State of Charge Evolution Equations for Flywheels

I. INTRODUCTION Flywheel energy storage devices are composed of a spinning composite disk in an low-pressure enclosure designed to contain the debris in the case of operation failure [18]. An electric motor-generator connector is used to convert the electric energy to kinetic energy by applying a torque on the flywheel. The torque can be positive (charge) or negative (discharge). The flywheel levitates over magnetic bearings in order to reduce friction. A cooling system must be used to remove the heat produced by the remaining friction and the electric components. To reduce drag losses, a vacuum pump is used to create a low-pressure vacuum in the enclosure. Flywheel technology has greatly improved in the past few years such that modern flywheels can rotate up to 100,000 round per minute (rpm) in a low-pressure vacuum enclosure and achieve a very high energy density [16]. The fast response time of flywheels makes this technology well suited for grid support (especially frequency regulation) [12], [4], [19]. Comprehensive overviews of flywheel energy storage devices are provided in [17], [8]. Extensive work has been done on flywheel energy storage devices and their modeling, but most of these works rely on simulation and circuit models [21], [2], [9], [3]. Nassar et al. [15] propose a simple algorithm for simulating flywheel energy storage devices. However, this model does not include the charging/discharging inefficiencies and the self-discharge of the flywheel. Flywheels can suffer from significant leakage rates, due to frictional windage and magnetic losses from the bearings and motor-generator components [14], [7]. Hearn et al. [10] make a conservative simplification on the core losses, and model the state of charge (SoC) evolution of a flywheel with a first-order differential equation. A flywheel energy storage system based on a doubly-fed induction motor-generator is composed of a wound-rotor induction machine and a cycloconverter. These storage devices are capable of both active and reactive power control while the conventional synchronous-speed rotary condenser is only capable of reactive power control. By adjusting the rotor speed of the doubly-fed induction machine, the motor-generator can either provide the electric power to the grid or draw it from the grid. Even though the model of Hearn et al. [10] includes a detailed description of the frictional windage and magnetic losses, it neglects the impact of the active power controller on the supplied/drawn energy from the flywheel. The purpose of this study is to obtain formulas for …