °C 76.8 U J s -1 m -2 K -1 482.3±9.7 T2 (3),0 °C 48.7 superscripts (1) – (3) denote individual HEs and the superscripts S and 0 denote the reference value and the initial value, respectively. Furthermore, two interval parametric uncertainties are considered – the heat-transfer coefficient U changes as the flow rate of the cooling medium changes, and the density of the petroleum ρ2 depends on the temperature in the HEs (Table 1). 3. Robust MPC For the robust MPC design, the mathematical model of the heat exchangers was derived using the enthalpy balances. The linearized time-invariant state-space model in the discrete-time domain is given by                kCxky xxkuBkxAkx vv   0 0,1 (1) where k represents the discrete time. The used sampling period was ts = 25 s. Further, x(k) is the vector of states represented by the temperatures T1 (1)–(3) and T2 (1)–(3) (Table 1), u(k) is the control input represented by the volumetric flow rate of the cooling medium q1, y(k) is the vector of the system outputs. The matrices A (v) , B (v) , C have appropriate dimensions. The model in Eq(1) is an uncertain system with interval polytopic uncertainty. For the uncertain model of the HEN one can obtain four vertices computed as the combination of boundary values of uncertain parameters. Hence, the matrices A (v) , B (v) , v = 1,…,4, describe the vertex systems of the uncertain system Eq(1). The 5 th considered system is the nominal system calculated for the mean values of the uncertain parameters (Table 1). Then the robust static state-feedback control problem in the discrete-time domain can be formulated as follows: find the state-feedback control law    kxFku k (2) for the system described by Eq(1). The matrix Fk in Eq(2) represents the static state-feedback robust controller for the k-th control step. The quality of the control performance can be described using the quadratic cost function            k 0 ux n k TT kuWkukxWkxJ (3) where nk is the total number of control steps. For the design purposes the infinity control horizon is assumed, and Wx, Wu are the real square symmetric positive-definite weight matrices of the states x(k) and the system inputs u(k). The aim is to design the controller Fk that ensures robust stability of all considered vertex systems and minimizes the cost function J in Eq(3). The control performance can be improved by taking into account the symmetric constraints on the system outputs y(k) and inputs u(k) in the form     2 max 22 max 2 , utuyty  (4) 147 Following conditions hold for the symmetric positively defined Lyapunov matrix Pk and the feedback controller Fk 11 , ,   kkkkkkkkk QYFQFYQP  (5) where k is the auxiliary optimization parameter, Qk is the symmetric positively defined matrix, and Yk represents the auxiliary matrix enabling the evaluation of the robust feedback controller Fk (Cuzzola et al., 2002). Several strategies were used to investigate the robust MPC of the HEN. RMPC1 denotes the control strategy described in the paper (Kothare et al., 1996). The algorithm for the controller design by the RMPC1 was presented in the paper Bakošová and Oravec (2013). The approach denoted RMPC2 was introduced in Cuzzola et al. (2002), and refined in Mao (2003). The robust stabilization problem can be solved as the robust MPC convex optimization problem based on the LMIs as follows (Cuzzola et al., 2002) kYX kkk  ,,min (6) subject to            0 *** 0** 00* ,0 * 1 u T x TTT T                       I I X WYWQYBQAXQQ X x k k v k kkk v k vv kkk v k k   (7) where v = 1,…, nv. The symbol * denotes a symmetric structure of the matrix, and I, 0 are the identity and zero matrices of appropriate dimensions. Xk (v) are the symmetric positively defined matrices. The symmetric constraints on control inputs and outputs in the form of Eq(4) can be added to the optimisation problem Eq(6) – Eq(7) in the following LMI form          0 * ,0 * 2 max TTT T 2 max                  Iy CYBQAXQQ XQQ YIu k v k vv kkk v kkk k (8) where v = 1,…, nv. The algorithm for the RMPC2 can be formulated in following eight steps (Cuzzola et al., 2002). Step 1: Set parameter k = 0. Step 2: Set number of control steps N, initial conditions of states x(0), values of the symmetric constraints on control input umax and output ymax. Step 3: Set parameter k = k + 1. Step 4: Set the values of states x(k). Step 5: Solve optimization problem described by Eq(6) – Eq(8) to evaluate Qk, Xk (v) and Yk. Step 6: Design the matrix Fk of the feedback controller using Eq(5). Step 7: Calculate the control input u(k) using the control law Eq(2). Step 8: If the parameter k < N then go to the Step 3 else Stop. The third considered strategy, denoted as RMPC3, is the robust MPC approach presented in Cao et al. (2005). In this approach, the single Lyapunov function is considered and the maintenance of input constraints is modified. This procedure reduces the conservativeness of control input evaluation and all at once ensures the robust stability. On the other hand, the additional saturation of computed values of control inputs is necessary. In the optimisation problem in Eq(6) – Eq(8) the LMIs presented in Eq(7) are replaced using 148         0 *** 0** 00* 0 * 1 u T x TT T                        