**Analysis and Control of Flywheel Energy Storage Systems**

Yong Xiao, Xiaoyu Ge and Zhe Zheng

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52412

## **1. Introduction**

18 Energy Storage

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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).

© 2013 Xiao et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Xiao et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Analysis and Control of Flywheel Energy Storage Systems 133

(1)

*b a*

are the

(2)

*a b <sup>y</sup> <sup>y</sup> l l*

,

Let *<sup>c</sup> x* and *<sup>c</sup> y* denote the displacements of the mass center of the rotor in the *x* and *y* -

bearings. It is also assumed that the rotor is rigid with its inertia perfectly balanced about the *z* axis 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.

*c xa xb x c ya yb y x yb b ya a z y xa a xb b z*

*J f l f l Jw e J f l f l Jw e*

, *m* is the mass of the rotor, *xJ* , *yJ* and *zJ* are the moments of inertia

about *x* -axis, *y* -axis and *z* -axis respectively, *w* is the spinning rate about *z* -axis, *xa f* , *xb f* ,

According to the Maxwell's law, the magnetic forces *xa f* , *xb f* , *ya f* and *yb f* have nonlinear relationships with the control currents and displacements of the rotor. Then, the magnetic

> *xa xa a xb xb b p c ya ya a yb b yb*

axis, respectively, 0 *h* is the nominal air gap at equilibrium, *G* is an electromagnet constant

*f x i f x i K K <sup>f</sup> <sup>y</sup> <sup>i</sup> f y i*

 

 

*mx f f e my f f e*

, *b a*

*ya f* and *yb f* are the magnetic forces along the radial directions, *xe* , *ye* , *e*

forces at equilibriums can be linearized with Taylor's method (Zhu et al., 2009),

The mass center of the rotor in the radial direction can be described by

 

*c ab ab ab l l xxx ll ll* 

the roll angles of rotation about *x* -axis and *y* -axis, respectively. Note

 

 

*c ab ab ab l l <sup>y</sup> y y ll ll* 

, <sup>1</sup>

is a constant angle corresponding to the structure of

*i* and *yi* are the control currents near *x* -axis and *y* -

is the air permeability, *Ag* is the cross-sectional area of air gap,

 and *e*

are assumed to be small since the air gap is very narrow within the magnetic

**3. System dynamics** 

 and 

where *b a*

0 1 4 *G ANg* 

*<sup>h</sup>* , <sup>0</sup>

*c GI <sup>K</sup>*

electromagnets, 0*I* is the bias current, *<sup>x</sup>*

, 0

2 0

,

8 sin

*h*

and *N* is the number of turns of the winding circuit.

8 *p GI <sup>K</sup>*

given by <sup>2</sup>

 <sup>1</sup> *a b*

*x x*

*a b*

disturbances.

where

*l l*

directions, and

that and 

## **2. Flywheel structure**

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.

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).

**Figure 1.** The flywheel energy storage system

#### **3. System dynamics**

132 Energy Storage – Technologies and Applications

optimal way.

**2. Flywheel structure** 

in magnetic levitation.

**Figure 1.** The flywheel energy storage system

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

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

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).

Let *<sup>c</sup> x* and *<sup>c</sup> y* denote the displacements of the mass center of the rotor in the *x* and *y* directions, 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 *z* axis 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

$$\begin{cases} m\ddot{\mathbf{x}}\_c = f\_{xa} + f\_{xb} + e\_x\\ m\ddot{y}\_c = f\_{ya} + f\_{yb} + e\_y\\ f\_x\dot{\theta} = f\_{yb}l\_b - f\_{ya}l\_a - f\_z w \dot{\phi} + e\_\theta\\ f\_y\ddot{\phi} = f\_{xa}l\_a - f\_{xb}l\_b + f\_z w \dot{\phi} + e\_\phi \end{cases} \tag{1}$$

$$\text{where}\\
\qquad \qquad \qquad \mathbf{x}\_c = \frac{l\_b}{l\_a + l\_b} \mathbf{x}\_a + \frac{l\_a}{l\_a + l\_b} \mathbf{x}\_b \\
\quad \text{+} \quad y\_c = \frac{l\_b}{l\_a + l\_b} y\_a + \frac{l\_a}{l\_a + l\_b} y\_b \\
\qquad \qquad \qquad \theta = \frac{1}{l\_a + l\_b} \left( y\_b - y\_a \right) \\
\text{-} \quad \mathbf{x} = \frac{l\_b}{l\_a + l\_b} \mathbf{x}\_b \quad \text{-} \quad \mathbf{x} = \frac{l\_b}{l\_a + l\_b} \mathbf{x}\_b \quad \text{-} \quad \mathbf{x} = \frac{l\_b}{l\_a + l\_b} \mathbf{x}\_b$$

 <sup>1</sup> *a b a b x x l l* , *m* is the mass of the rotor, *xJ* , *yJ* and *zJ* are the moments of inertia about *x* -axis, *y* -axis and *z* -axis respectively, *w* is the spinning rate about *z* -axis, *xa f* , *xb f* , *ya f* and *yb f* are the magnetic forces along the radial directions, *xe* , *ye* , *e* and *e* are the disturbances.

According to the Maxwell's law, the magnetic forces *xa f* , *xb f* , *ya f* and *yb f* have nonlinear relationships with the control currents and displacements of the rotor. Then, the magnetic forces at equilibriums can be linearized with Taylor's method (Zhu et al., 2009),

$$
\begin{bmatrix} f\_{xa} \\ f\_{xb} \\ f\_{ya} \\ f\_{yb} \end{bmatrix} \approx K\_p \begin{bmatrix} \chi\_a \\ \chi\_b \\ \chi\_a \\ \chi\_b \end{bmatrix} + K\_c \begin{bmatrix} i\_{xa} \\ i\_{xb} \\ i\_{ya} \\ i\_{yb} \end{bmatrix} \tag{2}
$$

where 2 0 3 0 8 *p GI <sup>K</sup> <sup>h</sup>* , <sup>0</sup> 2 0 8 sin *c GI <sup>K</sup> h* , is a constant angle corresponding to the structure of electromagnets, 0*I* is the bias current, *<sup>x</sup> i* and *<sup>y</sup> i* are the control currents near *x* -axis and *y* axis, respectively, 0 *h* is the nominal air gap at equilibrium, *G* is an electromagnet constant given by <sup>2</sup> 0 1 4 *G ANg* , 0 is the air permeability, *Ag* is the cross-sectional area of air gap, and *N* is the number of turns of the winding circuit.

Then, the state-space model of (1) is obtained,

$$\begin{cases} \dot{X} = A\_c X + B\_c \mu + B\_x d\_x \\ Z = C\_c X + D\_z d\_z \end{cases} \tag{3}$$

,

67

87

42

84

back.

where

1 *a b c p x*

1 *<sup>b</sup> c p x*

1 *<sup>b</sup> c c y l B K m J* 

1 *<sup>b</sup> c c x l B K m J* 

2

2

2

**4. Controller design** 

,

,

.

, <sup>82</sup>

*J wl <sup>A</sup>*

*c*

21

63

Let he discrete-time model of (3) be described by

By introducing the following synchronization errors,

1

*p*

*H T*

*i*

 

it has the modified cost function,

*z b*

*Jl l* , <sup>84</sup>

, 22

, 64

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

*X k AX k Bu k* 1

*Z k CX k* 

where *k* denotes the discrete time. Note that the disturbance term is ignored.

*a b b a a b b a*

*k ik k ik*

1 1

*P c*

*H H T T*

*i i*

ˆ ˆ | |

*x x x y k L y y y x*

  *c*

*J wl <sup>A</sup>*

*xa b*

2

2

1 *<sup>a</sup> c c y l B K m J* 

1 *<sup>a</sup> c c x l B K m J* 

*l l A K m J* 

*<sup>l</sup> <sup>A</sup> <sup>K</sup> m J* 

Analysis and Control of Flywheel Energy Storage Systems 135

, 41

, 83

1 *a b c p x*

1 *a b c c y l l B K m J* 

1 *a b c c x l l B K m J* 

,

,

,

(4)

(6)

*l l A K m J* 

*z b*

*Jl l* , 85

*k LCX k* (5)

1 10 0

10 0 1

0 1 10 , , 001 1

*Jk Z k ikZk ik u k i kuk i k*

ˆ ˆ | | ˆ ˆ 1| 1|

*xa b*

1 *a b c c y l l B K m J* 

1 *a b c c x l l B K m J* 

where ,,,,,,, *<sup>T</sup> X xxxxyyyy aabbaabb* is the state variable, ,,, *<sup>T</sup> xa xb ya yb u iiii* is the forcing vector, ,,, *<sup>T</sup> Z xxyy abab* is the output vector, and 1234 *TTTT* 1,0 are the output transition matrices, *<sup>x</sup> d* and *<sup>z</sup> d* denote model uncertainties or system disturbances with appropriate matrices *<sup>x</sup> B* and *Dz* ,

$$A\_c = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ A\_{c21} & 0 & A\_{c23} & 0 & 0 & A\_{c26} & 0 & A\_{c28} \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ A\_{c41} & 0 & A\_{c43} & 0 & 0 & A\_{c46} & 0 & A\_{c48} \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & A\_{c62} & 0 & A\_{c64} & A\_{c65} & 0 & A\_{c67} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & A\_{c82} & 0 & A\_{c84} & A\_{c85} & 0 & A\_{c87} & 0 \end{bmatrix}$$

$$B\_c = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ B\_{c21} & B\_{c22} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ B\_{c41} & B\_{c42} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \quad C\_c = \text{diag}\left(T\_1, T\_2, \cdots, T\_4\right).$$

where

$$A\_{c21} = \left(\frac{1}{m} + \frac{l\_a^2}{l\_y}\right) K\_{p's}, \quad A\_{c23} = \left(\frac{1}{m} - \frac{l\_al\_b}{l\_y}\right) K\_{p'}, \quad A\_{c2b} = -\frac{J\_z z l\_a}{J\_y \left(l\_a + l\_b\right)},$$

$$A\_{c28} = \frac{J\_z z l l\_a}{J\_y \left(l\_a + l\_b\right)}, \qquad \qquad A\_{c41} = \left(\frac{1}{m} - \frac{l\_al\_b}{J\_y}\right) K\_{p'}, \qquad \qquad A\_{c43} = \left(\frac{1}{m} + \frac{l\_b^2}{l\_y}\right) K\_{p'}, \qquad \qquad A\_{c4b} = \frac{J\_z z l l\_b}{J\_y \left(l\_a + l\_b\right)},$$

$$A\_{c48} = -\frac{J\_z z l l\_b}{J\_y \left(l\_a + l\_b\right)}, \qquad \qquad A\_{c62} = \frac{J\_z z l l\_a}{J\_x \left(l\_a + l\_b\right)}, \qquad \qquad A\_{c64} = -\frac{J\_z z l l\_a}{J\_x \left(l\_a + l\_b\right)}, \qquad \qquad A\_{c65} = \left(\frac{1}{m} + \frac{l\_a^2}{l\_x}\right) K\_{p'}, \qquad \qquad A\_{c66} = \frac{J\_z z l l\_a}{J\_x \left(l\_a + l\_b\right)}, \qquad \qquad A\_{c67} = \frac{J\_z z l l\_a}{J\_x \left(l\_a + l\_b\right)}$$

Analysis and Control of Flywheel Energy Storage Systems 135

$$\begin{split} &A\_{c\ominus 2} = \left(\frac{1}{m} - \frac{l\_{a}l\_{b}}{f\_{x}}\right) \mathbf{K}\_{p} \,, \qquad A\_{c\ominus 2} = -\frac{f\_{z}z\mathbf{u}\_{b}}{f\_{x}\left(l\_{a} + l\_{b}\right)} \mathbf{ \prime} \qquad A\_{c\ominus 4} = \frac{f\_{z}z\mathbf{u}\_{b}}{f\_{x}\left(l\_{a} + l\_{b}\right)} \mathbf{ \prime} \qquad A\_{c\ominus 5} = \left(\frac{1}{m} - \frac{l\_{a}l\_{b}}{f\_{x}}\right) \mathbf{K}\_{p} \,, \\ &A\_{c\ominus 6} = \left(\frac{1}{m} + \frac{l\_{b}^{2}}{f\_{y}}\right) \mathbf{K}\_{p} \,, \qquad B\_{c21} = \left(\frac{1}{m} + \frac{l\_{a}^{2}}{f\_{y}}\right) \mathbf{K}\_{c} \,, \qquad B\_{c41} = \left(\frac{1}{m} - \frac{l\_{a}l\_{b}}{f\_{y}}\right) \mathbf{K}\_{c} \,, \\ &B\_{c42} = \left(\frac{1}{m} + \frac{l\_{a}^{2}}{f\_{y}}\right) \mathbf{K}\_{c} \,, \qquad B\_{c63} = \left(\frac{1}{m} - \frac{l\_{a}l\_{b}}{f\_{x}}\right) \mathbf{K}\_{c} \,, \qquad B\_{c\ominus 1} = \left(\frac{1}{m} - \frac{l\_{a}l\_{b}}{f\_{x}}\right) \mathbf{K}\_{c} \,, \\ &B\_{c\ominus 4} = \left(\frac{1}{m} + \frac{l\_{a}^{2}}{f\_{y}}\right) \mathbf{K}\_{c} \,. \end{split}$$

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.

#### **4. Controller design**

Let he discrete-time model of (3) be described by

$$\begin{cases} X\left(k+1\right) = AX\left(k\right) + Bu\left(k\right) \\ Z\left(k\right) = CX\left(k\right) \end{cases} \tag{4}$$

where *k* denotes the discrete time. Note that the disturbance term is ignored.

