State of the Art in Global Optimization pp 563-583 | Cite as

# Global Optimization of Chemical Processes using Stochastic Algorithms

## Abstract

Many systems in chemical engineering are difficult to optimize using gradient-based algorithms. These include process models with multimodal objective functions and discontinuities. Herein, a stochastic algorithm is applied for the optimal design of a fermentation process, to determine multiphase equilibria, for the optimal control of a penicillin reactor, for the optimal control of a non-differentiable system, and for the optimization of a catalyst blend in a tubular reactor. The advantages of the algorithm for the efficient and reliable location of global optima are examined. The properties of these algorithms, as applied to chemical processes, are considered, with emphasis on the ease of handling constraints and the ease of implementation and interpretation of results. For the five processes, the efficiency of computation is improved compared with selected stochastic and deterministic algorithms. Results closer to the global optimum are reported for the optimal control of the penicillin reactor and the non-differentiable system.

## Keywords

Simulated Annealing Global Optimum Success Ratio Tubular Reactor Stochastic Algorithm## Preview

Unable to display preview. Download preview PDF.

## References

- Banga, J.R. and J.J. Casares, “Integrated Controlled Random Search: application to a wastewater treatment plant model,” IChemE Symp. Ser. 100, 183–192 (1987).Google Scholar
- Banga, J.R., A.A. Alonso, R.I. Perez-Martin and R. P. Singh, “Optimal control of heat and mass transfer in food and bioproducts processing,” Comput. Chem, Eng. 18, S699–S705 (1994).CrossRefGoogle Scholar
- Banga, J.R., R. I. Perez-Martin, J.M. Gallardo and J.J. Casares, “Optimization of the thermal processing of conduction-heated canned foods: study of several objective functions,” J. Food Eng. 14, 25–51 (1991).CrossRefGoogle Scholar
- Biegler, L.T.,“Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation,” Comput. Chem. Eng. 8, 243–248 (1984).CrossRefGoogle Scholar
- Bojkov, B., R. Hansel and R. Luus, “Application of direct search optimization to optimal control problems,” Hung. J. Ind. Chem. 21, 177–185 (1993).Google Scholar
- Book, N.L. and W.F. Ramirez, “The selection of design variables in systems of algebraic equations,” AIChE J. 22, 55–66 (1976).CrossRefGoogle Scholar
- Book, N.L. and W.F. Ramirez, “Structural analysis and solution of systems of algebraic equations,” AIChE J. 30, 609–622 (1984).CrossRefGoogle Scholar
- Brenan, K.E., S.L. Campbell and L.R. Petzold, “Numerical solution of initial-value problems in differential-algebraic equations,” North-Holland, New York (1989).zbMATHGoogle Scholar
- Brengel, D. D. and W. D. Seider, “Coordinated design and control optimization of nonlinear processes,” Comput. Chem. Eng. 16, 861–886 (1992).CrossRefGoogle Scholar
- Casares, J.J. and J.R. Banga, “Analysis and evaluation of a wastewater treatment plant model by stochastic optimization,” Appl. Math. Model. 13, 420–424 (1989).CrossRefGoogle Scholar
- Chen, C.T. and C. Hwang, “Optimal control computation for differential-algebraic process systems with general constraints,” Chem. Eng. Comm. 97, 9–26 (1990).CrossRefGoogle Scholar
- Corana, A., M. Marchesi, C. Martini and S. Ridella, “Minimizing multimodal functions of continuous variables with the simulated annealing algorithm,” ACM Transac. Math. Soft. 13, 262–280 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
- Cuthrell, J.E. and L.T. Biegler, “Simultaneous optimization and solution methods for batch reactor control profiles,” Comput. Chem. Eng. 13, 49–6 (1989).CrossRefGoogle Scholar
- Davis, L. L., “Genetic Algorithms and Simulated Annealing,” Ed. Morgan Kaufmann. (1989).Google Scholar
- Devroye, L. P., “Progressive global random search of continuous functions,” Math. Program. 15, 330–342 (1978).MathSciNetzbMATHCrossRefGoogle Scholar
- Dolan, W.B., P.T. Cummings and M.D. LeVan, “Process optimization via simulated annealing: application to network design,” AIChE J. 35, 725–736 (1989).CrossRefGoogle Scholar
