CompositionâTemperature Cascade Control of Dividing-Wall

Article Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

pubs.acs.org/IECR

Composition−Temperature Cascade Control of Dividing-Wall Distillation Columns by Combining Model Predictive and Proportional−Integral Controllers Xing Qian,† Kejin Huang,*,† Shengkun Jia,*,‡ Haisheng Chen,† Yang Yuan,† Liang Zhang,† and Shaofeng Wang† †

College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China School of Chemical Engineering and Technology, Chemical Engineering Research Center, Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), State Key Laboratory of Chemical Engineering, Tianjin University, Tianjin 300350, China

Ind. Eng. Chem. Res. Downloaded from pubs.acs.org by UNIV OF TEXAS AT DALLAS on 03/12/19. For personal use only.

‡

S Supporting Information *

ABSTRACT: Although dividing wall technology is very effective for process intensification of distillation, the resultant dividing-wall distillation columns (DWDCs) exhibit strongly interactive and highly nonlinear behaviors, which may pose great challenges to process operations. Proportional−integral (PI) controllers can stabilize the operation of the DWDC, but their performances are frequently far from satisfactory. Model predictive control (MPC) can potentially improve performance sharply, but the stringent requirements on the qualities of the embedded model considerably lower its robustness. To suppress these two deficiencies, in this work a cascaded control structure by combining the MPC and PI controllers is proposed for the DWDC. The PI controllers regulate the stage temperature in the inner layer, serving to quickly stabilize the DWDC, while the MPC directly controls the product compositions in the outer layer, addressing the strongly interactive and highly nonlinear behaviors of the DWDC. Two typical cases, that is, a three-product Petlyuk DWDC and a four-product Kaibel DWDC, and five control schemes, that is, two impurity composition controls (PI-ICC and MPC-ICC), two purity composition controls (PI-PCC and MPC-PCC), and one pure temperature control PI-TC (which represents the case wherein MPC in the outer layer fails to work), are studied to evaluate the feasibility and effectiveness of the proposed control structure. It is found that the cascaded MPC/PI controllers (i.e., MPC-ICC and MPC-PCC) substantially outperform the pure PI controllers (i.e., PI-ICC and PI-PCC), with lower maximum and steady-state deviations, shortened settling times, and suppressed oscillations. In the case of failure of the MPC, the cascaded MPC/PI controller can still maintain stable operation of the DWDC. These outcomes highlight that the combination of MPC and PI controllers can be a potential method for the operation of highly interactive and complicated DWDCs.

1. INTRODUCTION Distillation is one of the most widely used thermal separation technologies in the chemical and petrochemical industries. However, in spite of their simplicity and widespread application, distillation processes account for more than half of the operating costs of a plant.1 Among the advanced process integration and intensification technologies for distillation, dividingwall distillation column (DWDC) is one of the most effective as it has been industrialized.2 The DWDC is a thermally coupled single-shell distillation column that requires considerably less space and entails lower operating expenses and capital investment than its conventional counterpart.3 It is frequently reported that the three-product Petlyuk DWDC and fourproduct Kaibel DWDC can lower the utility requirement by about 30% and 40%, respectively.4−6 Because of the complex structure and the complicated interactive nature of the DWDC, controllability and operability are © XXXX American Chemical Society

the main obstacles to its large-scale industrialization. Most researches have focused on proportional−integral−derivative (PID) control. Wolff and Skogestad7 studied the operability and controllability of a DWDC separating an ethanol/propanol/ butanol mixture. They noted that the two impurities in the side product stream should not be specified independently. Kiss and Rewagad8 compared four composition control structures for a three-product DWDC separating a benzene/toluene/ xylene mixture, and their results showed that the DB/LSV and LB/DSV control structures were better than the others. D, S, B, L, and V refer to the distillate, side product flow rate, bottom product flow rate, reflux flow rate, and vapor boil up, Received: December 13, 2018 Revised: February 7, 2019 Accepted: February 27, 2019

A

DOI: 10.1021/acs.iecr.8b06179 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

Figure 1. Novel composition−temperature cascade control scheme.

