MULTIVARIABLE OPTIMAL CONTROL OF A HEAT EXCHANGER …

ISSN 0104-6632 Printed in Brazil

www.abeq.org.br/bjche

Vol. 33, No. 01, pp. 133 - 143, January - March, 2016 dx.doi.org/10.1590/0104-6632.20160331s00002780

*To whom correspondence should be addressed

Brazilian Journal of Chemical Engineering

MULTIVARIABLE OPTIMAL CONTROL OF A HEAT EXCHANGER NETWORK

WITH BYPASSES

F. Delatore1*, L. F. Novazzi3, F. Leonardi2 and J. J. da Cruz4

1Department of Electrical Engineering, FEI University, Av. Humberto de Alencar Castelo Branco 3972, Zip Code: 09850-901, Sao Paulo - SP, Brazil.

Phone: +55 11 43532900 E-mail: [email protected]

2Department of Mechanical Engineering, FEI University, Sao Paulo - SP, Brazil. 3Department of Chemical Engineering, FEI University, Sao Paulo - SP, Brazil.

4Department of Telecommunications and Control, University of Sao Paulo, Sao Paulo - SP, Brazil.

(Submitted: June 18, 2013 ; Revised: December 23, 2014 ; Accepted: February 2, 2015)

Abstract - Heat exchanger networks present an interesting control problem due to coupling among process streams. In this work, the linear quadratic regulator (LQR), a feedback optimal control technique, is used to control stream temperatures on a laboratory scale heat exchanger network, through bypass manipulation, in a multivariable system. The LQR design was based on a mathematical model of the plant and its performance was compared to traditional PID control and to dynamical decoupling. Experimental tests were performed to evaluate the controllers, involving regulatory and servo problems. The performance of the different controllers was quantitatively compared by using the integral absolute error. Although LQR is not a new control methodology, the results obtained in this work suggest that LQR is an interesting alternative to control HEN when compared to the PID and to the dynamic decoupler. Moreover, one of the main advantages of the LQR is its tuning simplicity, since only one parameter is sufficient for this application. Keywords: Heat exchanger network; Optimal control; LQR.

INTRODUCTION

Due to the oil price rise since the seventies and to environmental issues, efficient use of energy in chemical processes is very important. Almost thirty years ago the theoretical foundations of Process Inte-gration for the efficient use of energy were estab-lished, with Pinch Technology, an elegant approach to set energy / cost targets for heat exchanger networks (HEN), as well as rules to design such networks (Linhoff et al., 1982). Nowadays these synthesis tech-niques, including some which are based on mathe-matical programming, are well established in Process

Design and are easily found in many Chemical Engi-neering textbooks.

The design of a HEN depends on nominal stream supply temperatures and flowrates. However, during plant operation such nominal operating conditions can change, influencing stream target temperatures and propagating in the network, since the heat ex-changers introduce coupling among different parts of the process. Therefore, HEN control is an interesting issue and has been addressed in the literature since the eighties. A preliminary important contribution can be found in Marselle et al. (1982), where the authors proposed a HEN design technique by consid-

134 F. Delatore, L. F. Novazzi, F. Leonardi and J. J. da Cruz


ering process operability and controllability, which they called process resilience. Operability is defined as the ability of the network to remain steady-state feasible when subjected to process disturbances, whereas controllability is regarded as the network capacity to go from one steady-state to a different one, in a finite time.

Later on, Calandranis and Stephanopoulos (1988) proposed a sequence of control actions of the loops in a network to solve the regulatory and servo control problems in a HEN, exploiting its structural charac-teristics. The idea of the strategy was the identifica-tion of routes through the HEN that could allocate disturbances or setpoint changes to available sinks, i.e., utility heat exchangers.

Based on a previous work on HEN control by Ma-thisen (1994), Glemmestad et al. (1996) applied a method for optimal operation of the network and studied the coupling of manipulated variables, repre-sented by bypasses positions on the exchangers, with controlled variables, represented by stream target temperatures. In addition to the input / output pairing in the suggested decentralized control scheme, the proposed approach also contemplated the optimiza-tion of utility consumption in the HEN, since the number of manipulated variables is greater than the number of controlled parameters, which resulted in a positive degree of freedom. More recently and due to this positive degree of freedom, Sun et al. (2013) used non-square relative gain arrays to choose which bypasses should be selected to control a HEN.

Glemmestad et al. (1999) presented an alternative approach to the optimal operation of HEN systems based on on-line optimization of a steady state func-tion and a fixed control structure. Later, Giovanini and Marchetti (2003) showed that a low-level Dis-tributed Control System is also capable of handling HEN control problems when a flexible control loop structure is provided.

In the work of Lersbamrungsuk et al. (2008), a linear programming (LP) problem for the optimal operation of a HEN was formulated. As a conse-quence of the LP problem, the optimal point of oper-ation of a HEN remained at some of its constraints. The authors also proposed an offline strategy to switch between active constraints, identifying possi-ble operational regions, and combined this with de-centralized control. Previously, Aguilera and Mar-chetti (1998) developed a procedure for optimization and control of a HEN, in a more complex approach than the latter, since a nonlinear programming problem (NLP) needs to be solved online, during operation.

