Relaxation/Newton methods for concurrent time step solution of differential-algebraic equations in power system dynamic simulations

Many problems in transient network analysis are characterized by the solution of a simultaneous set of coupled algebraic and ordinary differential equations. The urgent need for on-line applications has motivated power system researchers to develop algorithms which can be implemented on parallel computers. Traditional algorithms for simulating power system dynamics do not readily lend themselves to parallel processing and only a limited amount of parallelism can be achieved. In this paper, a class of algorithms which exploits the concurrent solution of many time steps is presented. By applying a stable integration method, the overall algebraic-differential set of equations can be transformed into a unique algebraic problem at each time step. The dynamic behavior of the system can be obtained by solving an enlarged set of algebraic equations relative to the simultaneous solution of many time steps. A class of relaxation/Newton algorithms can be used to solve this problem efficiently. Furthermore, this formulation permits, easily, the implementation of multigrid techniques. Theoretical aspects about the convergence rates and computational complexity of the proposed algorithms are discussed in the paper. Test results on realistic power systems confirm theoretical expectations and show the promise of a several-fold increase speed over that obtainable by traditional parallel-in-space approaches. The synergism obtainable by parallelism in time and in space can allow high speed-up adequate for on-line implementations of transient stability analysis.