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Robust model predictive control for heat exchanger network
Applied Thermal Engineering 73 (2014) 922e928
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Robust model predictive control for heat exchanger network *, Juraj Oravec Monika Bakosova Slovak University of Technology in Bratislava, Faculty of Chemical and Food Technology, Institute of Information Engineering, Automation and Mathematics, Radlinsk eho 9, 812 37 Bratislava, Slovak Republic
h i g h l i g h t s The robust and the optimal controllers control the heat exchanger network. The robust model predictive controller is a solution of linear matrix inequalities. Robust control assures smaller offsets and lower coolant consumption.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 January 2014 Accepted 8 August 2014 Available online 28 August 2014
Optimal operation of heat exchangers represents a challenging task from the control viewpoint, due to the presence of system nonlinearities, varying process parameters, internal and external disturbances and measurement noise. Various robust control strategies were developed to overcome all these problems. The robust model predictive control (RMPC) represents one of suitable approaches. It enables to design effective control algorithms for optimization of the control performance subject to the process uncertainties and the input and output constraints. The possibility to implement the RMPC for control of a heat exchanger network is investigated in this paper, where three counter-current heat exchangers with uncertain parameters connected in series represent the controlled process. The efficiency of the advanced RMPC algorithm was verified by simulation experiments realized in the MATLAB/Simulink environment. The results confirmed that using the RMPC for the controlled process modelled as a system with uncertain parameters led to less consumption of cooling medium compared with the consumption achieved by using the optimal linear quadratic (LQ) control. © 2014 Published by Elsevier Ltd.
Keywords: Heat exchanger Uncertain system Robust model-based predictive control Linear matrix inequalities Energy saving
1. Introduction Heat exchangers (HEs) belong to the standard equipment in the chemical and process industries and the heat exchanger networks (HENs) are important processes in the petrochemical industry. As nearly 80% of the total energy consumption is related to the heat transfer, it is necessary to optimize the heat exchanger utilization, and to retrofit the heat exchanger networks [1]. Optimization and optimal control are of great importance for reducing energy consumption and for eliminating energy losses. The energy supply in most industrial devices was recognized as the second highest operating cost after primary feedstock costs. Moreover, since the energy prices tend to steadily increase, it is necessary to improve the efficiency of energy utilization and to reduce the energy
* Corresponding author. ), juraj.oravec@stuba. E-mail addresses: [email protected] (M. Bakosova sk (J. Oravec). http://dx.doi.org/10.1016/j.applthermaleng.2014.08.023 1359-4311/© 2014 Published by Elsevier Ltd.
consumption. One way is to use the advanced control strategies and to optimize the control performance [2]. Various papers deal with advanced control of thermal processes. The outlet temperature control of a counter-current tubular heat exchanger in heater configuration with the predictive functional control is presented in Ref. [3]. This approach is based on an internal adaptive model that corresponds to the response of the flow rate. Model predictive control (MPC) is one of control strategies satisfying demands on energy savings. MPC design is based on the solution of an optimization problem [4]. The quadratic cost function that penalizes the low quality of control performance is minimized over the control horizon in each control step to determine the optimal sequence of control inputs. Explicit MPC design based on a piecewise affine model of a boiler-turbine plant is presented in Ref. [5]. A nonlinear MPC configuration for hyperbolic distributed thermal systems is presented in Ref. [6]. This strategy also requires the development of a neural network function. Neural network predictive control (NNPC) is used for control of a co-current tubular heat exchanger in Ref. [7] and the presented results show that the
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
application of the NNPC can lead to significant energy savings in comparison with the classical PID control. The non-optimized and the optimized fuzzy controls of the heat pump are compared in Ref. [8]. Experimental case-study in Ref. [9] validates the balance-based adaptive control. An on-line adaptive optimal control approach is presented in Ref. [10]. Uncertainty in the controlled process, disturbances, measurement noise and the constraints on the control inputs and the controlled outputs play an important role in finding the proper solution of the control problem. The advantages of the RMPC originate from its flexibility to take into account the system constraints. Furthermore, the repetitive redesign of the controller in each control step can minimize the influence of neglected or unpredicted negative effects and improve the final control performance [11]. It was shown in Ref. [12] that the RMPC strategy is able to provide energy savings in control of a chemical reactor. The RMPC strategy can decrease energy consumption in the tubular and the jacketed HEs, see e.g. Ref. [13]. In Ref. [14] the RMPC was investigated using the real laboratory device AMIRA DST200. In Ref. [15] the RMPC of the HEN was designed to increase the energy savings. In Refs. [12,13] and [15] the results obtained by the RMPC
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2. Controlled heat exchangers Three counter-current shell-and-tube heat exchangers (HEs) connected in series form the controlled heat exchanger network (HEN) (Fig. 1), in which petroleum as a product of the distillation in a refinery is cooled. Petroleum flows through the inner tubes and the cooling water flows over the outside of the tubes inside the shell of the heat exchanger. The tubes of the HEs are made from steel. The objective is to cool the outlet stream of the petroleum from the 3rd HE to the reference value 45.3 C and to minimize the energy utilization measured by the total consumption of coolant. The manipulated input is the volumetric flow rate of the inlet stream of the cold water to the 3rd HE. The mathematical model of the HEN was derived using the heat balances under some simplifications [17]: the thermal capacities of the metal walls are neglected; the HEs are well insulated, and heat loss to the surroundings and mechanical work effects are negligible; the technological parameters are either constant or vary in some intervals. The heat balances for the HEN lead to six first-order differential equations in the form
ðjÞ dT1 ðtÞ A U ðjÞ ðjÞ ðjþ1Þ ðjþ1Þ ðj1Þ ðjÞ T2 ðtÞ T1 ¼ q1 ðtÞr1 cp;1 T1 ðtÞ þ q1 ðtÞr1 cp;1 T1 ðtÞ þ h ðtÞ þ T2 ðtÞ T1 ðtÞ 2 dt ðjÞ dT ðtÞ A U ðjÞ ðj1Þ ðjÞ ðjþ1Þ ðj1Þ ðjÞ T2 ðtÞ T1 ðtÞ q2 ðtÞr2 cp;2 T2 ðtÞ h ðtÞ þ T2 ðtÞ T1 ðtÞ ¼ q2 ðtÞr2 cp;2 T2 V2 r2 cp;2 2 2 dt
V1 r1 cp;1
ðjÞ
ðjÞ;0
T1 ð0Þ ¼ T1
ðjÞ
(1)
ðjÞ;0
; T2 ð0Þ ¼ T2
were compared with the results ensured by the LQ optimal controller. The LQ optimal control is well-known and widely-used control approach [4], and serves as the reference strategy. The aim of this paper is to present implementation of the RMPC for the HEN control. According to the authors' knowledge the case-study of RMPC control of HEN with uncertainty was not published yet by the other authors. The designed RMPC was based on the formulation of the optimization problem with constraints in the form of linear matrix inequalities (LMI) and this convex optimization problem was solved using the semi-definite programming (SDP) [16]. Therefore, the computational burden was quite high. Advantage of the designed RMPC was investigated using the theoretical case study of the HEN control. Controlled process was the complex uncertain and the non-linear model of three counter-current shelland-tube heat exchangers in series. Quality of the HEN control was assessed by comparing the consumption of coolant. Presented results were obtained by simulations, and are valid in the range of assumed conditions. They can be considered a basis for the next step, which is a practical implementation of the RMPC on a real heat exchanger.