From the point of view of performance, there are a lot of different control techniques that could be

used in heat exchanger networks. These techniques range from methodologies that have a complex and highly engineered design, typically with a superior performance, down to methodologies that are easy and effortless to design, but normally not capable to lead to a desirable performance. As an example, the PID is the most common controller due to its straight-forward design procedure and easy implementation.

Nonetheless, in multivariable control problems, as is the case of a HEN, the PID design will demand a supplementary engineering effort to tune well the controllers. In this situation, model predictive control (MPC) may be considered to be a suitable control strategy to be used within industrial process, as it can deal with multivariable systems, complex dynamics and constraints on input and/or output variables. In this approach the future moves of the manipulated variables depend on the model and on plant meas-ured output variables, in such a way that an on-line constrained optimization is performed.

Gonzalez et al. (2006) presented an application of optimization and control of heat exchanger networks, through a two-level control structure. In the lower level, a constrained MPC was used and the higher level was supervised by an online optimizer. The MPC was based on a linear approximate plant model whereas the optimizer was based on a rigorous model. By using a moving horizon, hard constraints on the manipulated variables were dealt with in a straightforward way. Although the proposed method-ology uses a consolidated control strategy, just simu-lated results were presented. Besides, modeling er-rors were not explicitly included in the design.

In the range of suitable controllers for the control of a HEN, one can also point out the Linear Quadratic Regulator (LQR), which has a very simple design in the time domain. During the design of a LQR, the expected dynamic response is explicitly not taken into account, and the closed-loop response is checked afterwards. If the time response does not present an acceptable performance, it is possible to try a new controller by changing the penalty matri-ces, resulting in an extraordinary ease to design, and normally a good performance (Delatore et al., 2009).

In order to reduce the difficulties involved in con-troller design, in this work a control solution for a HEN based on optimal linear control is proposed. The LQR controller performance was successfully checked by experimental results obtained in a pilot plant. It must be pointed out that LQR control ap-plied to a HEN is not commonly found in the scien-tific literature (Delatore et al., 2009). The objective of this work is to illustrate the proposed design pro-cedure, as well as to show the reasonable perfor-

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hot cold 55.0 28.5 55.0 26.9

rimental testssystem: two

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SULTS AND

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s were perforo of them invsturbances inTHIN2, and ininvolving a satory one, iables TCOUT

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) DisturbanTHIN2

To evaluawith disturban

.0 ºC was apTHIN2 was incesponses of Ters, where thated to LQRively. Figurehree controllhe same colo

Figure 9, the ions until 508.5 ºC and 3ypass valve

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Novazzi, F. Leon

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028

28.5

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29.5

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30.5

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Tem

pera

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(ºC

)

TCOUT1

LQR TCOUT2

L

00

1

2

3

4

5

6

7

8

9

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Val

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2

Multivariable Optimal Control of a Heat Exchanger Network with Bypasses 141

Brazilian Journal of Chemical Engineering Vol. 33, No. 01, pp. 133 - 143, January - March, 2016

Table 4: Controller performance under disturbance in THIN2.

TCOUT1 TCOUT2

LQR 169.6 268.1 PID 132.8 187.2

Decoupling 202.5 364.9

By observing the performance of the Integral Ab-

solute Error for the three controllers (Table 4), one can see that the PID has the best response for TCOUT1 and also for TCOUT2. However, even with a more vigorous action, the LQR controller also showed a good performance when compared to the PID. These results suggest that the tuning parameter in Q, which multiplies the identity matrix, and chosen as Q equal to 10-4·I for the LQR design, seems to be high and leded to a faster control action when compared to the PID and dynamic decoupler.

It is worth noting that the PID and the decoupler controllers have more tuning parameters than LQR, which has only one, as considered in this work. Since this value is the unique design parameter in the controller, it is simpler to tune than the PID and the dynamic decoupler.

b) Servo and Regulatory Problem

To evaluate the servo problem in the experimental HEN, the setpoints of TCOUT1 and TCOUT2 were re-duced 0.5 ºC at a time t equal to 500 s, i.e., TCOUT1 and TCOUT2 were set to 28.0 ºC and 29.5 ºC, respec-tively. Besides, to evaluate the servo problem and also the regulatory problem, an increase in cold stream flowrate mC of 0.010 kg/s was imposed at 1000 s. Figure 11 shows the responses of TCOUT1 and TCOUT2, where the black, green and red curves are related to LQR, PID and dynamic decoupler, respec-tively. Figure 12 indicates the control effort of the three controllers.

By observing Figures 11 and 12, one can see that until 500 s the plant was under nominal operating con-ditions, with TCOUT1 and TCOUT2 equal to 28.5 ºC and 30.0 ºC, and the bypasses valves completely closed.

After the setpoint change at 500 s, the plots in Figures 11 and 12 suggest a relatively smooth perfor-mance for setpoint tracking, as well as for disturb-ance rejection, which was imposed at 1000 s.