respectively. Dwivedi et al.9 proposed feasible composition control structures for a three-product DWDC with or without the vapor split ratio as a manipulated variable. Huang et al.10−12 investigated temperature inferential control for a three-product DWDC including temperature difference control, simplified temperature difference control, double temperature difference control, and simplified double temperature difference control schemes. Wang13 simulated and analyzed multiple steady states (MSSs) in the DWDC, and their results showed that MSSs inevitably exist in the DWDC and considerably influence its design, control, and operation. Qian et al.14 developed temperature control structures for a three-product DWDC with the vapor split ratio or liquid split ratio as a manipulated variable. The PID control is a mature technology but fails to adequately address interactions between the controlled variables. In comparison with the traditional multiloop PID control, model predictive control (MPC) has several advantages. It anticipates the future values of the controlled variables and determines their corrections by solving an optimization problem. The inherently strong interactions and highly nonlinear behaviors of the DWDC make MPC a suitable vehicle for process operation. Buck et al.15 proposed a method for the design of model predictive controllers for DWDC and confirmed its effectiveness though experimental investigation. Rewagad and Kiss16 compared MPC with traditional multiloop PID control and showed that the former significantly improved the dynamic performance of DWDC. Hernandez and ChineaHerranz17 presented a multiloop PID control and equivalent MPC for a DWDC separating an n-pentane/n-hexane/ n-heptane mixture. They demonstrated that MPC was superior to PID control in addressing coupling between the controlled variables. Qian et al.18 studied MPC for a reactive DWDC and again demonstrated the advantages of MPC. Sun et al.19 proposed composition and temperature control for a DWDC with MPC and an equivalent PID control and found that the former led to more stable responses than the latter. It is noteworthy that little research has been conducted so far on the cascade control of composition and temperature by combining model predictive and PI controllers. In the proposed composition−temperature cascade control scheme, the inner layer is temperature control using PI controllers and the outer layer is composition control using MPC. The scheme combines the advantages of both composition control and temperature control. The objective of this paper is to evaluate the feasibility and effectiveness of the proposed

control structure for the operation of a DWDC through a thorough comparison with the PI control structure. Two typical cases, a three-product Petlyuk DWDC and a four-product Kaibel DWDC, are studied. Further discussions on the control structure proposed and conclusions drawn appear in this paper.

2. COMPOSITION−TEMPERATURE CASCADE CONTROL SCHEME A novel composition−temperature cascade control scheme is proposed for the operation of a DWDC. The composition− temperature cascade control is in accordance with the control procedure suggested by Skogestad.20 The control system can be divided into two layers. The outer layer (the main control loop) is a supervisory layer that controls the primary variables (CV1), while the inner layer (the deputy control loop) is a stabilizing layer that controls drifting or sensitive secondary variables (CV2) that are easy to measure and control. For distillation, the primary variables (CV1) are usually compositions and the secondary controlled variables (CV2) are normally temperatures. The acceptance of MPC for DWDC can be enhanced through the proposed composition−temperature cascade control. The control scheme, as shown in Figure 1, features a PI controller in the inner layer and an MPC in the outer layer. The controlled object is a complex process such as a DWDC. For the main control loop using MPC, the main controlled variable is the product stream composition y(k). For the deputy control loop employing PI control, the deputy controlled variable is the sensitive tray temperature T(k). The parameter d(k) is the vector of disturbances for, e.g., the feed flow rate disturbances, and r(k) is the vector of output references, i.e., the expected product stream compositions. The deputy control loop contains the controlled object DWDC, temperature sensor, and PI controller. The main control loop includes the controlled object DWDC, composition sensor, and MPC. For the main control loop, u(k) is the manipulated variable, which is the set point of the temperature controller (TC). In other words, the output of the MPC is the set point of the PI controller. The MPC is shown in the dotted box in Figure 1. The MPC has three components: prediction model, feedback correction, and online optimization. The core of MPC is the prediction model. The MPC calculates the controlled variables from time k to time k + j based on the prediction model. The state space model, one of the linear time invariant models, is employed as B


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Industrial & Engineering Chemistry Research the prediction model in this paper. The expressions are as follows: x′(k) = Ax(k) + Bu(k)

(1)

ym(k|k) = Cx(k) + Dd(k)

(2)

where x(k) represents the vector of plant state variables; u(k) denotes the vector of manipulated variables; ym(k | k) refers to the predictive vector of controlled variables, and d(k) indicates the vector of disturbances. The rigorous nonlinear DWDC plant model employs the MESH equations. The model matrices A, B, C, and D can be achieved by linearizing the DWDC plant model near its stable operation state. The Taylor expansion is used in the linearization. The linearized model is of higher order, which would result in heavy computational burden. Therefore, a reduction of the state space model is necessary to accelerate the calculations at the expense of prediction accuracy. To determine the states to be removed and the reduced model order, the Hankel singular values21 are computed for the state space model. The Hankel singular values measure the contribution of each state to the input−output behaviors. The Hankel singular values for the model order are like the singular values for the matrix rank. Small Hankel singular values can be discarded to simplify the model. The Hankel singular values can be calculated in Matlab. The feedback correction helps to correct the model-calculated values ym(k + j | k) based on their deviations from the detected process outputs y(k). The model prediction values y(k + j | k) are obtained through the model-calculated values ym(k + j | k) and feedback corrections. The differences can be obtained between the model prediction values y(k + j | k) and reference values r(k), which are the inputs to the online optimization. Using the online optimization, the optimal values for the manipulated variables can be achieved. The optimization problem that needs to be solved at every sampling instant k is expressed as follows:

