On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulators

Technical Summary: Approximating the Dynamic Response of Synchronous Generators via Operator Learning

Problem Statement
The paper addresses the computational challenges associated with simulating the dynamic response of synchronous generators (SGs) within power grids. Traditional numerical simulation schemes, such as partitioned-solution approaches or implicit integration, often incur high computational costs, particularly when multiple forward simulations are required for control and optimization tasks in future smart grids. While deep learning offers potential alternatives, existing methods often struggle with generalization to unseen operating conditions, require large datasets to prevent overfitting, and fail to account for the infinite-dimensional nature of the solution operators governing these dynamical systems. Furthermore, purely data-driven models may lack the physical consistency provided by decades of established mathematical modeling in the power systems community.

Methodology
The authors propose an Operator Learning framework based on Deep Operator Networks (DeepONet) to approximate the solution operator of SGs interacting with a power grid or a numerical simulator. The methodology is structured into three primary components:

1. Data-Driven DeepONet: The authors design a DeepONet to approximate the local solution operator of an SG over an arbitrary time interval $[t_n, t_n + h_n]$. Unlike previous DeepONet applications that require the entire trajectory as input, this framework accepts the current state $x(t_n)$, a discretized sequence of interacting variables (e.g., stator currents and terminal voltages) $\tilde{y}_m^n$, and a flexible prediction timestep $h_n$. The network consists of a Branch Net (processing state and interacting variables) and a Trunk Net (processing the time step), outputting the future state via a linear combination of trainable coefficients and basis functions. This local approximation is then recursively employed to simulate the generator's response over a long-term horizon.

2. Residual DeepONet: To incorporate existing mathematical knowledge, the authors propose a residual learning scheme. Instead of learning the full solution operator, the network learns the "residual" or error correction between the true solution operator and an approximate operator derived from a known mathematical model (e.g., a simplified differential-algebraic equation or a physics-informed neural network). The final prediction is the sum of the approximate model's output and the residual DeepONet's correction. The paper provides a theoretical estimate of the cumulative error for this scheme, attributing error accumulation to both input approximation and neural network approximation errors.

3. Data Aggregation (DAgger) Strategy: To mitigate distributional drift and error accumulation when the DeepONet interacts with a simulator, the authors introduce a DAgger algorithm. This iterative strategy trains the DeepONet on an initial dataset, then uses the trained model to roll out simulations and collect new input data that the model is likely to encounter during interaction. This new data is labeled using the true solution operator and aggregated with the original training set for fine-tuning. This process is repeated to ensure the model generalizes to the specific distribution of states it will encounter in interactive simulations.

Key Contributions

• Operator Learning for Power Grids: The paper presents the first application of DeepONet to approximate the solution operator for dynamic power grid components, specifically addressing the infinite-dimensional nature of the problem and the need for recursive, multi-step prediction.
• Recursive Numerical Scheme: A novel numerical scheme is developed that recursively utilizes the trained DeepONet to simulate generator responses over short-to-medium time horizons with arbitrary time steps.
• Hybrid Modeling: The introduction of a residual DeepONet framework that effectively combines data-driven learning with established mathematical models, allowing the system to leverage prior physics knowledge while correcting for model inaccuracies.
• Error Analysis: Theoretical bounds are derived for the cumulative error of the residual DeepONet scheme, distinguishing between errors from input discretization and neural network approximation.
• DAgger for Interactive Simulation: A data aggregation algorithm is proposed to fine-tune DeepONets using aggregated training data, specifically designed to reduce error propagation when the model interacts with other grid components in a simulator.

Experimental Results
The proposed frameworks were validated using a two-axis transient generator model interacting with an infinite bus, simulated via the Power Systems Toolbox (PST).

• Supervised Training: In supervised training experiments, the residual DeepONet outperformed both the purely data-driven DeepONet and a standard fully-connected neural network (FNN). The residual DeepONet achieved mean L2-relative errors consistently below 1% for generator states and network inputs, whereas the FNN exhibited significant error accumulation (mean errors up to 37.8%). The data-driven DeepONet maintained errors below 5%.
• Sensitivity Analysis: The study demonstrated that increasing the number of training samples reduced error accumulation for both frameworks, with the residual DeepONet consistently showing superior generalization and lower error accumulation regardless of dataset size.
• DAgger Performance: When trained with limited data (100 samples) and fine-tuned using the DAgger algorithm, the residual DeepONet successfully generalized to unseen disturbance trajectories. The mean L2-relative error for all states and network inputs remained below 0.01%, indicating that the DAgger strategy effectively mitigated distributional drift and error propagation.

Significance and Claims
The paper claims that the proposed DeepONet framework offers a viable path toward building deep operator-based power grid simulators and digital twins. By approximating the solution operator rather than just the next state, the framework allows for flexible time-stepping and interaction with grid simulators. The authors emphasize that the residual approach allows the integration of decades of power system modeling expertise, improving generalization even with limited data. Furthermore, the DAgger strategy addresses a critical challenge in deploying machine learning in interactive environments: the accumulation of errors due to distributional drift. The authors position this work as a step toward composable, large-scale simulators capable of optimizing and controlling future smart grids with high renewable penetration, while noting that future work will focus on online calibration, privacy-preserving federated learning, and scaling to large networks.