Analysis and Control of Flywheel Energy Storage Systems

Since a few years ago, electrical energy storage has been attractive as an effective use of electricity and coping with the momentary voltage drop. Above all, flywheel energy storage systems (FESS) using superconductor have advantages of long life, high energy density, and high efficiency (Subkhan & Komori, 2011), and is now considered as enabling technology for many applications, such as space satellites and hybrid electric vehicles (Samineni et al., 2006; Suvire & Mercado, 2012). Also, the contactless nature of magnetic bearings brings up low wear, absence of lubrication and mechanical maintenance, and wide range of work temperature (Bitterly, 1998; Beach & Christopher, 1998). Moreover, the closed-loop control of magnetic bearings enables active vibration suppression and on-line control of bearing stiffness (Cimuca et al., 2006; Park et al., 2008).


Introduction
Since a few years ago, electrical energy storage has been attractive as an effective use of electricity and coping with the momentary voltage drop. Above all, flywheel energy storage systems (FESS) using superconductor have advantages of long life, high energy density, and high efficiency (Subkhan & Komori, 2011), and is now considered as enabling technology for many applications, such as space satellites and hybrid electric vehicles (Samineni et al., 2006;Suvire & Mercado, 2012). Also, the contactless nature of magnetic bearings brings up low wear, absence of lubrication and mechanical maintenance, and wide range of work temperature (Bitterly, 1998;Beach & Christopher, 1998). Moreover, the closed-loop control of magnetic bearings enables active vibration suppression and on-line control of bearing stiffness (Cimuca et al., 2006;Park et al., 2008).
Active magnetic bearing is an open-loop unstable control problem. Therefore, an initial controller based on a rigid rotor model has to be introduced to levitate the rotor. In reality, the spinning rotor under the magnetic suspension may experience two kinds of whirl modes. The conical whirl mode gives rise to the gyroscopic forces to twist the rotor, thereby severely affecting stability of the rotor if not properly controlled (Okada et al, 1992;Williams et al., 1990). The translatory whirl mode constrains the rotor to synchronous motion in the radial direction so as to suppress the gyroscopic rotation, which has been extensively used in industry (Tomizuka et al, 1992;Tsao et al., 2000). The synchronization control has also been shown to be very capable in dealing with nonlinear uncertain models, and to be very effective in disturbance rejection for systems subject to synchronous motion. Until the advent of synchronization control, the prevalent use of the synchronization controller has been limited to stable mechanical systems and therefore is not readily applicable to magnetic systems which are unstable in nature and highly nonlinear (Yang & Chang, 1996).
In the past three decades the theory of optimal control has been well developed in nearly all aspects, such as stability, nonlinearity, and robustness (Summers et al., 2011;Rawlings et al., 2008;Mayne, et al., 2000). It is known that multivariable constrained control problems in state-space can be effectively handled using Linear Quadratic Gaussian (LQG). An application of the optimal control to synchronize multiple motion axes has been reported in (Zhu & Chen, 2001;Xiao & Zhu, 2006), where cross-coupling design of generalized predictive control was presented by compensating both the tracking error and the synchronous error. In this chapter, robust MPC control algorithms for the flywheel energy storage system with magnetically assisted bearings are developed. The controllers are derived through minimization of a modified cost function, in which the synchronization errors are embedded so as to reduce the synchronization errors in an optimal way. Fig.1 illustrates the basic structure of a flywheel system with integrated magnetic bearings. The motor and generator with disk-type geometry are combined into a single electric machine, and the rotor is sandwiched between two stators. Each of the stators carries a set of three-phase copper winding to be fed with sinusoidal currents. Furthermore, both axial faces of the rotor contain rare-earth permanent magnets embedded beneath the surfaces. The radial magnetic bearing which consists of eight pairs of electromagnets is constructed around the circumference of hollow center. A combination of active and passive magnetic bearings allows the rotor to spin and remain in magnetic levitation.

