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Physics-Informed Neural Networks for Power Systems

George S. Misyris, Andreas Venzke, Spyros Chatzivasileiadis

arXiv:1911.03737v3eess.SYcs.LGeess.SP

TL;DR

The paper addresses power-system modeling and estimation by introducing physics-informed neural networks that incorporate governing equations into neural-network training. In a single-machine infinite-bus study, the approach accurately determines dynamic states and identifies uncertain parameters with substantially lower computational time and limited data.

  • Problem

    Power-system applications lack a physics-informed neural-network framework for exploiting governing equations to determine dynamic states and uncertain parameters efficiently.

  • Method

    The framework incorporates power-system differential and algebraic equations into neural-network training for steady-state and dynamic modeling, including system identification with unknown parameters.

  • Results

    In the single-machine infinite-bus system, physics-informed neural networks accurately determine rotor angle and frequency up to 87 times faster than conventional numerical methods and identify inertia and damping from limited input data.

  • Takeaways & Limitations

    The results showcase potential for applying physics-informed neural networks to larger power-system problems, including security and optimization tasks requiring accurate dynamic-state and parameter estimation.

  • Takeaways & Limitations

    The case studies used 40 input data points and 8,000 collocation points, with larger systems expected to require more collocation points and longer training times.

Abstract

from arXiv · show

This paper introduces for the first time, to our knowledge, a framework for physics-informed neural networks in power system applications. Exploiting the underlying physical laws governing power systems, and inspired by recent developments in the field of machine learning, this paper proposes a neural network training procedure that can make use of the wide range of mathematical models describing power system behavior, both in steady-state and in dynamics. Physics-informed neural networks require substantially less training data and can result in simpler neural network structures, while achieving high accuracy. This work unlocks a range of opportunities in power systems, being able to determine dynamic states, such as rotor angles and frequency, and uncertain parameters such as inertia and damping at a fraction of the computational time required by conventional methods. This paper focuses on introducing the framework and showcases its potential using a single-machine infinite bus system as a guiding example. Physics-informed neural networks are shown to accurately determine rotor angle and frequency up to 87 times faster than conventional methods.

I. INTRODUCTION

The paper addresses the data and model complexity demands of conventional machine-learning approaches in power systems by embedding physical laws into neural-network training. It introduces physics-informed neural networks for power-system steady-state and dynamic models, targeting accurate states and parameters with less data and simpler structures.

  • Motivation: Machine-learning methods for power systems can be substantially faster than traditional approaches but were largely agnostic to the underlying physical model.Prior methods were reported as up to three orders of magnitude faster, while remaining dependent on training data quality.
  • Framework: Physics-informed neural networks incorporate power-system differential and algebraic equations directly into the training procedure.The framework applies to both steady-state and dynamic mathematical models.
  • Framework: The approach requires less initial training data and can produce smaller neural networks while maintaining high performance.Automatic differentiation computes derivatives of neural-network outputs, including rotor angle, during training.
  • Opportunities: Physics-informed neural networks can directly determine dynamic states at specified times and handle higher-order differential equations without auxiliary first-order variables.The paper identifies these capabilities as opportunities for numerical solvers and dynamic state estimation.
  • Contributions: The paper demonstrates accurate dynamic-state prediction and uncertain-parameter identification, reporting 28–87 times faster computation for the single-machine infinite-bus system.The targeted states include rotor angle and frequency, while uncertain parameters include inertia and damping.

II. METHODOLOGY

Power-system dynamics are represented by swing equations that relate generator motion to inertia, damping, network susceptance, power, voltages, and angles.

  • Physical model: The swing equation provides a common simplified representation of power-system dynamics for each generator.The formulation neglects transmission losses and bus-voltage deviations.
  • Physical model: Generator inertia and damping, network susceptance, mechanical power, voltage magnitudes, and voltage angles determine the modeled dynamics.The angular-frequency variable is the time derivative of the generator angle.

1) Single Machine Infinite Bus (SMIB) System:

The single-machine infinite-bus system is used as the paper’s guiding example for evaluating physics-informed neural networks. The framework estimates rotor angle and frequency under varying active power and identifies uncertain inertia and damping parameters.

