Gradient Networks for Universal Magnetic Modeling of Synchronous Machines

Abstract

This paper presents a physics-informed neural network approach for dynamic modeling of saturable synchronous machines, including cases with spatial harmonics. We introduce an architecture that incorporates gradient networks directly into the fundamental machine equations, enabling accurate modeling of the nonlinear and coupled electromagnetic constitutive relationship. By learning the gradient of the magnetic field energy, the model inherently satisfies energy balance (reciprocity conditions). The proposed architecture can universally approximate any physically feasible magnetic behavior and offers several advantages over lookup tables and standard machine learning models: it requires less training data, ensures monotonicity and reliable extrapolation, and produces smooth outputs. These properties further enable robust model inversion and optimal trajectory generation, often needed in control applications. We validate the proposed approach using measured and finite-element method (FEM) datasets from a 5.6-kW permanent-magnet (PM) synchronous reluctance machine. Results demonstrate accurate and physically consistent models, even with limited training data.

Figures (11)

• Figure 1: Electromechanical dynamics of a generic synchronous machine: (a) stator coordinates; (b) rotor coordinates. The blocks $1/s$ denote integration in time.
• Figure 2: Gradient network used for the magnetic models.
• Figure 3: Elementwise squareplus activation $\sigma$ in \ref{['eq:squareplus']} and its derivative $\mathrm{d} \sigma/\mathrm{d} x$ at different values of parameter $\beta$. The shape of the algebraic sigmoid \ref{['eq:sigmoid']} is the same as the derivative of the squareplus, but shifted vertically and scaled.
• Figure 4: Vector activation $\sigma_1(z_1, z_2)$ visualized in two-dimensional case: (a) softmax \ref{['eq:softmax']} with $\beta=1$; (b) $p$-norm gradient \ref{['eq:pnormgrad']} with $p=4$ and $\beta=1$.
• Figure 5: Proposed magnetic models in rotor coordinates: (a) without spatial harmonics, q-axis symmetry by construction; (b) with spatial harmonics. In both cases, the function $\mathbf{g}(\cdot)$ is given by \ref{['eq:gradnet']}.