Initialization and reformulation strategies for improved solving of nonlinear algebraic equation systems tested on chemical process models

Abstract

Solving nonlinear algebraic systems of equations by numerical methods is often a time-consuming challenge in chemical engineering, especially when systems are ill-conditioned and/or no well estimated initial values for the numerical solver are available. In this work, a hybrid method is developed to solve such systems independently of an insufficient initialization. The hybrid method makes use of methods from interval arithmetic to exclude infeasible ranges of values of the unkown variables and to efficiently locate solutions in the remaining feasible region by Newton-based methods. The user only has to set the bounds of the unknown variables in advance. To ensure the independence of the approach from Newton-based methods, several of these are applied, namely: A self implemented Newton method, Scipy's SLSQP and Fsolve as well as Ipopt. The hybrid method is implemented in Python and tested on process engineering examples. These systems are all complex, but differ in dimension, condition, and nonlinearity. For all systems at least one physically feasible solution is found in a few minutes. All solutions in the unrestricted variable space can even be found for some systems. The interval arithmetic offers here the possibility to prove mathematically that there can be no further solutions. This is theoretically possible for all other test examples as well, but in the larger systems the interval arithmetic based reduction requires too many box reduction steps to get close to the real-valued solution(s). The effectiveness of the reduction of variable bounds is particularly dependent on the initialization of these and the formulation of the equations. As part of this work, a wide variety of initializations and formulations of the equations were examined and the most important findings were collected in the form of guidelines. Furthermore, a first classification of the investigated systems of equations was carried out, measured by their complexity. Based on this, it can be estimated which of the three solution strategies (interval arithmetic method, Newton-based method or hybrid approach) is most suitable in the individual case. The problem-independent applicability of the hybrid approach should be verified on further large, complex, nonlinear algebraic process models. Many steps within the procedure offer the possibility to be performed in parallel and could contribute significantly to its acceleration. Thus, the approach could also become interesting for solving optimization problems or discretized, differential algebraic equation systems.