Graphical Processing Unit Based Time-Parallel Numerical Method for Ordinary Differential Equations

Abstract

On-line transient stability analysis of a power grid is crucial in determining whether the power grid will traverse to a steady state stable operating point after a disturbance. The transient stability analysis involves computing the solutions of the algebraic equations modeling the grid network and the ordinary differential equations modeling the dynamics of the electrical components like synchronous generators, exciters, governors, etc., of the grid in near real-time. In this research, we investigate the use of time-parallel approach in particular the Parareal algorithm implementation on Graphical Processing Unit using Compute Unified Device Architecture to compute solutions of ordinary differential equations. The numerical solution accuracy and computation time of the Parareal algorithm executing on the GPU are demonstrated on the single machine infinite bus test system. Two types of dynamic model of the single synchronous generator namely the classical and detailed models are studied. The numerical solutions of the ordinary differential equations computed by the Parareal algorithm are compared to that computed using the modified Euler's method demonstrating the accuracy of the Parareal algorithm executing on GPU. Simulations are performed with varying numerical integration time steps, and the suitability of Parareal algorithm in computing near real-time solutions of ordinary different equations is presented. A speedup of 25× and 31× is achieved with the Parareal algorithm for classical and detailed dynamic models of the synchronous generator respectively compared to the sequential modified Euler's method. The weak scaling efficiency of the Parareal algorithm when required to solve a large number of ordinary differential equations at each time step due to the increase in sequential computations and associated memory transfer latency between the CPU and GPU is discussed.

Modified Parareal Algorithm for Solving Time-Dependent Differential Equations

“…The time domain simulations of the test systems discussed in [10] is performed using the modified PRA implementation. In figures 4a and 5a, the variation of the rotor angle for cases 1 and 2 are presented.…”

“…The results in [10] [11] showed that a significant amount of time is spent on the predictor step computing ̃ values. This step impacted the overall performance of the original PRA and showed poor scaling when implemented on many-core and multi-core architectures for a system of inter-dependent ODEs.…”