Use of Regularization Functions in Problems of Dynamic Optimization. Prof. Evaristo Biscaia Jr. PEQ/COPPE/UFRJ

Transcription

1 Use of Regularization Functions in Problems of Dynamic Optimization Prof. Evaristo Biscaia Jr. PEQ/COPPE/UFRJ

2 Original Idea: Consistent initialization of differential algebraic systems - R.C. Vieira & E.C. Biscaia, Jr. Computers and Chemical Engineering 25 (2001) A fictitious steady-state condition can be considered as a consistent initial condition t dz t, t,, t dt F z u 0 t0 DAE system: at t=t 0 t dz u uss 0 dt tt 0 and F,,, ss t0 zss 0 uss 0 is a consistent initial condition 2

3 0 when t ureg t uss u t uss freg t, where freg t, 1 when t t t 0 0 First order exponential function: tt0 freg t, 1 exp with 0 1 m th order exponential function: f reg t, m, 1exp m1 j mt t0 1 mt t0 j! j0 3

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5 Sinusoidal regularization function 1 tt0 1 tt0 freg t, 1+sin arctan tt0 Hyperbolic Tangent regularization function f reg 1 2 tt 0 t, 1tanh 5

6 Dynamic simulation of dryers (fixed bed) Relative humidity : 0<RU1 reg, 1, 1 RU RU RU f RU RU reg 6

7 Numerical Applications 1. Condenser-Tank - Pantelides(1988) Kröner et al. (1997) 2. Batch adsorption process 3. Ion exchange resin column 4. Chromatography Column 5. Chromatography Column with reaction 6. Fixed Bed dryer 7. Cross-flow dryer 7

8 Switching between models allowing continuous integration of the problem 8

9 Dynamic optimization: Example Schechter - Maximize x t ; 0u u final ut dx dt dx dt 1 2 max u x1 ux2 x2 A B C (1) (2) u ux x with x 0 =1 3 with 0 =0 3 H u ux x u x ux d1 H u t dt x1 d2 H u 2 t with = with 2 final =1 dt x final H u x1 12 2u x2 x11x16u x22 0u u 2 3 x uu

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11 Optimal Control Problem with a State Variable Inequality Constraint Jacobson and Lele (1969) :Minimize state variable x 3 at t final =1 through manipulation of control variable u(t) 3 3 Minimize x 1 dx dt ut dx1 x2 with x100 dt dx2 2 ux2 with x201 and x2 8t dt x + x u with x

12 Regularized problem: Use of Regularization Functions in Problems of Dynamic Optimization Regularized problem: dx1 x2, reg with x1 00 dt dx2 u x2 with x201 dt dx x1+ x2, reg ureg with x300 dt 2 2 x2, reg x2 freg x2 8 t t x ureg u freg x2 8 t t0.5 16t u 12

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14 HONG, J., 1986, Optimal Substrate Feeding Policy for a Fed Batch Fermentation with Substrate and Product Inhibition Kinetics, Biotechnology and Bioengineering, v. 28, pp Minimize P t final ut 0 S SP, P 1 K1s S K 1p dx 1 SP, x1 with x100 dt g0 S SP, dx2 P SP, x1 with x201 where : 1 K2s S dt K 2 p dx3 te u with x3 Vmax and 0 u u C x max 1 dt S Sf Y x3 x2 P x 3 Singular control arc: Fx1, x2, x3 0 x3 x3 Index 2 DAE system 14

15 F u uext F F z 1 if z where = 0 otherwise Regularized control variable: 1 u u f tt f V V u u reg ext reg s reg max ext 15

16 Control variable u (liter/h) u min u * u min time (h) 16

17 UBRICH, O., SRINIVASAN, B., STOESSEL, F., BONVIN, D., 1999, Optimization of a semi-batch reaction system under safety constraints. minimizet ut final dna kcacbv where na VcA dt dv u dt c c c V c c u u u V V c V Vmax A0 cc0 V0 cav cc V H Tcf t T t CB max C p min max B0 A0 Bin 0 A Bin cb restricted to: Tcf Tmax Regularized control variable: u reg u f reg V Vmax umin u 17