By introducing the following synchronization errors,

$$
\delta\delta(k) = \text{LCX}(k)\tag{5}
$$

where

134 Energy Storage – Technologies and Applications

where ,,,,,,, *<sup>T</sup> X xxxxyyyy aabbaabb*

appropriate matrices *<sup>x</sup> B* and *Dz* ,

*c*

*c*

*B*

21

where

*c*

*c*

 <sup>28</sup> *z a*

 <sup>48</sup> *z b*

*ya b*

*J wl <sup>A</sup>*

*J wl <sup>A</sup>*

*ya b*

*Jl l* , 41

21 22

*c c*

*B B*

0000

0000

0000

0000

, 23

1 *a b c p y*

*z a*

*xa b*

*l l <sup>A</sup> <sup>K</sup> m J* 

*J wl <sup>A</sup>*

41 42

*c c*

*B B*

0 0

0 0

2

1 *<sup>a</sup> c p y*

*<sup>l</sup> A K m J* 

*Jl l* , <sup>62</sup>

*c*

vector, ,,, *<sup>T</sup> Z xxyy abab*

Then, the state-space model of (1) is obtained,

*c c xx c zz X AX Bu Bd Z CX Dd* 

is the output vector, and 1234 *TTTT* 1,0 are the output

21 23 26 28

01000000 0 00 0 00010000 0 00 0 00000100 00 00 00000001 00 00

*cc cc*

*AA AA*

41 43 46 48

*cc cc*

*AA AA <sup>A</sup>*

63 64

*c c*

*B B*

0 0

0 0

83 84

1 *a b c p y*

*Jl l* , <sup>64</sup>

*c*

43

*J wl <sup>A</sup>*

*l l <sup>A</sup> <sup>K</sup> m J* 

,

*c c*

*B B*

62 64 65 67

*c cc c*

*A AA A*

82 84 85 87

, 12 4 ,,, *C diag T T T <sup>c</sup>* .

, <sup>26</sup>

*J wl <sup>A</sup>*

2

1 *<sup>b</sup> c p y*

*z a*

*xa b*

*Jl l* ,

*<sup>l</sup> <sup>A</sup> <sup>K</sup> m J* 

*c*

*z a*

*ya b*

, <sup>46</sup>

*J wl <sup>A</sup>*

*<sup>l</sup> <sup>A</sup> <sup>K</sup> m J* 

*c*

65

*z b*

*ya b*

2

,

1 *<sup>a</sup> c p x*

*Jl l* ,

*Jl l* ,

*c cc c*

*A AA A*

is the state variable, ,,, *<sup>T</sup>*

transition matrices, *<sup>x</sup> d* and *<sup>z</sup> d* denote model uncertainties or system disturbances with

(3)

*xa xb ya yb u iiii* is the forcing

,

$$
\mathcal{S}(k) = \begin{bmatrix}
\boldsymbol{\chi}\_a - \boldsymbol{\chi}\_b \\
\boldsymbol{\chi}\_b - \boldsymbol{\chi}\_a \\
\boldsymbol{\chi}\_a - \boldsymbol{\chi}\_b \\
\boldsymbol{\chi}\_b - \boldsymbol{\chi}\_a
\end{bmatrix}, \boldsymbol{L} = \begin{bmatrix}
1 & -1 & 0 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 1 & -1 \\
\end{bmatrix}'
$$

it has the modified cost function,

$$\begin{split} J\{\boldsymbol{k}\} &= \sum\_{i=1}^{H\_{\mathcal{E}}} \hat{\boldsymbol{Z}}^{T} \left(\boldsymbol{k} + \boldsymbol{i} \mid \boldsymbol{k}\right) \hat{\boldsymbol{Z}} \left(\boldsymbol{k} + \boldsymbol{i} \mid \boldsymbol{k}\right) + \lambda \sum\_{i=1}^{H\_{\mathcal{E}}} \hat{\boldsymbol{u}}^{T} \left(\boldsymbol{k} + \boldsymbol{i} - \boldsymbol{1} \mid \boldsymbol{k}\right) \hat{\boldsymbol{u}} \left(\boldsymbol{k} + \boldsymbol{i} - \boldsymbol{1} \mid \boldsymbol{k}\right) \\ &+ \nu \sum\_{i=1}^{H\_{\mathcal{E}}} \hat{\boldsymbol{\delta}}^{T} \left(\boldsymbol{k} + \boldsymbol{i} \mid \boldsymbol{k}\right) \hat{\boldsymbol{\Delta}} \left(\boldsymbol{k} + \boldsymbol{i} \mid \boldsymbol{k}\right) \end{split} \tag{6}$$

where ˆ *Zk ik* | is the future output vector, *uk i k* ˆ 1| is the future control input vector, ˆ *k ik* | is the future synchronization errors, *Hp* is the prediction horizon, *Hc* is the control horizon, is the positive weighting factor used to adjust the control action, *v* is the non-negative weighting factor for the synchronization error.

Rewrite (6) as,

$$J\{k\} = \sum\_{j=0}^{H\_p-1} \Theta(\hat{X}, \hat{u}) + \hat{X}^T \left(k + H\_p \mid k\right) P\_0 \hat{X}^T \left(k + H\_p \mid k\right) \tag{7}$$

where

$$\begin{aligned} \Theta\left(\hat{X}, \boldsymbol{\mu}\right) &= \hat{X}^T \left(\boldsymbol{k} + \boldsymbol{H}\_p - \mathbf{1} + \boldsymbol{j} \mid \boldsymbol{k}\right) \mathbf{Q}\_j \hat{X} (\boldsymbol{k} + \boldsymbol{H}\_p - \mathbf{1} + \boldsymbol{j} \mid \boldsymbol{k}) \\ &+ \hat{\boldsymbol{u}}^T \left(\boldsymbol{k} + \boldsymbol{H}\_p - \mathbf{1} + \boldsymbol{j} \mid \boldsymbol{k}\right) \mathbf{R}\_j \hat{\boldsymbol{u}} \left(\boldsymbol{k} + \boldsymbol{H}\_p - \mathbf{1} + \boldsymbol{j} \mid \boldsymbol{k}\right) \end{aligned} \tag{8}$$

$$P\_0 = \mathbf{C}^T \mathbf{C} + \nu \mathbf{C}^T \mathbf{L}^T \mathbf{L} \mathbf{C} \tag{9}$$

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

*Lemma 1.* Consider the following ARE with an infinite-horizon linear quadratic control

there exists a unique, maximal, non-negative definite symmetric solution *P* .

*P* is a unique stabilizing solution, *i e*. ., <sup>1</sup> ( ) *T T A B B PB R B PA* has all the eigenvalues

In order to connect the RDE (14) to the ARE (15), the Fake Algebraic Riccati Technique

where *QQ P P jjj j* <sup>1</sup> . Clearly, while one has not altered the RDE in viewing it as a

( )( ) *T T*

masquerading ARE, the immediate result from Lemma 1 and (17) can be obtained.

<sup>1</sup> ( ) *T TT T P A PA A PB B PB R B PA Q* (15)

( )( ) *T T P A BK P A BK K RK Q* (16)

*<sup>j</sup> j j j j jj j P A BK P A BK K R K Q* (17)

necessitating the abandonment of a specific control performance.

**5. Stability analysis** 

*A*,*B* is stabilizable,

1/2 *A Q*, is detectable,

strictly within the unit circle.

*Q R* 0 and 0.

Rewrite (15) as

(FART) is used as follows:

*A*,*B* is stabilizable,

1/2 , *A Qj* is detectable,

0 and 0. *Q R j j*

*Theorem 1.* Consider (17) with *Qj* . If

then *<sup>j</sup> P* is stabilizing, *i e*. . the closed-loop transition matrix

(Souza et al., 1996),

where

Then

$$\mathbf{Q}\_{j} = \begin{cases} \mathbf{C}^{T}\mathbf{C} + \nu \mathbf{C}^{T}\mathbf{L}^{T}\mathbf{L}\mathbf{C} & \text{if} \quad j = 0, 1, \dots, H\_{p} - 2 \\ \mathbf{0} & \text{if} \quad j = H\_{p} - 1 \end{cases} \tag{10}$$

$$R\_j = \begin{cases} \infty I, & \text{if} \quad j = 0, 1..., H\_p - H\_c - 1\\ \lambda I, & \text{if} \quad j = H\_p - H\_{c'} ..., H\_p - 1' \end{cases} \tag{11}$$

where *I* is the unit matrix with appropriate dimension.

Hence, minimization of the cost function (7) results in the synchronization control law,

$$\mu\left(k\right) = -K\_{H\_p-1}X\left(k\right)\tag{12}$$

where

$$\mathbf{K}\_{H\_{p}-1} = \left(\mathbf{B}^{T}P\_{H\_{p}-1}\mathbf{B} + \mathcal{X}I\right)^{-1}\mathbf{B}^{T}P\_{H\_{p}-1}A\tag{13}$$

$$P\_{j+1} = A^T P\_j A - A^T P\_j B (B^T P\_j B + R\_j)^{-1} B^T P\_j A + Q\_j \tag{14}$$

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 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.

## **5. Stability analysis**

*Lemma 1.* Consider the following ARE with an infinite-horizon linear quadratic control (Souza et al., 1996),

$$P = A^T P A - A^T P B (B^T P B + R)^{-1} B^T P A + Q \tag{15}$$

where

136 Energy Storage – Technologies and Applications

non-negative weighting factor for the synchronization error.

1

*Hp*

*j*

*j*

 

*j*

where *I* is the unit matrix with appropriate dimension.

0

0

*T TT*

*T*

*T*

*Zk ik* | is the future output vector, *uk i k* ˆ 1| is the future control input vector,

is the positive weighting factor used to adjust the control action, *v* is the

0

*p p*

(7)

0,1, , 2

*p*

*p c*

*pc p*

1

1

(14)

*p*

*uk K Xk* (12)

(13)

(8)

(10)

(11)

(9)

*k ik* | is the future synchronization errors, *Hp* is the prediction horizon, *Hc* is the

ˆ ˆ ˆ , 1 | ( 1 |)

*Xu X k H j k QXk H j k*

*T TT P C C C L LC* 

0 1

 

> *I if j H H H*

> > <sup>1</sup> *Hp*

11 1 ( ) *pp p T T K BP B I BP A HH H* 

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

<sup>1</sup> ( ) *T TT T j j j jj j j P A PA A PB B PB R B PA Q*

, 0,1 , 1 , , ,, 1

*C C C L LC if j H <sup>Q</sup> if j H*

*I if j H H <sup>R</sup>*

Hence, minimization of the cost function (7) results in the synchronization control law,

*J k Xu X k H k PX k H k*

ˆˆ ˆ ( ,) | ˆ |

*T T*

ˆ ˆ 1 | 1 |

*p jp*

*u k H j k Ru k H j k*

*p jp*

where ˆ

control horizon,

Rewrite (6) as,

where

where

from the initial condition 0*P* .

 ˆ


Then


Rewrite (15) as

$$P = \left(A - BK\right)^T P \left(A - BK\right) + K^T RK + Q \tag{16}$$

In order to connect the RDE (14) to the ARE (15), the Fake Algebraic Riccati Technique (FART) is used as follows:

$$P\_j = \left(A - BK\_j\right)^T P\_j \left(A - BK\_j\right) + K\_j^T R\_j K\_j + \overline{Q}\_j \tag{17}$$

where *QQ P P jjj j* <sup>1</sup> . Clearly, while one has not altered the RDE in viewing it as a masquerading ARE, the immediate result from Lemma 1 and (17) can be obtained.

*Theorem 1.* Consider (17) with *Qj* . If


then *<sup>j</sup> P* is stabilizing, *i e*. . the closed-loop transition matrix

$$\overline{A}\_{j} = A - \mathcal{B}\left(\mathbf{B}^{T}\mathbf{P}\_{j}\mathbf{B} + \mathbf{R}\_{j}\right)^{-1}\mathbf{B}^{T}\mathbf{P}\_{j}A\tag{18}$$

Analysis and Control of Flywheel Energy Storage Systems 139

1

is the sensitivity

*p*

10 ~ 1 Hz. Then, the performance of the proposed

*jT jT K e Se <sup>H</sup>* 

*p*

*jT jT K e Se <sup>H</sup>* 

0.01 and *v* 0

 

  , and

Because one does not know exactly what is, various assumptions can be made about the nature of : nonlinear, linear time-varying, linear parameter-varying and linear timeinvariant being the most common ones. Also, various norms can be used, and the most commonly used one is the 'H-infinity' norm , which is defined as the worst-case 'energy gain' of an operator even for nonlinear systems. It then follows from the small-gain theorem that the feedback combination of this system with the uncertainty block *<sup>A</sup>* will

<sup>1</sup> 1 *s s*

(22)

<sup>1</sup> <sup>ˆ</sup> *Hp Sz I PzK z*

 

*jT jT K e Se H A* 

function. Note that (22) is only a sufficient condition for robust stability; if it is not satisfied, robust stability may nevertheless have been obtained. In practice, when tuning a controller, one can try to influence the frequency response properties in such a way as to make (22) hold.

Stability robustness with respect to variable control parameters will first be carried out. The

controller will be demonstrated in the presence of external disturbances and model

Consider the flywheel system with parameters given in (Zhu & Xiao, 2009), and assume that the rotor is spinning at a constant speed. As the eigenvalues of *Ac* are: 2.0353*i* , 10.4*i* , 149.3 , 149.3 , the open-loop continuous system is obviously unstable. With appropriate control parameters for the discrete-time model (sampling period 0.008 *<sup>s</sup> T* s), such as

*AHp*1 are within the unit circle, which are: 0.782 0.555*i* , 0.379 , 0.481 , 0.378 , 0.127 0.195 *i* and 0.027 respectively. In another word, the system can be stabilized with

The prediction and control horizons are closely related to the stability of the closed-loop system. In the case of additive uncertainties, the maximum singular value 1 *s s*

are set. It can be seen that a larger prediction horizon results in a smaller singular value, which

0.01 , *v* 10 , all of the eigenvalues of the closed-loop transition matrix

*y* -axis of each graph indicates the maximum singular value of 1 *s s*

*p*

denotes the largest singular value,

remain stable if

uncertainties.

6 *Hp* , 1 *Hc* ,

this feedback controller.

**6. Simulation study** 

the *x* -axis is the frequency range, <sup>2</sup>

**6.1. Stability robustness against control parameters** 

against variation of prediction horizon is illustrated in Fig. 2, while 1 *Hc* ,

where

has all its eigenvalues strictly within the unit circle.

Regarding the receding horizon strategy, only *<sup>j</sup> P* with 1 *<sup>p</sup> j H* will be applied. This leads to

$$P\_{H\_p-1} = A^T P\_{H\_p-1} A - A^T P\_{H\_p-1} B \left( B^T P\_{H\_p-1} B + \lambda I \right)^{-1} B^T P\_{H\_p-1} A + \overline{Q}\_{H\_p-1} \tag{19}$$

where 1 1 *H HH p pp Q PP* . Then, the stability result of the control system can be given by the following theorem.

*Theorem 2.* Consider (19) with the weighting matrix 1 *Hp Q* . If


then the controller (12) is stabilizing, i.e., the closed-loop transition matrix <sup>1</sup> 111 *ppp T T AHHH A BBP B I BP A* has all its eigenvalues strictly within the unit circle.

Proof. The proof is completed by setting 1 *<sup>p</sup> j H* in Theorem 1.