Figure 2. Three-product Petlyuk DWDC.

minimize the three parts, and their importance is determined from their corresponding weights. In our work, the third part is not required because there is no target manipulated variable. Therefore, the weights of the manipulated variables (wu) are zero, and the weights of the manipulated variable variations (wΔu) and the weights of the controlled variables (wy) should be tuned. The tuning of wy and wΔu can be solved by genetic algorithm to obtain the Pareto optimum solutions22,23 for a multiobjective optimization problem. Pareto optimization is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Pareto optimum solutions can be graphically shown in the Pareto front. A final solution for the weights of the manipulated variables and controlled variables can be selected from the Pareto front (the discrete set of the Pareto optimum solutions obtained). Two objective functions are as follows:

min

Δu(k|k), ···, Δu(m − 1 + k|k), ε ÄÅ ny ÉÑ | l ÅÅ ÑÑ o o o o ÅÅ o 2Ñ y o ÑÑ o o Å | w ( y ( k + i + 1 | k ) − r ( k + i + 1)) | ∑ o o Å Ñ 1, i j j + o o j Å Ñ o o Å Ñ o o Å Ñ o o 1 = j Å Ñ o o Å Ñ o o Å Ñ o o Å Ñ o o Å Ñ n 1 − p o o Å Ñ u o o Å Ñ o o ÅÅ ÑÑ 2 Δu o o 2 + | w Δ u k + i | k | ( ) Å Ñ ∑ ∑ + ρ ε m } i,j j Å Ñ ε o o Å Ñ o o Å Ñ o o Å Ñ j=1 0 = i o o Å Ñ o o Å Ñ o o Å Ñ o o Å Ñ o o Å Ñ nu o o Å Ñ o o Å Ñ o o Å Ñ u 2 o o Å Ñ + | w u ( k + i | k ) − u ( k + i | k ) | o o ∑ Å Ñ i j j j , target o o Å Ñ o o Å Ñ o o Å Ñ o o Å Ñ j=1 Å Ñ Ç Ö n ~ (3)

where p is the prediction horizon and m is the control horizon. The subscript j indicates the component number of a vector. The term (k + i|k) refers to the value for future time k + i predicted at time k, and r(k) represents the output references (set points) of the controlled variables. The prediction horizon p and the control horizon m are set to maintain the stability of the DWDC system with fewer oscillations. The sampling time Δk is determined to be small enough to reproduce the system. The objective function in eq 3 contains three parts: the sum of the squared deviations of the controlled variables and the corresponding references, the sum of the squared manipulated variable variations, and the sum of the squared deviations between the manipulated variables and the corresponding target values. The purpose of the online optimization is to

min obj1 =

∑ [y(k) − r(k)]2

(4)

min obj2 =

∑ [Δu(k)]2

(5)

The outputs of the MPC are the set points of the PI controllers, which can maintain stable operation of the DWDC. The main control loop using MPC is a fixed value system to reduce the deviations of product compositions and suppress the influence of coupling between the controlled variables. The deputy control loop employing PI control is a servo system to C


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Table 1. Controller Tuning Parameters of PI-ICC for Case 1 control loop

controlled variable

TC1 TC2 TC3 CCDB

TP6 TM9 TM47 xD,B

CCSC

xS,C

CCBB CCP1

xB,B yDp,A+B

CCP2

xBp,B+C

manipulated variable LP L S set point of TC2 set point of TC3 QR set point of TC1 Vp

controller gain

controller integral time (min)

4.464 1.298 5.093 0.049

13.20 10.56 13.20 60.72

0.059

27.72

0.112 0.720

67.72 30.36

46.54

27.72

Table 2. Controller Tuning Parameters of PI-PCC for Case 1 control loop

controlled variable

CCDA

xD,A

CCSB

xS,B

CCBC CCP1

xB,C yDp,A+B

CCP2

xBp,B+C

manipulated variable set point of TC2 set point of TC3 QR set point of TC1 Vp

controller gain


0.049

59.40

0.167

38.28

0.114 0.077

26.40 20.64

9.851

19.80

the PI controller can quickly stabilize the DWDC and help enhance the applicability of the MPC.

