Mario Francisco a , Silvana Revollar b , Pastora Vega a, Rosalba Lamanna b

1. A Comparative Study Of Deterministic And Stochastic Optimization Methods For Integrated Design Of Processes Mario Franciscoa, Silvana Revollarb, Pastora Vegaa, Rosalba Lamannab aDepartamento de Informática y Automática. Universidad de Salamanca. Spain b Universidad Simón Bolívar. Dpto. de Procesos y Sistemas. Venezuela

2. Schedule • Introduction • Description of the process and plant controller • Formulation of the optimization problem • Process constraints • Controllability constraints • Solving the problem by deterministic and stochastic methods • Sequential Quadratic Programming • Genetic algorithms • Simulated annealing • Hybrid method • Integrated design results • Open loop design • Closed loop design • Conclusions

3. $ $$ T P V Introduction Classical process design: Sequential procedure Selection of the optimal process structure Dimensioning, and determination of working point Synthesis and Design Design might result in plants difficult to control Control system design

4. Process Synthesis Process Design Controllability Analysis Introduction Integrated design The integrated-process-and-control-system-design lies in the systematic study of the influence of the process design on the stability and controllability of the system, even before the process flowsheet is defined. Open loop and closed loop indices are considered for design Better controllable plants: Trade off between design and control

5. Introduction Open loop controlability contraints • Open loop eigenvalues analysis • Analysis of controllability indices derived from system linearized model to determine disturbance rejection capability Closed loop criteria • Proper tuning of the controller parameters to ensure: • closed loop stability • good disturbance rejection • optimization of dynamical performance indexes Min f (x,y) Constraints: h(x) = 0 g(x)  0 g(t,x)  0 x   The mathematical formulation for the integrated design results into a non-linear dynamical optimisation problem which considers controllability constraints and dynamical performance indices.

6. Introduction Objective • Perform the Integrated Design of an activated sludge process considering controllability indices such as disturbance sensitivity gains, the H norm, and dynamical performance indices as the ISE norm. • Apply and compare stochastic and deterministic optimization methods to solve the dynamical optimisation non-linear problem that emerges from the Integrated Design. • Propose an hybrid methodology that combines both deterministic and stochastic optimisation methods for the solution of the optimisation problem.

7. Schedule • Introduction • Description of the process and plant controller • Formulation of the optimization problem • Process constraints • Controllability constraints • Solving the problem by deterministic and stochastic methods • Sequential Quadratic Programming • Genetic algorithms • Simulated annealing • Hybrid method • Integrated design results • Open loop design • Closed loop design • Conclusions

8. Nitrate internal recycle ASU5 ASU2 ASU4 EFFLUENT ASU3 ASU1 waste RASS • Physical characteristics • 5 biological tanks in series with a secondary settler • Operational characteristics • ASU1 and ASU2 unaereated but fully mixed • Nitrate internal recycle • RAS recycle from the underflow of the secondary settler Formulation of the OptimizationProblem

9. The control of this process aims to keep the substrate at the output (s1) below a legal value despite the large variations of the flow rate and the substrate concentration of the incoming water (qi and si). A PI controller was chosen Si disturbances Qi disturbances Formulation of the OptimizationProblem

10. Schedule • Introduction • Description of the process and plant controller • Formulation of the optimization problem • Process constraints • Controllability constraints • Solving the problem by deterministic and stochastic methods • Sequential Quadratic Programming • Genetic algorithms • Simulated annealing • Hybrid method • Integrated design results • Open loop design • Closed loop design • Conclusions

11. Mass balances constraints Formulation of the OptimizationProblem Objective function: Investment and operation cost Activated sludge process superstructure

12. Limits in hydraulic capacity and sludge age in the settler Limits in the relationship between the input, recycled and purge flow rates: Residence times and mass loads in the aeration tanks: Formulation of the OptimizationProblem Objective function: Investment and operation cost Activated sludge process superstructure

13. Objective function: Investment and operation cost Activated sludge process superstructure Controllability Constraints: The H∞ norm The disturbance transfer function: Good disturbance rejection For the closed loop design: The ISE norm as a dynamical performance index Formulation of the OptimizationProblem

