Gradient Networks for Universal Magnetic Modeling of Synchronous Machines

Junyi Li; Tim Foissner; Floran Martin; Antti Piippo; Marko Hinkkanen

eess.SYFeb 16, 2026

Gradient Networks for Universal Magnetic Modeling of Synchronous Machines

Authors: Junyi Li, Tim Foissner, Floran Martin, Antti Piippo, Marko Hinkkanen

Organizations: Department of Electrical Engineering and Automation, Aalto University, 02150 Espoo, Finland · ABB Oy, Drives, 00380 Helsinki, Finland

Abstract

This paper presents a physics-constrained neural network framework for magnetic modeling of saturable synchronous machines, including spatial harmonics. By embedding gradient networks into the machine equations to model conservative electromagnetic behavior, the framework satisfies reciprocity and energy conservation by construction, while universally approximating any physically feasible magnetic characteristic. Unlike lookup tables and black-box neural networks, it guarantees monotonicity, invertibility, and smooth outputs, and remains highly data efficient. The method is validated using measured and finite-element method (FEM) data from a 5.6-kW permanent-magnet (PM) synchronous reluctance machine, and is demonstrated in real-time closed-loop control on an embedded platform. The results confirm accurate, physically consistent, and computationally efficient performance.

Figures & tables

Panel of Figure 1 — Fig. 1: Electromechanical dynamics of a generic synchronous machine: (a) stator coordinates; (b) rotor coordinates. The blocks $1/s$ denote integration in time.

Fig. 2: Gradient network used for the magnetic models.

Fig. 3: Elementwise squareplus activation

\sigma

in ( 11 ) and its derivative

\mathrm{d}\sigma/\mathrm{d}x

at different values of parameter

\beta

. The shape of the algebraic sigmoid ( 12 ) is the same as the derivative of the squareplus, but shifted vertically and scaled.

Panel of Figure 4 — Fig. 4: Vector activation $\sigma_{1}(z_{1},z_{2})$ visualized in two-dimensional case: (a) softmax ( 13 ) with $\beta=1$ ; (b) $p$ -norm gradient ( 14 ) with $p=4$ and $\beta=1$ .

Panel of Figure 5 — Fig. 5: Proposed magnetic models in rotor coordinates: (a) without spatial harmonics, q-axis symmetry by construction; (b) with spatial harmonics, shown without symmetrization for simplicity. In both cases, the function $\mathbf{g}(\cdot)$ is given by ( 10 ).

Voltage (line-to-neutral, peak value)	$\sqrt{2/3}\cdot 460$ V	1.00 p.u.
Current (peak value)	$\sqrt{2}\cdot 8.8$ A	1.00 p.u.
Frequency	60 Hz	1.00 p.u.
Speed	1 800 r/min	1.00 p.u.
Power	5.6 kW	0.80 p.u.
Torque	29.7 Nm	0.80 p.u.

TABLE I: Rated Values of the 5.6-kW PM Synchronous Reluctance Machine

Fig. 6: Test bench including a 5.6-kW PM synchronous reluctance machine (left), load machine (right), and their inverter cabinet (background).

Panel of Figure 8 — Fig. 7: Current maps: (a) $i_{\mathrm{d}}(\psi_{\mathrm{d}},\psi_{\mathrm{q}})$ ; (b) $i_{\mathrm{q}}(\psi_{\mathrm{d}},\psi_{\mathrm{q}})$ . The surfaces show the predicted maps from the squareplus gradient model ( 11 ) with $N=12$ hidden units, trained on a 10% subset. Markers show the measured dataset: red indicates the 10% training subset and blue the remaining points. Gray lines show constant-current contours corresponding to the measured dataset.

Panel of Figure 9 — Fig. 8: Flux-linkage maps: (a) $\psi_{\mathrm{d}}(i_{\mathrm{d}},i_{\mathrm{q}})$ ; (b) $\psi_{\mathrm{q}}(i_{\mathrm{d}},i_{\mathrm{q}})$ . The surfaces show the predicted maps from the $p$ -norm gradient model ( 14 ) with $p=8$ and $N=12$ hidden units, trained on a 10% subset. Markers show the same measured dataset as in Fig. 7 : red indicates the 10% training subset and blue the remaining points. Gray lines show constant-current contours corresponding to the measured dataset.

