Dynamic process simulation and optimization have emerged as valuable tools for plant design, analysis and operation. The usage of these tools in chemical process industry is on the rise, as is the demand for their increased efficiency and robustness. My Ph.D. research targets two aspects of this area: treatment of nonsmooth dynamic simulation problems through an adaptation of Successive Linear Programming (SLP) and the solution of large structured optimization problems using interior point methods. The work accomplished over the past three years includes theoretical considerations, software implementations as well as numerous applications in the chemical process industry.
The first part of this thesis develops a new framework for the solution of constrained simulation problems. The equation solving problem is converted to an optimization problem and the resulting nonlinear program is solved using SLP. A standard Newton's method often fails when encountered with poor starting points, ill-conditioning and singularities. Also, in the context of process simulation, problems invariably have conditionals, inequalities as well as variables which are bounded within physically well defined regions. The optimization framework provides a natural way for dealing with these issues. Our work proposes a new descent strategy which combines the two major classes of strategies for ensuring descent: line search and trust region methods. Numerical results using OPTLP, an implementation of this algorithm, on a variety of test problems show that the combined line search - trust region method performs at least as well as either the trust region or the line search methods individually. The next part of the thesis addresses the issues related to discontinuities occurring in dynamic simulation problems due to a change in the modeling equations. A new algorithm has been presented for the detection and location of events which cause a discrete change in the modeling equations. The test problems solved include complex flow networks with checkvalves exhibiting hysteresis behavior. Phase equilibrium problems were considered as an important and relevant practical application of implicit nonsmooth systems. Traditional equation - based systems require clumsy swapping of equation sets, each of which is valid at particular phase combinations, when there is a change in the number of phases present. Our approach introduces a single unified optimization formulation which automatically solves the relevant set of equations and thus avoids the swapping of equation sets. The approach has been applied to nonideal multiphase systems involving liquid-liquid-vapor equilibrium and calculations in the critical region. Currently, application of efficient smoothing methods to directly handle the complementarity equations arising from the optimality conditions of the above problems, particularly phase equilibrium, is being investigated. Simple representations based on smoothing techniques, of complex nested discontinuities occurring in process engineering problems have also been developed.
Dynamic simulation and optimization problems in chemical process engineering are most often modeled by Differential Algebraic Equations (DAEs). Determination of consistent initial conditions for a system of DAEs is a vital aspect of its solution and has been an active research area in the past decade. An SLP formulation with the DAE derivative array equations as the constraints in the optimization problem has been derived and tested on numerous examples. This approach identifies inconsistent specifications and easily accommodates the singularities and overdetermined nature of the problem and is an attractive alternative to expensive SVD based least square methods. A theoretical contribution to the problem of reinitialization after discontinuities of systems modeled by DAEs, is also made in this thesis. A mathematical criteria is derived to identify the subset of variables which are continuous across a discontinuity caused by a step change in the input variables, for problems of arbitrary index. This has relevance in control problems and shows that the currently used assumption of the continuity of the differential variables is not always valid.
Current research is focused on developing efficient primal - dual interior point successive quadratic programming (SQP) methods for the solution of structured process optimization problems. The solution strategy here is two pronged: utilize SQP strategies to exploit the structure of the problem coupled with an interior point approach to efficiently treat the inequality constraints, which when otherwise treated with an active set strategy often handicaps SQP methods for large problems. A variation of the well studied Mehrotra predictor-corrector interior point method has been implemented within an SQP framework.