where the superscripts j ¼ 1, 2, 3 represent the 1st, 2nd and the 3rd heat exchanger, respectively. The subscripts 1 and 2 indicate the water and petroleum, respectively. In (1), V is the volume, r is the density, cp is the specific heat capacity, t is the time, T(t) is the timevarying temperature, q(t) is the time-varying volumetric flow rate, Ah is the heat transfer area and U is the overall heat transfer coefficient. The initial conditions T1(j),0 and T2(j),0 in (1) are given in Table 1 for two control scenarios. These scenarios e Case 1 and Case 2 differing in initial conditions were investigated to confirm the efficiency of the designed RMPC approach. The values of technological parameters and the steady-state values of temperatures are summarized in Table 2. Here n is the number of the 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 shell, T1,in ¼ T1(4) is the temperature of the inlet stream of water and T2,in ¼ T2(0) is the temperature of the inlet stream of petroleum. The superscript S denotes the steady-state value, and the steady-state temperatures T1(j),S, T2(j),S, j ¼ 1, 2, 3,
Table 1 Initial conditions in HEN for two control cases. Case 1
Fig. 1. Scheme of the counter-current shell-and-tube heat exchanger network; 1 is the cooling water, 2 is the hot petroleum, and (1)e(3) are the heat exchangers.
Case 2
Variable
Unit
Value
T1(1),0 T1(2),0 T1(3),0 T2(1),0 T2(2),0 T2(3),0
C
87.1
C
55.7
C
34.4
C
118.4
C
76.8
C
48.7
Variable
Unit
T1(1),0 T1(2),0 T1(3),0 T2(1),0 T2(2),0 T2(3),0
Value
C
87.6
C
55.7
C
34.3
C
117.3
C
75.4
C
47.5
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Table 2 Technological parameters and steady-state values of variables in HEs. Variable
Unit
Value
n l din,1 dout,1 din,2 Ah V1 V2 cp,1 cp,2
1 m m m m m2 m3 m3 J kg1 K1 J kg1 K1
40 6 19 103 25 103 414 103 16.6 91.2 103 716.5 103 4.186 103 2.140 103
q1S
m3 s1
5.93 103
q2S
m3 s1
5.80 103
C
180.0
T2(1),S
C
113.0
T2(2),S
C
71.3
T2(3),S
C
45.3
C
20.0
C C C
30.8 48.0 75.8
T2,in ¼
T1,in ¼
T1(0)
T1(4)
T1(3),S T1(2),S T1(1),S
The linear discrete-time state-space model has the form
DTðk þ 1Þ ¼ Av DTðkÞ þ Bv Dq2 ðkÞ; DTð0Þ ¼ DT0 DTðkÞ ¼ CDTðkÞ
(2)
where the subscript v ¼ 0 represents the nominal system, the subscripts v ¼ 1, …,4 indicate the vertex systems, and k is the discrete time. The vector of the control inputs Dq2(k) and the vector of the controlled outputs DT(k) are defined as the differences between the actual values, i.e. q2(k), T1(j)(k), T2(j)(k), and the steadystate values, i.e. q2S, T1(j),S, T2(j),S, j ¼ 1, 2, 3, as it is shown in (3)
3 2 ð1Þ;S 3 ð1 Þ T1 T1 ðkÞ 7 6 7 6 ð1 Þ 6 T2 ðkÞ 7 6 T2ð1Þ;S 7 7 6 7 6 6 T ð2Þ ðkÞ 7 6 T ð2Þ;S 7 7 6 1 7 6 DT ðkÞ ¼ 6 1ð2Þ 7 6 ð2Þ;S 7; 7 6 T ðkÞ 7 6 T 7 6 2 7 6 2 7 6 ð3Þ;S 7 6 ð3 Þ 5 4 T1 ðkÞ 5 4 T1 2
ð3 Þ
T2 ðkÞ
h i Dq2 ðkÞ ¼ ½q2 ðkÞ qS2
(3)
ð3Þ;S
T2
The matrices Av, Bv, C of the uncertain system (2) have the form were computed for the volumetric flow-rates of water q1s ¼ 5.93 103 m3 s1 and of petroleum q2S ¼ 5.80 103 m3 s1 from (1) with zero derivatives. These steady state temperatures represent the reference values for control of the HEs. Further, two uncertain parameters are considered in the HEs. The heat-transfer coefficient changes as the flow rate of the cooling medium changes, and the density of the petroleum depends on the temperature in the HEs [17]. The values of these parameters are given in Table 3, where U is the heat transfer coefficient and r2 is the density of petroleum. The well-known approach of parametric uncertainties handling was used to describe the HEN in the form of a polytopic uncertain system [11]. Therefore the set of four vertex systems was generated for all variations of boundary values of two uncertain parameters (Table 3). Each vertex system was described by 6 ordinary differential equations (1). The control performance of the controlled process was investigated using these four limitbehaviour models. The nominal system was also created using the mean values of the uncertain parameters. This model served as the reference system. The non-linear state-space model of the controlled process (1) was linearized for the robust controller design using the 1st order Taylor expansion of nonlinear terms, and the linear state-space model of the HEN was obtained for the nominal system and each vertex system in the form of six ordinary linear differential equations. As the RMPC is a discrete-time control strategy, the linear continuous-time models were transformed into the discrete-time domain using the MATLAB's command c2dm [18] and the sampling time ts ¼ 25 s. The value of sampling time does not directly influence the RMPC design. It has to be chosen so that obtained discrete-time model matches the behaviour of the nonlinear model with sufficient accuracy.