I I X WUEYEWXUEYEBXAX X x k k k kjkjkkjkj v k v k k k   (9) Instead of LMIs in Eq(8) the constraints are handled by following LMIs       0 * ,0 * 2 max TT2 max                  Iy CUEYEBQAX X UIu kjkj v k v k k k (10) for v = 1,…, nv, j = 1,…, nu. The matrices Ej are the diagonal matrices with all variations of 1 and 0 on principal diagonal and zeroes elsewhere. Then the Ej – are the complement matrices obtained as Ej – = I – Ej. The idea of this extension is to take into account all variations of constrained and unconstrained control inputs. Then the algorithm for the RMPC3 can be formulated in following eight steps (Cao et al., 2005). Step 1: Set parameter k = 0. Step 2: Set number of control steps N, initial conditions of states x(0), values of the symmetric constraints on control input umax and output ymax. Step 3: Set parameter k = k + 1. Step 4: Set the values of states x(k). Step 5: Solve optimization problem described by Eq(6), Eq(9), Eq(10) to evaluate Xk, Yk and Uk. Step 6: Design the matrix Fk of the feedback controller using Eq(5). Step 7: Calculate the control input u(k) using the control law Eq(2). Step 8: If the parameter k < nk then go to the Step 3 else Stop. 4. Results and discussion The designed robust MPC strategies RMPC1 – RMPC3 were investigated via simulations of control of the non-linear model of HEN using 2.8 GHz CPU and 4 GB RAM in the MATLAB-Simulink environment using the toolbox YALMIP (Löfberg, 2004) and the solver SeDuMi (Sturm, 1999). The robust state-feedback controllers were designed using the weight matrices Wx, Wu in the cost function described by Eq(3) in the form diag(Wx)=[100,100,100,100,100,100] T , diag(Wu)=[100], where diag denotes the diagonal matrix with the given elements on the principal diagonal and zero elsewhere. These weight matrices were considered in all RMPCi algorithms to make the obtained results fully comparable. The RMPCi strategies were analyzed by evaluating the offset of the petroleum temperature ΔT2 (3) , and consumption of the cooling medium VC. The aim of control was to cool down the petroleum temperature from 118.4 °C to 45.3 °C during the control running 2,100 s. The Figures 1, 2 show just first 1,500 s to show the dynamics clearly. Figure 1 presents the control performance of the outlet petroleum temperature assured by RMPC1 (dotted line), RMPC2 (solid line), RMPC3 (dashed line) strategies in the worst-case (●) and the best-case (□) scenarios. The reference is denoted by the dashed-dotted line. The worst-case scenario represents the vertex system with the maximal value of analyzed criterion Eq(3). The best-case scenario is the vertex system with the minimal value of analyzed criterion Eq(3). Figure 2 shows the associated control inputs. The computed values of the cost function are presented in Table 2. As can be seen, the PDLF-based RMPC2 approach assured better value of temperature in the best-case scenario in comparison with the original RMPC1 strategy. But the worst-case scenario was not very efficient. Although RMPC3 results were not the best, this strategy ensured the tightest range of the temperature offset. Contrary to the best-case behaviour, the worst-case temperature trajectories indicate slight overshoot at the beginning. The fastest convergence to the reference temperature was assured by the RMPC3 procedure. The PDLF-based 149 Table 2: Results of the RMPC approaches of HEN control Figure 1: Control performance of the outlet petroleum temperature assured by RMPC1 (dotted), RMPC2 (solid), RMPC3 (dashed) strategies in the worst-case (●) and the best-case (□) scenarios. Figure 2: Control-input trajectories of the volumetric flow-rate generated by RMPC1 (dotted), RMPC2 (solid), RMPC3 (dashed) strategies in the worst-case (●) and the best-case (□) scenarios. approach led to the minimal consumption of cooling medium in the worst-case scenario and to the satisfactory consumption in the best-case scenario (Table 2). Hence, the RMPC2 is the most suitable strategy for minimisation of the energy demands and the RMPC3 is the most suitable strategy for the precise temperature control. 5. Conclusion The paper demonstrates on the simulation case-study of the non-linear HEN control the possibility to implement various robust MPC strategies. The obtained results were analysed according to the control trajectories and consumption of cooling medium. In comparison with the other investigated procedures, the PDLF-based approach ensured the highest energy savings of the worst-case scenario, meanwhile providing the satisfying control performance. The tightest range of the worst-case and best-case scenarios was ensured by the robust MPC with improved handling of control inputs. Acknowledgement The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grants 1/0973/12 and the Slovak Research and Development Agency APVV 0551-11. J. Oravec was also supported by the internal STU grant no. 1323. References Bakošová, M., Oravec, J., 2013, Robust Model Predictive Control of Heat Exchanger Network. Chemical Engineering Transactions 35, 241-246, DOI:10.3303/CET1335040. Bakošová, M. and Oravec, J., 2012, Robust Model Predictive Control of Heat Exchangers. 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