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Design of a new PID controller using predictive functional control optimization for chamber pressure in a coke furnace

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An improved proportional integral derivative (PID) controller based on predictive functional control (PFC) is proposed and tested on the chamber pressure in an industrial coke furnace. The proposed design is motivated by the fact that PID controllers for industrial processes with time delay may not achieve the desired control performance because of the unavoidable model/plant mismatches, while model predictive control (MPC) is suitable for such situations. In this paper, PID control and PFC algorithm are combined to form a new PID controller that has the basic characteristic of PFC algorithm and at the same time, the simple structure of traditional PID controller. The proposed controller was tested in terms of set-point tracking and disturbance rejection, where the obtained results showed that the proposed controller had the better ensemble performance compared with traditional PID controllers.

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Design of a new PID controller using predictive functional control optimization for chamber pressure in a coke furnace

  1. 1. Research article Design of a new PID controller using predictive functional control optimization for chamber pressure in a coke furnace$ Jianming Zhang 1 State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou 310027, PR China a r t i c l e i n f o Article history: Received 19 August 2016 Received in revised form 31 October 2016 Accepted 15 November 2016 Available online 25 November 2016 Keywords: PID control Predictive functional control time delay processes ensemble performance a b s t r a c t An improved proportional-integral-derivative (PID) controller based on predictive functional control (PFC) is proposed and tested on the chamber pressure in an industrial coke furnace. The proposed design is motivated by the fact that PID controllers for industrial processes with time delay may not achieve the desired control performance because of the unavoidable model/plant mismatches, while model pre- dictive control (MPC) is suitable for such situations. In this paper, PID control and PFC algorithm are combined to form a new PID controller that has the basic characteristic of PFC algorithm and at the same time, the simple structure of traditional PID controller. The proposed controller was tested in terms of set-point tracking and disturbance rejection, where the obtained results showed that the proposed controller had the better ensemble performance compared with traditional PID controllers. & 2016 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Coke furnace plays an important role in the petroleum re- fining industry by providing different oil products for in- dustrial needs [1–4]. Since the yield, quality and safety of the process are associated with proper operation, modeling based control of relevant process variables are very important [5,6]. However, due to the interactions in the whole coke process and batch operation, accurate modeling may result in complex model structure or higher order and poses some complexity for the subsequent controller design [7,8]. From control point of view, PID controllers are the most common ones with wide- spread applications. It is well-known that good performance of a PID controller depends on the tuning of its parameters, therefore, a lot of tuning methods have been proposed. Ziegler and Nichols first proposed the famous tuning guidelines called Z-N method and had been referred to numerous times in practical applications [9]. In order to compensate for the in- sufficiency of Z-N method in dealing with time delay processes, Cohen and Coon proposed an improved parameters tuning approach [10]. Tyreus and Luyben came up with a special parameters tuning method for first-order plus dead time pro- cesses (FOPDT) [11]. Later, internal model control based PID tuning method was proposed to improve the dynamic re- sponses of the closed-loop system [12]. Nowadays, substantial research results on PID tuning for different process models are available [13–17]. However, for industrial coke processes with long time delay and nonlinearity, the application of the aforementioned traditional PID controllers may not achieve the desired effect [18–21]. In order to solve this problem, smith predictor was proposed and known as an advanced PID control algorithm to deal