- Eaton, J.W., “Octave: a high-level interactive language for numerical computations, Edition 1.0,” Department of Chemical Engineering, The University of Texas at Austin (1994).Google Scholar
- Floquet, P., L. Pibouleau and S. Domenech, “Separation sequence synthesis: how to use the simulated annealing procedure?,” Comput. Chem. Eng. 18, 1141–1148 (1994).Google Scholar
- Gelfand, S. B. and S. K. Mitter, “Recursive stochastic algorithms for global optimization in R,” SIAM J. Control Optim. 29: 999–1018 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
- Gill, P.E., W. Murray and M.H. Wright, “Practical Optimization,” Academic Press, London (1981).zbMATHGoogle Scholar
- Goldberg, D. D., “Genetic Algorithms in Search, Optimization and Machine Learning,” Ed. Addison-Wesley (1989).zbMATHGoogle Scholar
- Goulcher, R. and J.J. Casares, “The solution of steady-state chemical engineering optimization problems using a random search technique,” Comput. Chem. Eng. 2, 33–36 (1978).CrossRefGoogle Scholar
- Han, S. P., “A globally convergent method for nonlinear programming,” J. Optim. Theory Applic. 22, 297 (1977).zbMATHCrossRefGoogle Scholar
- Huber, M.L., “Structural optimization of vapor pressure correlations using simulated annealing and threshold accepting: application to R134a,” Comput. Chem. Eng. 18, 929–932 (1994).CrossRefGoogle Scholar
- Ingber, L., “Very fast simulated re-annealing,” J. Math. Comput. Model. 12, 967–973. (1989).MathSciNetzbMATHCrossRefGoogle Scholar
- Ingber, L., “Simulated annealing: practice versus theory,” J. Math. Comput. Model. 11, 29–57 (1993).MathSciNetCrossRefGoogle Scholar
- Kirkpatrick, S., C.D. Gelatt and M.P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).MathSciNetCrossRefGoogle Scholar
- Ku, H.M., and I.A. Karimi, “An evaluation of simulated annealing for batch process scheduling,” Ind. Eng. Chem. Res. 30, 163 (1991).CrossRefGoogle Scholar
- Lim, H.C., Y.J. Tayeb, J.M. Mo dak and P. Bonte, “Computational algorithms for optimal feed rates for a class of fed-batch fermentation: numerical results for penicillin and cell mass production,” Biotechnol. Bioeng. 28, 1408–1420 (1986).CrossRefGoogle Scholar
- Luus, R.,“Application of dynamic programming to differential-algebraic process systems,” Comput. Chem. Eng. 17, 373–377 (1993a).CrossRefGoogle Scholar
- Luus, R.,“Optimization of fed-batch fermentors by Iterative Dynamic Programming,” Biotechnol. Bioeng. 41, 599–602 (1993b).CrossRefGoogle Scholar
- Luus, R.,“Piecewise linear continuous optimal control by iterative dynamic programming,” Ind. Eng. Chem. Res. 32, 859–865 (1993c).CrossRefGoogle Scholar
- Luus R.,“Optimization of heat exchanger networks,” Ind. Eng. Chem. Res. 32, 2633–2635. (1993d).CrossRefGoogle Scholar
- Luus, R. and B. Bojkov,“Global optimization of the bifunctional catalyst problem,” Canad. J. Chem. Eng. 72, 160–163 (1994).CrossRefGoogle Scholar
- Luus, R., J. Dittrich and F. J. Keil, “Multiplicity of solutions in the optimization of a bifunctional catalyst blend in a tubular reactor,” Canad. J. Chem. Eng. 70, 780–785 (1992).CrossRefGoogle Scholar
- Luus, R. and T.H.I. Jaakola, “Optimization by direct search and systematic reduction of the size of search region,” AIChE J. 19, 760–766 (1973).CrossRefGoogle Scholar
- Martin, D.L. and J.L. Gaddy, “Process optimization with the adaptive randomly directed search,” AIChE Symp. Ser. 78, 99 (1982).Google Scholar
- Masri, S.F., G.A. Bekeyand F.B. Safford, “A global optimization algorithm using adpative random search,” Appl. Math, and Comput. 7, 353–375 (1980).MathSciNetzbMATHCrossRefGoogle Scholar
- McDonald, C.M. and C.A. Floudas,“Global optimization for the phase and chemical equilibrium problem: application to the NRTL equation,” Comput. Chem. Eng., in press (1994a).Google Scholar
- McDonald, C.M. and C.A. Floudas, “Decomposition based and branch and bound global optimization approaches for the phase equilibrium problem,” J. Global Optim., in press (1994b).Google Scholar
- McDonald, C.M. and C.A. Floudas, “Global optimization for the phase stability problem,” AIChE J., in press (1994c).Google Scholar