It can be seen from the above theorem that the prediction horizon *Hp* is a key parameter for stability, and an increasing *Hp* 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 *P z* 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) can be obtained by evaluating:

$$
\hat{P}\left(z\right) = \mathbb{C}\left(zI - A\right)^{-1}B\tag{20}
$$

Then, the real system *P z* is given by a 'norm-bounded' description:

, for additive model uncertaint es <sup>ˆ</sup> <sup>i</sup> *<sup>A</sup> Pz Pz* (21)

where *<sup>A</sup>* is stable bounded operator, and *P z* is often normalized in such a way that 1 .

Because one does not know exactly what is, various assumptions can be made about the nature of : nonlinear, linear time-varying, linear parameter-varying and linear timeinvariant being the most common ones. Also, various norms can be used, and the most commonly used one is the 'H-infinity' norm , which is defined as the worst-case 'energy gain' of an operator even for nonlinear systems. It then follows from the small-gain theorem that the feedback combination of this system with the uncertainty block *<sup>A</sup>* will remain stable if

$$\overline{\sigma} \left[ K\_{H\_p - 1} \left( e^{j o T\_s} \right) \mathcal{S} \left( e^{j o T\_s} \right) \right] \left\| \Lambda\_A \right\|\_{\infty} < 1 \tag{22}$$

where denotes the largest singular value, 1 <sup>1</sup> <sup>ˆ</sup> *Hp Sz I PzK z* is the sensitivity function. Note that (22) is only a sufficient condition for robust stability; if it is not satisfied, robust stability may nevertheless have been obtained. In practice, when tuning a controller, one can try to influence the frequency response properties in such a way as to make (22) hold.

### **6. Simulation study**

138 Energy Storage – Technologies and Applications

where 1 1 *H HH p pp*

*A*,*B* is stabilizable,

is detectable,

*P* is non-increasing,

 <sup>1</sup> 111 *ppp T T AHHH A BBP B I BP A*

nominal model (3) can be obtained by evaluating:

<sup>1</sup> , *Hp A Q* 

following theorem.

1/2

<sup>1</sup> *Hp*

circle.

1 .

has all its eigenvalues strictly within the unit circle.

*Theorem 2.* Consider (19) with the weighting matrix 1 *Hp*

0 ,

Proof. The proof is completed by setting 1 *<sup>p</sup> j H* in Theorem 1.

Then, the real system *P z* is given by a 'norm-bounded' description:

<sup>1</sup> *T T A A B B PB R B PA <sup>j</sup> jj j*

Regarding the receding horizon strategy, only *<sup>j</sup> P* with 1 *<sup>p</sup> j H* will be applied. This leads to

then the controller (12) is stabilizing, i.e., the closed-loop transition matrix

It can be seen from the above theorem that the prediction horizon *Hp* is a key parameter for stability, and an increasing *Hp* 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 *P z* 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

where *<sup>A</sup>* is stable bounded operator, and *P z* is often normalized in such a way that

has all its eigenvalues strictly within the unit

 <sup>1</sup> 1 1 1 1 11 *p p pP p P T TT T H H H H HH P AP A AP BBP B I BP A Q*

*Q PP* . Then, the stability result of the control system can be given by the

 (19)

*Q* . If

(18)

<sup>1</sup> *P z C zI A B* <sup>ˆ</sup> (20)

, for additive model uncertaint es <sup>ˆ</sup> <sup>i</sup> *<sup>A</sup> Pz Pz* (21)

Stability robustness with respect to variable control parameters will first be carried out. The *y* -axis of each graph indicates the maximum singular value of 1 *s s p jT jT K e Se <sup>H</sup>* , and the *x* -axis is the frequency range, <sup>2</sup> 10 ~ 1 Hz. Then, the performance of the proposed controller will be demonstrated in the presence of external disturbances and model uncertainties.

Consider the flywheel system with parameters given in (Zhu & Xiao, 2009), and assume that the rotor is spinning at a constant speed. As the eigenvalues of *Ac* are: 2.0353*i* , 10.4*i* , 149.3 , 149.3 , the open-loop continuous system is obviously unstable. With appropriate control parameters for the discrete-time model (sampling period 0.008 *<sup>s</sup> T* s), such as 6 *Hp* , 1 *Hc* , 0.01 , *v* 10 , all of the eigenvalues of the closed-loop transition matrix *AHp*1 are within the unit circle, which are: 0.782 0.555*i* , 0.379 , 0.481 , 0.378 , 0.127 0.195 *i* and 0.027 respectively. In another word, the system can be stabilized with this feedback controller.

#### **6.1. Stability robustness against control parameters**

The prediction and control horizons are closely related to the stability of the closed-loop system. In the case of additive uncertainties, the maximum singular value 1 *s s p jT jT K e Se <sup>H</sup>* against variation of prediction horizon is illustrated in Fig. 2, while 1 *Hc* , 0.01 and *v* 0 are set. It can be seen that a larger prediction horizon results in a smaller singular value, which means that the stability robustness of the control system can be improved. As a rule of thumb, *Hp* can be chosen according to int 2 / *Hp s b* , where *<sup>s</sup>* is the sampling frequency and *<sup>b</sup>* is the bandwidth of the process. Fig. 3 shows the singular value when the control horizon is varying. Clearly, a smaller control horizon *Hc* may enhance the stability robustness of the control system. However, if the nominal model of the process is accurate enough, and the influence of model uncertainties is negligible, then 1 *Hc* is preferred for faster system responses.

Analysis and Control of Flywheel Energy Storage Systems 141

will reduce the control action and

, the feedback action

should be chosen when the

can improve the stability robustness of

would then be expected as the system

, while 6 *Hp* ,

The stability robustness bounds shown in Fig. 4 is obtained by varying

the influence of the model uncertainties on system stability will become less important.

system stability might be degraded due to significant model uncertainty. However, if the

response can be improved in this case, i.e., a decrease in the response time. In practice, a

The synchronization factor *v* is introduced to compensate the synchronization error of the rotor in radial direction. Fig. 5 shows that the influence of *v* on stability robustness is not consistent over frequency. In particular, a lower value of *v* can enhance the stability robustness at certain frequencies, but the performance will be degraded at higher frequencies. Another interesting observation is that the two boundaries for *v* 5 and *v* 10 are almost overlapping. It means that the stability robustness of the control system will not be affected if a further increase of *v* is applied. In general, one can increase the prediction horizon and the synchronization control weighting factor so that the stability of the control system is maintained while the synchronization performance can be

is necessary as it may have a large range of the values and is difficult to

1 *Hc* and *v* 0 are set. Clearly, a larger value of

model uncertainty is insignificant, a smaller

careful choice of

predetermine it.

improved.

the control system. This is because that the increasing

**Figure 4.** Maximum singular value 1 *s s*

*p*

*jT jT K e Se <sup>H</sup>* 

  against weighting factor

Consequently, the stability robustness can be enhanced. If

disappears and the closed loop is broken. In general, a larger

**Figure 2.** Maximum singular value 1 *s s p jT jT K e Se <sup>H</sup>* against prediction horizon *Hp*

**Figure 3.** Maximum singular value 1 *s s p jT jT K e Se <sup>H</sup>* against control horizon *Hc*

The stability robustness bounds shown in Fig. 4 is obtained by varying , while 6 *Hp* , 1 *Hc* and *v* 0 are set. Clearly, a larger value of can improve the stability robustness of the control system. This is because that the increasing will reduce the control action and the influence of the model uncertainties on system stability will become less important. Consequently, the stability robustness can be enhanced. If , the feedback action disappears and the closed loop is broken. In general, a larger should be chosen when the system stability might be degraded due to significant model uncertainty. However, if the model uncertainty is insignificant, a smaller would then be expected as the system response can be improved in this case, i.e., a decrease in the response time. In practice, a careful choice of is necessary as it may have a large range of the values and is difficult to predetermine it.

140 Energy Storage – Technologies and Applications

responses.

*Hp* can be chosen according to int 2 / *Hp*

**Figure 2.** Maximum singular value 1 *s s*

**Figure 3.** Maximum singular value 1 *s s*

*p*

*p*

*jT jT K e Se <sup>H</sup>* 

 

*jT jT K e Se <sup>H</sup>* 

  against prediction horizon *Hp*

against control horizon *Hc*

means that the stability robustness of the control system can be improved. As a rule of thumb,

*<sup>b</sup>* is the bandwidth of the process. Fig. 3 shows the singular value when the control horizon is varying. Clearly, a smaller control horizon *Hc* may enhance the stability robustness of the control system. However, if the nominal model of the process is accurate enough, and the influence of model uncertainties is negligible, then 1 *Hc* is preferred for faster system

, where

*<sup>s</sup>* is the sampling frequency and

*s b* 

> The synchronization factor *v* is introduced to compensate the synchronization error of the rotor in radial direction. Fig. 5 shows that the influence of *v* on stability robustness is not consistent over frequency. In particular, a lower value of *v* can enhance the stability robustness at certain frequencies, but the performance will be degraded at higher frequencies. Another interesting observation is that the two boundaries for *v* 5 and *v* 10 are almost overlapping. It means that the stability robustness of the control system will not be affected if a further increase of *v* is applied. In general, one can increase the prediction horizon and the synchronization control weighting factor so that the stability of the control system is maintained while the synchronization performance can be improved.

**Figure 4.** Maximum singular value 1 *s s p jT jT K e Se <sup>H</sup>* against weighting factor 

Analysis and Control of Flywheel Energy Storage Systems 143

**Figure 6.** Radial displacements of the rotor along x-axis

**Figure 7.** Radial displacements of the rotor along y-axis

**Figure 5.** Maximum singular value 1 *s s p jT jT K e Se <sup>H</sup>* against synchronization factor *v*

#### **6.2. Disturbances on magnetic forces**

In this simulation, force disturbances are introduced to the bearings of the rotor at different time instants, and amplitudes are 0.5N, -0.5N, 0.5N and -0.5N on xa-axis, xb-axis, ya-axis and yb-axis respectively. The duration of 0.2 seconds for each disturbance is assumed. Figs. 6-11 show the numerical results of the control algorithm when 10 *Hp* , 1 *Hc* , 0.01 are set for the two cases: with *v* 0 , and *v* 10 . Clearly, without cross-coupling control action due to *v* 0 , evident synchronization errors and a conical whirl mode during the transient responses are resulted. However, when *v* 10 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.

**Figure 6.** Radial displacements of the rotor along x-axis

**Figure 5.** Maximum singular value 1 *s s*

**6.2. Disturbances on magnetic forces** 

near the nominal position.

*p*

6-11 show the numerical results of the control algorithm when 10 *Hp* , 1 *Hc* ,

*jT jT K e Se <sup>H</sup>* 

In this simulation, force disturbances are introduced to the bearings of the rotor at different time instants, and amplitudes are 0.5N, -0.5N, 0.5N and -0.5N on xa-axis, xb-axis, ya-axis and yb-axis respectively. The duration of 0.2 seconds for each disturbance is assumed. Figs.

set for the two cases: with *v* 0 , and *v* 10 . Clearly, without cross-coupling control action due to *v* 0 , evident synchronization errors and a conical whirl mode during the transient responses are resulted. However, when *v* 10 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

  against synchronization factor *v*

0.01 are

**Figure 7.** Radial displacements of the rotor along y-axis

Analysis and Control of Flywheel Energy Storage Systems 145

**Figure 10.** Displacements of rotor mass center

**Figure 8.** Control currents to bearings along x-axis

**Figure 9.** Control currents to bearings along y-axis

**Figure 10.** Displacements of rotor mass center

**Figure 8.** Control currents to bearings along x-axis

**Figure 9.** Control currents to bearings along y-axis

Analysis and Control of Flywheel Energy Storage Systems 147

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**Figure 11.** Rolling angles of rotor mass center

## **7. 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* 

#### **8. References**

146 Energy Storage – Technologies and Applications

**Figure 11.** Rolling angles of rotor mass center

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

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

*College of Information Engineering, Shenyang University of Chemical Technology, China* 

**7. Conclusion** 

uncertainties.

**Author details** 

Yong Xiao, Xiaoyu Ge and Zhe Zheng

be effectively suppressed.


Zhu K. & Chen B. (2001). Cross-coupling design of generalized predictive control with reference models. *Proc. IMechE Part I: Journal of Systems Control Engineering*, 215: 375- 384.

**Chapter 7** 

© 2013 Uno, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Uno, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Single- and Double-Switch Cell Voltage** 

**Lithium-Ion Cells and Supercapacitors** 

As demands for energy-efficient electrical devices and equipment continue to increase, the role of energy storage devices and systems becomes more and more important. Applications of such energy storage devices range from portable electronic devices, where a single cell is sufficient to provide adequate run time, to electric vehicles that require more than 100 cells in series to produce a sufficient high voltage to drive motors. Lithium-ion batteries (LIBs) are the most prevalent and promising because of their highest specific energy among

Supercapacitor (SC) technologies, including traditional electric double-layer capacitors and lithium-ion capacitors (hybrid capacitors that combine features of double-layer capacitors and LIBs) are also drawing significant attention, because of their outstanding service life over a wide temperature range, high-power capability, and high-energy efficiency performance. The use of such SC technologies has traditionally been limited to high-power applications such as hybrid electric vehicles and regenerative systems in industries, where high-power energy buffers are needed to meet short-term large power demands. But it is found that SC technologies also have a great potential to be alternative energy storage sources to traditional secondary batteries once their superior life performance over a wide temperature range is factored in (Uno, 2011; Uno &

The voltage of single cells is inherently low, typically lower than 4.2, 2.7, and 3.8 V for lithium-ion cells, traditional electric double-layer capacitors, and lithium-ion capacitors, respectively. Hence in most practical uses, a number of single cells need to be connected in series to produce a high voltage level to meet the load voltage requirement. Voltages of

**Equalizers for Series-Connected** 

Additional information is available at the end of the chapter

commercially available secondary battery technologies.

Masatoshi Uno

**1. Introduction** 

Tanaka, 2011).

http://dx.doi.org/10.5772/52215


Masatoshi Uno

148 Energy Storage – Technologies and Applications

384.

219: 499-510.

*Automatic Control*, AC-31:831-838.

