Systematic approach in tuning PID controller for a high purity control of Binary Distillation column based on reduced order model: A Case study Vetriselvi V1, N. Pappa2, Anuj Abraham3 Department of Instrumentation Engineering MIT Campus, Anna University Chennai, India 1 [email protected], [email protected], [email protected] Abstract— This paper describes a systematic way for tuning PID control structure for a Binary Distillation column process. An approach to various Model Order Reduction (MOR) techniques is compared with the original distillation column process. The modeling and simulation of nonlinear model of multivariable is implemented in MATLAB and a comparative study is analyzed for full order linearised model using Taylor’s Series expansion, Jacobian linearization method and reduced order linear model namely Balanced Truncation, Singular perturbation, Hankel Norm approximation methods. Also controller performance evaluated for conventional PID controller for simulated model. The result shows consistently good servo and regulatory response. distillate flow rate, bottom flow rate, cooling water flow rate and control variables are concentration of distillate and bottom, level of condenser and reboiler. Index Terms—Model Order Reduction, Binary Distillation Column, Balanced Truncation, Singular perturbation, Hankel Norm. I. INTRODUCTION Distillation column plays a very important role in chemical processes mainly petro-chemical and refinery industries, for an effective way to separate mixtures of liquid or vapor mixture of two or more substances into its component fractions of desired purity. Binary distillation column is used to separate two components. The schematic diagram of a general distillation column is shown in Fig. 1. Distillation column consists of a vertical column, where plates or trays are used to increase the component separations. Distillation column is separated into two sections. They are stripping section and rectification section. The trays above the feed tray is called stripping section. The trays below the feed tray are called rectification section. Reboiler and condenser are used as heat duties. Condenser is used to condense distillate vapor and reboiler is used to provide heat for the necessary vaporization from the bottom of the column. Condensed vapor is collected in reflux drum and require amount of it is used as a reflux. The L-V (Liquid-Vapor) structure  is known as the energy balance structure and can be considered as the standard control structure for a dual composition control distillation. The various manipulated variables considered for distillation process in literature are reflux rate, boilup rate, Fig. 1. Schematic diagram of distillation column In this paper the three most important manipulated variables taken are reflux rate, boilup rate. Also, the control variables chosen are concentration of distillate and bottom. The disturbances associated with distillation column are feed flow rate and concentration of feed. The vertical column is designed for 14 trays and the list of process parameters considered is shown in Fig. 2. Tray n (n=2 to 7): Mx n = V(y n-1 - y n ) + (L + L F )(x n+1 - x n ) (6) Re-boiler (n=1): M B x1 = (L + L F )x 2 - Vy1 - Bx1 (7) C. Process data for distillation column under normal operating condition The process data are based on a real petroleum project Petro Vietnam Gas Company . The input feed consist of LPG and Naphtha. The relative volatility α under operating conditions is 5.68. The properties and variations of the feed includes molar weight, liquid density, feed composition of feed under operating conditions are listed in Table I. TABLE I. PROPERTIES OF FEED Properties Fig. 2. Process parameters of Distillation Column II. MATHEMATICAL MODELING A. Assumptions The various assumptions considered for the distillation column modeling are given below:  a) The relative volatility α is constant throughout the column. This means the vapor liquid equilibrium relationship can be expressed