Performance enhancement of the simultaneous solvers for non-convex and ill-conditioned vapor compression systems utilizing gradient methods

Abstract

Simultaneous solvers using gradient methods are desirable for solving complex vapor compression system (VCS) models due to their flexibility and computational efficiency; however, they often lack robustness. Recent research has contributed towards improving the robustness of the simultaneous solvers, but there is still a need to understand why these solvers lack robustness and how to improve their convergence as the solver variables are increased. Furthermore, for a thorough system analysis, it is important to decrease the computational cost while increasing prediction accuracy. In this study, the performance of simultaneous solvers in terms of robustness, accuracy, and computational cost is evaluated using the standard ejector cycle as a representative example of advanced VCS. To ensure the focus of the study is on the solver's performance, computationally efficient, and accurate component models are considered. Utilizing optimization theory and problem formulation with minimum variables, this study finds for the first time that the non-convex and ill-conditioned behavior of the residual equations causes the simultaneous solvers to be less robust. Non-convexity primarily comes from the enthalpy residual, while the ill-conditioned behavior comes from the pressure residuals and the difference in the order of magnitude of the residuals. A novel method of the failure grid is introduced for a fair comparison of different solvers’ robustness. A few methods to improve the solver’s performance using gradient methods are introduced. The robustness of unconstrained algorithms can be improved by almost four times using scaling factors in the residual equations. However, the improvement in convergence of constrained algorithms is found to be almost twice more than the unconstrained algorithms using the same scaling factors, proving that the constrained algorithms result in a more robust solver. The study also finds that the computational cost is reduced by almost half when using a problem formulation without nested loops. However, this results in a higher number of solver variables for which constrained algorithms are found to be more robust, indicating that robustness remains important in determining the most computationally efficient solver. The accuracy of all the solvers evaluated is found to be comparable. Overall, these findings can help in developing flexible, robust, and computationally efficient simultaneous solvers with a higher number of variables for advanced thermal systems that include both vapor compression systems and thermal power systems. The findings can also be extended to develop parallelized-finite-volume heat exchanger models to reduce their computational cost.