Zhu K. & Chen B. (2001). Cross-coupling design of generalized predictive control with reference models. *Proc. IMechE Part I: Journal of Systems Control Engineering*, 215: 375-

Xiao Y. & Zhu K. (2006). Optimal synchronization control of high-precision motion systems,

Zhang C. & Tseng K. (2007). A novel flywheel energy storage system with partially-selfbearing flywheel-rotor, *IEEE Transactions on Energy Conversion*, 22(2): 477-487. Xiao Y., Zhu K., Zhang C., Tseng K. & Ling K. (2005). Stabilizing synchronization control of rotor-magnetic bearing system, *Proc IMechE Part I: Journal of Systems Control Engineering*,

Zhu K., Xiao Y. & Rajendra A. (2009). Optimal control of the magnetic bearings for a

Souza, C., Gevers, M., Goodwin, G. (1986). Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices. *IEEE Transactions on* 

*IEEE Transactions on Industrial Electronics*, 53(4): 1160-1169.

flywheel energy storage system, *Mechatronics*, 19: 1221–1235.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52215

## **1. Introduction**

As demands for energy-efficient electrical devices and equipment continue to increase, the role of energy storage devices and systems becomes more and more important. Applications of such energy storage devices range from portable electronic devices, where a single cell is sufficient to provide adequate run time, to electric vehicles that require more than 100 cells in series to produce a sufficient high voltage to drive motors. Lithium-ion batteries (LIBs) are the most prevalent and promising because of their highest specific energy among commercially available secondary battery technologies.

Supercapacitor (SC) technologies, including traditional electric double-layer capacitors and lithium-ion capacitors (hybrid capacitors that combine features of double-layer capacitors and LIBs) are also drawing significant attention, because of their outstanding service life over a wide temperature range, high-power capability, and high-energy efficiency performance. The use of such SC technologies has traditionally been limited to high-power applications such as hybrid electric vehicles and regenerative systems in industries, where high-power energy buffers are needed to meet short-term large power demands. But it is found that SC technologies also have a great potential to be alternative energy storage sources to traditional secondary batteries once their superior life performance over a wide temperature range is factored in (Uno, 2011; Uno & Tanaka, 2011).

The voltage of single cells is inherently low, typically lower than 4.2, 2.7, and 3.8 V for lithium-ion cells, traditional electric double-layer capacitors, and lithium-ion capacitors, respectively. Hence in most practical uses, a number of single cells need to be connected in series to produce a high voltage level to meet the load voltage requirement. Voltages of

© 2013 Uno, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Uno, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

series-connected cells are gradually imbalanced because their individual properties, such as capacity/capacitance, self-discharge rate, and internal impedance, are different from each other. Nonuniform temperature gradient among cells in a battery pack/module also lead to nonuniform self-discharging that accelerates voltage imbalance. In a voltage-imbalanced battery/module, some cells in the series connection may be overcharged and overdischarged during the charging and discharging processes, respectively, even though the average voltage of the series-connected cells is within the safety boundary. Using LIBs/SCs beyond the safety boundary not only curtails their operational life but also undermines their electrical characteristics. Overcharging must be prevented especially for LIBs since it may result in fire or even an explosion in the worst situation.

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 151

The most common and traditional approach involves the use of dissipative equalizers, which do not require high-frequency switching operations. With dissipative equalizers, the voltage of series-connected cells can be equalized by removing stored energy or by shunting charge current from the cells with higher voltage. During the equalization process, the excess energy or current is inevitably dissipated in the form of heat, negatively influencing the thermal system of batteries/modules. Dissipative equalizers introduce advantages over nondissipative equalizers in terms of circuit simplicity and cost. The dissipative equalizers

Fig. 1(a) shows the simplest solution using passive resistors. Voltage imbalance gradually decreases because of different self-discharge through resistors depending on cell voltages. Although simple, the relentless power loss in resistors reduces the energy efficiency, depending on the resistance values, and hence, this equalizer is rarely used in practice.

Another concept of a passive dissipative equalizer is the use of Zener diodes, as shown in Fig. 1(b). Cell voltages exceeding a Zener voltage level are cramped by Zener diodes. The power loss in the Zener diodes during the rest period is negligibly low depending on their leakage current. However, these diodes must be chosen to be capable of the largest possible charge current because the charge current flows through them when the cell voltage reaches or exceeds the Zener voltage level. In addition, a great temperature dependency of the Zener

**Figure 1.** Passive dissipative equalizers using (a) resistors and (b) Zener diodes, and (c) active shunting

Fig. 1(c) shows a schematic drawing of shunting equalizers (Isaacson et al., 2000; Uno, 2009). Cell voltages are monitored and compared with a preset voltage level (shunt voltage level). When the cell voltage reaches or exceeds the shunt voltage level, the charge current is bypassed through a transistor to reduce the net charge current. The product of cell voltage and shunt current is the power dissipation in the equalizer. The shunting equalizer needs as

voltage, which may not be acceptable in most applications, should be factored in.

**2. Conventional equalization techniques** 

can be categorized into two groups: passive and active equalizers.

**2.1. Dissipative equalizers** 

equalizer.

In addition to the safety issues mentioned above, the voltage imbalance also reduces the available energies of cells. When charging the cells in series, charging processes must be halted as soon as the most charged cell reaches the upper voltage limit, above which accelerated irreversible deterioration is very likely. Similarly, in order to avoid overdischarging during discharging processes, the least charged cell in the series connection limits the discharging time as a whole. Thus, voltage imbalance should be minimized in order to prolong life time as well as to maximize the available energies.

Various kind of equalization techniques have been proposed, demonstrated, and implemented for LIBs and SCs (Cao et al., 2008; Guo et al., 2006). However, conventional equalization techniques have one of the following major drawbacks:


This chapter presents single- and double-switch cell voltage equalizers for series-connected lithium-ion cells and SCs. The equalization process of the equalizers is nondissipative, and a multi-winding transformer is not necessary. Hence, all the issues underlying the conventional equalizers listed above can be addressed by the presented equalizers. In Section 2, the above-mentioned issues are discussed in detail, and conventional cell voltage equalizers are briefly reviewed. In Section 3, single-switch cell voltage equalizers based on multi-stacked buck–boost converters are presented. In the single-switch equalizers, although multiple inductors are required, the circuitry can be very simple because of the single-switch configuration. Section 4 introduces double-switch cell voltage equalizers using a resonant inverter and a voltage multiplier. Although the circuitry is slightly more complex than the single-switch equalizers, its single-magnetic configuration minimizes circuit size and cost. Detailed operation analyses are mathematically made, and experimental equalization tests performed for series-connected SCs and lithium-ion cells using the prototype of the singleand double-switch equalizers are shown. Finally, in Section 5, the presented single- and double-switch equalizers are compared with conventional equalizers in terms of the required number of circuit components.

## **2. Conventional equalization techniques**

## **2.1. Dissipative equalizers**

150 Energy Storage – Technologies and Applications

result in fire or even an explosion in the worst situation.

order to prolong life time as well as to maximize the available energies.

equalization techniques have one of the following major drawbacks:

2. Complex circuitry and control because of high switch count.

windings.

required number of circuit components.

1. Low energy efficiency because of the dissipative equalization mechanism.

series-connected cells are gradually imbalanced because their individual properties, such as capacity/capacitance, self-discharge rate, and internal impedance, are different from each other. Nonuniform temperature gradient among cells in a battery pack/module also lead to nonuniform self-discharging that accelerates voltage imbalance. In a voltage-imbalanced battery/module, some cells in the series connection may be overcharged and overdischarged during the charging and discharging processes, respectively, even though the average voltage of the series-connected cells is within the safety boundary. Using LIBs/SCs beyond the safety boundary not only curtails their operational life but also undermines their electrical characteristics. Overcharging must be prevented especially for LIBs since it may

In addition to the safety issues mentioned above, the voltage imbalance also reduces the available energies of cells. When charging the cells in series, charging processes must be halted as soon as the most charged cell reaches the upper voltage limit, above which accelerated irreversible deterioration is very likely. Similarly, in order to avoid overdischarging during discharging processes, the least charged cell in the series connection limits the discharging time as a whole. Thus, voltage imbalance should be minimized in

Various kind of equalization techniques have been proposed, demonstrated, and implemented for LIBs and SCs (Cao et al., 2008; Guo et al., 2006). However, conventional

3. Design difficulty and poor modularity because of the need for a multi-winding transformer that imposes strict parameter matching among multiple secondary

This chapter presents single- and double-switch cell voltage equalizers for series-connected lithium-ion cells and SCs. The equalization process of the equalizers is nondissipative, and a multi-winding transformer is not necessary. Hence, all the issues underlying the conventional equalizers listed above can be addressed by the presented equalizers. In Section 2, the above-mentioned issues are discussed in detail, and conventional cell voltage equalizers are briefly reviewed. In Section 3, single-switch cell voltage equalizers based on multi-stacked buck–boost converters are presented. In the single-switch equalizers, although multiple inductors are required, the circuitry can be very simple because of the single-switch configuration. Section 4 introduces double-switch cell voltage equalizers using a resonant inverter and a voltage multiplier. Although the circuitry is slightly more complex than the single-switch equalizers, its single-magnetic configuration minimizes circuit size and cost. Detailed operation analyses are mathematically made, and experimental equalization tests performed for series-connected SCs and lithium-ion cells using the prototype of the singleand double-switch equalizers are shown. Finally, in Section 5, the presented single- and double-switch equalizers are compared with conventional equalizers in terms of the The most common and traditional approach involves the use of dissipative equalizers, which do not require high-frequency switching operations. With dissipative equalizers, the voltage of series-connected cells can be equalized by removing stored energy or by shunting charge current from the cells with higher voltage. During the equalization process, the excess energy or current is inevitably dissipated in the form of heat, negatively influencing the thermal system of batteries/modules. Dissipative equalizers introduce advantages over nondissipative equalizers in terms of circuit simplicity and cost. The dissipative equalizers can be categorized into two groups: passive and active equalizers.

Fig. 1(a) shows the simplest solution using passive resistors. Voltage imbalance gradually decreases because of different self-discharge through resistors depending on cell voltages. Although simple, the relentless power loss in resistors reduces the energy efficiency, depending on the resistance values, and hence, this equalizer is rarely used in practice.

Another concept of a passive dissipative equalizer is the use of Zener diodes, as shown in Fig. 1(b). Cell voltages exceeding a Zener voltage level are cramped by Zener diodes. The power loss in the Zener diodes during the rest period is negligibly low depending on their leakage current. However, these diodes must be chosen to be capable of the largest possible charge current because the charge current flows through them when the cell voltage reaches or exceeds the Zener voltage level. In addition, a great temperature dependency of the Zener voltage, which may not be acceptable in most applications, should be factored in.

**Figure 1.** Passive dissipative equalizers using (a) resistors and (b) Zener diodes, and (c) active shunting equalizer.

Fig. 1(c) shows a schematic drawing of shunting equalizers (Isaacson et al., 2000; Uno, 2009). Cell voltages are monitored and compared with a preset voltage level (shunt voltage level). When the cell voltage reaches or exceeds the shunt voltage level, the charge current is bypassed through a transistor to reduce the net charge current. The product of cell voltage and shunt current is the power dissipation in the equalizer. The shunting equalizer needs as

many switches, voltage sensors, and comparators as the number of series connection of cells. In addition, this equalizer inevitably causes energy loss in the form of heat generation during the equalization process. The operation flexibility is also poor because cells are equalized only during the charging process, especially at fully charged states.

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 153

connected cells can be redistributed via a multi-winding transformer to the cell(s) having the lowest voltage (Kutkut, et al., 1995). The required number of switches in the multiwinding transformer-based equalizers is significantly less than those required in ICE topologies. However, these topologies need a multi-winding transformer that must be customized depending on the number of series connections, and hence, the modularity is not good. In addition, since parameter mismatching among multiple secondary windings results in voltage imbalance that can never be compensated by control, multi-winding transformers must be designed and made with great care (Cao et al., 2008; Guo et al., 2006). In general, the difficulty of parameter matching significantly increases with the number of windings. Therefore, their applications are limited to modules/batteries with a

**Figure 2.** (a) Generic configuration of individual cell equalizer, (b) switched capacitor-based equalizer,

**Figure 3.** Equalizers using a multi-winding transformer: (a) flyback converter-based and (b) forward

few series connections.

and (c) buck–boost converter-based equalizer.

converter-based equalizers.

Although dissipative equalization techniques seem less effective compared with nondissipative equalizers, which are reviewed in the following subsection, the shunting equalizers are widely used in various applications, and a number of battery management ICs that include shunting equalizers are available because of their simplicity, good modularity (or extendibility), and cost effectiveness.

## **2.2. Nondissipative equalizers**

Nondissipative equalizers that transfer charges or energies among series-connected cells are considered more suitable and promising than dissipative equalizers in terms of energy efficiency and thermal management. In addition, nondissipative equalizers (including single- and double-switch equalizers presented in this chapter) are usually operational during both charging and discharging, and hence, operation flexibility can be improved compared with dissipative equalizers. Numerous nondissipative equalization techniques have been proposed and demonstrated. Representative nondissipative equalizer topologies are reviewed in the following subsections.

## *2.2.1. Individual cell equalizer*

Fig. 2(a) depicts a schematic drawing of the individual cell equalizer (ICE) (Lee & Cheng, 2005). ICEs are typically based on individual bidirectional dc–dc converters such as switched capacitor converters (Pascual & Krein, 1997; Uno & Tanaka, 2011) and buck–boost converters (Nishijima et al., 2000), as shown in Figs. 2(b) and (c), respectively. Other types of bidirectional converters, such as resonant switched capacitor converters and Ćuk converters (Lee & Cheng, 2005), can also be used for improving equalization efficiencies. In ICE topologies, the charges or energies of the series-connected cells can be transferred between adjacent cells to eliminate cell voltage imbalance. The number of series connection of cells can be arbitrary extended by adding the number of ICEs.

Since these ICE topologies are derived from multiple individual bidirectional dc–dc converters, numerous switches, sensors, and switch drivers are required in proportion to the number of series-connected energy storage cells. Therefore, their circuit complexity and cost are prone to increase, especially for applications needing a large number of series connections, and their reliability decreases as the number of series connections increases.

## *2.2.2. Equalizers using a multi-winding transformer*

In cell voltage equalizers using a multi-winding transformer based on flyback and forward converters, as shown in Figs. 3(a) and (b), respectively, the energies of series-

**2.2. Nondissipative equalizers** 

are reviewed in the following subsections.

can be arbitrary extended by adding the number of ICEs.