by, αx n yn = (1) 1 + (α -1)x n th b) c) d) e) f) xn is liquid composition on n stage, yn is vapor composition on nth stage, α is the relative volatility. The overhead vapor is totally condensed in a condenser. The liquid holdups on each tray, condenser, and the reboiler are constant and perfectly mixed The holdup of vapor is negligible throughout the system. The molar flow rates of the vapor and liquid through the stripping and rectifying Sections are constant. The column is numbered from bottom (n=1 for the reboiler, n=2 for the first tray, n=f for the feed tray, n=N+1 for the top tray and n=N+2 for the condenser) B. Dynamic model of distillation column process Based on the assumptions described in section 2.1, the dynamic models of distillation process are expressed by the following component material balance equations: Condenser (n=16): M D x n = (V + VF ) y n-1 - Lx n - Dx n (2) Tray n (n=10 to 15): Mx n = (V + VF )(y n-1 - y n ) + L(x n+1 - x n ) (3) Tray above the feed flow (n=9): Mx n = V(y n-1 - y n ) + L(x n+1 - x n ) + VF (y F - y n ) Tray below the feed flow Mx n = V(y n-1 - y n ) + L(x n+1 - x n ) + L F (x F - x n ) (4) (n=8): (5) LPG Naphtha Molar weight 54.4-55.6 84.1-86.3 Liquid density (kg/m3) 570-575 725-735 Feed composition (vol %) 38-42 58-62 TABLE II. OPERATING CONDITIONS OF DISTILLATION COLUMN PROCESS Stream Feed LPG Naphtha 118 46 144 Pressure (atm) 4.6 4 4.6 Density (kg/m3) 670 585 727 Volume flow rate (m3/h) 22.76 8.78 21.88 Mass flow rate (kg/h) 15480 5061 10405 Plant capacity (ton/year) 130000 43000 87000 Temperature (oC) TABLE III. NOMINAL VALUES DISTILLATION COLUMN PROCESS Variable FOR Stream PROCESS PARAMETERS Molar flow Unit 31.11 Kmole 5.8 Kmole 13.07 Kmole VF Liquid holdup in the column base Liquid holdup on a tray Liquid holdup in the reflux drum Vapor rate in feed 98.5152 kmole/h LF Liquid rate in feed 104.2491 kmole/h V Internal vapor rate 66.3407 kmole/h MB M MD L Internal liquid rate 75.638 kmole/h D Distillate flow rate 92.7597 kmole/h B Bottoms flow rate 110.9235 kmole/h OF The operating conditions of distillation column process and nominal values of process parameters are summarized in Table II and Table III respectively. D. Linearization of nonlinear differential equation a) Taylor’s Series Expansion: Multivariable binary distillation column is nonlinear model, which are linearized to perform a simulation and stability analysis. The nonlinear part of the model equations is yn. It can be approximated using Taylor series approximation around the steady-state operating point ( x n =x n ) f(x n )=f(x n )+f (x n )(x n -x n )+ f (x n ) 2 (x n -x n ) +... (8) 2! For x sufficiently close to x , higher order terms will be very close to zero, so we neglect the quadratic and higher order terms to obtain the approximation. y n f(x n )+K n (x n -x n ) where, f(x n )= αx n 1+(α-1)x n , Kn = (9) A ( A, B, C ) 11 A21 By truncating the discard able states, the truncated reduced system is then given by (A11 ,B1 ,C1 ) B. Singular perturbation (SP) The full-order linear model which represents a two inputstwo outputs plant in equation can be expressed as a reduced order linear model as, 1 x D L = G(0) x B 1+τ s V c α (1+(α-1)x n ) A12 B1 , , C1 C2 A22 B2 2 (14) where, G(0) is the steady state gain defined by, b) Jacobian Linearization Process: -1 G(0)=-CA B Vector form of nonlinear model x=f(x,u) (10) z=g(x,u) Nonlinear model of distillation process has 16 state variables (x1 ,x 2 ,...,x16 ) , 2 input variables (L,V) , and 2 output τc = MI + Is lnS M D (1-x D )x D + M B (1-x B )x B Is (15) Is where, τ c is the time constant, M I is the total holdup of liquid inside the column defined by, variables (x D ,x B ) . 