with the time delay processes [22–26]. Unlike PID controllers, smith predictor converts the closed-loop system transfer function into a formula in which time delay is separated alone, which results in the fact that the response curve of the entire closed-loop system is the same as a corresponding closed-loop system without time delay. However, it is not very easy to enhance process performance with smith pre- dictor due to the fact that accurate process models cannot be ob- tained for industrial processes. With the development of computer control technologies, model predictive control (MPC) was proposed as an effective advanced algorithm and significantly researched in handling chemical processes with time delays, together with the relevant issues of input-output modeling and state space model identi- fication or estimation [27–40]. MPC is successful because it bears the prediction and an effective feedback correction me- chanism in designing a control law, which can consider the complex process behavior with uncertainties. However, How- ever, limited by the cost, hardware and so on, the application of MPC is more troublesome than PID, which makes it necessary to Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2016.11.006 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved. ☆ A novel PID controller optimized by predictive functional control (PFC) is pro- posed and tested on chamber pressure in the industrial coke furnace in this paper. The resulting controller displays the performance of both PFC and PID control with easy implementation in practice. The performance of the proposed PID controller is compared with traditional PID controller, from which results show that the pro- posed PID controller provides better performance than traditional PID controller. E-mail address: jmzhang@csc.zju.edu.cn 1 Tel.: +86-571-87952233. ISA Transactions 67 (2017) 208–214
  2. 2. find a compromise between MPC and PID. Some representative strategies are as follows. Xu, Li, and Cai proposed a novel PID controller that was optimized by generalized predictive control (GPC) and obtained the similar performance as GPC [41], how- ever, the resultant PID is obtained through linear approxima- tion. A novel multivariable predictive fuzzy-PID control system was developed by incorporating the fuzzy and PID control ap- proaches into the predictive control framework by Savran [42]. Lee and Yeo developed a new PID controller on the basis of simplified GPC [43]. Many other advanced control algorithms have also been introduced to improve the performance of tra- ditional PID controllers in dealing with time delay processes [44–48]. In this study, a predictive functional control (PFC) algorithm is used to derive a new PID controller, which has the excellent performance of PFC algorithm and simultaneously, the same structure as traditional PID controllers. The chamber pressure in a coke furnace is considered as a case study, where the well- known PID controllers tuned by the Z-N method, T-L method and IMC method are used for comparison with the proposed PID controller. 2. Process model for the chamber pressure and conventional PID control strategies Though industrial processes are generally nonlinear with un- certainties, linear models based on step-response test can gen- erally grasp the main process characteristics [49,50]. For process uncertainties, the subsequent controller design and tuning will play an important role. For simplicity and in accordance with the chamber pressure control case study later in this article, we choose the FOPDT model as follows. ( ) = + ( ) τ− G s Ke Ts 1 1a s where K is the steady process gain, T is the time constant and τ is the time delay of the process. In this section, the conventional PID controller tuning methods used in this paper are introduced as follows. In the following, K T T, ,p i d will be denoted as the proportional coefficient, integral time and differential time of the PID controller respectively. For the Z-N method, its formulas are = = = ( )K K T P T P0.6 , 0.5 , 0.125 1bp u i u d u where K P,u u are the critical gain and the cycle of critical oscillation. The T-L method can be described as = = = ( )K K T P T P/2.2, 2.2 , /6.3 1cp u i u d u where Ku is the critical gain. Pu is the cycle of critical oscillation. The detailed calculation formulas of IMC-PID are as follows. τ λ τ τ τ τ = ( + ) ( + ) = + = + ( ) K T K T T T T T 0.5 0.5 , 0.5 , 2 1d p i d where λ is a tunable parameter, and usually λ τ> 0.8 . Note that the process is modeled as the common first order