*2.2.2. Equalizers using a multi-winding transformer* 

*2.2.1. Individual cell equalizer* 

increases.

modularity (or extendibility), and cost effectiveness.

many switches, voltage sensors, and comparators as the number of series connection of cells. In addition, this equalizer inevitably causes energy loss in the form of heat generation during the equalization process. The operation flexibility is also poor because cells are

Although dissipative equalization techniques seem less effective compared with nondissipative equalizers, which are reviewed in the following subsection, the shunting equalizers are widely used in various applications, and a number of battery management ICs that include shunting equalizers are available because of their simplicity, good

Nondissipative equalizers that transfer charges or energies among series-connected cells are considered more suitable and promising than dissipative equalizers in terms of energy efficiency and thermal management. In addition, nondissipative equalizers (including single- and double-switch equalizers presented in this chapter) are usually operational during both charging and discharging, and hence, operation flexibility can be improved compared with dissipative equalizers. Numerous nondissipative equalization techniques have been proposed and demonstrated. Representative nondissipative equalizer topologies

Fig. 2(a) depicts a schematic drawing of the individual cell equalizer (ICE) (Lee & Cheng, 2005). ICEs are typically based on individual bidirectional dc–dc converters such as switched capacitor converters (Pascual & Krein, 1997; Uno & Tanaka, 2011) and buck–boost converters (Nishijima et al., 2000), as shown in Figs. 2(b) and (c), respectively. Other types of bidirectional converters, such as resonant switched capacitor converters and Ćuk converters (Lee & Cheng, 2005), can also be used for improving equalization efficiencies. In ICE topologies, the charges or energies of the series-connected cells can be transferred between adjacent cells to eliminate cell voltage imbalance. The number of series connection of cells

Since these ICE topologies are derived from multiple individual bidirectional dc–dc converters, numerous switches, sensors, and switch drivers are required in proportion to the number of series-connected energy storage cells. Therefore, their circuit complexity and cost are prone to increase, especially for applications needing a large number of series connections, and their reliability decreases as the number of series connections

In cell voltage equalizers using a multi-winding transformer based on flyback and forward converters, as shown in Figs. 3(a) and (b), respectively, the energies of series-

equalized only during the charging process, especially at fully charged states.

connected cells can be redistributed via a multi-winding transformer to the cell(s) having the lowest voltage (Kutkut, et al., 1995). The required number of switches in the multiwinding transformer-based equalizers is significantly less than those required in ICE topologies. However, these topologies need a multi-winding transformer that must be customized depending on the number of series connections, and hence, the modularity is not good. In addition, since parameter mismatching among multiple secondary windings results in voltage imbalance that can never be compensated by control, multi-winding transformers must be designed and made with great care (Cao et al., 2008; Guo et al., 2006). In general, the difficulty of parameter matching significantly increases with the number of windings. Therefore, their applications are limited to modules/batteries with a few series connections.

**Figure 2.** (a) Generic configuration of individual cell equalizer, (b) switched capacitor-based equalizer, and (c) buck–boost converter-based equalizer.

**Figure 3.** Equalizers using a multi-winding transformer: (a) flyback converter-based and (b) forward converter-based equalizers.

## *2.2.3. Equalizers using a single converter with selection switches*

Figs. 4(a) and (b) show equalizers using a single converter and selection switches based on the flying capacitor and the flyback converter, respectively (Kim et al., 2011). In these topologies, individual cell voltages are monitored, and the cell(s) with the lowest and/or highest voltages are determined. In the flying capacitor-based equalizer, shown in Fig. 4(a), the energy of the most charged cell is delivered to the least charged cell via the flying capacitor C by selecting proper switches. In the flyback converter-based equalizer, shown in Fig. 4(b), the energy of series-connected cells is redistributed to the least charged cell via the flyback converter and properly selected switches.

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 155

without a transformer because of its inverting property. Out of the three candidates shown in Fig. 5, the Ćuk converter is the best in terms of current pulsation at its input and output because inductors Lin and Lout are connected in series to the input and output, respectively. Although the SEPIC and Zeta converter can be adapted without a transformer, transformer-less operation may suffer from duty cycle limitations when the number of series connection is large, as will be discussed in the following subsection. Therefore, the need for the transformer in the Ćuk converter is justified, although comparative analysis is necessary to determine the best topology to achieve sufficient performance at reasonable size for a given application. The isolated Ćuk converter-based

**Figure 5.** Traditional buck–boost converters that can be used as the basic topology for single-switch cell

The derived isolated Ćuk converter-based single-switch equalizer for four cells connected in series is shown in Fig. 6. The circuit consisting of Cin, Lin, Q, Ca, transformer, C3, D3, and L3 is identical to the circuit shown in Fig. 5(c), while the circuit consisting of Ci–Di–Li (i = 1…4) is multi-stacked to the isolated Ćuk converter. Hence, the derived equalizer can be regarded as

**N1 T1 N2**

*VP VS*

**Ca**

+

**L1**

**D2**

**D1**

*i D1*

*i L1*

*i D2*

*i L2*

*i D3*

*i L3*

**B1**

*V1 i B1*

**B2**

*i B2*

*V2*

*V3*

*V4*

*i L4*

*i D4*

**B3**

*i B3*

*i B4*

**B4**

**C1**

+

*i C1*

**C2**

+

*i C2*

**C3**

+

*i C3*

**C4**

+

*i C4*

**L2**

**L3**

**D3**

**L4**

**D4**

voltage equalizers. (a) SEPIC , (b) Zeta, and (c) isolated Ćuk converters.

**Lin**

**Q** *i Lin*

**Cin**

**Figure 6.** Single-switch cell voltage equalizer based on isolated Ćuk converter.

equalizer is focused in the following sections.

multi-stacked Ćuk converters.

**Figure 4.** Equalizers using a single converter with selection switches based on (a) flying capacitor and (b) flyback converter.

These topologies can reduce the number of passive components significantly when compared with ICE topologies shown in Fig. 2, and do not need a multi-winding transformer. However, the required number of switches is still large (proportional to the number of series connections), and furthermore, microcontroller- or DSP-based intelligent management is mandatory because target cell(s) (i.e., the least and/or most charged cells) must be determined for equalization to be implemented.

## **3. Single-switch cell voltage equalizer using multi-stacked buck–boost converters**

## **3.1. Circuit description and major benefits**

The single-switch cell voltage equalizers are derived by multi-stacking traditional buck– boost converters that consist of two inductors and one coupling capacitor (Uno & Tanaka, 2011). Candidate topologies, which can be used as a basic topology, are shown in Fig. 5. The single-ended primary inductor converter (SEPIC) and Zeta converters can be simply adapted as the basic topology. On the other hand, the Ćuk topology cannot be used

(b) flyback converter.

**converters** 

*2.2.3. Equalizers using a single converter with selection switches* 

flyback converter and properly selected switches.

must be determined for equalization to be implemented.

**3.1. Circuit description and major benefits** 

Figs. 4(a) and (b) show equalizers using a single converter and selection switches based on the flying capacitor and the flyback converter, respectively (Kim et al., 2011). In these topologies, individual cell voltages are monitored, and the cell(s) with the lowest and/or highest voltages are determined. In the flying capacitor-based equalizer, shown in Fig. 4(a), the energy of the most charged cell is delivered to the least charged cell via the flying capacitor C by selecting proper switches. In the flyback converter-based equalizer, shown in Fig. 4(b), the energy of series-connected cells is redistributed to the least charged cell via the

**Figure 4.** Equalizers using a single converter with selection switches based on (a) flying capacitor and

These topologies can reduce the number of passive components significantly when compared with ICE topologies shown in Fig. 2, and do not need a multi-winding transformer. However, the required number of switches is still large (proportional to the number of series connections), and furthermore, microcontroller- or DSP-based intelligent management is mandatory because target cell(s) (i.e., the least and/or most charged cells)

**3. Single-switch cell voltage equalizer using multi-stacked buck–boost** 

The single-switch cell voltage equalizers are derived by multi-stacking traditional buck– boost converters that consist of two inductors and one coupling capacitor (Uno & Tanaka, 2011). Candidate topologies, which can be used as a basic topology, are shown in Fig. 5. The single-ended primary inductor converter (SEPIC) and Zeta converters can be simply adapted as the basic topology. On the other hand, the Ćuk topology cannot be used without a transformer because of its inverting property. Out of the three candidates shown in Fig. 5, the Ćuk converter is the best in terms of current pulsation at its input and output because inductors Lin and Lout are connected in series to the input and output, respectively. Although the SEPIC and Zeta converter can be adapted without a transformer, transformer-less operation may suffer from duty cycle limitations when the number of series connection is large, as will be discussed in the following subsection. Therefore, the need for the transformer in the Ćuk converter is justified, although comparative analysis is necessary to determine the best topology to achieve sufficient performance at reasonable size for a given application. The isolated Ćuk converter-based equalizer is focused in the following sections.

**Figure 5.** Traditional buck–boost converters that can be used as the basic topology for single-switch cell voltage equalizers. (a) SEPIC , (b) Zeta, and (c) isolated Ćuk converters.

The derived isolated Ćuk converter-based single-switch equalizer for four cells connected in series is shown in Fig. 6. The circuit consisting of Cin, Lin, Q, Ca, transformer, C3, D3, and L3 is identical to the circuit shown in Fig. 5(c), while the circuit consisting of Ci–Di–Li (i = 1…4) is multi-stacked to the isolated Ćuk converter. Hence, the derived equalizer can be regarded as multi-stacked Ćuk converters.

**Figure 6.** Single-switch cell voltage equalizer based on isolated Ćuk converter.

The required number of switches is only one; thus, reducing the circuit complexity significantly when compared with conventional equalizers that need numerous switches proportional to the number of series connections, as explained in the previous section. In addition, a multi-winding transformer is not necessary, and the number of series connections can be arbitrarily extended by stacking the circuit consisting of Ci–Di–Li. Therefore, in addition to the reduced circuit complexity, the single-switch equalizer offers a good modularity as well. Furthermore, as will be mathematically indicated, feedback control is not necessary when it is operated in discontinuous conduction mode (DCM), further simplifying the circuitry by removing the feedback control loop.

#### **3.2. Operation analysis**

#### *3.2.1. Operation under voltage-balanced condition*

Traditional buck–boost converters, including the isolated Ćuk converter, operate in either continuous conduction mode (CCM) or DCM. The boundary between CCM and DCM is the discontinuity of the diode current during the off-period. Although ripple currents of inductors tend to be large in DCM, currents in the circuit can be limited under desired levels without feedback control, as will be mathematically indicated later. The following analysis focuses on DCM operation. Key operation waveforms and current flow directions in DCM under the voltage-balanced condition are shown in Figs. 7 and 8, respectively. The fundamental operation is similar to the traditional isolated Ćuk converter, and the DCM operation can be divided into three periods: *Ton*, *Toff-a*, and *Toff-b*.

Under a steady-state condition, average voltages of inductors and transformer windings are zero, and hence, the average voltages of C1–C4 and Ca, *VC1*–*V*C4 and *VCa*, can be expressed as

$$\begin{cases} V\_{C1} = -V\_2 \\ V\_{C2} = 0 \\ V\_{C3} = V\_3 \\ V\_{C4} = V\_3 + V\_4 \\ V\_{Ca} = V\_1 + V\_2 + V\_3 + V\_4 \end{cases} \tag{1}$$

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 157

TON Toff-a Toff-b TOFF

> *ILin-b I Li-b*

*I Lin*

0 iLin

0 i

0 i

0 vDS

i

L

C

D

**Figure 7.** Key operation waveforms of isolated Ćuk converter-based single-switch equalizer under a

*Vin Vin+Vi*

<sup>0</sup> Time

*I Li*

**Figure 8.** Current flow directions during periods (a) *Ton*, (b) *Toff-a*, and (c) *Toff-b* under a voltage-balanced

 

where *D* and *Da* are the duty ratio of *Ton* and *Toff-a*, respectively, *VD* is the forward voltage of

*P* 1234 *in*

, (3)

*NNN*

*V VVVV V*

*DV V V D V V DV V D V V DV V V D V V DV V V V D V V*

 

 

diodes, and *VS* is the transformer secondary voltage designated in Fig. 6 expressed as

*S*

*V*

where *N* is the transformer turn ratio. From Eqs. (1) and (2),

1 2 1 2 2 3 3 3 434 4

(2)

*S C a D S C a D S C a D S C a D*

The voltage-time product of inductors under a steady-state condition is zero, yielding

voltage-balanced condition.

condition.

where *V1*–*V4* are the voltages designated in Fig. 6.

During *Ton* period, as shown in Fig. 8(a), all the inductors are energized and their currents increase linearly. When Q is turned off, *Toff-a* period begins and diodes start to conduct, as shown in Fig. 8(b). As the inductors release stored energies, the inductor currents as well as the diode currents decrease linearly. When the diode currents fall to zero, period *Toff-b* begins. In this period, all the currents in the equalizer are constant because the voltages across inductors are zero.

156 Energy Storage – Technologies and Applications

**3.2. Operation analysis** 

*Toff-a*, and *Toff-b*.

expressed as

across inductors are zero.

The required number of switches is only one; thus, reducing the circuit complexity significantly when compared with conventional equalizers that need numerous switches proportional to the number of series connections, as explained in the previous section. In addition, a multi-winding transformer is not necessary, and the number of series connections can be arbitrarily extended by stacking the circuit consisting of Ci–Di–Li. Therefore, in addition to the reduced circuit complexity, the single-switch equalizer offers a good modularity as well. Furthermore, as will be mathematically indicated, feedback control is not necessary when it is operated in discontinuous conduction mode (DCM),

Traditional buck–boost converters, including the isolated Ćuk converter, operate in either continuous conduction mode (CCM) or DCM. The boundary between CCM and DCM is the discontinuity of the diode current during the off-period. Although ripple currents of inductors tend to be large in DCM, currents in the circuit can be limited under desired levels without feedback control, as will be mathematically indicated later. The following analysis focuses on DCM operation. Key operation waveforms and current flow directions in DCM under the voltage-balanced condition are shown in Figs. 7 and 8, respectively. The fundamental operation is similar to the traditional isolated Ćuk converter, and the DCM operation can be divided into three periods: *Ton*,

Under a steady-state condition, average voltages of inductors and transformer windings are zero, and hence, the average voltages of C1–C4 and Ca, *VC1*–*V*C4 and *VCa*, can be

1234

, (1)

*V VVVV*

During *Ton* period, as shown in Fig. 8(a), all the inductors are energized and their currents increase linearly. When Q is turned off, *Toff-a* period begins and diodes start to conduct, as shown in Fig. 8(b). As the inductors release stored energies, the inductor currents as well as the diode currents decrease linearly. When the diode currents fall to zero, period *Toff-b* begins. In this period, all the currents in the equalizer are constant because the voltages

*V V*

*C C C C Ca*

 

*V V V V VV*

where *V1*–*V4* are the voltages designated in Fig. 6.