14 M I = Mi x1 =f(x1 ,...,x16 ,L,V) (16) i=1 I S is the Impurity sum defined by, x16 =f(x1 ,...,x16 ,L,V) IS =D(1-x D )x D +B(1-x B )x B (11) z1 =x D z 2 =x B State space form of linearized model x=Ax+Bu z=Cx+Du Elements of the linearization matrices A ij f i x j g Cij i x j Bij x ,u x ,u f i u j g Dij i u j S is the Separation factor defined by, (1-x B )x D S= (1-x D )x B (17) (18) C. Hankel norm approximation (12) The hankel norm of a system is defined, as in Eq.19, G= A,B,C,D H x ,u (13) G x ,u 2 y (t) dt 2 H sup 0 2 u (t) dt (19) 0 III. MODEL ORDER REDUCTION (MOR) The requirement of model order reduction is that the reduced order model so obtained should retain the important and key qualitative and quantitative properties such as stability, transient and steady state response etc. of the original system. A. Balanced Truncation (BT) An arbitrary system (A,B,C,D) can be transformed into a balanced system (A,B,C,D) via a state-space transformation. The balanced systems states are ordered (descending) by how controllable and observable they are, thus allowing a portion of the form: 0 where, y(t) C e A(t-s) B u(s) Hankel norm gives how much energy can be transferred from past inputs into future outputs through the system G. In control theory, eigenvalues define system stability and Hankel singular values define the "energy" of each state in the system. Its characteristics in terms of stability, frequency, and time responses are preserved by keeping larger energy states of a system based on the Hankel singular values. They can achieve a reduced-order model that preserves the majority of the system characteristics. Mathematically, given a stable statespace system (A,B,C,D) its Hankel singular values are defined as in Eq.20, G 2 H = λ max (PQ)=σ i (20) A. Decoupling control system As shown in Fig.4, to compensate for process interactions additional controllers are used and reduces control loop interaction. where, σi is the hankel singular values, The controllability and observability grammians P and Q respectively satisfies, T T AP+PA =-BB T (21) T (22) A Q+QA=-C C One defines the Hankel operator ΓG of the system G(s) by, 0 ΓG :L2 (, 0] : (ΓG u(t)) C e A(t-s) B u(s) ds, t>0 (23) This method also guarantees an error bound on the infinity norm of the additive error G-G red for well-conditioned Fig. 4. Example of a figure caption. (figure caption) model reduced problems as in balanced truncation method. n G-G red 2 σi (24) K+1 where, σ i are singular values of a given system G(s). IV. CONTROLLER DESIGN OF BINARY DISTILLATION COLUMN Figure 3 shows the composition control diagram of binary distillation column. In this control configuration, the vapour flow rate V and the liquid flow rate L are the control inputs to maintain the specification of the product concentration outputs XB and XD (controlled variable) due to disturbance F (feed flow) and XF (feed concentration). B. Decoupler design equations For multi-loop control, cross-controller T12 is used to cancel the effect of U2 on Y1. T12 GP11 U22 + GP12 U22 = 0 Because U22≠0 in general, then G T12 =- P12 (25) G P11 4 T12 = 3 2 0.0053s +0.2816s +3.8374s +17.5437s+6.2479 4 3 4 3 2 0.0024s +0.1298s +1.558s +7.6499+4.152 Similarly, cross controller T 12 is used to cancel the effect of U1 on Y2. T21 GP22 U11 + GP21 U11 = 0 G T21 =- P21 (26) G P22 T21 = 2 0.0735s +2.1553s +16.4563s +22.4s+4.932 4 3 2 0.0753s +2.2082s +17.0774s 25.1257s+7.4866 V. RESULTS AND DISCUSSION A. Quality of top and bottom product Fig. 5 and Fig. 6 shows the liquid and vapor concentration of top product on each tray for nominal operating conditions of the binary distillation column. Fig. 3. Composition control of binary distillation column Fig. 5 Response of Liquid concentration of LPG on each tray Fig.8 Product quality for nominal and ±10% change Reflux rate When reflux rate is decreased by 10% from its nominal value, the quality of the distillate product increases by 2.5% and the quality of the bottoms product increases by 2.1%. In contrast, when reflux rate is increased by 10% from its nominal value, the quality of the distillate product decreases by 4.3% the quality of the bottoms product decreases by 3.6%. 