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Integration of Design and Control : Robust approach using MPC and PI controllers

Integration of Design and Control : Robust approach using MPC and PI controllers. N. Chawankul, H. M. Budman and P. L. Douglas Department of Chemical Engineering University of Waterloo. Outline. Introduction Objectives Methodology Case study Results Conclusions. Introduction.

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Integration of Design and Control : Robust approach using MPC and PI controllers

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  1. Integration of Design and Control :Robust approach using MPC and PI controllers N. Chawankul, H. M. Budman and P. L. Douglas Department of Chemical Engineering University of Waterloo

  2. Outline Introduction Objectives Methodology Case study Results Conclusions

  3. Introduction Integrated Approach 1- Design+Control min (capital +operating +variability costs) st stability actuator constraints Traditional Approach 1- Design: min (capital + operating costs) 2- Control for designed plant -stability -actuator constraints -performance specs: Small Overshoot Short Settling Time Large closed loop bandwidth

  4. disturbance output Variability Cost • Variability Cost = Cost of Imperfect Control For a disturbance d (green): What is the cost due to off-spec product (blue)?

  5. Robust Control Approach • To test stability and calculate variability cost we need a model. Nonlinear model: stability (Lyapunov-difficult) variability (numerically- difficult). “Robust” linear model: nonlinear model= family of linear models family of linear models= nominal model +model uncertainty (error)

  6. Introduction Previous Approaches Our approach • Linear Dynamic Nominal Model + Model Uncertainty (Simple optimization problem) • Variability cost into cost function : One objective function • Centralized Control : MPC • Nonlinear Dynamic Model (difficult optimization problem) • Variability cost not into cost function: Multi-objective optimization • Decentralized Control : PI /PID

  7. Objectives of the current work • Variability using MPC based on a nominal model and model error. • Cost of variability in one objective function together with the design cost. • Model uncertainty (as a function of design variables) into the objective function. • The robust stability criteria as a process constraint. • Compare the traditional method to integrated method. • Preliminary study on SISO system (distillation column) with MPC.

  8. Methodology • Model Predictive Control (MPC) • Nominal Step Response and Uncertainty • Process Variability • Optimization - Objective Function - Constraints

  9. MPC Controller d Simplified MPC block diagram W Sd  r u y + + MPC Process + + + - u(k) past future target    y(k)   y(k+1/k)  k k-1 k+1 k+n k+3 k+2

  10. Nominal Step Response and Uncertainty Step Response Model, Sn y u Sn S6 S5 S4 1 S3 S2 S1 t 0 t u Nominal step response model, Sn,nom 1 t 0 -1 Uncertainty, Upper bound Nominal step response Actual Sn Lower bound t Sn-Sn,nom

  11. Process variability-1y = f(W) W (Sinusoid unmeasured disturbance) r=0 y u + MPC Process + + - Substitute (k), u(k-1) into the first equation and apply Z-transform

  12. disturbance output Process variability-2 Assume, W is sinusoidal disturbance with specific d. (alternatively, superposition of sinusoids) Amplitude of output,y With phase lag Amplitude of disturbance,W Consider worst case variability :

  13. Optimization Minimize Cost(u,c) = Capital Cost + Operating Cost + Variability Cost u,c Such that h(u,c) = 0 (equality constraints) g(u,c)  0 (inequality constraints) u is a vector of design variables. c is a vector of control variables.

  14. Constraints • h(u,c) = 0 (equality constraints) • steady state empirical correlations • g(u,c)  0 (inequality constraints) • manipulated variable constraint • robust stability

  15. Inequality Constraints- 1 Consider the MPC controller gain, KMPC: Manipulated variable constraint where  is a manipulated variable weight The infinity norm of A is the amplitude of the disturbance.

  16. Inequality Constraints- 2 Block diagram of the MPC and the interconnection M- M Z(k) w(k) U(k) W1  W2 Z-1I (k+1/k) u(k) U(k-1) + T2 N1 + Li Kmpc T1 + + H + - 2. Robust stability constraint (Zanovello and Budman, 1999) H N1 + + Mp H - + N2 Z-1I U(k+1) U(k) M z w 

  17. Case study- Distillation ColumnPreliminary study: SISO system  Depropanizer column from Lee, 1994  adjust reflux ratioto control the mole fraction of propane in distillate RR A + Feed - MPC = 0.783 (propane) XD* Ethane Propane Isobutane N-Butane N-Pentane N-Hexane Q

  18. (Equality Constraints) Process Model The mathematic expressions of the process variables (N, D, Q) as functions of RR  RadFrac model in ASPEN PLUS  different column design, 19 – 59 stages  design variables (number of stages and column diameter) are functions of nominal RR Number of stages VS. RR

  19. Input/Output Model (Equality Constraints) y Sn S3 • First Order Model S2 S1 t Dynamic simulation using ASPEN DYNAMICS Step change on RR by  10 % (19- 59 stages) Upper bound Nominal step response Lower bound 63.2 %

  20. Objective Function Cost = CC(u) + OC(u) + VC(u,c) Annualized capital cost, CC, (Luyben and Floudas, 1994) ($/day) Operating cost, OC ($/day) where Q = reboiler duty (GJ/hr) OP = operating period (hrs) UC = Utility cost ($/GJ)

  21. Variability Cost (Inventory cost) - 1 V1 Variability cost, VC ($/day) - assume sinusoid unmeasured disturbance, W disturbance induces process variability consider a holding tank to attenuate the product variation calculate the volume of the holding tank - calculate the loss due to the product held in the tank V2 where P = product price, N = payoff period (10 years), i = interest rate (10%) and V = volume of the holding tank

  22. Variability Cost (Inventory cost) - 2  The worst case variability: A simple mass balance Feed disturbance Distillation Column Cin Cout Holding V F out F in The required volume of the holding tank spec

  23. Two different approaches Traditional Method • Integrated Method Robust Performance (Morari, 1989)

  24. W RR  D (m) N Capital cost ($/day) Operating cost ($/day) Variability cost ($/day) Total cost ($/day) 1 2.4 0.1849 5.8 31 551 654 68 1273 3 2.8 0.1705 6.23 28 530 703 168 1401 5 3.0 0.1653 6.5 27 529 726 257 1512 7 3.4 0.1357 6.9 25 538 774 284 1596 Results-1 Results from Integrated Method: W is a product price multiplier.

  25. W Total Cost of integrated method ($/day) Total Cost of traditional method ($/day) Saving ($/day) % Saving 1 1273 1297 24 1.8 3 1401 1481 80 5.4 5 1512 1665 153 9.2 7 1596 1849 253 13.7 Results-2 Comparison using Traditional and Integrated methods

  26. IMC Control (Morari and Zafiriou, 1989) is used. • Internal Model Control, IMC Gd(s) d + r y + + C(s) Gp(s) - + y’ - Gp(s) F(s)  IMC-based PID parameters for

  27. Results Comparison using both methods

  28. Conclusions • single objective function • linear dynamic model + model uncertainty • MPC variability cost is explicitly incorporated in the objectivefunction • integrated approach results in lower costs • - savings can be significant; >13% for high value products On-going work: Formulate the MIMO problem with MPC

  29. Acknowledgement Funding was provided by The Natural Sciences and Engineering Research Council (NSERC)

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