0

further simplifying the circuitry by removing the feedback control loop.

*3.2.1. Operation under voltage-balanced condition* 

**Figure 7.** Key operation waveforms of isolated Ćuk converter-based single-switch equalizer under a voltage-balanced condition.

**Figure 8.** Current flow directions during periods (a) *Ton*, (b) *Toff-a*, and (c) *Toff-b* under a voltage-balanced condition.

The voltage-time product of inductors under a steady-state condition is zero, yielding

$$\begin{cases} D\left(V\_S + V\_{C1} + V\_2\right) &= D\_a \left(V\_1 + V\_D\right) \\ D\left(V\_S + V\_{C2}\right) &= D\_a \left(V\_2 + V\_D\right) \\ D\left(V\_S + V\_{C3} - V\_3\right) &= D\_a \left(V\_3 + V\_D\right) \\ D\left(V\_S + V\_{C4} - V\_3 - V\_4\right) = D\_a \left(V\_4 + V\_D\right) \end{cases} \tag{2}$$

where *D* and *Da* are the duty ratio of *Ton* and *Toff-a*, respectively, *VD* is the forward voltage of diodes, and *VS* is the transformer secondary voltage designated in Fig. 6 expressed as

$$V\_S = \frac{V\_P}{N} = \frac{V\_1 + V\_2 + V\_3 + V\_4}{N} = \frac{V\_{in}}{N} \,\tag{3}$$

where *N* is the transformer turn ratio. From Eqs. (1) and (2),

$$V\_i = \frac{D}{D\_a} V\_S - V\_{D'} \tag{4}$$

*i in in i*

*L L L L* 

2 2 4

*in i*

*S S in i*

2 24 2 4 *S S S S S a S in i*

*V DT N V DT V DD T L N L I D*

*VDT L N L <sup>I</sup> N LL* 

Eqs. (10)–(12) imply that currents in the equalizer under a voltage-balanced condition can be limited under a desired level as long as a variation range of *VS* is known. In Eq. (12), for example, *VS* is variable and *D* is determinable, while others are fixed values, and hence, with a known variation range of *VS*, *ILin* can be designed limited under the desired level by properly determining *D*. *ILi* and *IDi* (that can be expressed by Eqs. (10) and (11)) can be similarly designed because *Da* is a predictable variable given by Eq. (4). Thus, the currents in the equalizer operating in DCM can be limited under desired levels even in fixed duty cycle operations, and feedback control is not necessary for the single-switch equalizer, further

As expressed by Eq. (4), the single-switch equalizer inherently produces the uniform output voltages to the cells. This characteristic implies that in the case where voltages of cells are imbalanced, the currents from the equalizer tend to concentrate to a cell having the lowest voltage. Fig. 9 shows the key operation waveforms under a voltage-imbalanced condition. Asterisks added to the symbols in Fig. 9 correspond to the cell with the lowest voltage, B\*. As shown in Fig. 9, when there is voltage imbalance, only *iD\** flows, whereas the other diode currents (*iDi*) are zero for the entire periods. Since all the currents concentrate to D\*, the

> 4 4 22 2

which is fourfold larger than that of Eq. (11). Since *IDi* is zero under the voltage-imbalanced condition, *ILi* is also zero according to Eq. (10), although ripples exist. *ILin* under the voltageimbalanced condition is identical to that under the voltage-balanced condition because *ILin* is

A 5-W prototype of the isolated Ćuk converter-based single-switch equalizer was built for 12 cells connected in series, as shown in Fig. 10. Component values are listed in Table 1.

*V DT N V DT V DD T L N L I D*

2 2

*S S S S S a S in i*

*i in in i*

*L L L L* 

2

*Lin*

simplifying the circuit by eliminating the feedback control loop.

average current of D\* is obtained by transforming Eq. (11) as

\*

**3.3. Experimental** 

*D a*

independent of cell voltages, as expressed by Eq. (12).

*3.3.1. Prototype and its fundamental performance* 

*3.2.2. Operation under voltage-imbalanced condition* 

*Di a*

Substituting Eqs. (10) and (11) into Eq. (9) produces

2 2 4

. (11)

, (13)

. (12)

where *i* = 1…4. This equation means that the equalizer produces the uniform output voltages to the cells, and all the cell voltages can eventually become uniform.

In order for the equalizer to operate in DCM, *Toff-b* period must exist, meaning *Da* < (1 − *D*). From Eq. (4), the critical duty cycle to ensure DCM operation, *Dcritical*, is given by

$$D\_{critical} < \frac{V\_i + V\_{\,\,D}}{V\_S + V\_i + V\_D}.\tag{5}$$

Eqs. (3) and (5) imply that without the variable *N*, as *Vin* increases, *D* must be lowered for a given value of *Vi*. In other words, the duty cycle limitation confronts when the number of series connection is large. On the other hand, with the introduction of the transformer, the issue on the duty cycle limitation can be overcome by properly determining *N*.

According to Fig. 7, the average currents of Li and Lin, *ILi* and *ILin*, are expressed as

$$\begin{cases} I\_{Li} = \left(D + D\_a\right) \frac{V\_S D T\_S}{2L\_i} + I\_{Li-b} \\\\ I\_{Lin} = \left(D + D\_a\right) \frac{V\_S D T\_S}{2L\_{in}} + I\_{Lin-b} \end{cases} \tag{6}$$

where *TS* is the switching period, and *ILi-b* and *ILin-b* are the currents flowing through Li and Lin, respectively, during period *Toff-b* as designated in Fig. 7. Assuming that impedances of C1–C4 are equal, *iC1*–*iC4* as well as *IL1-b*–*IL4-b* can be uniform, as expressed by

$$NI\_{Lin-b} = -I\_{L1-b} - I\_{L2-b} - I\_{L3-b} - I\_{L4-b} = -4I\_{Li-b} \, . \tag{7}$$

The average current of Ci, *ICi*, is expressed as

$$I\_{Ci} = I\_{Li-b} + D \frac{V\_S D T\_S}{2L\_i} - D\_a \frac{N^2 V\_S D T\_S}{2 \cdot 4L\_{in}} = 0 \quad . \tag{8}$$

From Eqs. (6)–(8),

$$\frac{I\_{Li}}{I\_{Lit}} = \frac{\text{ND}\_a}{4D} \,. \tag{9}$$

With *ICi* = 0, Kirchhoff's current law in Fig. 6 yields

$$I\_{Li} = I\_{Di} \tag{10}$$

where *IDi* is the average current of Di. From Figs. 7 and 8(b), *IDi* is expressed as

$$I\_{Di} = D\_a \left(\frac{V\_S D T\_S}{2L\_i} + \frac{N^2 V\_S D T\_S}{2 \cdot 4 L\_{in}}\right) = \frac{V\_S D D\_a T\_S}{2} \left(\frac{4L\_{in} + N^2 L\_i}{4L\_{in} L\_i}\right). \tag{11}$$

Substituting Eqs. (10) and (11) into Eq. (9) produces

158 Energy Storage – Technologies and Applications

*i SD a <sup>D</sup> V VV*

where *i* = 1…4. This equation means that the equalizer produces the uniform output

In order for the equalizer to operate in DCM, *Toff-b* period must exist, meaning *Da* < (1 − *D*).

Eqs. (3) and (5) imply that without the variable *N*, as *Vin* increases, *D* must be lowered for a given value of *Vi*. In other words, the duty cycle limitation confronts when the number of series connection is large. On the other hand, with the introduction of the transformer, the

2

*S S Li a Li b i S S Lin a Lin b in*

<sup>1234</sup> 4 *NI I I I I I Lin b L b L b L b L b Li b* . (7)

0

*I D* . (9)

*Li Di I I* , (10)

*L L* . (8)

2

2 24 *SS SS*

4 *Li a*

*I ND*

*i in*

*L*

2

where *TS* is the switching period, and *ILi-b* and *ILin-b* are the currents flowing through Li and Lin, respectively, during period *Toff-b* as designated in Fig. 7. Assuming that impedances of

*L*

*i D*

*SiD V V*

*V VV* 

voltages to the cells, and all the cell voltages can eventually become uniform.

From Eq. (4), the critical duty cycle to ensure DCM operation, *Dcritical*, is given by

*critical*

issue on the duty cycle limitation can be overcome by properly determining *N*.

 

The average current of Ci, *ICi*, is expressed as

With *ICi* = 0, Kirchhoff's current law in Fig. 6 yields

From Eqs. (6)–(8),

C1–C4 are equal, *iC1*–*iC4* as well as *IL1-b*–*IL4-b* can be uniform, as expressed by

*Ci Li b a*

*V DT N V DT II D D*

*Lin*

where *IDi* is the average current of Di. From Figs. 7 and 8(b), *IDi* is expressed as

According to Fig. 7, the average currents of Li and Lin, *ILi* and *ILin*, are expressed as

*V DT I DD I*

*V DT I DD I*

*D*

*D* , (4)

. (5)

, (6)

$$I\_{L\text{in}} = \frac{V\_S D^2 T\_S}{2N} \left(\frac{4L\_{\text{in}} + N^2 L\_i}{L\_{\text{in}} L\_i}\right). \tag{12}$$

Eqs. (10)–(12) imply that currents in the equalizer under a voltage-balanced condition can be limited under a desired level as long as a variation range of *VS* is known. In Eq. (12), for example, *VS* is variable and *D* is determinable, while others are fixed values, and hence, with a known variation range of *VS*, *ILin* can be designed limited under the desired level by properly determining *D*. *ILi* and *IDi* (that can be expressed by Eqs. (10) and (11)) can be similarly designed because *Da* is a predictable variable given by Eq. (4). Thus, the currents in the equalizer operating in DCM can be limited under desired levels even in fixed duty cycle operations, and feedback control is not necessary for the single-switch equalizer, further simplifying the circuit by eliminating the feedback control loop.

#### *3.2.2. Operation under voltage-imbalanced condition*

As expressed by Eq. (4), the single-switch equalizer inherently produces the uniform output voltages to the cells. This characteristic implies that in the case where voltages of cells are imbalanced, the currents from the equalizer tend to concentrate to a cell having the lowest voltage. Fig. 9 shows the key operation waveforms under a voltage-imbalanced condition. Asterisks added to the symbols in Fig. 9 correspond to the cell with the lowest voltage, B\*. As shown in Fig. 9, when there is voltage imbalance, only *iD\** flows, whereas the other diode currents (*iDi*) are zero for the entire periods. Since all the currents concentrate to D\*, the average current of D\* is obtained by transforming Eq. (11) as

$$I\_{D^{\*}} = D\_a \left(\frac{4V\_S D T\_S}{2L\_i} + \frac{N^2 V\_S D T\_S}{2L\_{in}}\right) = \frac{V\_S D D\_a T\_S}{2} \left(\frac{4L\_{in} + N^2 L\_i}{L\_{in} L\_i}\right) \tag{13}$$

which is fourfold larger than that of Eq. (11). Since *IDi* is zero under the voltage-imbalanced condition, *ILi* is also zero according to Eq. (10), although ripples exist. *ILin* under the voltageimbalanced condition is identical to that under the voltage-balanced condition because *ILin* is independent of cell voltages, as expressed by Eq. (12).

#### **3.3. Experimental**

#### *3.3.1. Prototype and its fundamental performance*

A 5-W prototype of the isolated Ćuk converter-based single-switch equalizer was built for 12 cells connected in series, as shown in Fig. 10. Component values are listed in Table 1.

Cout1–Cout12, which were not depicted in figures for the sake of simplicity, are smoothing capacitors connected to the cells in parallel. The RCD snubber was added at the primary winding in order to protect the switch from surge voltages generated by the transformer leakage inductance. The prototype was operated with a fixed *D* = 0.3 at *f* = 150 kHz.

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 161

the peak efficiencies under the voltage-balanced and -imbalanced conditions were approximately 70% and 65%, respectively. The lower efficiencies under the voltageimbalanced condition were due to the current concentration to C1, D1, and L1, which caused

Component Value

**C R D**

Z

Cin Ceramic Capacitor, 20 μF Ca Ceramic Capacitor, 22 μF Cout 1–Cout 12 Ceramic Capacitor, 200 μF C1–C12 Ceramic Capacitor, 22 μF Lin 1 mH L1–L12 47 μH

Transformer *N <sup>1</sup>* :*N <sup>2</sup>* = 24:4, *L kg* = 8.7 μH, *Lm g* = 2.03 m H Q N-Ch MOSFET, IRFR13N20D, *Ro n* = 235 mΩ D1–D12 Schottky Diode, CRS08, *V <sup>D</sup>* = 0.36 V RCD Snubber *R* = 2.2 kΩ, *C* = 470 pF

**Ca C7**

increased Joule losses in resistive components.

**Table 1.** Component values used for the prototype.

**Figure 11.** Experimental setup for efficiency measurement.

**Lin**

**Cin Q**

**Vext**

Figs. 13(a) and (b) show typical operation waveforms measured under the voltage-balanced and -imbalanced conditions, respectively. Under the voltage-balanced condition, as shown in Fig. 13(a), all the inductor currents were uniform, although the oscillations caused by interactions between inductors and parasitic capacitance of the MOSFET were observed. Under the voltage-imbalanced condition, *iL2*–*iL12* were uniform and their averages were zero, whereas only *iL1* showed an average higher than zero because of the current concentration.

**L1**

**D1**

**B1 Rv ar**

X Y

**C1**

**L6**

**D6**

**C6**

**L7**

**D7**

**L6**

**D6**

**B6**

**B6**

**B7**

**C6**

**Figure 9.** Key operation waveforms under a voltage-imbalanced condition.

The experimental setup for power conversion efficiency measurement is shown in Fig. 11. The tap Y and X in Fig. 11 were selected to emulate the voltage-balanced and -imbalanced (*V1* < *Vi* (*i* = 2…12)) conditions, respectively. The external power supply, Vext, was used, and the input and output of the equalizer were broken at point Z in order to measure efficiencies. The efficiencies were measured by changing the ratio of *V1*/*Vin* between approximately 1/12 and 1/16. During the efficiency measurement, cells were removed and only smoothing capacitors (Cout1–Cout12) were used to sustain the voltages of *V1*–*V12*.

**Figure 10.** Photograph of a 5-W prototype of the isolated Ćuk converter-based single-switch cell voltage equalizer for 12 cells connected in series.

The measured power conversion efficiencies and output power characteristics as a function of *V1* are shown in Fig. 12. The efficiencies increased with *V1* because the diode voltage drop represented a lesser portion of the output voltage (i.e., *V1*). The efficiencies under the voltage-balanced condition were higher than those under the voltage-imbalanced condition; the peak efficiencies under the voltage-balanced and -imbalanced conditions were approximately 70% and 65%, respectively. The lower efficiencies under the voltageimbalanced condition were due to the current concentration to C1, D1, and L1, which caused increased Joule losses in resistive components.