plus dead time model, where this will facilitate the corresponding controller design. The controller design will be based on this simple model to cope with the uncertainty caused by modeling error using the PFC optimization based PID. 3. Proposed parameters tuning method of the PID controller In this section, PFC algorithm is introduced to optimize the parameters of the PID controller, where the obtained PID con- troller has the same performance as PFC algorithm and the simple structure as traditional PID controllers simultaneously. The aforementioned process model can be transformed into a corresponding differenced equation model by adding a zero-order holder at the sample time Ts. ( ) = ( − ) + ( − ) ( − − ) ( )y k a y k K a u k L1 1 1 2m m m m where ( )y km is the output of the process model at time instant k, = −a em T T/s , ( − − )u k L1 is the input of the process model at time instant − −k L1 , L is the time delay with τ=L T/ s. Based on Eq. (2) and to obtain the performance of PFC, we need to construct a process model without time delay, which results in the following formulation ( ) = ( − ) + ( − ) ( − ) ( )y k a y k K a u k1 1 1 3mav m mav m where ( )y kmav is the output of the process model that is free of time delay at time instant k. The output prediction based on Eq. (3) after P steps is ( + ) = ( ) + ( − ) ( ) ( )y k P a y k K a u k1 4mav m P mav m P where ( + )y k Pmav is the output prediction of the process model that is free of time delay for time instant +k P made at time in- stant k, P is the prediction horizon. Then we need to calculate the corrected process output as follows ( ) = ( ) + ( ) − ( − ) ( )y k y k y k y k L 5pav p mav mav where ( )y kp is the actual output of the process at time instant k, ( )y kpav is the corrected output of the process at time instant k. We choose the reference trajectory ( + )y k Pr and the cost function J1 as β β( + ) = ( ) + ( − ) ( ) = ( ( + ) − ( + ) − ( )) ( ) = ( ) − ( ) ( ) y k P y k c k J y k P y k P e k e k y k y k 1 min 6 r P p P r mav pav mav 1 2 where β is the smoothing factor, ( )c k is the set-point at time instant k, ( )e k is the error between the corrected output and the process model that is free of time delay at time instant k. Note that the prediction horizon P and the smoothing factor β both have the impact on the final control input calculation, which are deemed as the tuning parameters. They will be chosen to result in a stable and acceptable closed-loop control system. The PID controller used here is an incremental PID controller as follows. ( ) = ( − ) + ( )( ( ) − ( − )) + ( ) ( ) + ( )( ( ) − ( − ) + ( − )) ( ) = ( ) − ( ) ( ) u k u k K k e k e k K k e k K k e k e k e k e k c k y k 1 1 2 1 2 7 p i d 1 1 1 1 1 1 1 where ( ) ( ) ( )K k K k K k, ,p i d are the proportional coefficient, the in- tegral coefficient and differential coefficient at time instant k re- spectively, ( )e k1 is the error between the set-point and the process output at time instant k. Eq.(7) can be converted into the following ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ( ) = ( − ) + ( ) ( ) ( ) = ( ) ( ) ( ) ( ) = ( ) + ( ) + ( ) ( ) = − ( ) − ( ) ( ) = ( ) ( ) = ( ) ( − ) ( − ) ( ) Τ Τ Τ u k u k w k E k w k w k w k w k w k K k K k K k w k K k K k w k K k E k e k e k e k 1 , , , 2 , , 1 , 2 8 p i d p d d 1 2 3 1 2 3 1 1 1 From Eqs. (3)–(8) and taking a derivative of the cost function J1, we can obtain the optimal control law J. Zhang / ISA Transactions 67 (2017) 208–214 209
  3. 3. ( ) α α α ( ) = ( ( + ) − ( ) − ( − ) ( − ) − ( )) ( ) ( − ) ( ) ( )Τ 9 w k y k P y k K u k e k E k K E k E k 1 1 1 r m P mav m m P m m P then ( ) ( ) ( )K k K k K k, ,p i d are obtained ( ) = − ( ) − ( ) ( ) = ( ) − ( ) − ( ) ( ) = ( ) ( ) K k w k K k K k w k K k K k K k w k 2 10 p d i P d d 2 1 3 It can be easily seen that wwill be infinite if ( ) ( )ΤE k E k is ap- proximating zero, that is, ( ) ( ) ( )K k K k K k, ,p i d will be infinite and is unrealistic for the PID controller when the control system has reached the steady state. Thus it is necessary to set a small per- missible error limitation δ, where ( ) ( ) ( )K k K k K k, ,p i d remain un- changed as the previous sampling instant ⎧ ⎨ ⎪ ⎩ ⎪ ⎧ ⎨ ⎪ ⎩ ⎪ δ δ ( ) = ( − ) ( ) = ( − ) ( ) = ( − ) ⋯⋯ ( ) ≤ ( ) = − ( ) − ( ) ( ) = ( ) − ( ) − ( ) ( ) = ( ) ⋯⋯ ( ) > ( ) K k K k K k K k K k K k e k K k w k K k K k w k K k K k K k w k e k 1 1 1 2 10a p p i i d d p d i P d d 1 2 1 3 1 The control input at sampling instant k is ( ) = ( − ) + ( )( ( ) − ( − )) + ( ) ( ) + ( )( ( ) − ( − ) + ( − )) ( ) u