3. CASE 1: OPERATION OF A THREE-PRODUCT PETLYUK DWDC The fully thermally coupled three-product Petlyuk configuration of DWDC contains a prefractionator and a main section, as shown in Figure 2b. If the heat transfer across the vertical dividing wall is considered negligible, the thermally coupled configuration in Figure 2b is equivalent to the DWDC configuration in Figure 2a. The atmospheric separation of a threecomponent liquid mixture of ethanol (A), n-propanol (B), and n-butanol (C) is used as the illustrative case, which is also studied in our previous work,24 to test the feasibility of the proposed novel composition−temperature cascade control scheme with PI controllers in the inner layer and MPC in the outer layer. Ethanol is widely employed as a solvent, and it is also used as a clean-burning fuel. Propanol can be used as a solvent in the pharmaceutical industry. Butanol is primarily used as a solvent, as an intermediate in chemical synthesis, and as a fuel. The feed is atmospheric equimolar saturated liquid. The feed flow rate is 1 kmol/h. The molar purities of all the three products are 99%. The relative volatilities of the three components are 4.46, 2.16, and 1, respectively. The thermodynamic model uses the NRTL liquid activity equation. Nominal steady-state data of the three-product DWDC are shown in Table 1 in the Supporting Information. All compositions are expressed in mole fractions. The vapor split ratio (RV) and liquid split ratio (RL) are defined as the fractions of the vapor or liquid that are sent to the prefractionator. The steady-state design of the three-product Petlyuk DWDC is shown in Figure 2b. 3.1. PI Control. Three PI control structures are studied in case 1 in this paper. The impurity composition control (PI-ICC) and purity composition control (PI-PCC) structures are shown in panels a and b of Figure 3, respectively. The PI-PCC is proposed to improve the dynamic performance of PI-ICC. The PI controllers are employed in PI-ICC and PI-PCC, whereas

Figure 3. Composition−temperature cascade control of the threeproduct Petlyuk DWDC using pure PI control.

respond fast and enhance the robustness of the system. Hence, the MPC can deal with the complicated dynamics of the DWDC and serve to improve the control performance, while D


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Figure 4. Comparison of three pure PI controls for case 1: dynamic responses to ±20% feed flow rate disturbances

Figure 6. Pareto optimum solution set for case 1.

sump level of the prefractionator is controlled by its bottom product flow rate. For the pressure controllers and level controllers, the gains and integral times used are the values recommended by Luyben.25 Composition−temperature cascade controls are used in the proposed PI-ICC and PI-PCC structures in this work. These are in accordance with the two-layer control procedure recommended by Skogestad.20 Temperature control in the inner layer is faster and less expensive than composition control in the outer layer, while composition control is better at reducing the steady-state discrepancies in product purities. Therefore, the composition−temperature cascade control combines the advantages of both composition control and temperature control. The inner stabilizing layer (temperature control layer) is identical in PI-ICC and PI-PCC. Sensitivity analysis is used to achieve the sensitive tray temperatures. The sensitive trays are the sixth tray in the prefractionator (TP6), the ninth tray in the main section (TM9), and the 47th tray in the main section (TM47). The thermally coupled liquid stream flowing back to the prefractionator (Lp) is the manipulated variable of the temperature controller 1 (TC1) to control TP6. The reflux flow rate (L) of the main section is manipulated for the temperature controller 2 (TC2) to control TM9. The side product flow rate (S) is varied for the temperature controller 3 (TC3) to control TM47. According to the control rules suggested by Skogestad,26 one stabilizing temperature control loop is needed for each split. Three temperature control loops are designed for the three sharp splits in the three-product DWDC. The prefractionator performs a sharp A/C split and needs one

Figure 5. Composition−temperature cascade control of the threeproduct Petlyuk DWDC using MPC and PI control.

proportional (P) controllers are used for level controls. Derivative (D) control is not used in this work because it is very sensitive to measurement noise. The process is simulated using Aspen Dynamics. The pressure control and level control are not illustrated in the figures. The pressure of the main section is controlled by the condenser duty. The pressure of the prefractionator is controlled by its overhead vapor flow rate. The reflux drum level of the main section is controlled by the distillate product flow rate. The sump level of the main section is controlled by the bottom product flow rate. The E