**Table 1.** Component values used for the prototype.

160 Energy Storage – Technologies and Applications

Cout1–Cout12, which were not depicted in figures for the sake of simplicity, are smoothing capacitors connected to the cells in parallel. The RCD snubber was added at the primary winding in order to protect the switch from surge voltages generated by the transformer

TON Toff-a

*ILin*

*I Lin-b*

*i Ci*

*i Li*

*<sup>I</sup> <sup>I</sup> L\*-b Li-b*

Toff-b TOFF

The experimental setup for power conversion efficiency measurement is shown in Fig. 11. The tap Y and X in Fig. 11 were selected to emulate the voltage-balanced and -imbalanced (*V1* < *Vi* (*i* = 2…12)) conditions, respectively. The external power supply, Vext, was used, and the input and output of the equalizer were broken at point Z in order to measure efficiencies. The efficiencies were measured by changing the ratio of *V1*/*Vin* between approximately 1/12 and 1/16. During the efficiency measurement, cells were removed and

<sup>0</sup> Time

*i Di i D\**

*Vin Vin+V\**

only smoothing capacitors (Cout1–Cout12) were used to sustain the voltages of *V1*–*V12*.

**Figure 10.** Photograph of a 5-W prototype of the isolated Ćuk converter-based single-switch cell

The measured power conversion efficiencies and output power characteristics as a function of *V1* are shown in Fig. 12. The efficiencies increased with *V1* because the diode voltage drop represented a lesser portion of the output voltage (i.e., *V1*). The efficiencies under the voltage-balanced condition were higher than those under the voltage-imbalanced condition;

voltage equalizer for 12 cells connected in series.

leakage inductance. The prototype was operated with a fixed *D* = 0.3 at *f* = 150 kHz.

*i C\**

*IL\* i L\**

0 iLin

0 i

0 i

0 i

0 vDS

L

C

D

**Figure 9.** Key operation waveforms under a voltage-imbalanced condition.

**Figure 11.** Experimental setup for efficiency measurement.

Figs. 13(a) and (b) show typical operation waveforms measured under the voltage-balanced and -imbalanced conditions, respectively. Under the voltage-balanced condition, as shown in Fig. 13(a), all the inductor currents were uniform, although the oscillations caused by interactions between inductors and parasitic capacitance of the MOSFET were observed. Under the voltage-imbalanced condition, *iL2*–*iL12* were uniform and their averages were zero, whereas only *iL1* showed an average higher than zero because of the current concentration.

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 163

decreased. The voltage imbalance was gradually eliminated as time elapsed, and the standard deviation of cell voltages eventually decreased down to approximately 7 mV at the end of the experiment; thus, demonstrating the equalizer's equalization performance. The cell voltages kept decreasing even after the voltage imbalance disappeared. This decrease was due to the power conversion loss in the equalizer. After the cell voltages were balanced, the energies of the cells were meaninglessly circulated by the equalizer, and therefore, the equalizer should be disabled after cell voltages are sufficiently balanced in order not to

Another experimental equalization was performed for 12-series lithium-ion cells with a capacity of 2200 mAh at a rated charge voltage of 4.2 V. The state of charges (SOCs) of the cells were initially imbalanced between 0%–100%. The experimental results are shown in Fig. 14(b). Although the resultant profiles were somewhat elusive because of the nonlinear characteristics of the lithium-ion chemistry, the voltage imbalance was successfully

**Figure 14.** Experimental profiles of 12 series-connected (a) SCs and (b) lithium-ion cells equalized by

The double-switch resonant equalizer is essentially a combination of a conventional series resonant inverter and a voltage multiplier, shown in Figs. 15(a) and (b), respectively. The voltage multiplier shown in Fig. 15(b) is an example circuit that can produce a 4 times higher voltage than the amplitude of the input. The voltages of the stationary capacitors C'1– C'4 automatically become uniform as the amplitude of the input square wave under a steady-state condition (when diode voltage drops are neglected). Detailed operation

**4. Double-switch resonant cell voltage equalizer using a voltage** 

the isolated Ćuk converter-based single-switch equalizer.

**4.1. Circuit description and major benefits** 

**multiplier** 

eliminated, and all the cell voltage converged to a uniform voltage level.

waste the stored energies of cells.

**Figure 12.** Measured power conversion efficiencies and output powers as a function of *V1* under (a) voltage-balanced and (b) -imbalanced conditions.

**Figure 13.** Measured waveforms under (a) voltage-balanced and (b) -imbalanced conditions.

#### *3.3.2. Equalization*

The experimental equalization test was performed for 12-series SCs with a capacitance of 500 F at a rated charge voltage of 2.5 V. The voltages of SCs were initially imbalanced between 0.85–2.5 V. The resultant equalization profiles are shown in Fig. 14(a). As the equalizer redistributed energies from the series connection to cells with low voltages, voltages of cells with low initial voltages increased while those with high initial voltages decreased. The voltage imbalance was gradually eliminated as time elapsed, and the standard deviation of cell voltages eventually decreased down to approximately 7 mV at the end of the experiment; thus, demonstrating the equalizer's equalization performance. The cell voltages kept decreasing even after the voltage imbalance disappeared. This decrease was due to the power conversion loss in the equalizer. After the cell voltages were balanced, the energies of the cells were meaninglessly circulated by the equalizer, and therefore, the equalizer should be disabled after cell voltages are sufficiently balanced in order not to waste the stored energies of cells.

Another experimental equalization was performed for 12-series lithium-ion cells with a capacity of 2200 mAh at a rated charge voltage of 4.2 V. The state of charges (SOCs) of the cells were initially imbalanced between 0%–100%. The experimental results are shown in Fig. 14(b). Although the resultant profiles were somewhat elusive because of the nonlinear characteristics of the lithium-ion chemistry, the voltage imbalance was successfully eliminated, and all the cell voltage converged to a uniform voltage level.

**Figure 14.** Experimental profiles of 12 series-connected (a) SCs and (b) lithium-ion cells equalized by the isolated Ćuk converter-based single-switch equalizer.

## **4. Double-switch resonant cell voltage equalizer using a voltage multiplier**

#### **4.1. Circuit description and major benefits**

162 Energy Storage – Technologies and Applications

voltage-balanced and (b) -imbalanced conditions.

*3.3.2. Equalization* 

**Figure 12.** Measured power conversion efficiencies and output powers as a function of *V1* under (a)

**Figure 13.** Measured waveforms under (a) voltage-balanced and (b) -imbalanced conditions.

The experimental equalization test was performed for 12-series SCs with a capacitance of 500 F at a rated charge voltage of 2.5 V. The voltages of SCs were initially imbalanced between 0.85–2.5 V. The resultant equalization profiles are shown in Fig. 14(a). As the equalizer redistributed energies from the series connection to cells with low voltages, voltages of cells with low initial voltages increased while those with high initial voltages The double-switch resonant equalizer is essentially a combination of a conventional series resonant inverter and a voltage multiplier, shown in Figs. 15(a) and (b), respectively. The voltage multiplier shown in Fig. 15(b) is an example circuit that can produce a 4 times higher voltage than the amplitude of the input. The voltages of the stationary capacitors C'1– C'4 automatically become uniform as the amplitude of the input square wave under a steady-state condition (when diode voltage drops are neglected). Detailed operation analyses on both the resonant inverter and the voltage multiplier are separately made in the following section.

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 165

currents under the desired current levels, and therefore, the circuit can be further simplified

Although the series-resonant inverter with a transformer is used for the proposed equalizer, other types of resonant inverters, such as parallel-, series-parallel-, and LLC-resonant

Similar to traditional resonant inverters, the double-switch resonant equalizer is operated at the switching frequency *f* higher than the resonant frequency, *fr*, above which the seriesresonant circuit consisting of Lr and Cr represents an inductive load. The key operation

1

*r r*

*vGSa vGSb Vin*

*L C* , (14)

2 *<sup>r</sup>*

Mode 1 Mode 2 Mode 3 Mode 4

where *Lr* is the inductance of the resonant inductor Lr, and *Cr* is the capacitance of the resonant capacitor Cr. MOSFETs Qa and Qb are complementarily operated with a fixed duty cycle slightly less than 50% in order to provide adequate dead times to prevent a shortthrough current. Above *fr*, the current of the resonant circuit *iLr* lags behind the fundamental component of the voltage *vDSb*, which corresponds to the input voltage for the resonant

*f*

**Figure 17.** Key operation waveforms of resonant equalizer above resonant frequency (*f* > *fr*).

iSa

vGS

vDSb

iLr

iSb

iD(2i-1)

i

C

iD(2i)

The operation of the series-resonant equalizer can be divided into four modes: Mode 1–4, and the current flow directions under a voltage-balanced condition during each mode are shown in Fig. 18. In Mode 1, the current of the resonant circuit, *iLr*, flows through the antiparallel diode of Sa, Da, toward the series connection of B1–B4, and the energy transfer

Time

inverters, can be used instead, and may even offer better performance.

waveforms at *f* > *fr* are shown in Fig. 17, and *fr* is given by

circuit, and Qa and Qb are turned on at zero-voltage.

by removing the feedback control loop.

**4.2. Operation analysis** 

*4.2.1. Fundamental operation* 

By combining the series-resonant inverter and the voltage multiplier, the double-switch equalizer can be synthesized as shown in Fig. 16. The leakage inductance of the transformer is used as the resonant inductor, Lr. The magnetizing inductance of the transformer is not depicted in Fig. 16 for the sake of simplicity. The stationary capacitors C'1–C'4 in the voltage multiplier in Fig. 15(b) are replaced with energy storage cells B1–B4.

**Figure 15.** (a) Series-resonant inverter and (b) 4x-voltage multiplier.

**Figure 16.** Double-switch series-resonant equalizer using voltage multiplier.

The required number of switches and magnetic components are only two and one, respectively, and hence, the circuit complexity as well as the size and cost of the circuit can be significantly reduced when compared with conventional equalizers, which need multiple switches and/or magnetic components in proportion to the number of series connections. In addition, the proposed resonant equalizer can be configured without a multi-winding transformer, and the number of series connections can be readily extended by stacking a capacitor and diodes, thus offering a good modularity (or extendibility). As mathematically explained in the following analysis, feedback control is not necessarily needed to limit currents under the desired current levels, and therefore, the circuit can be further simplified by removing the feedback control loop.

Although the series-resonant inverter with a transformer is used for the proposed equalizer, other types of resonant inverters, such as parallel-, series-parallel-, and LLC-resonant inverters, can be used instead, and may even offer better performance.

#### **4.2. Operation analysis**

164 Energy Storage – Technologies and Applications

following section.

analyses on both the resonant inverter and the voltage multiplier are separately made in the

By combining the series-resonant inverter and the voltage multiplier, the double-switch equalizer can be synthesized as shown in Fig. 16. The leakage inductance of the transformer is used as the resonant inductor, Lr. The magnetizing inductance of the transformer is not depicted in Fig. 16 for the sake of simplicity. The stationary capacitors C'1–C'4 in the voltage

multiplier in Fig. 15(b) are replaced with energy storage cells B1–B4.

**Figure 15.** (a) Series-resonant inverter and (b) 4x-voltage multiplier.

**Da**

**Cr**

*vDSa*

**Qa**

*i Sa*

*i Sb*

Sa

**Qb**

Sb

**Db**

*vDSb*

**Figure 16.** Double-switch series-resonant equalizer using voltage multiplier.

The required number of switches and magnetic components are only two and one, respectively, and hence, the circuit complexity as well as the size and cost of the circuit can be significantly reduced when compared with conventional equalizers, which need multiple switches and/or magnetic components in proportion to the number of series connections. In addition, the proposed resonant equalizer can be configured without a multi-winding transformer, and the number of series connections can be readily extended by stacking a capacitor and diodes, thus offering a good modularity (or extendibility). As mathematically explained in the following analysis, feedback control is not necessarily needed to limit

**N1 N2**

*vP vS*

**B2**

**D4 D5 D6 D7 D8**

**C2**

*i C2* +

*i C1* +

**C3**

*i C3* +

**C4**

*i C4* +

**D3**

**D1 C1 D2** **B3**

*i B3*

*i B2*

*i B1*

*V3*

*V2*

*V1*

**B4**

*V4 i B4*

**B1**

**Lkg**

**Lr**

*i Lr*

#### *4.2.1. Fundamental operation*

Similar to traditional resonant inverters, the double-switch resonant equalizer is operated at the switching frequency *f* higher than the resonant frequency, *fr*, above which the seriesresonant circuit consisting of Lr and Cr represents an inductive load. The key operation waveforms at *f* > *fr* are shown in Fig. 17, and *fr* is given by

$$f\_r = \frac{1}{2\pi\sqrt{L\_r\mathcal{C}\_r}},\tag{14}$$

where *Lr* is the inductance of the resonant inductor Lr, and *Cr* is the capacitance of the resonant capacitor Cr. MOSFETs Qa and Qb are complementarily operated with a fixed duty cycle slightly less than 50% in order to provide adequate dead times to prevent a shortthrough current. Above *fr*, the current of the resonant circuit *iLr* lags behind the fundamental component of the voltage *vDSb*, which corresponds to the input voltage for the resonant circuit, and Qa and Qb are turned on at zero-voltage.

**Figure 17.** Key operation waveforms of resonant equalizer above resonant frequency (*f* > *fr*).

The operation of the series-resonant equalizer can be divided into four modes: Mode 1–4, and the current flow directions under a voltage-balanced condition during each mode are shown in Fig. 18. In Mode 1, the current of the resonant circuit, *iLr*, flows through the antiparallel diode of Sa, Da, toward the series connection of B1–B4, and the energy transfer capacitors in the voltage multiplier, Ci (i = 1…4), discharge through even-numbered diodes, D(2i), as shown in Fig. 18(a). Before *iSa* reaches zero, the gate signal for Qa, *vGSa*, is applied. Since the voltage across Qa, *vDSa*, is zero at this moment, Qa is turned on at zerovoltage. After *iSa* is reversed, as shown in Fig. 17, Qa starts to conduct, and Mode 2 begins. In Mode 2, the resonant circuit is energized by the series connection of B1–B4, and Ci is charged via odd-numbered diodes, D(2i−1). As Qa is turned off, the current is diverted from Qa to the anti-parallel diode of Sb, Db, and Mode 3 begins. Ci is still being charged. The gate signal for Qb, *vGSb*, is applied and Qb is turned on at zero-voltage, before the current of Sb, *iSb*, is reversed. As *iSb* reaches zero, the operation shifts to Mode 4, in which Ci discharges through D(2i). When Qb is turned off, the current is diverted from Qb to Da, and the operation returns to Mode 1. Thus, similar to the conventional resonant inverters, the double-switch resonant equalizer achieves zero-voltage switching (ZVS) operation when Qa and Qb are turned on.