k u k K k e k e k K k e k K k e k e k e k 1 1 2 1 2 11 p i d 1 1 1 1 1 1 At each time instant, if δ( ) ≤e k1 , ( ) ( ) ( )K k K k K k, ,p i d will keep the same value as the previous time instant. If δ( ) >e k1 , ( )w k is obtained by Eq. (9), then ( ) ( ) ( )K k K k K k, ,p i d can be acquired form Eq. (10). Remark 1. It is shown that the proposed PID parameters are op- timized, where in fact a PFC is designed for the considered process and the final PID controller is implemented on the process. The advantages are that the process controller is designed based ad- vanced control using PFC theory and three PID parameters are obtained but not fixed through optimization for the controller to achieve improved control performance. 4. Case study: furnace pressure system control in the coke furnace 4.1. Coke unit description In Fig.1, a sketch of the unit is given. The residual oil is sepa- rated into two ramifications (FRC8103,FRC8104) firstly and sent into the furnace (F101/3) to be heated to about 330°C, then they go to the fractionating tower (T102) to exchange heat with gas oil. After the heat exchange, the mixed oil is called circulating oil and is sent to enter the furnace (F101/3) to be heated to about 495 °C. The two branches add together and go to the coke towers (T101/ 5,6) for coke removing finally. 4.2. The control target The chamber pressure in the coke furnace is one of the im- portant output variables. Here the manipulated variable is the opening the flue damper and the steady state operating point is summarized in Table 1. 4.3. The process model Under the conditions of Table 1, the process model is roughly modeled as residual oil FRC8104 36.9 t/h FRC8103 37.2 t/h Circulating oil from pumps 102/1,2,3 To T102 To T101/5,6 FRC8107 43.5 t/h FRC8108 43.6 t/h P2 Furnace 3 P1PC Pressure Set-point PC Pressure Set-pointy1 y2 u1 u2 Flue damper 1 Flue damper 2 Fig. 1. Overall flow of coke unit. Table 1 Steady state operating conditions. Coke furnaces Radiation output temperature 495 °C Convection output temperature 330 °C Chamber temperature 800 °C Oxygen content 5% Circulating oil flow 35 t/h Coke fractionating tower Tower bottom temperature 350 °C Tower liquid level 70% Coke towers Tower top temperature 415 °C Tower bottom temperature 300 °C Temperature after cooling 85 °C Tower top pressure 0.25 Mpa Table 2 Tuning parameters for servo problem. Parameters IMC-PID T-L Z-N Kp À125 À119 À157 Ti 170 397 90 Td 17.65 28.57 22.5 Parameters PFC-PID P 2 β 0 J. Zhang / ISA Transactions 67 (2017) 208–214210
  4. 4. 0 50 100 150 200 0 0.5 1 1.5 2 k y(k) Output disturbance responses Set-point PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 -400 -300 -200 -100 0 100 200 k u(k) PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 0 0.5 1 1.5 2 k Input disturbance responses y(k) Set-point PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 -400 -300 -200 -100 0 100 200 u(k) k PFC-PID IMC-PID Z-N T-L Fig. 2. (a) Closed-loop servo response under output disturbance for Case 1 (sampling time Ts ¼20 s). (b) Closed-loop servo response under input disturbance for Case 1 (sampling time Ts ¼20 s). 0 50 100 150 200 0 0.5 1 1.5 2 Input disturbance responses k y(k) Set-point PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 -400 -300 -200 -100 0 100 200 u(k) k PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 0 0.5 1 1.5 2 Input disturbance responses k y(k) Set-point PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 -400 -300 -200 -100 0 100 200 u(k) k PFC-PID IMC-PID Z-N T-L Fig. 3. (a) Closed-loop servo response under output disturbance for Case 2 (sampling time Ts ¼20 s). (b) Closed-loop servo response under input disturbance for Case 2 (sampling time Ts ¼20 s). J. Zhang / ISA Transactions 67 (2017) 208–214 211
  5. 5. ( ) = − + ( ) − G s e s 0.02 150 1 12 s40 4.4. Performance of the proposed PID controller and discussions A comparison with the aforementioned PID controllers tuned by Z-N method, T-L method and IMC method was made to verify the performance of the proposed PID controller. In this study, we obtain the parameters of model/plant mis- matches by Monte Carlo simulation. For the chamber pressure process, we assume the maximum of 30% uncertainty from the original process model, which are τ= − = =K T0.02, 150, 40. The three cases of parameter uncertainties are generated as below: Case 1. the mismatched process parameters are estimated as τ= − = =K T0.025, 187, 49. Case 2. the mismatched process parameters are estimated as τ= − = =K T0.016, 113, 51. Case 3. the