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Industrial & Engineering Chemistry Research Table 3. Controller Tuning Parameters of MPC for Case 1 manipulated variables weights

TC1SP

TC2SP

controlled variables

TC3SP

Vp

0 1.992

0 0.411

y

w wu wΔu

0 0 1.947 2.820 prediction horizon p 100

QR

yDp,C

36.283 0 3.499 control horizon m 4

xD,B

xS,C

xBp,A

xB,B

32.136

4.775

37.979

21.968

sampling time Δk 0.01 h

Figure 7. PI-PCC vs MPC-PCC for case 1: dynamic responses to ±20% feed flow rate disturbances or ±20% feed composition disturbances

temperature controller, TC1. The upper part of the main section carries out a sharp A/B split and needs one temperature

controller, TC2. The lower part of the main section accomplishes a sharp B/C split and needs one temperature controller, F


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Figure 8. PI-TC vs MPC-PCC for case 1: dynamic responses to ±20% feed flow rate disturbances or ±20% feed composition disturbances.

TC3. The dead time of temperature controllers is 1 min. The dead time of composition controllers is 3 min. The gains and integral times are obtained through the Tyreus−Luyben method. The tuning parameters of the temperature controllers are shown in Table 1. The outer supervisory layer uses composition control. Two composition controls (CCP1 and CCP2) are used at the two ends of the prefractionator. In CCP1, the controlled variable is the sum of the compositions of components A and B in the top overhead vapor of the prefractionator, and the manipulated

variable is the set point of TC1. In CCP2, the controlled variable is the sum of the compositions of components B and C in the bottom “product” of the prefractionator, and the manipulated variable is the flow rate of the thermally coupled vapor flowing back to the prefractionator. The impurity compositions of the prefractionator are not used as the controlled variables in PI control because they are very sensitive to feed disturbances and result in excessive oscillations. In the PI-ICC structure, as shown in Figure 3a, the three impurity compositions of the three product streams are G


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Industrial & Engineering Chemistry Research Table 4. Comparison between Dynamic Performances of MPC-PCC and PI-PCC for Case 1 in Terms of ISE disturbance +20% F −20% F +20% A −20% A +20% B −20% B +20% C −20% C