18 Control variable u (liter/h) u * u min time (h) 18

19 LOGSDON, J. S.; BIEGLER, L. T., 1992, Decomposition strategies for large-scale dynamic optimization problems. Chemical Engineering Science, v. 47, n. 4. minimize dx dt x3 tfinal ut 2 dx dt 1 x x u x 10 with x 10x u 1u x with x 0 0 restricted to u u u min max x 1x x Regularized control variable: 1 max 1 2 min ureg 1 freg tts u freg tts u freg tts u u 19

20 Control variable u u * Residence time 20

21 VISSER, E., SRINIVASAN, B., PALANKI, S., BONVIN, D., 2000, A feedback-based implementation scheme for batch process optimization. Journal of Process Control, v.10, pp ˆ 1 Regularized control variable: u 1 f tt f SS u 1 f SS u f tt f X X u f X X u reg reg s reg e reg e max reg s reg max reg max min Control variable u (liter/h) u û time (h) 21

22 SRINIVASAN, B.; PALANKI, S.; BONVIN, D., 2003 Dynamic optimization of batch processes: I. characterization of the nominal solution Computers and Chemical Engineering, v. 27, pp Two regularized control variables. Regularized control variables: dt dt u u f V Vmax umin u and f tt dt reg reg reg temperature dt reg max Control variable u (liter/h) time (h) 22

23 Control variable T ( o C) time (h) 23

24 GENERAL CONCLUSIONS Dynamic optimization problems with inequality constraints appear frequently in process system engineering applications. These constraints usually describe the conditions when control variables or state variables operate in their extreme values, due to economic or security limits. Normally, some inequality constraints are activated along the optimal trajectory, remaining active during a period of time. This behavior could provoke a change in the differential index of the system. This kind of dynamic system is called of varying or floating index system. The proposed methodology incorporates the elimination of the adjoint variables, related with rigorous approach of the optimal dynamic problem, with a regularization technique applied to the constrained variables. This procedure can be easily implemented and presents low computational costs in comparison with traditional techniques, avoiding the boundary value problem associated with the adjoint variables. Benchmark examples have been considered to validate the methodology, and the obtained results were successfully compared with reported results from the literature. 24

25 REFERENCES BIEGLER, L. T., 2007, An overview of simultaneous strategies for dynamic optimization. Chemical Engineering and Processing, vol. 46, pp BIEGLER, L. T., CERVANTES, A. M., WÄCHTER, A., 2002, Advanced in Simultaneous Strategies for Dynamic Process Optimization, Chemical Engineering Science, v. 57, pp BIEGLER, L. T., GROSSMANN, I. E., 2004, "Retrospective on Optimization", Computers and Chemical Engineering, v. 28, pp CHACHUAT, B. C., NONLINEAR AND DYNAMIC OPTIMIZATION From Theory to PracticeIC- 32: Winter Semester 2006/2007. Laboratoire d'automatique, Ecole Polytechnique Federale de Lausanne FEEHERY, W. F., 1998, Dynamic optimization with path constraints. Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. FEEHERY, W. F., BARTON, P. I., 1996, Dynamic Simulation and Optimization with Inequality Path Constraints. Computers and Chemical Engineering, vol. 20, suppl., pp. S707-S712. FEEHERY, W. F.; BARTON, P. I., 1998, "Dynamic optimization with state variable path constraints". Computers and Chemical Engineering, v. 22, n. 9, pp GANI, R., CAMERON, T., 1992, "Modelling for dynamic simulation of chemical processes- the index problem. Chemical Engineering Science, 47, pp

26 REFERENCES GOMES, E. O., 2000, Abordagem algébrico diferencial na solução de problemas de controle ótimo. (In Portuguese), Master of Science Dissertation, Universidade Federal de Uberlândia, Uberlândia, MG, Brasil. HONG, J., 1986, "Optimal Substrate Feeding Policy for a Fed Batch Fermentation with Substrate and Product Inhibition Kinetics. Biotechnology and Bioengineering, v. 28, pp JACOBSON,D.H.,LELE,M.M.,1969,"ATransformation Technique for Optimal Control Problems with a State Variable Inequality Constraint. IEEE Trans. Autom. Control, v. 14, pp LOBATO, F. S., 2004, Abordagem mista para problemas de otimização dinâmica. (In Portuguese), Master of Science Dissertation, Universidade Federal de Uberlândia, Uberlândia, MG, Brasil. LOGSDON,J.S.;BIEGLER,L.T.,1989,"Accuratesolution of diferential-algebraic optimization problems". Ind. Eng. Chem. Res., v. 28, pp LOGSDON, J. S.; BIEGLER, L. T., 1992, "Decomposition strategies for large-scale dynamic optimization problems". Chemical Engineering Science, v. 47, n. 4, pp MODAK, J. L., LIM, H. C., TAYLEB, Y. J., 1986, "General characteristics of optimal feed rate profiles for various fed-batch fermentation process". Biotechnology and Bioengineering, XXVIII, pp

