(28f) Dynamic Optimization of Constrained SEMI-Batch Processes Using Pontryagin’S Minimum Principle – an Effective Quasi-Newton Based Approach

Erdal Aydin^a, Kai Sundmacher^a,b,*

^aMax Planck Institute for Dynamics of Complex Technical Systems, SandtorstraÃ?e 1, 39106 Magdeburg, Germany

^bOtto-von-Guericke University Magdeburg, Universitätplatz 2, 39106 Magdeburg, Germany

^*Corresponding author: sundmacher@mpi-magdeburg.mpg.de; Tel: +49 391 6110 351, Fax: +49 391 6110 353

Semi-batch processing is employed in many chemical sectors including food, pharmaceuticals, polymers and specialty chemical industries. Semi-batch processes possess a flexible operation scheme for the production of high-value and low volume products. Dynamic optimization of semi-batch processes has been attracting significant attention because of the competing environment in the industry, the requirement of higher product qualities and strict environmental regulations. Direct methods are usually stated as promising techniques for the solution of the dynamic optimization problems. However, depending on the type of the problem, these methods could be inefficient in terms of the time required for the solution. In addition, the usage of the direct techniques may result in unstabilities or infeasible solutions due to required approximations[1]. Indirect methods, such as Pontryaginâ??s Minimum Principle, are the alternative approaches which transcript the problem into a different structure. Although Pontryaginâ??s Minimum Principle was extensively studied for unconstrained problems and considered as an effective method, its application to complex and constrained dynamic optimization problems is limited [2],[3]. In this study, we present an alternative indirect solution technique for the dynamic optimization of constrained batch and fed-batch processes. The algorithm depends on the Pontryaginâ??s Minimum Principle and the necessary conditions of optimality, and uses first order gradients with a Hessian approximation algorithm to update the corresponding control variables at each iteration step. Nonlinear constraints are augmented into Hamiltonian function using indirect adjoining and corresponding constraints are saturated at each iteration step in order to increase the efficiency of the method [3],[4]. The fact that the system equations can be integrated at each iteration step prevents the necessity of any approximation in the model equations. The performance of the proposed strategy was tested through different three case studies, which includes fed-batch reactors and a batch distillation column with complex path and terminal constraints. The performance of the algorithm was also compared with the simultaneous and sequential algorithms. Results show that the proposed solution algorithm is effective in dealing with the complex constraints and it is competing in solving the corresponding problems in small and acceptable computational time.

1. Srinivasan, B., S. Palanki, and D. Bonvin, Dynamic optimization of batch processes: I. Characterization of the nominal solution. Computers & Chemical Engineering, 2003. 27(1): p. 1-26.

2. Chachuat, B., Nonlinear and dynamic optimization: from theory to practice. 2007.

3. Hartl, R.F., S.P. Sethi, and R.G. Vickson, A survey of the maximum principles for optimal control problems with state constraints. SIAM review, 1995. 37(2): p. 181-218.

4. Harvey Jr, P.S., H.P. Gavin, and J.T. Scruggs, Optimal performance of constrained control systems. Smart Materials and Structures, 2012. 21(8): p. 085001.

Keywords: Constrained dynamic optimization, optimal control, Pontryaginâ??s Minimum Principle, Gradient-based optimization, semi-batch process