- McDonald, C.M. and C.A. Floudas, “GLOPEQ: A new computational tool for the phase and chemical equilibrium problem,” Annual Meeting of AIChE, San Francisco (1994d).Google Scholar
- McDonald, C.M. and C.A. Floudas, “Global optimization and analysis for the Gibbs free energy function using the UNIFAC, Wilson, and ASOG equations,” submitted to Ind. Eng. Chem. Res. (1994e).Google Scholar
- Michelsen, M.L., “The isothermal flash problem, I: stability,” Fluid Phase Equil. 9, 1. (1982a).CrossRefGoogle Scholar
- Michelsen, M.L., “The isothermal flash problem, II: phase split calculations,” Fluid Phase Equil. 9, 21 (1982b).CrossRefGoogle Scholar
- Mihail, R. and G. Maria, “A modified Matyas algorithm (MMA) for random process optimization,” Comput. Chem. Eng. 10, 539–544 (1986).CrossRefGoogle Scholar
- Moscato, P. and J.F. Fontanari, “Stochastic versus deterministic update in simulated annealing,” Phys. Lett. A 146, 204–208 (1990).CrossRefGoogle Scholar
- Petzold, L., “Differential/algebraic equations are not ODEs,” SIAM J. Sci. Stat. Comput. 3, 367–384 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
- Powell, M.J.D., “A fast algorithm for nonlineary constrained optimization calculations,” in Numerical Analysis, Ed. G.A. Watson, Springer Verlag (1978).Google Scholar
- Pronzato, L., E. Walter, A. Venor and J.-F. Lebruchec, “A general purpose global optimizer: implementation and applications,” Math. Compute. Simul XXVI, 412–422 (1984).Google Scholar
- Reklaitis, G.V., A. Ravindran and K.M. Ragsdell, “Engineering Optimization: Methods and Applications,” Wiley, New York, p. 277–287 (1983).Google Scholar
- Salcedo, R.L., M.J. Gonalves and S. F. de Azevedo. F. de Azevedo, “An improved random search algorithm for non-linear optimization,” Comput. Chem. Eng. 14, 1111–1126 (1990).Google Scholar
- Salcedo, R.L., “Solving nonconvex nonlinear programming and mixed-integer nonlinear programming problems with adaptive random search,” Ind. Eng. Chem. Res. 31, 262–273 (1992).CrossRefGoogle Scholar
- Schiesser, W.E, W.E., “The Numerical Method of Lines: Integration of Partial Differential Equations,” Academic Press, Inc., San Diego (1991).Google Scholar
- Seider, W.D., D.D. Brengel and S. Widagdo, “Nonlinear analysis in process design,” AIChE J. 37, 1–38 (1991).CrossRefGoogle Scholar
- Solis, F. J. and R. J-B. Wets, “Minimization by random search techniques,” Math. Oper. Res. 6, 19–30 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
- Spaans, R. and R. Luus, “Importance of search-domain reduction in random optimization,” J. Optim. Theory Appl. 75, 635–638 (1992).MathSciNetzbMATHCrossRefGoogle Scholar
- Sun, A. C.-T., “Global optimization using the Newton homotopy-continuation method with ap¬plication to phase equilibria,” Ph.D. Thesis, University of Pennsylvania (1993).Google Scholar
- Sun, A. C.-T. and W. D. Seider, “Homotopy-continuation algorithm for global optimization,” in Recent Advances in Global Optimization, Eds C.A. Floudas and P.M. Pardalos, Princeton University Press, p. 561–592, (1992).Google Scholar
- Sun, A. C.-T. and W. D. Seider, “Homotopy-continuation method for stability analysis in the global minimization of the Gibbs free energy,” Fluid Phase Equil. 103, 213–249 (1995).CrossRefGoogle Scholar
- Umeda, T., A. Shinodo and A. Ichikawa, “Complex method for solving variational problems with state-variable inequality constraints,” Ind. Eng. Chem. Process Des. Dev. 11, 102–107 (1972).CrossRefGoogle Scholar
- Van Impe, J.F., B.M. Nicola, P.A. Vanrolleghem, J.A. Spriet, B. Moor and J. Vandewalle, “Optimal control of the penicillin G fed-batch fermentation: an analysis of a modified unstructured model,” Chem. Eng. Comm. 117, 337–353 (1992).CrossRefGoogle Scholar
- Vanderbilt, D. and S.G. Louie, “A Monte Carlo simulated annealing approach to optimization over continuous variables,” J. Comput. Phys. 56, 259–271 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
- Venkatasubramanian, V., K. Chan and J.M. Caruthers, “Computer-aided molecular design using genetic algorithms,” Comput. Chem. Eng. 18, 833–844 (1994).CrossRefGoogle Scholar
- Wang, B.-C. and R. Luus, “Reliability of optimization procedures for obtaining global optimum,” AIChE J. 24, 619–626 (1978)MathSciNetCrossRefGoogle Scholar