27 REFERENCES PANTELIDES, C. C., VASSILIADIS, V. S., SARGENT, R. W. H., 1994, "Optimal control of multistage systems described by high-index differential-algebraic equations". In: Computational Optimal Control, R. Bulirsch and D. Kraft, (eds.), Birkhäuser, Basel, Germany, pp PANTELIDES, C., 1988, "The consistent initialization of differential algebraic systems". SIAM J. SCI. STAT. Compt., v. 9, pp PANTELIDES, C., GRITSIS, D., MORISON, K., SARGENT, R., 1988, "The mathematical modelling of transient systems using differential-algebraic equations. Computers and Chemical Engineering,12, pp PFEIFER, A. A., 2007, "Controle Ótimo de Sistemas Algébrico-Diferenciais com Flutuação do Índice Diferencial. (In Portuguese), Master of Science Dissertation, Universidade Federal de Uberlândia, Uberlândia, MG, Brasil. QUINTO, T. S., 2010, Abordagem Algébrico-Diferencial da Otimização Dinâmica de Processos com Índice Flutuante. (In Portuguese), Master of Science Dissertation, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brasil. SOUZA, D. F. S., 2007, Abordagem algébrico-diferencial da otimização dinâmica de processos. (In Portuguese). Doctor on Science Thesis, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brasil. 27

28 REFERENCES SOUZA, D. F. S., VIEIRA, R. C., BISCAIA, E. C., Floating Index of Inequality Constrained DAE Systems. Computer Aided Chemical Engineering- Volume 21, 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering , Pages SRINIVASAN, B., AMRHEIN, M., BONVIN, D., 1998, Reaction and flow variants/invariants in chemical reaction systems with inlet and outlet streams. American Institute of Chemical Engineers Journal, 44 (8), pp SRINIVASAN, B., PRIMUS, C. J., BONVIN, D., RICKER, N. L., 2001, "Run-to-run optimization via generalized constraint control". Control Engineering Practice, v. 9, pp SRINIVASAN, B.; PALANKI, S.; BONVIN, D., 2003, "Dynamic optimization of batch processes: I. characterization of the nominal solution". Computers and Chemical Engineering, v. 27, pp UBRICH, O., SRINIVASAN, B., STOESSEL, F., BONVIN, D., 1999, "Optimization of a semi-batch reaction system under safety constraints". In: European Control Conference (pp. F306.1-F306.6), Karlsruhe, Germany. VASANTHARAJAN, S., BIEGLER, L. T., 1990, "Simultaneous strategies for optimization of differential-algebraic systems with enforcement error criteria". Computers and Chemical Engineering, v. 14, n. 10, pp

29 REFERENCES VASSILIADIS, V. S., SARGENT, R. W. H., PANTELIDES, C. C., 1994a, "Solution of a Class of Multistage Dynamic Optimization Problems. 1. Problems without Path Constraints", Process Design and Control by Ind. Eng. Chem. Res., v. 33, pp VASSILIADIS, V. S., SARGENT, R. W. H., PANTELIDES, C. C., 1994b, "Solution of a Class of Multistage Dynamic Optimization Problems 2. Problems with Path Constraints", Process Design and Control - Ind. Eng. Chem. Res., v. 33, pp VASSILIADIS, V., 1993, Computational Solution of Dynamic Optimization Problems with General Differential-Algebraic Constraints. Ph.D. Thesis, University of London, London, UK. VIEIRA, R. C, 2001, Técnicas de Inicialização de Sistemas Algébrico-Diferenciais. (In Portuguese), Doctor on Science Thesis, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brasil. VIEIRA, R. C., BISCAIA JR, E. C., 2001, "Direct methods for consistent initialization of DAE systems. Computers and Chemical Engineering, v. 25, pp VISSER, E., SRINIVASAN, B., PALANKI, S., BONVIN, D., 2000, "A feedback-based implementation scheme for batch process optimization". Journal of Process Control, v.10, pp