This behavior is quantitatively indicated in Table 5, by using the IAE criteria. Since there is a setpoint decrease at 500 s, the bypass valves tend to open until 1000 s, when an increase in mC takes place, disturbing both TCOUT1 and TCOUT2, and then the bypasses begin to close. It can be observed that LQR action is fast when compared to the PID and the de-

coupler controller, leading to a quick setpoint track-ing and also disturbance rejection. This fact can be noted in Table 5, where LQR presents the smallest IAE among the three controllers.

Figure 11: Plots of TCOUT responses under setpoint change and flowrate disturbance.

Figure 12: Control effort of bypass valves under set-point change and flowrate disturbance. Table 5: Controller performance under setpoint change and flowrate disturbance.

TCOUT1 TCOUT2

LQR 371.9 340.9 PID 431.8 374.3


c) Disturbance in Cold Stream Flowrate mC

To evaluate a disturbance rejection in cold stream flowrate mC, a variation from 0.165 kg/s to 0.135 kg/s was imposed at a time t equal to 500 s and then re-moved at 1200 s. The responses of TCOUT1 and TCOUT2 for the three controllers are shown in Figure 13, where the black, green and red curves are related to LQR, PID and dynamic decoupler, respectively. As previ-ously mentioned in Table 3, setpoints of TCOUT1 and

0 500 1000 150027.5

28

28.5

29

29.5

30

30.5

Time(s)

Tem

pera

ture

(ºC

)

TCOUT1

LQR TCOUT2

LQR TCOUT1

PID TCOUT2

PID TCOUT1

Decoupling TCOUT2

Decoupling

0 500 1000 15000

1

2

3

4

5

6

7

8

9

10

Time (s)

Va

lve

Opp

eni

ng (

Vo

lts)

LQR - fci1

LQR - fci2

PID - fci1

PID - fci2

Decoupling - fci1

Decoupling - fci2

142 F. Delatore, L. F. Novazzi, F. Leonardi and J. J. da Cruz


TCOUT2 are equal to 28.5 ºC and 30.0 ºC. Figure 14 indicates the control effort of the three controllers, using the same color patterns as in Figure 13.

It can be seen from the plots in Figure 13 and 14 that the LQR controller reacted fast, and opened both bypass valves near 500 s, when the cold stream flow-rate increased. When the flowrate in C1 returned to its nominal operating value, at 1200 s, once more the LQR was fast and drove the controlled variables to 28.5 ºC and 30.0 ºC. Table 6 presents the controllers performance based on IAE for LQR, PID and the dynamic decoupler.

Figure 13: Plots of TCOUT responses under step dis-turbance in mC.

Figure 14: Control effort of bypass valves under dis-turbance in mC. Table 6: Controller performance under disturbance in mC.

TCOUT1 TCOUT2

LQR 403.9 145.9 PID 431.3 374.3


For the first controlled variable, TCOUT1, the de-

coupling technique shows the best performance and

the LQR is slightly better than the PID. As for TCOUT2, the decoupler has once more the best perfor-mance and the LQR is much better than the PID. Although the LQR was not the best option in this test, it is still a reasonable choice, due to its tuning simplicity.

CONCLUSIONS

In this work the use of a linear quadratic regulator to control heat exchanger networks was studied. This control technique is an alternative approach which presents advantages in HEN control: the LQR con-troller is easily designed and its performance can be even better than more common control techniques, such as PID and the dynamic decoupler. The aim of the proposed approach is not to accomplish a better performance than the one typically obtained with model predictive control techniques. On the other hand, the LQR can be designed just by using an ap-proximate plant model and only one tuning parame-ter, which can be iteratively chosen.

The approximate dynamic plant model used in the design of the LQR was based on energy balance equa-tions, which were linearized and their order reduced. However, this dynamic model could also be obtained directly from plant operation, in a more straightfor-ward way and probably with equivalent results.

The LQR was applied to a lab scale heat ex-changer network, constituted by two heat exchangers with two hot and one cold stream. Manipulated vari-ables were the bypass valve positions and the con-trolled outputs were outlet temperatures. The LQR tuning parameter was determined by simulation and the controller performance was compared to the PID and the dynamic decoupler. The controllers’ perfor-mance was quantitatively assessed by the integral ab-solute error and this index showed that the LQR per-formed well both in regulatory and servo problems.

ACKNOWLEDGEMENTS

The authors wish to thank all the institutional support of FEI University.

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Calandranis, J. and Stephanopoulos, G., A structural approach to the design of control systems in heat

0 500 1000 150027.5

28

28.5

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29.5

30

30.5

31

Time(s)

Te

mp

era

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(ºC

)

TCOUT1

LQR TCOUT2

LQR TCOUT1

PID TCOUT2

PID TCOUT1

Decoupling TCOUT2

Decoupling

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1

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6

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9

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(V

olts

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LQR - fci1

LQR - fci2

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PID - fci2

Decoupling - fci1

Decoupling - fci2

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Brazilian Journal of Chemical Engineering Vol. 33, No. 01, pp. 133 - 143, January - March, 2016

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