Table 3 Uncertain parameters of HEs. Variable
Unit
Minimum value
Maximum value
U
J s1 m2 K1 kg m3
472.8 793.8
491.3 826.2
r2
2
a1;1 6a 6 2;1 6 6 0 Av ¼ 6 6 0 6 6 4 0
a1;2 a1;;3
0
0
a2;2 a2;1 0 0 a1;2 a1;1 a1;2 a1;3 a4;2 a2;1 a2;2 a2;1 0 0 a1;2 a1;1
2 3 b1 7 607 0 7 6 7 7 6 7 6b 7 0 7 7; Bv ¼ 6 2 7; 607 0 7 7 6 7 7 6 7 4 b3 5 a1;2 5 0
3
0 0 0 a4;2 a2;1 a2;2 3 1 0 0 0 0 0 60 1 0 0 0 07 7 6 7 6 60 0 1 0 0 07 7 Cv ¼ 6 60 0 0 1 0 07 7 6 7 6 40 0 0 0 1 05
0
2
(4)
0 0 0 0 0 1 where v ¼ 0,1, …,4. The matrix elements of Av, Bv in the continuoustime representation of (4) before the transformation into the discrete-time domain are expressed in Table 4. The specific matrices describing the discrete-time nominal model of the heat exchanger (v ¼ 0) were calculated from the continuous-time representation (Table 4) for the nominal values of the uncertain parameters and the sampling time ts ¼ 25 s, and are as follows Table 4 Matrix elements of the linear continuous-time statespace model of the HEN. Element
Value
a1,1
2qS1 r1 cp;1 Ah U 2V1 r1 cp;1
a1,2
Ah U 2V1 r1 cp;1
a1,3
2qS1 r1 cp;1 Ah U 2V1 r1 cp;1
a2,1
Ah U 2V2 r2 cp;2
a2,2
2qS2 r2 cp;2 Ah U 2V2 r2 cp;2
a4,2
2qS2 r2 cp;2 Ah U 2V2 r2 cp;2
b1 b2 b3
ð2Þ;S
T1
ð1Þ;S
T1 V1
ð3Þ;S ð2Þ;S T1 T1
V1 ð3Þ;S T1 V1
ð4Þ;S
T1
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
2 6 6 6 6 Av ¼ 6 6 6 6 4
6:4 0:3 0 0 0
1:0
4:2
1:1 0:3 1:0 6:4 0:5 0
0:3 0
0
0
0 1:0
0 4:2
1:1 0:3 1:0 6:4
0 0 0 3 2 3:4 6 0 7 7 6 7 6 6 2:3 7 2 7 Bv ¼ 6 6 0 7 10 7 6 7 6 4 1:6 5
0:5
0:3
0
3
be improved by taking into account the symmetric constraints on the system outputs DT(k) and inputs Dq2(k) in the form
7 7 7 7 7 102 ; 0 7 7 7 1:1 5 0 0
1:1
2 2 2 2 DT ðkÞ DTmax ; Dq2 ðkÞ Dq2;max 2
(5)
0 The matrices Av, Bv of the four vertex systems (v ¼ 1, …, 4) have the same dimensions. They were calculated from the continuoustime representation (Table 4) for all combinations of the boundary values of uncertain parameters and the sampling time ts ¼ 25 s. 3. Robust MPC The robust state-feedback control problem in the discrete-time domain can be formulated as follows: find a state-feedback control law (6)
Dq2 ðkÞ ¼ Fk DTðkÞ
(6)
for the system described by (2) so that the uncertain closed-loop system will be asymptotically stable [11], and will ensure satisfying control performance. The control law determines the relation between control inputs and outputs. The matrix Fk in (3) represents the gain matrix of the robust state-feedback controller in the k-th control step. To design the RMPC the approach described in Ref. [11] was adopted. The quality of the control performance can be expressed using the quadratic cost function J
J¼
N X
DTðkÞT WT DTðkÞ þ Dq2 ðkÞT Wq Dq2 ðkÞ
925
(7)
k¼0
where N is the number of control steps. For the design purposes the infinity control horizon is assumed (N / ∞), and WT, Wq are the real square symmetric positive-definite weight matrices of states DT(k) and the system inputs Dq2(k), respectively. The first term of the cost function J penalizes the control inaccuracy as the difference between the current and required values of outputs. Similarly, the second term penalizes the utilization of control action. The aim of control is to design such a controller Fk that minimizes the value of the objective J while providing the satisfaction of the robust stability condition for all vertex systems. The control performance can
2
2
(8)
2
where the values of the input constraints were set to prevent computing unreal negative values of volumetric flow-rates, i.e. Dq2,max ¼ 5.93 10 3 m3 s1. The output constraints were chosen such that the temperatures remained in the ±10 C surroundings of the steady-state values. The robust stability condition is raised from the Lyapunov stability theorem, see e.g. Ref. [11]. Then the gain matrix of the state-feedback control law Fk is designed as the solution of the optimization problem of SDP, where the constraints are in the form of LMIs [11].
4. Results and discussion The RMPC approach was designed to control three countercurrent shell-and-tube HEs in series. Two control scenarios were investigated to confirm the efficiency of the designed RMPC approach. These two scenarios e Case 1 and Case 2 differed in initial conditions (Table 1). The RMPC of HEs was analysed by simulation experiments in MATLAB/Simulink environment using 2.8 GHz CPU and 4 GB RAM. The optimization problem was formulated via the YALMIP toolbox [19] and solved by the SeDuMi solver [16]. We developed the MUP toolbox [20] to design RMPC. The toolboxes are freely available to the users. The obtained control performances were compared with the control performances assured by the discrete-time linearquadratic (LQ) optimal controller computed in the form FLQ ¼ [110.4, 31.2, 1.7, 21.1, 1.0, 8.5] 10 6 [4]. The gain matrix FLQ of the LQ controller was designed using the weight matrices WT, Wq of the cost function (7) in the form
2 6 6 6 WT ¼ 6 6 6 4
100 0 0 0 0 0
0 100 0 0 0 0
0 0 100 0 0 0
0 0 0 100 0 0
0 0 0 0 100 0
3 0 0 7 7 0 7 7; Wq ¼ ½100 0 7 7 0 5 100
(9)
The weight matrices (9) were also utilized in the RMPC algorithm to obtain fully comparable results. Both strategies were compared by evaluating the total consumption of coolant VC, which was needed for cooling the petroleum to the temperature 45.3 C in Case 1 and in Case 2 (Table 1). The simulations of control were stopped after 1200 s, i.e. after N ¼ 48 control steps. The simulation results were obtained using the more precise non-linear model of HEs (1). Table 5 shows obtained numerical results of the HEN control in both considered cases. Here, system represents the
Table 5 Results obtained by the RMPC and the discrete-time LQ optimal control. System
(3) DoffT2,RMPC ( C)
VC,RMPC (m 3)
(3) DoffT2,LQ ( C)
VC,LQ (m 3)
DrelT2(3) (%)
Case 1 Nominal 1st vertex 2nd vertex 3rd vertex 4th vertex
DrelVC (%)
0.01 0.48 0.51 0.48 0.50
7.200 7.410 6.702 7.411 6.706
0.00 0.97 0.93 0.98 0.93
7.313 7.247 7.377 7.247 7.377
e 50.5 45.2 51.0 46.2
1.5 2.2 9.2 2.3 9.1
Case 2 Nominal 1st vertex 2nd vertex 3rd vertex 4th vertex
0.01 0.48 0.55 0.48 0.54
7.166 7.385 6.640 7.386 6.644
0.00 0.97 0.94 0.98 0.93
7.316 7.250 7.380 7.250 7.380
e 50.5 41.5 51.0 41.9
2.1 1.9 10.0 1.9 10.0
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Fig. 2. Control performance of the petroleum temperature at the outlet from the 1st heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 1.
controlled non-linear model of HEs, which is either the nominal or (3) (3) the vertex system. Variables DoffT2,RMPC and DoffT2,LQ are the steadystate offsets of the outlet petroleum temperature assured using the (3) RMPC and the LQ control, respectively. DoffT2,LQ was zero when the nominal system was controlled by the LQ optimal controller. But when the vertex systems were controlled, the RMPC ensured lower absolute values of the offsets than the LQ optimal controller, i.e. the outlet petroleum temperature was closer to the reference when the RMPC was used. The LQ optimal controller was able to assure better results only for the ideal nominal system. The RMPC significantly reduced the steady-state offsets in the real situations represented by control of the vertex systems. The total consumption of VC is also presented in Table 5 for the nominal system and four vertex systems. In Table 5, VC,RMPC, VC,LQ are the total volumes of the coolant consumed during the robust control and the LQ optimal control, respectively. The 2nd and the 4th vertex systems had significantly lower consumption of the coolant during the robust control than during the LQ control. Robust control ensured smaller consumption of cold water even for the nominal system. But the 1st and the 3rd vertex systems had slightly higher consumption of the coolant during the robust control then during the LQ control. The RMPC and the LQ control were compared also using the following quality criteria:
Fig. 3. Control performance of the petroleum temperature at the outlet from the 2nd heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 1.
Fig. 4. Control performance of the petroleum temperature at the outlet from the 3rd heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 1.
