Recent advances in deep learning have allowed neural networks (NNs) to successfully replace traditional numerical solvers in many applications, thus enabling impressive computing gains. One such application is time domain simulation, which is indispensable for the design, analysis and operation of many engineering systems. Simulating dynamical systems with implicit Newton-based solvers is a computationally heavy task, as it requires the solution of a parameterized system of differential and algebraic equations at each time step. A variety of NN-based methodologies have been shown to successfully approximate the trajectories computed by numerical solvers at a fraction of the time. However, few previous works have used NNs to model the numerical solver itself. For the express purpose of accelerating time domain simulation speeds, this paper proposes and explores two complementary alternatives for modeling numerical solvers. First, we use a NN to mimic the linear transformation provided by the inverse Jacobian in a single Newton step. Using this procedure, we evaluate and project the exact, physics-based residual error onto the NN mapping, thus leaving physics “in the loop”. The resulting tool, termed the Physics-pRojected Neural-Newton Solver (PRoNNS), is able to achieve an extremely high degree of numerical accuracy at speeds which were observed to be up to 31% faster than a Newton-based solver. In the second approach, we model the Newton solver at the heart of an implicit Runge-Kutta integrator as a contracting map iteratively seeking a fixed point on a time domain trajectory. The associated recurrent NN simulation tool, termed the Contracting Neural-Newton Solver (CoNNS), is embedded with training constraints (via CVXPY Layers) which guarantee the mapping provided by the NN satisfies the Banach fixed-point theorem; successive passes through the NN are therefore guaranteed to converge to a unique, fixed point. Explicitly capturing the contracting nature of Newton iterations leads to significantly increased NN accuracy relative to a vanilla NN. We test and evaluate the merits of both PRoNNS and CoNNS on three dynamical test systems.

In order to drastically reduce the heavy computational burden associated with time-domain simulations, this paper introduces a Physics-Informed Neural Network (PINN) to directly learn the solutions of power system dynamics. In contrast to the limitations of classical model order reduction approaches, commonly used to accelerate time-domain simulations, PINNs can universally approximate any continuous function with an arbitrary degree of accuracy. One of the novelties of this paper is that we avoid the need for any training data. We achieve this by incorporating the governing differential equations and an implicit Runge-Kutta (RK) integration scheme directly into the training process of the PINN; through this approach, PINNs can predict the trajectory of a dynamical power system at any discrete time step. The resulting Runge-Kutta-based physics-informed neural networks (RK-PINNs) can yield up to 100 times faster evaluations of the dynamics compared to standard time-domain simulations. We demonstrate the methodology on a single-machine infinite bus system governed by the swing equation. We show that RK-PINNs can accurately and quickly predict the solution trajectories.