xD,A_PI-PCC 4.99× 1.37× 1.63× 1.78× 1.89× 1.47× 3.87× 5.10×

10−4 10−4 10−4 10−4 10−5 10−5 10−5 10−5

xD,A_MPC-PCC 1.15× 1.24× 2.24× 5.01× 1.35× 1.83× 1.27× 1.40×

10−4 10−4 10−4 10−6 10−6 10−6 10−6 10−6

xS,B_PI-PCC 6.09× 3.26× 1.37× 3.97× 9.35× 1.23× 4.43× 4.43×

10−4 10−4 10−4 10−5 10−5 10−5 10−6 10−5

xS,B_MPC-PCC 2.11× 6.63× 2.65× 2.39× 2.55× 5.07× 3.39× 1.99×

10−5 10−5 10−4 10−5 10−5 10−6 10−6 10−5

xB,C_PI-PCC 1.38× 5.49× 6.85× 2.55× 1.14× 2.71× 3.54× 1.73×

10−1 10−3 10−3 10−3 10−4 10−4 10−4 10−3

xB,C_MPC-PCC 1.38× 4.68× 2.68× 9.46× 2.14× 2.11× 9.15× 8.05×

10−4 10−4 10−4 10−6 10−5 10−5 10−6 10−6

dynamic performances are satisfactory and the number of composition detectors can be reduced. In the case of MPCICC, as shown in Figure 5a, the five controlled variables are the two impurity compositions of the top and bottom “products” of the prefractionator and the three impurity compositions of the three product streams of the main section. (For the side stream, the heavy impurity C is controlled.) In the case of MPC-PCC, as shown in Figure 5b, the five controlled variables are the two impurity compositions of the top and bottom “products” of the prefractionator and the three product purity compositions in the main section. The MPC-PCC is proposed in order to improve the dynamic performances of MPC-ICC. The weights of MPC are determined by the Pareto front, as shown in Figure 6. This is a set of optimum solutions based on the two objective functions shown in eqs 4 and 5. The horizontal and vertical axis represent obj1 and obj2. The two objective functions refer to the minimization of the manipulated variable variations and the deviations between the controlled variables and the set points. For a specific obj1 value on the Pareto front, no smaller obj2 value exists. For a specific obj2 value on the Pareto front, no smaller obj1 value likewise exists. The hollow point with an obj1 value of 0.005 and an obj2 value of 2.157 is selected as the final optimum solution. The optimal tuning parameters of MPC, that is, the final selected optimum solution, are shown in Table 3. 3.3. Results and Discussion. The dynamic performances of the composition−temperature cascade control of DWDC using MPC and PI control are compared with those obtained by using pure PI control. The results show that the dynamic performances improve greatly through the introduction of MPC. The dynamic responses to ±20% feed flow rate disturbances and ±20% feed composition disturbances using PI-ICC and MPC-ICC are shown in Figure 2 in the Supporting Information. During the tuning of PI-ICC, the system is not stable when the integral times of the side product composition control and bottom product composition control are close in value. This is probably due to the strong interactions between the side and bottom product streams. The MPC is well-suited for the strongly interactive DWDC. The dynamic performances of MPC-ICC are much better than those of PI-ICC with fewer oscillations, shorter settling times, and reduced maximum and steady-state deviations. However, the steady-state deviations are relatively large in the case of +20% disturbance to the feed composition of A or −20% disturbance to the feed composition of C. Therefore, MPC-PCC is proposed in this work to further reduce the steady-state deviations. The dynamic responses to ±20% feed flow rate disturbances or ±20% feed composition disturbances using PI-PCC and MPC-PCC for case 1 are shown in Figure 7. The steady-state deviations of PI-PCC and MPC-PCC are both very small. The

controlled. (For the side stream, the heavy impurity C is controlled.) In the PI-PCC structure, as show in Figure 3b, the three product purity compositions are controlled. The three manipulated variables for the two control structures are the same; these are the set point of TC2, set point of TC3, and reboiler duty. The values of gains and integral times are achieved through the Tyreus−Luyben method. The tuning parameters of PI-ICC for case 1 are shown in Table 1. The tuning parameters of PI-PCC for case 1 are shown in Table 2. The tuning parameters of the three temperature controllers are the same for both PI-PCC and PI-ICC and hence are not displayed in Table 2. The temperature control structure of PI control (PI-TC) is shown in Figure 3c. The PI-TC is employed as a comparison to evaluate the dynamic performances of PI-ICC and PI-PCC. The dynamic responses to ±20% feed flow rate disturbances using the three pure PI control structures (PI-ICC, PI-PCC, and PI-TC) are shown in Figure 4. The dynamic responses to ±20% feed composition disturbances using the three pure PI control structures (PI-ICC, PI-PCC, and PI-TC) are shown in Figure 1 in the Supporting Information. The dynamic performances of PI-ICC are not satisfactory. Among PI-ICC, PI-PCC, and PI-TC, PI-ICC has the largest number of oscillations and the longest settling time. Additionally, when PI-ICC is employed, the side product composition decreases rapidly when −20% disturbance in the feed composition of C occurs because the light impurity in the side product stream is not controlled in PI-ICC. The PI-TC has the shortest settling time and the minimum steady-state deviation. These dynamic performances prove that PI-TC can yield a fast response while PI-PCC is able to ensure product purities. 3.2. Model Predictive Control. Given its lack of industrial application, a complex DWDC using pure MPC may be not acceptable to the industry. Therefore, a composition−temperature cascade control is proposed in this paper with PI control for the inner layer and MPC for the outer layer. In this way, the control structure is workable even if the MPC fails to work. This enhances the acceptance of MPC for DWDC. The impurity composition control structure of MPC (MPC-ICC) and purity composition control structure of MPC (MPC-PCC) are shown in panels a and b of Figure 5, respectively. The inner stabilizing layer performs temperature control with PI controllers. The temperature controllers are the same as those employed in pure PI control. The outer supervisory layer uses composition control with MPC. The outputs of MPC are five manipulated variables: the set points of TC1, TC2, and TC3; reboiler duty (QR); and the flow rate of thermally coupled vapor flowing back to the prefractionator (Vp). The inputs to MPC are five controlled variables. The two impurity compositions of the top and bottom “products” of the prefractionator are used as the controlled variables in MPC because the H


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Industrial & Engineering Chemistry Research MPC-PCC is superior to PI-PCC with fewer oscillations, shorter settling time, and reduced maximum deviations. The dynamic responses to ±20% feed flow rate disturbances or ±20% feed composition disturbances using PI-TC and MPC-PCC for case 1 are shown in Figure 8. The PI-TC exhibits an advantage of fast response over PI-ICC and PI-PCC in Figure 4. The results show that PI-TC is still workable when MPC in the outer layer of MPC-PCC fails to work. The settling times using PI-TC are comparable to those using MPCPCC in Figure 8. Nevertheless, with MPC-PCC, the maximum deviations are not large and the steady-state deviations are very small. Therefore, MPC-PCC is the best among the five control structures studied in case 1. The dynamic performances can be evaluated by the integral of the squared error (ISE), which is defined as follows: ISE =