ð3Þ
Drel T2
ð3Þ ð3Þ Doff T2;RMPC Doff T2;LQ 100% ¼ ð3Þ Doff T2;LQ
Drel VC ¼
VC;RMPC VC;LQ 100% VC;LQ
(10)
(11)
The negative values of DrelT2(3) or DrelVC mean that the RMPC improved the control results in comparison with the LQ control, and the steady-state offset and the coolant consumption were smaller during the robust control. The higher the absolute values of DrelT2(3) or DrelVC were calculated, the higher was the improvement. According to DrelT2(3), the RMPC improved the set-point tracking from 45.2% up to 51.0% in the Case 1 and from 41.5% up to 51% in the Case 2. Better results obtained using the LQ control for the nominal system follow from the fact that the LQ controller was designed as an optimal controller for the nominal system. According to DrelVc, the RMPC decreased the coolant consumption by 9e10% in the 2nd and the 4th vertex system and by 1.5e2% even in the nominal system control. But the RMPC of the 1st and the 3rd vertex system increased the coolant consumption by approximately 2%. The increase is not as significant as the decrease of coolant consumption for the other two vertex systems. The control responses of the HEs in Case 1 are shown in Figs. 2e4 and the associated control inputs in Fig. 5. The control responses of the HEs in Case 2 are shown in
Fig. 5. Control inputs generated by the RMPC (vertex systems e solid, nominal system e dashed) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 1.
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
Fig. 6. Control performance of the petroleum temperature at the outlet from the 1st heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 2.
Fig. 7. Control performance of the petroleum temperature at the outlet from the 2nd heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 2.
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Fig. 9. Control inputs generated by the RMPC (vertex systems e solid, nominal system e dashed) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 2.
Figs. 6e8 and the associated control inputs in Fig 9. From the robust control viewpoint, the key analysis is based on the limit system behaviour. Therefore the crucial information in these figures is the range of the temperature that is bounded by the four vertexcontrol-trajectories. The real temperature is located inside this area. The input disturbance expressed by the difference of the initial conditions from the references is more significant in Case 1; it is seen in Table 5 that the consumption of coolant increased, while the accuracy of the temperature control was the same. The control responses obtained using the RMPC are in most cases slower than those obtained using the LQ optimal control. The reason is that the RMPC design takes into account the input constraints and generated control inputs are less aggressive than control inputs generated by the LQ controller. As the LQ optimal controller design does not include process uncertainties, this strategy does not assure the asymptotic stability of the uncertain controlled system. The disadvantage of the RMPC implementation is increase of the computational effort due to the necessity to solve the complex optimization problem.
5. Conclusions
Fig. 8. Control performance of the petroleum temperature at the outlet from the 3rd heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 2.
The possibility to improve the control performance and to increase energy savings by implementation of the RMPC is presented in the paper via control of the HEN. This is the main contribution of this paper as the implementation of the RMPC for control of heat exchangers has not been studied by many authors up till now. The robust control of the HEN was compared with the optimal LQ optimal control by simulation. Obtained results confirmed the effectiveness of the RMPC approach due to the smaller offsets and the decreased consumption of cold water used for petroleum cooling in the most of studied systems. The discrete-time LQ optimal controller ensured better quality only for the nominal system that represents the ideal system. In the presence of uncertainty and boundaries on control inputs and controlled outputs, the robust feedback control approach reduced the steadyestate offsets of the petroleum temperature in the outlet streams from the HEs by approximately 40e50% compared to the LQ optimal control. Consumption of the cooling water was reduced by up to 10% in about 20 min and is larger for greater disturbances (Case 1). Therefore it can be stated that the RMPC strategy used in practical implementations can lead to energy savings in comparison with the optimal LQ control. On the other hand, the overall computational
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
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effort increased. The further research will be focused on improvement of the RMPC algorithm so that the offset free control responses will be achieved. Acknowledgements
Superscripts S steady-state (1) 1st heat exchanger (2) 2nd heat exchanger (3) 3rd heat exchanger
The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grant 1/ 0973/12 and the contribution of the Slovak Research and Development Agency under the grant APVV 0551-11. J. Oravec was also supported by an internal STU grant no. 1356.
Abbreviations LQ linear-quadratic MPC model predictive control RMPC robust model predictive control
Nomenclature
References
Symbols Ah a A b B C cp d F J k N q Dq2 t ts T DT (3) DoffT2,LQ
heat transfer area (m2) element of the state matrix state matrix element of the input matrix input matrix output matrix specific heat capacity (kJ kg1 K1) diameter (m) gain matrix of the controller quadratic cost function control step control horizon volumetric flow rate (m3 s1) control input (m3 s1) time (s) sampling time (s) temperature ( C) controlled outputs ( C) steady-state offset in the LQ control ( C)
(3) DoffT2,RMPC steady-state offset in the RMPC ( C)
DrelT2(3) U Wq WT v V VC,LQ VC,RMPC DrelVC
relative steady-state offset (%) overall heat transfer coefficient (J s1 m2 K1) system-input weight matrix system-state weight matrix uncertain system vertex volume (m3) consumption of cooling water in the LQ control (m3) consumption of cooling water in the RMPC (m3) relative consumption of cooling water (%)
Greek letters r density (kg m3) Subscripts k associated with k-th control step LQ linear-quadratic optimal control max maximal value min minimal value RMPC robust model predictive control v uncertain system vertex 0 nominal system
[1] J.J. Klemes, P.S. Varbanov, Heat integration including heat exchangers, combined heat and power, heat pumps, separation processes and process control, Appl. Therm. Eng. 43 (2012) 1e6, http://dx.doi.org/10.1016/ j.applthermaleng.2012.03.044. [2] A.S. Morrison, M.J. Atkins, M.R.W. Walmsle, Ensuring cost-effective heat exchanger network design for non-continuous processes, Chem. Eng. Trans. 29 (2012) 295e300, http://dx.doi.org/10.3303/CET1229050. res-Hassimi, D. Seguin, M.A. Abdelghani-Idrissi, [3] M.A. Arbaoui, L. Vernie Counter-current tubular heat exchanger: modeling and adaptive predictive functional control, Appl. Therm. Eng. 27 (2007) 2332e2338, http://dx.doi.org/ 10.1016/j.applthermaleng.2006.12.027. [4] J. Mikles, M. Fikar, Process Modelling, Identification, and Control, Springer, Berlin, Germany, 2007. [5] M. Keshavarz, M.B. Yazdi, M.R. Jahed-Motlagh, Piecewise affine modeling and control of a boilereturbine unit, Appl. Therm. Eng. 30 (2010) 781e791, http:// dx.doi.org/10.1016/j.applthermaleng.2009.11.009. [6] E. Aggelongiannaki, H. Sarimveis, D. Koubogiannis, Model predictive temperature control in long ducts by means of a neural network approximation tool, Appl. Therm. Eng. 27 (2007) 2363e2369, http://dx.doi.org/10.1016/ j.applthermaleng.2007.03.005. , M. Bakosova , A. Me sza ros, J.J. Klemes, Neural network [7] A. Vasi ckaninova predictive control of a heat exchanger, Appl. Therm. Eng. 31 (2011) 2094e2100, http://dx.doi.org/10.1016/j.applthermaleng.2011.01.026. [8] J.W. Choi, G. Lee, M.S. Kim, Capacity control of a heat pump system applying a fuzzy control method, Appl. Therm. Eng. 31 (2011) 2332e2339, http:// dx.doi.org/10.1016/j.applthermaleng.2011.03.032. [9] J. Czeczot, P. Laszczyk, M. Metzger, Local balance-based adaptive control in the heat distribution system e practical validation, Appl. Therm. Eng. 30 (2010) 879e891, http://dx.doi.org/10.1016/j.applthermaleng.2009.12.021. [10] S. Wang, D.C. Gao, Y. Sun, F. Xiao, An online adaptive optimal control strategy for complex building chilled water systems involving intermediate heat exchangers, Appl. Therm. Eng. 50 (2013) 614e628, http://dx.doi.org/10.1016/ j.applthermaleng.2012.06.010. [11] M.V. Kothare, V. Balakrishnan, M. Morari, Robust constrained model predictive control using linear matrix inequalities, Automatica 32 (1996) 1361e1379. , J. Oravec, Comparison of robust and optimal approach to sta[12] M. Bakosova bilization of CSTRs, Chem. Eng. Trans. 25 (2011) 99e104. , J. Oravec, Robust model predictive control of heat exchangers, [13] M. Bakosova Chem. Eng. Trans. 29 (2012) 1465e1470, http://dx.doi.org/10.3303/ CET1229245. [14] M. Bakosov a, J. Oravec, Robust model predictive control of a laboratory twoetank system, in: Proc. Of the American Control Conference, 2014, pp. 5242e5247. Portland, Oregon, USA. , J. Oravec, Robust model predictive control of heat exchanger [15] M. Bakosova network, Chem. Eng. Trans. 35 (2013) 241e246, http://dx.doi.org/10.3303/ CET1335040. [16] J.F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw. 11 (1999) 625e653, http:// dx.doi.org/10.1080/10556789908805766. [17] J. Ingham, I.J. Dunn, E. Heinzle, J.E. Prenosil, J.B. Snape, Chemical Engineering Dynamics: An Introduction to Modelling and Computer Simulation, WileyVCH Verlag GmbH, Weinheim, Germany, 2007. [18] MathWorks Documentation Center, Available at: mathworks.com/help/index. html, 2014 (accessed 12 07 14). € fberg, Yalmip: a toolbox for modelling and optimization in MATLAB, in: [19] J. Lo Proc. of the CACSD Conference, 2004, pp. 284e289. Taipei, Taiwan. [20] J. Oravec, MUP Toolbox, 2014. Available at: bitbucket.org/oravec/mup/wiki/ Home (accessed 30 05 14).
