Adaptive Homotopy Based Coarse Solver for Parareal-in-Time Transient Stability Simulations

The Parareal-in-time algorithm belongs to a class of temporal decomposition methods for parallel solutions of differential equations. Parareal utilizes two solvers: a coarse solver and a fine solver. The Parareal algorithm's performance depends on selecting the suitable solvers and the best time step for the coarse solver. In this paper, a new adaptive coarse solver based on homotopy analysis, by controlling the local truncation error (LTE), has been proposed with an option to adapt either the time step or both the time step and the auxiliary parameter
$c$
of homotopy. Using two communication paradigms, Master Worker and Distributed, in message passing interface (MPI), the proposed coarse solvers are evaluated for transient stability simulations. Fixed step Runge-Kutta
$4^{th}$
order (RK4), modified Euler (ME) and homotopy analysis method (HM) are used as the fine solvers. In addition, comparisons are made with the ME, HM, and an adaptive homotopy-assisted ME as coarse solvers. Many disturbance scenarios have been simulated on the Polish 2383 bus system and PEGASE 13569 bus system using a 32-core server-grade machine. The Master Worker paradigm is found to be faster than the Distributed paradigm. Using lighter solvers (ME, HM) for coarse and fine solvers has shown the best execution times. The proposed coarse solver, notably with both the time step &
$c$
adaptation, using LTE specification higher than the Parareal convergence tolerance has shown better execution time and faster than real-time performance while maintaining acceptable accuracy in both the paradigms with lighter fine solvers consistently.