Flywheel structure
The control of such a system normally includes two steps. First, the spinning speed and the axial displacement of the rotor are properly regulated (Zhang & Tseng, 2007). Second, a synchronization controller is introduced to suppress the gyroscopic rotation of the rotor caused by the outside disturbance and model uncertainty (Xiao et al., 2005).

System dynamics
Let c x and c y denote the displacements of the mass center of the rotor in the x and ydirections, and  and  the roll angles of rotation about x -axis and y -axis, respectively. Note that  and  are assumed to be small since the air gap is very narrow within the magnetic bearings. It is also assumed that the rotor is rigid with its inertia perfectly balanced about the zaxis so that the flexibility and eccentricity of the rotor are not considered herein; thereby, the variation effects of tensor of inertia due to the roll motion of the rotor can be negligible.
The mass center of the rotor in the radial direction can be described by cx ax bx cy ay by   During a closed-loop control phase, the position and rate of the shaft are constantly monitored by contactless sensors, and are processed in a controller, so that a control current to the coils of electromagnets which attract or repel the shaft is amplified and fed back.

Controller design
Let he discrete-time model of (3) be described by where k denotes the discrete time. Note that the disturbance term is ignored.
By introducing the following synchronization errors,

 
where I is the unit matrix with appropriate dimension.
Hence, minimization of the cost function (7) from the initial condition 0 P .
Indeed, as receding horizon LQG control is a stationery feedback strategy, over an infinite interval, questions of stability naturally arise while solutions are slow to emerge. On the other hand, the stability of the proposed controller (12) can sometimes be guaranteed with finite horizons, even if there is no explicit terminal constraint. The finite horizon predictive control problem is normally associated with a time-varying RDE, which is related to the Analysis and Control of Flywheel Energy Storage Systems 137 optimal value of the cost function. Attempts at producing stability result for MPC on the basics of its explicit input-output description have been remarkably unsuccessful, usually necessitating the abandonment of a specific control performance.

Stability analysis
Lemma 1. Consider the following ARE with an infinite-horizon linear quadratic control (Souza et al., 1996), there exists a unique, maximal, non-negative definite symmetric solution P .  P is a unique stabilizing solution, . ie., has all the eigenvalues strictly within the unit circle.
Rewrite (15) as In order to connect the RDE (14) to the ARE (15), the Fake Algebraic Riccati Technique (FART) is used as follows: Proof. The proof is completed by setting 1 p jH   in Theorem 1.
It can be seen from the above theorem that the prediction horizon p H is a key parameter for stability, and an increasing p H is always favorable. This was the main motivation to extend the one-step-ahead control to long range predictive control. However, a stable linear feedback controller may not remain stable for a real system   Pz with model uncertainty, which is normally related to stability robustness of the system. The most common specification of model uncertainty is norm-bounded, and the frequency response of a nominal model (3)  10 v  is introduced, the synchronization performance can be improved significantly, especially in terms of the rolling angles, as shown in Fig. 11. Therefore, with adequately selected control parameters the improved synchronization performance as well as guaranteed stability of the FESS can be obtained, and in consequence, the whirling rotor in the presence of disturbances would be suppressed near the nominal position.

Conclusion
In this chapter, stability problem of magnetic bearings for a flywheel energy storage system has been formulated, and a synchronization design has been presented by incorporating cross-coupling technology into the optimal control architecture. The basic idea of the control strategy is to minimize a new cost function in which the synchronization errors are embedded, so that the gyro-dynamic rotation of the rotor can be effectively suppressed.
However, as optimal control, using receding horizon idea, is a feedback control, there is a risk that the resulting closed-loop system might be unstable. Then, stability of the control system based on the solution of the Riccati Difference Equation has also been analyzed, and some results are summarized. The illustrative example reveals that with adequately adjusted control parameters the resulting control system is very effective in recovering the unstable rotor and suppressing the coupling effects of the gyroscopic rotation at high spinning speeds as well as under external disturbances and model uncertainties.

Author details
Yong Xiao, Xiaoyu Ge and Zhe Zheng College of Information Engineering, Shenyang University of Chemical Technology, China