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Robust model predictive control for heat exchanger network *, Juraj Oravec Monika Bakosova Slovak University of Technology in Bratislava, Faculty of Chemical and Food Technology, Institute of Information Engineering, Automation and Mathematics, Radlinsk eho 9, 812 37 Bratislava, Slovak Republic
h i g h l i g h t s The robust and the optimal controllers control the heat exchanger network. The robust model predictive controller is a solution of linear matrix inequalities. Robust control assures smaller offsets and lower coolant consumption.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 January 2014 Accepted 8 August 2014 Available online 28 August 2014
Optimal operation of heat exchangers represents a challenging task from the control viewpoint, due to the presence of system nonlinearities, varying process parameters, internal and external disturbances and measurement noise. Various robust control strategies were developed to overcome all these problems. The robust model predictive control (RMPC) represents one of suitable approaches. It enables to design effective control algorithms for optimization of the control performance subject to the process uncertainties and the input and output constraints. The possibility to implement the RMPC for control of a heat exchanger network is investigated in this paper, where three counter-current heat exchangers with uncertain parameters connected in series represent the controlled process. The efficiency of the advanced RMPC algorithm was verified by simulation experiments realized in the MATLAB/Simulink environment. The results confirmed that using the RMPC for the controlled process modelled as a system with uncertain parameters led to less consumption of cooling medium compared with the consumption achieved by using the optimal linear quadratic (LQ) control. © 2014 Published by Elsevier Ltd.
Keywords: Heat exchanger Uncertain system Robust model-based predictive control Linear matrix inequalities Energy saving
1. Introduction Heat exchangers (HEs) belong to the standard equipment in the chemical and process industries and the heat exchanger networks (HENs) are important processes in the petrochemical industry. As nearly 80% of the total energy consumption is related to the heat transfer, it is necessary to optimize the heat exchanger utilization, and to retrofit the heat exchanger networks [1]. Optimization and optimal control are of great importance for reducing energy consumption and for eliminating energy losses. The energy supply in most industrial devices was recognized as the second highest operating cost after primary feedstock costs. Moreover, since the energy prices tend to steadily increase, it is necessary to improve the efficiency of energy utilization and to reduce the energy
* Corresponding author. ), juraj.oravec@stuba. E-mail addresses: [email protected] (M. Bakosova sk (J. Oravec). http://dx.doi.org/10.1016/j.applthermaleng.2014.08.023 1359-4311/© 2014 Published by Elsevier Ltd.
consumption. One way is to use the advanced control strategies and to optimize the control performance [2]. Various papers deal with advanced control of thermal processes. The outlet temperature control of a counter-current tubular heat exchanger in heater configuration with the predictive functional control is presented in Ref. [3]. This approach is based on an internal adaptive model that corresponds to the response of the flow rate. Model predictive control (MPC) is one of control strategies satisfying demands on energy savings. MPC design is based on the solution of an optimization problem [4]. The quadratic cost function that penalizes the low quality of control performance is minimized over the control horizon in each control step to determine the optimal sequence of control inputs. Explicit MPC design based on a piecewise affine model of a boiler-turbine plant is presented in Ref. [5]. A nonlinear MPC configuration for hyperbolic distributed thermal systems is presented in Ref. [6]. This strategy also requires the development of a neural network function. Neural network predictive control (NNPC) is used for control of a co-current tubular heat exchanger in Ref. [7] and the presented results show that the
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
application of the NNPC can lead to significant energy savings in comparison with the classical PID control. The non-optimized and the optimized fuzzy controls of the heat pump are compared in Ref. [8]. Experimental case-study in Ref. [9] validates the balance-based adaptive control. An on-line adaptive optimal control approach is presented in Ref. [10]. Uncertainty in the controlled process, disturbances, measurement noise and the constraints on the control inputs and the controlled outputs play an important role in finding the proper solution of the control problem. The advantages of the RMPC originate from its flexibility to take into account the system constraints. Furthermore, the repetitive redesign of the controller in each control step can minimize the influence of neglected or unpredicted negative effects and improve the final control performance [11]. It was shown in Ref. [12] that the RMPC strategy is able to provide energy savings in control of a chemical reactor. The RMPC strategy can decrease energy consumption in the tubular and the jacketed HEs, see e.g. Ref. [13]. In Ref. [14] the RMPC was investigated using the real laboratory device AMIRA DST200. In Ref. [15] the RMPC of the HEN was designed to increase the energy savings. In Refs. [12,13] and [15] the results obtained by the RMPC
923
2. Controlled heat exchangers Three counter-current shell-and-tube heat exchangers (HEs) connected in series form the controlled heat exchanger network (HEN) (Fig. 1), in which petroleum as a product of the distillation in a refinery is cooled. Petroleum flows through the inner tubes and the cooling water flows over the outside of the tubes inside the shell of the heat exchanger. The tubes of the HEs are made from steel. The objective is to cool the outlet stream of the petroleum from the 3rd HE to the reference value 45.3 C and to minimize the energy utilization measured by the total consumption of coolant. The manipulated input is the volumetric flow rate of the inlet stream of the cold water to the 3rd HE. The mathematical model of the HEN was derived using the heat balances under some simplifications [17]: the thermal capacities of the metal walls are neglected; the HEs are well insulated, and heat loss to the surroundings and mechanical work effects are negligible; the technological parameters are either constant or vary in some intervals. The heat balances for the HEN lead to six first-order differential equations in the form
ðjÞ dT1 ðtÞ A U ðjÞ ðjÞ ðjþ1Þ ðjþ1Þ ðj1Þ ðjÞ T2 ðtÞ T1 ¼ q1 ðtÞr1 cp;1 T1 ðtÞ þ q1 ðtÞr1 cp;1 T1 ðtÞ þ h ðtÞ þ T2 ðtÞ T1 ðtÞ 2 dt ðjÞ dT ðtÞ A U ðjÞ ðj1Þ ðjÞ ðjþ1Þ ðj1Þ ðjÞ T2 ðtÞ T1 ðtÞ q2 ðtÞr2 cp;2 T2 ðtÞ h ðtÞ þ T2 ðtÞ T1 ðtÞ ¼ q2 ðtÞr2 cp;2 T2 V2 r2 cp;2 2 2 dt
V1 r1 cp;1
ðjÞ
ðjÞ;0
T1 ð0Þ ¼ T1
ðjÞ
(1)
ðjÞ;0
; T2 ð0Þ ¼ T2
were compared with the results ensured by the LQ optimal controller. The LQ optimal control is well-known and widely-used control approach [4], and serves as the reference strategy. The aim of this paper is to present implementation of the RMPC for the HEN control. According to the authors' knowledge the case-study of RMPC control of HEN with uncertainty was not published yet by the other authors. The designed RMPC was based on the formulation of the optimization problem with constraints in the form of linear matrix inequalities (LMI) and this convex optimization problem was solved using the semi-definite programming (SDP) [16]. Therefore, the computational burden was quite high. Advantage of the designed RMPC was investigated using the theoretical case study of the HEN control. Controlled process was the complex uncertain and the non-linear model of three counter-current shelland-tube heat exchangers in series. Quality of the HEN control was assessed by comparing the consumption of coolant. Presented results were obtained by simulations, and are valid in the range of assumed conditions. They can be considered a basis for the next step, which is a practical implementation of the RMPC on a real heat exchanger.