8.4 Simulation with 10% decreasing and increasing Boilup rate: Fig. 6 Response of Vapor concentration of LPG on each tray From the simulation result the Purity of top product is obtained as 96.45% and Impurity of Bottom product as 3.13%. Simulation with the nominal values of stream, the purity of the distillate product is 96.45% and the impurity of the bottoms product is 3.13%. Fig. 9 shows the product quality for nominal and ±10% change in boilup rate to the binary distillation column. 8.2 Simulation with 10% decreasing and increasing feed flow rate: Fig. 7 shows the product quality for nominal and ±10% change in feed flow rate to the binary distillation column. Fig. 7 Product quality for nominal and ±10% change in feed rate When feed flow rate decreased by 10% from its nominal value, the quality of the distillate product decreases by 6.9% the quality of the bottoms product increases by 2.7%. In contrast, when feed flow rate increased by 10% from its nominal value, the quality of the distillate product increases by 9% and the quality of the bottoms product decreases by 8.6%. 8.3 Simulation with 10% decreasing and increasing Reflux rate: The product quality for nominal and ±10% change in reflux rate to the binary distillation column is shown in Fig.8. Fig. 9 Product quality for nominal and ±10% change Boilup rate When boilup rate is decreased by 10% from its nominal value, the quality of the distillate product decreases by 4.8% and the quality of the bottom product decreases by 4.01%. In contrast, When boilup rate is increased by 10% from its nominal value, the quality of the distillate product increases by 3.5% and the quality of the bottoms product increases by 2.9%. 8.5 Response of nonlinear and linearized model Based on the linearized model the simulation were carried out for nominal values. Response of Top product liquid concentrations of linearized model and simulated binary distillation column process are shown in Fig. 10 8.8 Reduced order linearised model (Gh) based on Hankel norm The Hankel singular values in decreasing order are 3 3 4 5.2172 10 , 2.606 103 , 1.038 10 , 1.4281 10 , 5 3.7081 10 and all remaining singular values are smaller than 106. It is observed from the Fig. 11, that the Hankel singular values that the 5th order of the system captures the majority of the input–output behavior of the system. Fig. 10 Response of nonlinear and linearised model The linearized model response tracks the simulated binary distillation, column process at steady state condition with small deviation in transient condition is shown in Fig. 9. Table 5 Steady state values of xn, yn, Kn on each tray Trays 16 15 14 13 12 11 10 9 xn 0.9645 0.9216 0.8316 0.6824 0.5110 0.3804 0.3087 0.2763 yn 0.9936 0.9853 0.9656 0.9243 0.8558 0.7772 0.7173 0.6844 Kn 0.1868 0.2012 0.2373 0.3230 0.4938 0.7347 0.9502 1.0803 Trays 8 7 6 5 4 3 2 1 xn 0.2657 0.2617 0.2525 0.2325 0.1944 0.1373 0.0765 0.0313 yn 0.6727 0.6682 0.6574 0.6325 0.5781 0.4747 0.3199 0.1551 Kn 1.1284 1.1473 1.1931 1.3027 1.5576 2.1054 3.0806 4.3214 8.6 Reduced order linearised model (Gb) based on BT in state space: The linearised fifth order model (Gb) based on BT in state space is obtained as, 5th order: 1.456 3.084 0.155 8.006 3.227 6.353 4.165 1.923 2.887 1.115 A 0.988 0.915 0.792 1.903 0.882 0.944 12.23 0.5977 3.185 3.061 0.9387 0.5377 0.6563 7.55 16.94 0.1918 0.2162 0.0567 0.2834 0.1192 0.0864 0.0669 0.1311 B 0.0403 0.0049 C 0.0211 0.0346 0.0367 0.0463 0.00315 0.059 0.0121 0.0333 0.0352 0.0043 D T Fig.11 Hankel Singular Values 5th order: 20.69 4.391 4.548 1.85 1.562 0 12.62 1.194 1.375 1.621 A 0 6.285 1.983 1.667 0 0 0 1.378 0.397 0 0 0 0 0 0.397 0.0568 0.9297 0.1871 0.1725 0 0 D 0 0 B 0.113 0.1035 0.03752 0.09185 0.0309 0.00242 C 0.03884 0.009042 0.007691 0.08733 0.006092 0.09442 0.2501 0.1421 0.0944 0.03215 8.9 Comparative study of reduced order model Based on the reduced order model the simulation is carried out for nominal values.Response of linearized model and reduced order model for top and bottom product is shown in Fig.12, Fig.13 0 0 0 0 8.7 Reduced order linearised model (Gs) based on SP in state space: The linearised