Theoretical Contribution to multiphysical modeling of flywheel energy storage systems with a focus on thermal effects in magnetic bearings

This paper gives a theoretical contribution to the multiphysical modeling of Flywheel Energy Storage Systems. In this work, a laboratory prototype of a flywheel consisting of a vertical rotor supported by one axial passive magnetic bearing and by two radial active magnetic bearings, is used as an example for modeling. The rotor is modeled as a flexible body accounting for thermal and centrifugal expansions. The forces from the passive magnetic bearing are evaluated using experimentally validated analytical expressions based on Lorentz's law, while the forces from the active magnetic bearings are calculated using the virtual work principle. The current dynamics are obtained through Ohm's, Faraday's, and Ampere's laws. The temperatures are found from the energy equation, in which the heat sources are the system's energy losses due to eddy currents and ohmic effects. The outcome of the multiphysical model is a set of nonlinear equations depending on the states: (i) the rotor lateral and axial displacements and velocities, (ii) the currents of active magnetic bearing stators, and (iii) the temperatures in the rotor and bearing stator. The system of equations is linearized around the operational point allowing for the calculation of a state matrix as a function of rotor and active magnetic bearing stator temperature, material, and geometrical properties of the rotor, passive magnetic bearing, and active magnetic bearings. The coupled linearized system is solved together such that it highlights the thermal effects on the natural frequencies, damping ratios, and time constants of the system. This holistic approach can be useful when optimizing and designing model-based controllers for flywheels.