where the superscripts j ¼ 1, 2, 3 represent the 1st, 2nd and the 3rd heat exchanger, respectively. The subscripts 1 and 2 indicate the water and petroleum, respectively. In (1), V is the volume, r is the density, cp is the specific heat capacity, t is the time, T(t) is the timevarying temperature, q(t) is the time-varying volumetric flow rate, Ah is the heat transfer area and U is the overall heat transfer coefficient. The initial conditions T1(j),0 and T2(j),0 in (1) are given in Table 1 for two control scenarios. These scenarios e Case 1 and Case 2 differing in initial conditions were investigated to confirm the efficiency of the designed RMPC approach. The values of technological parameters and the steady-state values of temperatures are summarized in Table 2. Here n is the number of the 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 shell, T1,in ¼ T1(4) is the temperature of the inlet stream of water and T2,in ¼ T2(0) is the temperature of the inlet stream of petroleum. The superscript S denotes the steady-state value, and the steady-state temperatures T1(j),S, T2(j),S, j ¼ 1, 2, 3,
Table 1 Initial conditions in HEN for two control cases. Case 1
Fig. 1. Scheme of the counter-current shell-and-tube heat exchanger network; 1 is the cooling water, 2 is the hot petroleum, and (1)e(3) are the heat exchangers.
Case 2
Variable
Unit
Value
T1(1),0 T1(2),0 T1(3),0 T2(1),0 T2(2),0 T2(3),0
C
87.1
C
55.7
C
34.4
C
118.4
C
76.8
C
48.7
Variable
Unit
T1(1),0 T1(2),0 T1(3),0 T2(1),0 T2(2),0 T2(3),0
Value
C
87.6
C
55.7
C
34.3
C
117.3
C
75.4
C
47.5
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
924
Table 2 Technological parameters and steady-state values of variables in HEs. Variable
Unit
Value
n l din,1 dout,1 din,2 Ah V1 V2 cp,1 cp,2
1 m m m m m2 m3 m3 J kg1 K1 J kg1 K1
40 6 19 103 25 103 414 103 16.6 91.2 103 716.5 103 4.186 103 2.140 103
q1S
m3 s1
5.93 103
q2S
m3 s1
5.80 103
C
180.0
T2(1),S
C
113.0
T2(2),S
C
71.3
T2(3),S
C
45.3
C
20.0
C C C
30.8 48.0 75.8
T2,in ¼
T1,in ¼
T1(0)
T1(4)
T1(3),S T1(2),S T1(1),S
The linear discrete-time state-space model has the form
DTðk þ 1Þ ¼ Av DTðkÞ þ Bv Dq2 ðkÞ; DTð0Þ ¼ DT0 DTðkÞ ¼ CDTðkÞ
(2)
where the subscript v ¼ 0 represents the nominal system, the subscripts v ¼ 1, …,4 indicate the vertex systems, and k is the discrete time. The vector of the control inputs Dq2(k) and the vector of the controlled outputs DT(k) are defined as the differences between the actual values, i.e. q2(k), T1(j)(k), T2(j)(k), and the steadystate values, i.e. q2S, T1(j),S, T2(j),S, j ¼ 1, 2, 3, as it is shown in (3)
3 2 ð1Þ;S 3 ð1 Þ T1 T1 ðkÞ 7 6 7 6 ð1 Þ 6 T2 ðkÞ 7 6 T2ð1Þ;S 7 7 6 7 6 6 T ð2Þ ðkÞ 7 6 T ð2Þ;S 7 7 6 1 7 6 DT ðkÞ ¼ 6 1ð2Þ 7 6 ð2Þ;S 7; 7 6 T ðkÞ 7 6 T 7 6 2 7 6 2 7 6 ð3Þ;S 7 6 ð3 Þ 5 4 T1 ðkÞ 5 4 T1 2
ð3 Þ
T2 ðkÞ
h i Dq2 ðkÞ ¼ ½q2 ðkÞ qS2
(3)
ð3Þ;S
T2
The matrices Av, Bv, C of the uncertain system (2) have the form were computed for the volumetric flow-rates of water q1s ¼ 5.93 103 m3 s1 and of petroleum q2S ¼ 5.80 103 m3 s1 from (1) with zero derivatives. These steady state temperatures represent the reference values for control of the HEs. Further, two uncertain parameters are considered in the HEs. The heat-transfer coefficient changes as the flow rate of the cooling medium changes, and the density of the petroleum depends on the temperature in the HEs [17]. The values of these parameters are given in Table 3, where U is the heat transfer coefficient and r2 is the density of petroleum. The well-known approach of parametric uncertainties handling was used to describe the HEN in the form of a polytopic uncertain system [11]. Therefore the set of four vertex systems was generated for all variations of boundary values of two uncertain parameters (Table 3). Each vertex system was described by 6 ordinary differential equations (1). The control performance of the controlled process was investigated using these four limitbehaviour models. The nominal system was also created using the mean values of the uncertain parameters. This model served as the reference system. The non-linear state-space model of the controlled process (1) was linearized for the robust controller design using the 1st order Taylor expansion of nonlinear terms, and the linear state-space model of the HEN was obtained for the nominal system and each vertex system in the form of six ordinary linear differential equations. As the RMPC is a discrete-time control strategy, the linear continuous-time models were transformed into the discrete-time domain using the MATLAB's command c2dm [18] and the sampling time ts ¼ 25 s. The value of sampling time does not directly influence the RMPC design. It has to be chosen so that obtained discrete-time model matches the behaviour of the nonlinear model with sufficient accuracy.