14. Schedule • Introduction • Description of the process and plant controller • Formulation of the optimization problem • Process constraints • Controllability constraints • Solving the problem by deterministic and stochastic methods • Sequential Quadratic Programming • Genetic algorithms • Simulated annealing • Hybrid method • Integrated design results • Open loop design • Closed loop design • Conclusions

15. Genetic Algorithms Genetic algorithms are general optimization methods which mimics principles of natural evolution Chromosome codification: Real coded -The variables are normalised Open loop Closed loop Techniques to deal with constraints: Stronger penalty function Crossover technique: Parameters used for solving the problem: Population size of 60 individuals and a maximum generation number of 300.

16. Simulated Annealing The simulated annealing is inspired in the annealing process to get minimum energy states in a solid.The states represent candidate solutions and the energy is the cost associated to each state Codification: Real coded -The variables are normalised Starting point Acceptance probability New state Parameters used for solving the problem: Linear cooling schedule for c, decreasing rate 0.88

17. For open loop design: Optimization of function f1 considering ISE< is sufficient For closed loop design: A methodology consisting of an iterative two steps approach is proposed to solve closed loop Integrated Design. (Suboptimal solution) • Step 1: Performs the plant design optimizing f1 • Step 2: Performs he controller tuning optimizing f2 Optimal plant parameters Controller parameters: Kp, Ti constant Optimal plant parameters with the best controller Optimal PI controller parameters: Kp Ti Plant parameters constant Sequential Quadratic Programming

18. Hybrid method • Genetic Algorithms have the advantage of avoiding local minima and the ability of providing solutions when dealing with complex problems, but sometimes, do not arrived to feasible solutions. • SQP have been broadly applied obtaining good solutions in a reasonable amount of computing time, mainly if the search starts near the optimum, but might not converge to any solution when dealing with complex problems. Hybrid method Step 1: Genetic Algorithm Step 2: SQP

19. Schedule • Introduction • Description of the process and plant controller • Formulation of the optimization problem • Process constraints • Controllability constraints • Solving the problem by deterministic and stochastic methods • Sequential Quadratic Programming • Genetic algorithms • Simulated annealing • Hybrid method • Integrated design results • Open loop design • Closed loop design • Conclusions

20. Integrated Design without controllability Open loop integrated design Norm H< Closed loop integrated design ISE< Results

21. V (m3): 5046 A (m2): 1885 S1 (mg/l): 87.5 ISE: 588790 Cost : 0.040 Ds (1): 2.342 Ds (2): 2.700 Norm H: 0.2900 V (m3): 7772 A (m2): 2172 S1 (mg/l): 51.26 ISE: 185350 Cost : 0.083 Ds (1): 1.349 Ds (2): 1.510 Norm H: 0.1600 Results Open loop Integrated Design Norm H< Integrated Design without controllability Closed Loop Integrated Design V (m3): 8611 A (m2): 3026.1 S1 (mg/l): 38.63 ISE: 79771 Cost : 0.1292 Kp: -7.33 Ti: 415.1 Norm H: 0.1080

22. Schedule • Introduction • Description of the process and plant controller • Formulation of the optimization problem • Process constraints • Controllability constraints • Solving the problem by deterministic and stochastic methods • Sequential Quadratic Programming • Genetic algorithms • Simulated annealing • Hybrid method • Integrated design results • Open loop design • Closed loop design • Conclusions

23. Conclusions • The Integrated Design of an activated sludge process considering controllability indices and dynamical performance indices as the ISE norm was successfully performed. • The stochastic methods (SA and GA) and deterministic (SQP) showed good results in open loop design and closed loop Integrated Design with PI controllers. • Hybrid optimization starting with GA and refining solutions with SQP has also been developed, combining advantages of both methods, and giving also good results for Integrated Design. • GA seems very suitable for solving MINLP problems, these results are encouraging for the application of the hybrid method to solve the problems derived from process synthesis, or Integrated Design with model predictive controllers, that also involves integer variables.

24. Disturbances Gains