State-Dependent Neural Flux Linkage Models of Synchronous Machines

Flux linkage maps (FLMs) are routinely used in high-precision control and modeling of synchronous machines. Common methods often consider only the dependence of the FLM on the stator currents, allowing for convenient representation in lookup tables or neural networks. However, the flux linkage also depends on speed, position, and other state variables. Although this is formally simple to add as an additional input to neural models of FLM, the estimation with additional inputs becomes more demanding. We demonstrate that the conventional approach of FLM training using the assumption of a steady-state regime is insufficient to learn the dependency on the rotor position. It is necessary to use the complete ordinary differential equation of the current to learn the FLM model. Even for a shallow neural model of the FLM, the estimation procedure yields a deep learning task known as neural ODE. This procedure essentially generates multistep ahead prediction of differential equations and minimizes the mismatch between the mathematical model and data. The efficiency of this approach is demonstrated on the FLM of a synchronous machine considering flux saturation, speed dependence, and slot harmonics. The proposed approach significantly improves current prediction, yielding improved deadbeat current control. The results are experimentally verified on a 4.5 kW laboratory prototype.