Table 3 Uncertain parameters of HEs. Variable
Unit
Minimum value
Maximum value
U
J s1 m2 K1 kg m3
472.8 793.8
491.3 826.2
r2
2
a1;1 6a 6 2;1 6 6 0 Av ¼ 6 6 0 6 6 4 0
a1;2 a1;;3
0
0
a2;2 a2;1 0 0 a1;2 a1;1 a1;2 a1;3 a4;2 a2;1 a2;2 a2;1 0 0 a1;2 a1;1
2 3 b1 7 607 0 7 6 7 7 6 7 6b 7 0 7 7; Bv ¼ 6 2 7; 607 0 7 7 6 7 7 6 7 4 b3 5 a1;2 5 0
3
0 0 0 a4;2 a2;1 a2;2 3 1 0 0 0 0 0 60 1 0 0 0 07 7 6 7 6 60 0 1 0 0 07 7 Cv ¼ 6 60 0 0 1 0 07 7 6 7 6 40 0 0 0 1 05
0
2
(4)
0 0 0 0 0 1 where v ¼ 0,1, …,4. The matrix elements of Av, Bv in the continuoustime representation of (4) before the transformation into the discrete-time domain are expressed in Table 4. The specific matrices describing the discrete-time nominal model of the heat exchanger (v ¼ 0) were calculated from the continuous-time representation (Table 4) for the nominal values of the uncertain parameters and the sampling time ts ¼ 25 s, and are as follows Table 4 Matrix elements of the linear continuous-time statespace model of the HEN. Element
Value
a1,1
2qS1 r1 cp;1 Ah U 2V1 r1 cp;1
a1,2
Ah U 2V1 r1 cp;1
a1,3
2qS1 r1 cp;1 Ah U 2V1 r1 cp;1
a2,1
Ah U 2V2 r2 cp;2
a2,2
2qS2 r2 cp;2 Ah U 2V2 r2 cp;2
a4,2
2qS2 r2 cp;2 Ah U 2V2 r2 cp;2
b1 b2 b3
ð2Þ;S
T1
ð1Þ;S
T1 V1
ð3Þ;S ð2Þ;S T1 T1
V1 ð3Þ;S T1 V1
ð4Þ;S
T1
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
2 6 6 6 6 Av ¼ 6 6 6 6 4
6:4 0:3 0 0 0
1:0
4:2
1:1 0:3 1:0 6:4 0:5 0
0:3 0
0
0
0 1:0
0 4:2
1:1 0:3 1:0 6:4
0 0 0 3 2 3:4 6 0 7 7 6 7 6 6 2:3 7 2 7 Bv ¼ 6 6 0 7 10 7 6 7 6 4 1:6 5
0:5
0:3
0
3
be improved by taking into account the symmetric constraints on the system outputs DT(k) and inputs Dq2(k) in the form
7 7 7 7 7 102 ; 0 7 7 7 1:1 5 0 0
1:1
2 2 2 2 DT ðkÞ DTmax ; Dq2 ðkÞ Dq2;max 2
(5)
0 The matrices Av, Bv of the four vertex systems (v ¼ 1, …, 4) have the same dimensions. They were calculated from the continuoustime representation (Table 4) for all combinations of the boundary values of uncertain parameters and the sampling time ts ¼ 25 s. 3. Robust MPC The robust state-feedback control problem in the discrete-time domain can be formulated as follows: find a state-feedback control law (6)
Dq2 ðkÞ ¼ Fk DTðkÞ
(6)
for the system described by (2) so that the uncertain closed-loop system will be asymptotically stable [11], and will ensure satisfying control performance. The control law determines the relation between control inputs and outputs. The matrix Fk in (3) represents the gain matrix of the robust state-feedback controller in the k-th control step. To design the RMPC the approach described in Ref. [11] was adopted. The quality of the control performance can be expressed using the quadratic cost function J
J¼
N X
DTðkÞT WT DTðkÞ þ Dq2 ðkÞT Wq Dq2 ðkÞ
925
(7)
k¼0
where N is the number of control steps. For the design purposes the infinity control horizon is assumed (N / ∞), and WT, Wq are the real square symmetric positive-definite weight matrices of states DT(k) and the system inputs Dq2(k), respectively. The first term of the cost function J penalizes the control inaccuracy as the difference between the current and required values of outputs. Similarly, the second term penalizes the utilization of control action. The aim of control is to design such a controller Fk that minimizes the value of the objective J while providing the satisfaction of the robust stability condition for all vertex systems. The control performance can
2
2
(8)
2
where the values of the input constraints were set to prevent computing unreal negative values of volumetric flow-rates, i.e. Dq2,max ¼ 5.93 10 3 m3 s1. The output constraints were chosen such that the temperatures remained in the ±10 C surroundings of the steady-state values. The robust stability condition is raised from the Lyapunov stability theorem, see e.g. Ref. [11]. Then the gain matrix of the state-feedback control law Fk is designed as the solution of the optimization problem of SDP, where the constraints are in the form of LMIs [11].
4. Results and discussion The RMPC approach was designed to control three countercurrent shell-and-tube HEs in series. Two control scenarios were investigated to confirm the efficiency of the designed RMPC approach. These two scenarios e Case 1 and Case 2 differed in initial conditions (Table 1). The RMPC of HEs was analysed by simulation experiments in MATLAB/Simulink environment using 2.8 GHz CPU and 4 GB RAM. The optimization problem was formulated via the YALMIP toolbox [19] and solved by the SeDuMi solver [16]. We developed the MUP toolbox [20] to design RMPC. The toolboxes are freely available to the users. The obtained control performances were compared with the control performances assured by the discrete-time linearquadratic (LQ) optimal controller computed in the form FLQ ¼ [110.4, 31.2, 1.7, 21.1, 1.0, 8.5] 10 6 [4]. The gain matrix FLQ of the LQ controller was designed using the weight matrices WT, Wq of the cost function (7) in the form
2 6 6 6 WT ¼ 6 6 6 4
100 0 0 0 0 0
0 100 0 0 0 0
0 0 100 0 0 0
0 0 0 100 0 0
0 0 0 0 100 0
3 0 0 7 7 0 7 7; Wq ¼ ½100 0 7 7 0 5 100
(9)
The weight matrices (9) were also utilized in the RMPC algorithm to obtain fully comparable results. Both strategies were compared by evaluating the total consumption of coolant VC, which was needed for cooling the petroleum to the temperature 45.3 C in Case 1 and in Case 2 (Table 1). The simulations of control were stopped after 1200 s, i.e. after N ¼ 48 control steps. The simulation results were obtained using the more precise non-linear model of HEs (1). Table 5 shows obtained numerical results of the HEN control in both considered cases. Here, system represents the
Table 5 Results obtained by the RMPC and the discrete-time LQ optimal control. System
(3) DoffT2,RMPC ( C)
VC,RMPC (m 3)
(3) DoffT2,LQ ( C)
VC,LQ (m 3)
DrelT2(3) (%)
Case 1 Nominal 1st vertex 2nd vertex 3rd vertex 4th vertex
DrelVC (%)
0.01 0.48 0.51 0.48 0.50
7.200 7.410 6.702 7.411 6.706
0.00 0.97 0.93 0.98 0.93
7.313 7.247 7.377 7.247 7.377
e 50.5 45.2 51.0 46.2
1.5 2.2 9.2 2.3 9.1
Case 2 Nominal 1st vertex 2nd vertex 3rd vertex 4th vertex
0.01 0.48 0.55 0.48 0.54
7.166 7.385 6.640 7.386 6.644
0.00 0.97 0.94 0.98 0.93
7.316 7.250 7.380 7.250 7.380
e 50.5 41.5 51.0 41.9
2.1 1.9 10.0 1.9 10.0
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Fig. 2. Control performance of the petroleum temperature at the outlet from the 1st heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 1.