second order linearised model (Gs) based on SP in state space:is obtained as, 2nd order: A 0 0 0.4639 0.0625 B 0 0 0.4639 0.0625 C 0.0289 0.0445 0 0 0.0349 0.0527 D 0 0 Fig.12 Comparison of linearized model and reduced order models for top product quality Fig.13 Comparison of linearized model and reduced order models for bottom product quality Based on the response of the reduced order model based on Hankel norm approximation captures the majority of the input output behavior of the system. Fig. 14 Servo response of the simulated model based PID controller for top product quality From the controller responses, it is observed that the PID controller is able to track the set point changes with smaller overshoot and lesser settling time. Table 6 ISE, IAE and MSE for original and reduced order systems Top Bottom Product Order 5th order HN 2nd order SP ISE 2.5*10-3 2.54 IAE 2.02 41.45 MSE 8.3*10-8 8.4*10-5 5th order BT 1.31 48.02 4.3*10-5 5th order HN 2nd order SP 5th order BT 2.02*10-4 5.11*10-1 8.10*10-1 0.61 18.13 383.18 6.7*10-9 1.7*10-5 2.6*10-3 From the Table 6 it is observed that the reduced order model obtained using Hankel norm has minimum ISE, IAE and MSE values for both top and bottom product quality.The transfer function of the reduced order model is given below. G11 0.0024 s 4 0.1298s 3 1.558s 2 7.6499 4.152 s 42.1s 4 579.8s 3 2853.3s 2 3697.9s 1049.3 G21 0.0735s 4 2.1553s 3 16.4563s 2 22.4 s 4.932 s 5 42.1s 4 579.8s 3 2853.3s 2 3697.9s 1049.3 G12 0.0053s 4 0.2816 s 3 3.8374s 2 17.5437 s 6.2479 s 5 42.1s 4 579.8s 3 2853.3s 2 3697.9s 1049.3 5 Fig. 15 Servo response of the simulated model based PID controller for bottom product quality The simulation experiment was carried by increasing feed flow rate by 3% from its nominal value. The responsed obtained for the decoupled system with simulated model are shown in Fig.16 and Fig.17 respectively. 0.0753s 4 2.2082 s 3 17.0774 s 2 25.1257 s 7.4866 8. G22 s 5 42.1s 4 579.8s3 2853.3s 2 3697.9s 1049.3 10 Analysis of controller performance Fig.16 Regulatory response of the simulated model PID controller for top product quality. The simulation was carried out for the setpoint change of Purity of top product from 0.964 to 0.99 and Impurity of bottom product from 0.031 to 0.01. Both the controllers are tuned based on ZN closed loop method. Servo response of the simulated model for purity of top product and impurity of bottom product based PID controller for binary distillation column process are shown in Fig.14 and Fig.15 respectively. Fig.17 Regulatory response of the simulated model based PID controller for bottom product quality. It is observed that the PID controller is able to reject the disturbance with smaller overshoot and lesser settling time. 9. Conclusions The first principle model of binary distillation column is developed using governing equations and parameter values. The simulated distillation column is validated under nominal and steady state operating conditions. A linearized model of order 16 is obtained using Taylor’s Series expansion, Jacobian linearization Process. Three different model order reduction techniques namely Balanced Truncation, Singular Perturbation, Hankel Norm approximation are obtained and it is observed that a 5th order reduced model obtained using Hankel Norm captures the majority of behavior of the system. PID controller was designed using simulated model. Designed controller is able to provide good servo and regulatory response. Acknowledgement The authors gratefully acknowledge Anna University, Chennai for providing financial support to carry out this research work under Anna Centenary Research Fellowship (ACRF) scheme. References  Vu TrieuMinhand John Pumwa “Modeling and Adaptive Control Simulation For A Distillation Column’’ 14th International Conference On Modelling And Simulation, 2012.  Pradeep B. Deshpande, Charles A. Plank, “Distillation Dynamics and Control”, Instrument Society of America, 1985.  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