  • Single Machine Infinite Bus (SMIB) System: The single-machine infinite-bus system is a standard setting for analyzing fundamental power-system dynamic phenomena.The paper uses it to introduce the framework while noting that the proposed framework is intended to be general.
  • Single Machine Infinite Bus (SMIB) System: The study estimates rotor angle and frequency while active power varies within a specified range.The same setup also supports identification of uncertain inertia and damping parameters.

B. Physics-Informed Neural Networks

Physics-informed neural networks incorporate power-system equations into neural-network training, constraining predictions with both data and physical-law residuals. This can reduce data and model-size requirements while supporting known or unknown system parameters.

  • Training objective: Considering physical laws bounds the space of admissible neural-network parameters, reducing required training data and network size.The loss balances deviations from initial data and deviations from the imposed physical law.
  • Architecture: Physics-informed neural networks combine feed-forward neural-network predictions with physical-system equations during training.The network predicts u(t, x), computes derivatives through automatic differentiation, and evaluates f(t, x).
  • Mathematical formulation: The network approximates u(t, x) governed by a nonlinear operator N[u; λ], with t denoting time, x system input, and λ system parameters.Unknown λ turns the approximation into system identification.
  • System identification: When λ is unknown, the training jointly seeks neural-network weights and biases and system parameters satisfying the governing expression.Known parameters simplify the nonlinear operator to N[u].
  • Training objective: The loss combines MSEu over training data with MSEf over collocation points enforcing the dynamical-system equation.Nu and Nf denote the numbers of training and collocation points, respectively.

1) Physics-informed neural networks capturing power system dynamics:

For a single-machine infinite-bus system, the physics-informed network predicts rotor angle over time and mechanical-power inputs, then derives frequency from the estimated angle. It directly incorporates the higher-order swing equation.

  • Physics-informed neural networks capturing power system dynamics: The network derives rotor angle δ and frequency ω = ˙δ from the swing equation for a range of mechanical power P1.The system parameters are assumed known and voltages V1 and V2 are fixed.
  • Physics-informed neural networks capturing power system dynamics: The system input is x := {P1}, while the network output is δ(t, P1) over a chosen time interval and bounded input domain.The mechanical-power input is restricted to [Pmin, Pmax].
  • Physics-informed neural networks capturing power system dynamics: Unlike conventional solvers, physics-informed neural networks directly incorporate higher-order ordinary differential equations without introducing additional variables.Conventional numerical solvers first convert higher-order ODEs to first-order form.
  • Physics-informed neural networks capturing power system dynamics: After training, frequency is extracted as ˙δ from the estimated angle, so frequency error depends on angle-prediction error and the differential method.The paper refers to this structure as NNδ.

2) Data-driven discovery of inertia and damping coefficients:

Physics-informed neural networks can identify uncertain power-system parameters alongside dynamic trajectories. The paper treats inertia and damping as trainable variables using scattered measurements.

  • Data-driven discovery of inertia and damping coefficients: System identification treats inertia m1 and damping d1 as unknown neural-network variables optimized during training.The same physics-informed structure is retained while a subset of λ is added to the minimization.
  • Data-driven discovery of inertia and damping coefficients: The approach targets inertia estimation because changing renewable generation makes power-system inertia uncertain and requires regular estimation or prediction.The stated operational relevance is preventing large frequency deviations and maintaining frequency stability.

III. SIMULATION & RESULTS

The simulation study generates training and test data with MATLAB’s ode45 solver and evaluates configurations using relative L2 error for angle and frequency trajectories. It includes known-parameter and unknown-parameter case studies.

  • Simulation setup: ode45 simulations use a 0.1s time step over T = [0, 20s], producing 201 time steps per trajectory.The setup fixes V1 and V2 at 1 p.u. and B12 at 0.2 p.u.
  • Simulation setup: The first case study assumes known inertia and damping, uncertain P1 ∈ [0.08, 0.18], and initial δ and ω values of 0.1 rad and 0.1 rad/s.The resulting combined training and test dataset contains 20′100 samples.
  • Case studies: The second case study treats inertia and damping as unknown and uses scattered active-power, frequency, and angle observations to identify m1 and d1 and estimate δ trajectories.Ten parameter pairs are generated with m1 in [0.1, 0.4] and d1 in [0.05, 0.15].
  • Configuration study: Model selection varies hidden layers, neurons per layer, training data Nu, and collocation points Nf, retaining configurations with the lowest relative L2 error.Errors are assessed for predicted and exact δ(t, P1) and ω(t, P1).