∫0

t

[e(t )]2 dt

The ISE reflects both settling times and maximum deviations. A comparison between the dynamic performances of MPC-PCC and PI-PCC in terms of ISE is shown in Table 4. The dynamic performances employing MPC-PCC are greatly improved as the ISE values are lower. Therefore, the dynamic performances of MPC-PCC are stable and superior to those of PI-PCC.

4. CASE 2: OPERATION OF A FOUR-PRODUCT KAIBEL DWDC The fully thermally coupled four-product Kaibel DWDC contains a prefractionator as well as a main section. A feed of equimolar saturated liquid mixture containing methanol (A), ethanol (B), n-propanol (C), and n-butanol (D) is used in case 2, which is also studied in our previous work,27 to verify the effectiveness of the proposed novel composition−temperature cascade control scheme with PI controllers in the inner layer and MPC in the outer layer. The feed flow rate is 1 kmol/h. The purities of the four products in the corresponding product streams (D, S1, S2, and B) are all 99%. The steady-state design of the four-product Kaibel DWDC is shown in Figure 9. Figure 10. Composition−temperature cascade control of the fourproduct Kaibel DWDC using pure PI control

Table 5. Controller Tuning Parameters of PI-PCC for Case 2 control loop

controlled variable

TCP TC1 TC2 TC3 CCDA

TP12 TM12 TM43 TM64 xD,B

CCS1B

xS1,B

CCS2C

xS2,C

CCBD CCP1

xB,D yDp,C

CCP2

xBp,B

Figure 9. Four-product Kaibel DWDC configuration.

4.1. PI Control. Two PI control structures are studied in case 2. The purity composition control (PI-PCC) and temperature control (PI-TC) are shown in panels a and b of Figure 10, respectively. Sensitivity analysis is used to achieve the sensitive tray temperatures. The sensitive trays are the 12th tray in the prefractionator (TP12), the 12th tray in the main section (TM12),

manipulated variable LP L S1 S2 set point TC1 set point TC2 set point TC3 QR/F set point TCP Vp

controller gain


of

4.330 2.457 6.321 6.198 0.236

17.16 10.56 14.52 15.84 34.32

of

0.812

19.80

of

0.418

30.36

of

0.074 0.031

58.08 26.40

0.264

38.28

the 43rd tray in the main section (TM43), and the 64th tray in the main section (TM64). The rate of the thermally coupled I


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Figure 12. Pareto optimum solution set for case 2.

The middle part of the main section finishes a sharp B/C split and needs one temperature controller, TC2. The lower part of the main section accomplishes a sharp C/D split and needs one temperature controller, TC3. The controller tuning parameters of PI-TC for case 2 are shown in Table 5. The gains and integral times are achieved through the Tyreus−Luyben method. The tuning parameters for purity composition control (PI-PCC) in case 2 are also provided in Table 5. However, the PI-PCC is not stable because the composition control of the four-product Kaibel DWDC is very interactive. Therefore, only PI-TC is used for a comparison with MPC-PCC. The PI-TC is also the control scheme when MPC fails to work. 4.2. Model Predictive Control. The purity composition control structure of the four-product Kaibel DWDC using MPC and PI control (MPC-PCC) is shown in Figure 11. The inputs to MPC are six controlled variables: four product compositions of the product streams and two impurity compositions of the top and bottom “products” of the prefractionator. The outputs of MPC are six manipulated variables: the set points of TCP, TC1, TC2, and TC3; reboiler duty; and the flow rate of the thermally coupled vapor flowing back to the prefractionator. The Pareto front for case 2 is shown in Figure 12. The horizontal and vertical axis represent obj1 and obj2. The hollow point with an obj1 value of 0.00014 and an obj2 value of 0.02060 is selected as the final optimum solution. The optimal tuning parameters of MPC are shown in Table 6. 4.3. Results and Discussion. The dynamic responses to ±10% feed flow rate disturbances or ±10% feed composition disturbances using PI-TC and MPC-PCC for case 2 are shown in Figure 13. The results prove that PI-TC remains workable for the four-product Kaibel DWDC when MPC in the outer layer of MPC-PCC fails to work. The settling times using PI-TC are comparable to those using MPC-PCC. Nevertheless, with MPC-PCC, the maximum deviations are not large and the steady-state deviations are very small. This proves that MPCPCC is the best among the three control structures studied in case 2.