controlled non-linear model of HEs, which is either the nominal or (3) (3) the vertex system. Variables DoffT2,RMPC and DoffT2,LQ are the steadystate offsets of the outlet petroleum temperature assured using the (3) RMPC and the LQ control, respectively. DoffT2,LQ was zero when the nominal system was controlled by the LQ optimal controller. But when the vertex systems were controlled, the RMPC ensured lower absolute values of the offsets than the LQ optimal controller, i.e. the outlet petroleum temperature was closer to the reference when the RMPC was used. The LQ optimal controller was able to assure better results only for the ideal nominal system. The RMPC significantly reduced the steady-state offsets in the real situations represented by control of the vertex systems. The total consumption of VC is also presented in Table 5 for the nominal system and four vertex systems. In Table 5, VC,RMPC, VC,LQ are the total volumes of the coolant consumed during the robust control and the LQ optimal control, respectively. The 2nd and the 4th vertex systems had significantly lower consumption of the coolant during the robust control than during the LQ control. Robust control ensured smaller consumption of cold water even for the nominal system. But the 1st and the 3rd vertex systems had slightly higher consumption of the coolant during the robust control then during the LQ control. The RMPC and the LQ control were compared also using the following quality criteria:
Fig. 3. Control performance of the petroleum temperature at the outlet from the 2nd heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 1.
Fig. 4. Control performance of the petroleum temperature at the outlet from the 3rd heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 1.
ð3Þ
Drel T2
ð3Þ ð3Þ Doff T2;RMPC Doff T2;LQ 100% ¼ ð3Þ Doff T2;LQ
Drel VC ¼
VC;RMPC VC;LQ 100% VC;LQ
(10)
(11)
The negative values of DrelT2(3) or DrelVC mean that the RMPC improved the control results in comparison with the LQ control, and the steady-state offset and the coolant consumption were smaller during the robust control. The higher the absolute values of DrelT2(3) or DrelVC were calculated, the higher was the improvement. According to DrelT2(3), the RMPC improved the set-point tracking from 45.2% up to 51.0% in the Case 1 and from 41.5% up to 51% in the Case 2. Better results obtained using the LQ control for the nominal system follow from the fact that the LQ controller was designed as an optimal controller for the nominal system. According to DrelVc, the RMPC decreased the coolant consumption by 9e10% in the 2nd and the 4th vertex system and by 1.5e2% even in the nominal system control. But the RMPC of the 1st and the 3rd vertex system increased the coolant consumption by approximately 2%. The increase is not as significant as the decrease of coolant consumption for the other two vertex systems. The control responses of the HEs in Case 1 are shown in Figs. 2e4 and the associated control inputs in Fig. 5. The control responses of the HEs in Case 2 are shown in
Fig. 5. Control inputs generated by the RMPC (vertex systems e solid, nominal system e dashed) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 1.
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
Fig. 6. Control performance of the petroleum temperature at the outlet from the 1st heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 2.
Fig. 7. Control performance of the petroleum temperature at the outlet from the 2nd heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 2.
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Fig. 9. Control inputs generated by the RMPC (vertex systems e solid, nominal system e dashed) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 2.
Figs. 6e8 and the associated control inputs in Fig 9. From the robust control viewpoint, the key analysis is based on the limit system behaviour. Therefore the crucial information in these figures is the range of the temperature that is bounded by the four vertexcontrol-trajectories. The real temperature is located inside this area. The input disturbance expressed by the difference of the initial conditions from the references is more significant in Case 1; it is seen in Table 5 that the consumption of coolant increased, while the accuracy of the temperature control was the same. The control responses obtained using the RMPC are in most cases slower than those obtained using the LQ optimal control. The reason is that the RMPC design takes into account the input constraints and generated control inputs are less aggressive than control inputs generated by the LQ controller. As the LQ optimal controller design does not include process uncertainties, this strategy does not assure the asymptotic stability of the uncertain controlled system. The disadvantage of the RMPC implementation is increase of the computational effort due to the necessity to solve the complex optimization problem.
5. Conclusions
Fig. 8. Control performance of the petroleum temperature at the outlet from the 3rd heat exchanger assured using the RMPC (vertex systems e solid, nominal system e dashed, reference e dashed-circled) and the LQ optimal control (vertex systems e dotted, nominal system e dashed-dotted) e Case 2.
The possibility to improve the control performance and to increase energy savings by implementation of the RMPC is presented in the paper via control of the HEN. This is the main contribution of this paper as the implementation of the RMPC for control of heat exchangers has not been studied by many authors up till now. The robust control of the HEN was compared with the optimal LQ optimal control by simulation. Obtained results confirmed the effectiveness of the RMPC approach due to the smaller offsets and the decreased consumption of cold water used for petroleum cooling in the most of studied systems. The discrete-time LQ optimal controller ensured better quality only for the nominal system that represents the ideal system. In the presence of uncertainty and boundaries on control inputs and controlled outputs, the robust feedback control approach reduced the steadyestate offsets of the petroleum temperature in the outlet streams from the HEs by approximately 40e50% compared to the LQ optimal control. Consumption of the cooling water was reduced by up to 10% in about 20 min and is larger for greater disturbances (Case 1). Therefore it can be stated that the RMPC strategy used in practical implementations can lead to energy savings in comparison with the optimal LQ control. On the other hand, the overall computational
, J. Oravec / Applied Thermal Engineering 73 (2014) 922e928 M. Bakosova
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effort increased. The further research will be focused on improvement of the RMPC algorithm so that the offset free control responses will be achieved. Acknowledgements
Superscripts S steady-state (1) 1st heat exchanger (2) 2nd heat exchanger (3) 3rd heat exchanger
The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grant 1/ 0973/12 and the contribution of the Slovak Research and Development Agency under the grant APVV 0551-11. J. Oravec was also supported by an internal STU grant no. 1356.
Abbreviations LQ linear-quadratic MPC model predictive control RMPC robust model predictive control
Nomenclature
References
Symbols Ah a A b B C cp d F J k N q Dq2 t ts T DT (3) DoffT2,LQ
heat transfer area (m2) element of the state matrix state matrix element of the input matrix input matrix output matrix specific heat capacity (kJ kg1 K1) diameter (m) gain matrix of the controller quadratic cost function control step control horizon volumetric flow rate (m3 s1) control input (m3 s1) time (s) sampling time (s) temperature ( C) controlled outputs ( C) steady-state offset in the LQ control ( C)
(3) DoffT2,RMPC steady-state offset in the RMPC ( C)
DrelT2(3) U Wq WT v V VC,LQ VC,RMPC DrelVC
relative steady-state offset (%) overall heat transfer coefficient (J s1 m2 K1) system-input weight matrix system-state weight matrix uncertain system vertex volume (m3) consumption of cooling water in the LQ control (m3) consumption of cooling water in the RMPC (m3) relative consumption of cooling water (%)
Greek letters r density (kg m3) Subscripts k associated with k-th control step LQ linear-quadratic optimal control max maximal value min minimal value RMPC robust model predictive control v uncertain system vertex 0 nominal system
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