B. Data-driven

The physics-informed neural network predicts rotor-angle dynamics accurately and recovers frequency through numerical differentiation, while offering substantial computational speedups over conventional integration. A single-output angle network is more accurate and faster to train than predicting angle and frequency as separate outputs.

  • Training setup: Nu = 40 initial and boundary samples and Nf = 8′000 collocation points were used with a 5-layer network containing 10 neurons per hidden layer.Increasing Nu led to over-fitting to the training data.
  • Trajectory prediction: The network predicts angle and frequency trajectories, recovering frequency by numerically differentiating the predicted angle with a Newton method.The study uses a fixed simulation step of h = 0.1s.
  • Computational speed: 28×: Neural-network evaluation solves the differential equation faster than ode45 across 100 different initial conditions.The average times are 0.016 s for the neural network and 0.45 s for ode45.
  • Computational speed: 87×: Evaluating δ(t1) at a specified time is faster than integrating from t0 to t1 with ode45.At t1 = 10 s, evaluation takes 4 · 10^-3 s versus 0.35 s for ode45.
  • Architecture comparison: A single-output angle network outperforms separate angle-and-frequency outputs in training time and predictive accuracy.The separate-output model required Nf = 50′000 collocation points, took approximately 30 minutes to train, and had relative L2 errors of 9.43·10^-2 and 1.51·10^-1 for δ and ω.

C. Data-driven discovery of inertia and damping coefficients through physics-informed neural networks

The physics-informed neural network estimates unknown inertia and damping from limited observed trajectories while simultaneously predicting the rotor-angle dynamics.

  • Parameter discovery: 0.74% and 1.28%: Average errors for predicting inertia m1 and damping d1 across 10 parameter pairs.Each model used Nu = 100 randomly distributed points and a 5-layer network with 30 neurons per hidden layer.
  • Parameter discovery: Less than 60 seconds: Average training time to identify system inertia and damping parameters.The parameters were inferred from limited datapoints containing t, P1, and δ.

IV. DISCUSSION AND OUTLOOK

The paper presents physics-informed neural networks as a power-system training framework that incorporates differential and algebraic equations. Its demonstrated benefits are accompanied by unresolved challenges in collocation cost, scalability, and application range.

  • Discussion and outlook: The framework explicitly incorporates power-system differential and algebraic equations into neural-network training.The paper presents this as the first such work in power systems, to the authors’ knowledge.
  • Discussion and outlook: Physics-informed neural networks may solve differential-algebraic equation systems substantially faster than traditional numerical integration and directly evaluate states at specified times.The paper identifies these capabilities as opportunities rather than established large-scale results.
  • Limitations: Nf = 8′000 collocation points were used alongside Nu = 40 data points, and larger systems are expected to require more collocation points and longer training.The authors propose Runge-Kutta-based methods as a possible way to eliminate collocation points.
  • Limitations: Scalability to large-scale power systems remains to be explored, including computational needs and neural-network training challenges.The authors also identify comparisons with numerical solvers and polynomial fits as needed benchmarks.
  • Limitations: The demonstrated study covers a single stable swing prediction, while broader regimes and phenomena such as unstable conditions, voltage stability, converter dynamics, and discrete events remain future application areas.Different regimes such as multiple oscillations or unstable conditions may require different physics-informed neural networks.

V. CONCLUSIONS

The paper proposes physics-informed neural networks for power systems and demonstrates their potential on a single-machine infinite-bus system. The method accurately determines dynamic states and identifies uncertain parameters while reducing computational time and training-data requirements.

  • Physics-informed neural networks explicitly incorporate power-system governing equations to solve differential-algebraic systems with substantially less training data than conventional approaches.The framework includes the underlying swing equation during training.
  • 87 times faster, physics-informed neural networks accurately determine rotor angle and frequency in the single-machine infinite-bus case study.The comparison is against conventional numerical methods.
  • The case studies also identify uncertain system parameters, including inertia and damping, from a limited set of input data.
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