Activation	Training data	$e_{\mathrm{rms}}$ (p.u.)	$e_{\mathrm{max}}$ (p.u.)	$e_{\mathrm{std}}$ (p.u.)
squareplus ( 11 )	10%	0.017	0.070	0.011
squareplus ( 11 )	2%	0.076	0.344	0.054
softmax ( 13 )	10%	0.031	0.226	0.021
softmax ( 13 )	2%	0.108	0.407	0.068
$p$ -norm	10%	0.021	0.110	0.012
gradient ( 14 )	2%	0.096	0.389	0.061

TABLE II: Comparison of Activation Functions in Current Maps Without Spatial Harmonics (

N=12

)

Activation	Training data	$e_{\mathrm{rms}}$ (p.u.)	$e_{\mathrm{max}}$ (p.u.)	$e_{\mathrm{std}}$ (p.u.)
algebraic	10%	0.016	0.044	0.010
sigmoid ( 12 )	2%	0.051	0.165	0.032
softmax ( 13 )	10%	0.007	0.033	0.004
softmax ( 13 )	2%	0.029	0.081	0.019
$p$ -norm	10%	0.004	0.022	0.003
gradient ( 14 )	2%	0.018	0.061	0.012

TABLE III: Comparison of Activation Functions in Flux-Linkage Maps Without Spatial Harmonics (

N=12

)

Panel of Figure 12 — Fig. 9: Electromagnetic torque from the $p$ -norm gradient model with $N=48$ hidden units: (a) $\tau_{\mathrm{m}}(i_{\mathrm{q}},\vartheta_{\mathrm{m}})$ at constant $i_{\mathrm{d}}$ corresponding to the rated MTPA current; (b) $\tau_{\mathrm{m}}(\vartheta_{\mathrm{m}})$ at the constant rated MTPA stator current. The curve and markers in (b) are a slice from (a). Red and blue markers show the full FEM dataset. Red markers indicate the 10% training subset.

Activation	Training data	$e_{\mathrm{rms}}$ (p.u.)	$e_{\mathrm{max}}$ (p.u.)	$e_{\mathrm{std}}$ (p.u.)
softmax ( 13 )	10%	0.011	0.062	0.007
softmax ( 13 )	2%	0.015	0.091	0.010
$p$ -norm	10%	0.015	0.106	0.010
gradient ( 14 )	2%	0.025	0.140	0.016

TABLE IV: Comparison of Activation Functions with Spatial Harmonics (

N=48

) for Torque Error

Panel of Figure 14 — Fig. 10: Closed-loop control responses: (a) simulation with the proposed magnetic model including spatial harmonics in the plant; (b) corresponding measurement. The same controller is used in both cases, parameterized entirely by the proposed model without spatial harmonics. Speed reference step from 0 to 2 p.u. at $t=0.25$ s and load-torque step at $t=1.25$ s. From top to bottom, the responses are speed, torque, current, and flux linkage.

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Incorporating hysteresis and eddy currents into finite element simulations of laminated-core electrical machines is computationally challenging. Resolving the fields inside the laminations at each integration point and at every nonlinear iteration leads to computational costs several orders of magnitude higher than anhysteretic simulations, making such approaches impractical for design applications. Conversely, simplified models accounting only for magnetic saturation are becoming increasingly inadequate as electrical machine topologies and operating conditions grow in complexity. In this context, machine learning surrogate modeling has emerged as a promising alternative, offering efficient and accurate approximations of complex electromagnetic behaviors. In this paper, a recurrent neural network is trained as a surrogate of a laminated-core material model for an isotropic laminated core, and is integrated into realistic two-dimensional magnetodynamic finite element simulations based on a magnetic vector potential formulation. The proposed approach achieves excellent agreement with the reference laminated-core model while limiting the computational cost to about twice that of an anhysteretic simulation. By training the recurrent neural network on a sufficiently diverse set of artificially generated magnetic field sequences designed to mimic those encountered in electrical machine simulations, the proposed approach can be readily applied across a wide range of finite element simulations. Furthermore, the trained surrogate model is provided as a standalone component that can be easily incorporated into existing computational frameworks. It is publicly available at https://gitlab.onelab.info/getdp/lamnet.

Abstract

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