liquid stream flowing back to the prefractionator (Lp) is the manipulated variable of the temperature controller for the prefractionator (TCP) to control TP12. The reflux flow rate (L) of the main section is manipulated for temperature controller 1 (TC1) to control TM12. The first side product flow rate (S1) is varied for temperature controller 2 (TC2) to control TM43. The second side product flow rate (S2) is manipulated for temperature controller 3 (TC3) to control TM64. According to the control rules suggested by Skogestad,26 one stabilizing temperature control loop is needed for each split. Four temperature control loops are designed for the four sharp splits in the four-product Kaibel DWDC. The prefractionator performs a sharp AB/CD split and needs one temperature controller, TCP. The upper part of the main section completes a sharp A/B split and needs one temperature controller, TC1.

5. CONCLUSIONS In this paper, the feasibility and effectiveness of the composition− temperature cascade control of DWDC with PI control for the inner layer and MPC for the outer layer were verified. In the cascade control structure, PI controllers can maintain stable operation of the DWDC and MPC can potentially improve performance sharply. Thus, the control structure remains workable in an industrial application even if the MPC in the outer layer fails to work. This enhances the acceptance of MPC for DWDC. Two typical cases are studied in this paper, that is, the threeproduct Petlyuk DWDC and four-product Kaibel DWDC. The PI-ICC and MPC-ICC schemes are proposed to control the impurity compositions of the product streams. The PI-PCC and MPC-PCC are proposed to control the purity compositions of the product streams. The PI-TC is proposed

Figure 11. Composition−temperature cascade control of the fourproduct Kaibel DWDC using MPC and PI control (MPC-PCC).

Table 6. Controller Tuning Parameters of MPC for Case 2 manipulated variables weights

TCPSP

Vp

TC1SP

TC2SP

TC3SP

0 0.108

0 0.653

y

w wu wΔu

0 0 0 0.672 3.609 0.779 prediction horizon p 100

controlled variables QR/F

yDp,C

5.989 0 7.228 control horizon m 4 J

xD,A

xS1,B

xS2,C

xBp,B

xB,B

8.063

4.456

5.777

6.354

4.174

sampling time Δk 0.01 h DOI: 10.1021/acs.iecr.8b06179 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 13. continued

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Figure 13. continued

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Figure 13. PI-TC vs MPC-PCC for case 2: dynamic responses to ±10% feed flow rate disturbances or ±10% feed composition disturbances

ORCID

in order to compare the dynamic performances when MPC in the outer layer of MPC-PCC fails to work. The dynamic performances of these five control structures are compared and evaluated. The dynamic performances using MPC are much better with fewer oscillations, shorter settling times, and reduced maximum and steady-state deviations. The MPC-PCC is the best of the five control structures studied in this work. This study proves that MPC is well-suited for the complicated, strongly interactive, and multi-input, multi-output DWDC.

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Xing Qian: 0000-0002-5208-2672 Kejin Huang: 0000-0003-2649-0223 Shengkun Jia: 0000-0003-4524-0588 Yang Yuan: 0000-0003-4947-2474 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the National Nature Science Foundation of China (21808007, 21878011, 21676011, and 21576014), Open Foundation of State Key Laboratory of Chemical Engineering (No. SKL-ChE-18B01), China Postdoctoral Science Foundation (No. 2017M620587), and Fundamental Research Funds for the Central Universities (ZY1837 and ZY1930) are gratefully acknowledged. The very constructive suggestions from Professor Sigurd Skogestad (Norwegian University of Science and Technology) during X.Q.’s visit to Process Control Group is also gratefully acknowledged.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.8b06179.

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Nominal steady-state data of the three-product DWDC, dynamic responses to ±20% feed flow rate disturbances or ±20% feed composition disturbances using three pure PI controls (PI-ICC, PI-PCC, and PI-TC), and the dynamic responses to ±20% feed flow rate disturbances or ±20% feed composition disturbances using PI-ICC and MPC-ICC (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*K.H.: College of Information Science and Technology, Beijing University of Chemical Technology, North Third Ring Road 15, Chaoyang District, Beijing 100029, China. E-mail: [email protected]. *S.J.: School of Chemical Engineering and Technology, Tianjin University, Beiyangyuan Campus, Yaguan Road 135, Jinnan District, Tianjin 300350, China. E-mail: [email protected]. M


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