Many practical chemical engineering processes involve a sequence of distinct transient operations, forming multistage systems in which each stage is described by mixed sets of differential and algebraic equations (DAEs) [1]. These models usually involve decision variables that must be chosen so as to optimize some performance subject to operational constraints, thus leading to dynamic optimization problems, as well as parameters whose values are not known accurately.

The sensitivity analysis of the solutions to these models is typically conducted via forward sensitivity analysis [2,3]. However, when sensitivities with respect to a large number of variables or parameters are required, the forward sensitivity approach may become intractable, especially if the number of state variables is also large. These problems can often be handled more efficiently via adjoint sensitivity analysis [4,5]. In the first part of the presentation, we propose an extension of the adjoint sensitivity approach to address index-1, multistage DAEs. We allow discontinuous junction conditions between the various stages of the system, as well as different number of equations in each stage. Moreover, we consider functionals depending not only on the differential states at stage end times, but also on the algebraic states and the differential state time derivatives. In particular, both end-point conditions and junction conditions at stage times will be discussed.

Next, we consider optimization problems embedding index-1 mutistage DAEs and show how the adjoint sensitivity results can be used for the post-optimal sensitivity analysis of an optimal solution with respect to parameter variations. An important application of post-sensitivity analysis, in the field of real-time optimization, is for the development of neighboring-extremal controllers for multistage DAE systems [6]. The proposed framework is illustrated via the numerical simulation and optimization of micro-scale chemical processes for portable power generation [7].

References

[1] Barton, P. I., Pantelides, C. C. (1994). Modelling of Combined Discrete/Continuous Processes. AIChE J, 40:966-979.

[2] Vassiliadis, V. S., Sargent, R. W. H., Pantelides, C. C. (1994) Solution of a Class of Multistage Dynamic Optimization Problems. 1. Problems without Path Constraints. Ind Eng Chem Res, 33(9):2111-2122.

[3] Galán, S., Feehery, W. F., Barton, P. I. (1999) Parametric Sensitivity Functions for Hybrid Discrete/Continuous Systems. Applied Numerical Mathematics, 31(1):17-47.

[4] Ruban, A. I. (1997). Sensitivity Coefficients for Discontinuous Dynamic Systems. J Comput Syst Sci Int, 36(4):536–542.

[5] Cao, Y., Li, S., Petzold, L. R., Serban, R. (2003). Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution. SIAM J Sci Comput, 24(3):1076-1089.

[6] Büskens, C., Maurer, H. (2001). Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems using Nonlinear Programming Methods. In: On-Line Optimization of Large-Scale Systems, Grötschel, Krumke, S. O., Rambau, J., Eds. Springer-Verlag, Berlin.

[7] Chachuat, B., Mitsos, A., and Barton, P. I. (2005). Optimal Design and Steady-state Operation of Micro Power Generation Employing Fuel Cells. Chem Eng Sci, 60(16):4535-4556.