mismatched process parameters are estimated as τ= − = =K T0.024, 121, 29. Simulations are as follows. The set-point is changed from 0 to 1 at time instant =k 0. A continuous disturbance with amplitude of À0.1 and an input disturbance with amplitude of 30 are added 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 k y(k) Output disturbance responses Set-point PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 -400 -300 -200 -100 0 100 200 300 k u(k) PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Input disturbance responses k y(k) Set-point PFC-PID IMC-PID Z-N T-L 0 50 100 150 200 -400 -300 -200 -100 0 100 200 300 u(k) k PFC-PID IMC-PID Z-N T-L Fig. 4. (a) Closed-loop servo response under output disturbance for Case 3 (sampling time Ts ¼20 s). (b) Closed-loop servo response under input disturbance for Case 3 (sampling time Ts ¼20 s). Table 3 Statistical results for each method. Case item Overshoot (%) Response time (s) Settling time (s) Case 1 PFC-PID 15.89 140 320 IMC-PID 36.13 120 420 T-L 20.59 120 1020 Z-N 97.3 100 1120 Case 2 PFC-PID 5.63 140 440 IMC-PID 26.73 120 500 T-L 13.29 120 1500 Z-N 86.4 100 1380 Case 3 PFC-PID 0.65 80 320 IMC-PID 30.39 80 420 T-L 25.9 80 1240 Z-N 78.54 80 1980 Table 4 Mean tracking error (MTE) for each method. Case item Output disturbance Input disturbance Case 1 PFC-PID 0.0410 0.0373 IMC-PID 0.0420 0.0387 T-L 0.0699 0.0621 Z-N 0.0732 0.0705 Case 2 PFC-PID 0.0399 0.0366 IMC-PID 0.0425 0.0396 T-L 0.0855 0.0786 Z-N 0.0807 0.0780 Case 3 PFC-PID 0.0268 0.0278 IMC-PID 0.0307 0.0316 T-L 0.0642 0.0664 Z-N 0.1419 0.1427 J. Zhang / ISA Transactions 67 (2017) 208–214212
  6. 6. to the output at time instant =k 100 respectively. Here if the tracking error is small enough and acceptable, we think that the controlled system is indeed well controlled. Compared with the set-point (here selected as 1), we deem that the permissible error as 10À4 is small enough. It means that the parameters of the proposed PID controller is invariable and remain the same as the previous sampling instant when the absolute value of the error between actual chamber pressure value and set-point is less-than 10À4 . The parameters of PFC algorithm and the traditional PID controllers are shown in Table 2. Figs. 2–4 show the responses of three cases. The ensemble performance of the proposed PID controller is better than the other three PID controllers. In Fig. 2, we can see that the responses of proposed method and IMC method are smoother than the other two methods that show more drastic oscillations. Specifically, the response of Z-N method shows an unacceptable overshoot, whereas the tracking performance of T-L method is worse than the other three methods. Compared with the proposed method, the performance of IMC method is inferior because its curve shows a bigger overshoot and a more drastic oscillation. In Figs. 3–4, the situation is the same as that in Fig. 2, where the performance of proposed method is better than the other three methods. Here Z-N method shows an unsatisfactory overshoot and oscillation which is more obvious in Fig.4, T-L method presents the bad tracking performance, whereas performance of the IMC method still loses to the proposed method in control performance. Table 3 shows the statistical results of overshoot, response time and settling time for each method. We can easily find that the ensemble control performance of the proposed PFC-PID control strategy is the best. The settling time of the proposed method is the smallest under all cases. The statistical results of mean tracking error (MTE) for each method are shown in Table 4. It is obvious that the MTE of the proposed PID controller is the smallest, which verifies the aforementioned results. 5. Conclusion An improved PID controller optimized by PFC algorithm in this paper. The case study on the chamber pressure of a coke furnace shows that the proposed PID controller shows improved perfor- mance than traditional PID controllers under conditions of various disturbances and model/plant mismatches, where simultaneously has the same structure as PID controller. References [1] Sawarkar AN, Pandit AB, Samant SD, Joshi JB. Petroleum residue upgrading via delayed coking: a review. Can J Chem Eng 2007;85(2):1–24. [2] Haseloff V, Friedman YZ, Goodhart SG. Implementing coker advanced process control. Hydrocarb Process 2007;86(6):99–103. [3] Zhang R, Cao Z, Lu R, Li P, Gao F. State-space predictive-p control for liquid level in an industrial coke fractionation tower. 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