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 167

1 1 2 2 2

, (15)

, (16)

, (18)

*V V V VV V V VV V VV V V VV*

*CO SO D CO SO D CO SO D CO SO D*

 

4 3

where *VS-O* is the peak voltage of the transformer secondary winding when D(2i−1) is on, and *VD* is the forward voltage drop of the diodes. Similarly, the bottom voltages of Ci when D(2i)

1 2

*CE SE D CE SE D CE SE D CE SE D*

 

*V V VV V VV V V VV V V V VV*

 

where *VS-E* is the bottom voltage of the transformer secondary winding when D(2i) is on.

3 3 4 3 4

Subtracting Eq. (16) from Eq. (15) yields the voltage variation of Ci during a single switching

Generally, an amount of charge delivered via a capacitor having a capacitance of *C*, and an

*It I <sup>V</sup> IR C Cf*

where *ΔV* is the voltage variation caused by charging/discharging. Substitution of Eq. (18)

Eq. (19) yields a dc equivalent circuit of the voltage multiplier, as shown in Fig. 19. All cells, B1–B4, are tied to a common dc-source Vdc, which provides a voltage of (*VS-O* + *VS-E*), via two diodes and one equivalent resistor, Reqi. When *V1*–*V4* are balanced, *IC1*–*IC4* can be uniform as long as *C1*–*C4* are designed so that all the equivalent resistances, *Req1*–*Req4*, are uniform. In the case of voltage imbalance, the current preferentially flows to the cell(s) having the lowest

*Q It C V*

2 *V V V VV Ci S O S E i D* . (17)

2 *Ci eqi S O S E D i IR V V V V* , (19)

*eq*

The peak voltage of Ci during the time D(2i−1) is on, *VCiO*, can be expressed as

3

2

equivalent resistance for the charge transfer, *Req*, are given by

where *ICi* is the average current flowing via Ci.

*4.2.2. Voltage multiplier* 

is on, *VCiE*, are

cycle, *ΔVCi*:

into Eq. (17) produces

**Figure 18.** Current flow directions during Mode (a) 1, (b) 2, (c) 3, and (d) 4.

Repeating the above sequence, energies of the series connection of B1–B4 are supplied to the resonant inverter, and then are transferred to the voltage multiplier that redistributes the energies to B1–B4. Thus, the energies of B1–B4 are redistributed via the resonant inverter and voltage multiplier. Throughout a single switching cycle, Ci as well as B1–B4 are charged and discharged via D(2i−1) and D(2i), and consequently, voltages of B1–B4, *V1*–*V4*, become automatically uniform. The voltage equalization mechanism by the voltage multiplier is discussed in detail in the following subsection.

#### *4.2.2. Voltage multiplier*

166 Energy Storage – Technologies and Applications

Qa and Qb are turned on.

capacitors in the voltage multiplier, Ci (i = 1…4), discharge through even-numbered diodes, D(2i), as shown in Fig. 18(a). Before *iSa* reaches zero, the gate signal for Qa, *vGSa*, is applied. Since the voltage across Qa, *vDSa*, is zero at this moment, Qa is turned on at zerovoltage. After *iSa* is reversed, as shown in Fig. 17, Qa starts to conduct, and Mode 2 begins. In Mode 2, the resonant circuit is energized by the series connection of B1–B4, and Ci is charged via odd-numbered diodes, D(2i−1). As Qa is turned off, the current is diverted from Qa to the anti-parallel diode of Sb, Db, and Mode 3 begins. Ci is still being charged. The gate signal for Qb, *vGSb*, is applied and Qb is turned on at zero-voltage, before the current of Sb, *iSb*, is reversed. As *iSb* reaches zero, the operation shifts to Mode 4, in which Ci discharges through D(2i). When Qb is turned off, the current is diverted from Qb to Da, and the operation returns to Mode 1. Thus, similar to the conventional resonant inverters, the double-switch resonant equalizer achieves zero-voltage switching (ZVS) operation when

**Figure 18.** Current flow directions during Mode (a) 1, (b) 2, (c) 3, and (d) 4.

discussed in detail in the following subsection.

Repeating the above sequence, energies of the series connection of B1–B4 are supplied to the resonant inverter, and then are transferred to the voltage multiplier that redistributes the energies to B1–B4. Thus, the energies of B1–B4 are redistributed via the resonant inverter and voltage multiplier. Throughout a single switching cycle, Ci as well as B1–B4 are charged and discharged via D(2i−1) and D(2i), and consequently, voltages of B1–B4, *V1*–*V4*, become automatically uniform. The voltage equalization mechanism by the voltage multiplier is

The peak voltage of Ci during the time D(2i−1) is on, *VCiO*, can be expressed as

$$\begin{cases} V\_{C1O} = V\_{S-O} - V\_D - V\_1 - V\_2 \\ V\_{C2O} = V\_{S-O} - V\_D - V\_2 \\ V\_{C3O} = V\_{S-O} - V\_D \\ V\_{C4O} = V\_{S-O} - V\_D + V\_3 \end{cases} \tag{15}$$

where *VS-O* is the peak voltage of the transformer secondary winding when D(2i−1) is on, and *VD* is the forward voltage drop of the diodes. Similarly, the bottom voltages of Ci when D(2i) is on, *VCiE*, are

$$\begin{cases} V\_{C1E} = -V\_{S-E} + V\_D - V\_2 \\ V\_{C2E} = -V\_{S-E} + V\_D \\ V\_{C3E} = -V\_{S-E} + V\_D + V\_3 \\ V\_{C4E} = -V\_{S-E} + V\_D + V\_3 + V\_4 \end{cases} \tag{16}$$

where *VS-E* is the bottom voltage of the transformer secondary winding when D(2i) is on.

Subtracting Eq. (16) from Eq. (15) yields the voltage variation of Ci during a single switching cycle, *ΔVCi*:

$$
\Delta V\_{Ci} = \left(V\_{S-O} + V\_{S-E}\right) - V\_i - \mathcal{D}V\_D \,. \tag{17}
$$

Generally, an amount of charge delivered via a capacitor having a capacitance of *C*, and an equivalent resistance for the charge transfer, *Req*, are given by

$$\begin{cases} Q = It = \mathbb{C}\Delta V\\ \Delta V = \frac{It}{C} = \frac{I}{Cf} = IR\_{eq} \,' \end{cases} \tag{18}$$

where *ΔV* is the voltage variation caused by charging/discharging. Substitution of Eq. (18) into Eq. (17) produces

$$I\_{\rm Ci}R\_{eqi} = \left(V\_{S-O} + V\_{S-E}\right) - \mathcal{D}V\_D - V\_{i\ \nu} \tag{19}$$

where *ICi* is the average current flowing via Ci.

Eq. (19) yields a dc equivalent circuit of the voltage multiplier, as shown in Fig. 19. All cells, B1–B4, are tied to a common dc-source Vdc, which provides a voltage of (*VS-O* + *VS-E*), via two diodes and one equivalent resistor, Reqi. When *V1*–*V4* are balanced, *IC1*–*IC4* can be uniform as long as *C1*–*C4* are designed so that all the equivalent resistances, *Req1*–*Req4*, are uniform. In the case of voltage imbalance, the current preferentially flows to the cell(s) having the lowest voltage, and its voltage increases more quickly than the others. Eventually, voltages *V1*–*V4* automatically reach a uniform voltage level.

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 169

, (25)

*Vi /2+VD*

*Vi /2+VD*

. (23)

. (24)

2 2

 

. (26)

0

*vP <sup>v</sup> vDSb <sup>S</sup>*

*N*(*Vi /2+VD*)

*Vm-P*

*–N*(*Vi /2+VD*)

*–Vm-P*

*<sup>r</sup> <sup>r</sup> r r*

  . (21)

Similarly, the amplitude of the fundamental component of *vDSb*, *Vm-in*, is

where *Z0* is the characteristic impedance of the resonant circuit given by

0

**Db**

**Da**

*Z L*

*Z*

**Figure 20.** Key waveforms and their fundamental components.

*Vin*

*Vm-in*

*–Vm-in*

**Q2**

**Q1**

be obtained from Eq. (22) with the Fourier transfer,

The amplitude of *iLr*, *Im*, is obtained as

*m*

*S*

*v*

112 22 3 43 2 1

*V V V V V V V V V V V V D are on*

2 *VV V Si D* . (22)

The amplitude of the fundamental component of the transformer primary winding, *Vm-P*, can

 <sup>2</sup> <sup>2</sup> *V NV V m P i D* 

2 2 *V V VVVV m in in*

 

*VV VV VV <sup>I</sup>*

*r r*

In order for the series-resonant inverter to transfer energies to the voltage multiplier connected to the secondary winding, *Vm-in* must be higher than *Vm-P*. Assuming that the

<sup>1234</sup>

1

*L Z C*

1

*r r*

*C*

**Lr Cr**

*iLr*

*m in m P m in m P m in m P*

*V V V V V V V V V V V V D are on*

12 2 33 434 2

*C DC DC DC D i*

*vS* is a square wave with an amplitude of *VS*, which is obtained from Eqs. (20) and (21) as

*C DC D C DC D i*

**Figure 19.** DC equivalent circuit for voltage multiplier.

Current flows in the dc equivalent circuit in Fig. 19 can then be transformed to those in the original circuit shown in Figs. 16 and 18. We consider the case that *V1* is the lowest and when no currents flow through Req2–Req4, as a simple example. Since any current flowing through Reqi represents charging/discharging the capacitor Ci, as indicated by Eq. (18), no currents in Req2–Req4 mean that no currents flow through C2–C4 as well as D3–D8 in the original circuit. Meanwhile, the current from the transformer secondary winding concentrates to C1 and D1–D2 in the original circuit. Thus, under a voltage-imbalanced condition, currents flow through only the capacitor(s) and diodes that are connected to the cell(s) having the lowest voltage, although practical current distribution tendencies are dependent on *Reqi* as well as the voltage conditions of *Vi*.

#### *4.2.3. Series-resonant circuit*

The average voltage across a transformer winding throughout a single switching cycle is zero, and the on-duties of D(2i−1) and D(2i) are both 50%. Therefore, the average voltages of C1– C4, *VC1*–*VC4*, can be obtained from Fig. 18, and are expressed as

$$\begin{cases} V\_{C1} = -\frac{V\_1}{2} - V\_2 \\ V\_{C2} = -\frac{V\_2}{2} \\ V\_{C3} = \frac{V\_3}{2} \\ V\_{C4} = V\_3 + \frac{V\_4}{2} \end{cases} \tag{20}$$

The Square voltage waves in the resonant circuit are approximated to the sinusoidal fundamental components, as shown in Fig. 20, in which key waveforms of the seriesresonant inverter and their fundamental components are sketched. By assuming that the voltage of Ci is constant as *VCi* throughout a single switching cycle, the voltage of the transformer secondary winding, *vS*, is

$$\boldsymbol{v}\_{S} = \begin{pmatrix} V\_{C1} + V\_{1} + V\_{2} + V\_{D} = V\_{C2} + V\_{2} + V\_{D} = V\_{C3} + V\_{D} = V\_{C4} - V\_{3} + V\_{D} \left( D\_{\{2i-1\}} \arccos \right) \\\\ V\_{C1} + V\_{2} - V\_{D} = V\_{C2} - V\_{D} = V\_{C3} - V\_{3} - V\_{D} = V\_{C4} - V\_{3} - V\_{4} - V\_{4} - V\_{D} \left( D\_{\{2i\}} \arccos \right) \end{pmatrix} \tag{21}$$

*vS* is a square wave with an amplitude of *VS*, which is obtained from Eqs. (20) and (21) as

$$V\_S = V\_i + 2V\_D \,. \tag{22}$$

The amplitude of the fundamental component of the transformer primary winding, *Vm-P*, can be obtained from Eq. (22) with the Fourier transfer,

$$V\_{m-P} = \frac{2}{\pi} N \left( V\_i + 2V\_D \right). \tag{23}$$

Similarly, the amplitude of the fundamental component of *vDSb*, *Vm-in*, is

$$V\_{m-in} = \frac{2}{\pi} V\_{in} = \frac{2}{\pi} \left( V\_1 + V\_2 + V\_3 + V\_4 \right). \tag{24}$$

The amplitude of *iLr*, *Im*, is obtained as

168 Energy Storage – Technologies and Applications

automatically reach a uniform voltage level.

**Figure 19.** DC equivalent circuit for voltage multiplier.

**Vdc** (*VS-O*+*VS-E*)

dependent on *Reqi* as well as the voltage conditions of *Vi*.

C4, *VC1*–*VC4*, can be obtained from Fig. 18, and are expressed as

*4.2.3. Series-resonant circuit* 

transformer secondary winding, *vS*, is

voltage, and its voltage increases more quickly than the others. Eventually, voltages *V1*–*V4*

**Req3 Req2 Req1**

**Req4 D5 D6 D3 D4 D1 D2**

**D7 D8**

Current flows in the dc equivalent circuit in Fig. 19 can then be transformed to those in the original circuit shown in Figs. 16 and 18. We consider the case that *V1* is the lowest and when no currents flow through Req2–Req4, as a simple example. Since any current flowing through Reqi represents charging/discharging the capacitor Ci, as indicated by Eq. (18), no currents in Req2–Req4 mean that no currents flow through C2–C4 as well as D3–D8 in the original circuit. Meanwhile, the current from the transformer secondary winding concentrates to C1 and D1–D2 in the original circuit. Thus, under a voltage-imbalanced condition, currents flow through only the capacitor(s) and diodes that are connected to the cell(s) having the lowest voltage, although practical current distribution tendencies are

*IC1 IC2 IC3 IC4*

**B1 B2 B3 B4**

The average voltage across a transformer winding throughout a single switching cycle is zero, and the on-duties of D(2i−1) and D(2i) are both 50%. Therefore, the average voltages of C1–

> 1 1 2

2

*<sup>V</sup> V V*

2

2

4

. (20)

2

3

2

The Square voltage waves in the resonant circuit are approximated to the sinusoidal fundamental components, as shown in Fig. 20, in which key waveforms of the seriesresonant inverter and their fundamental components are sketched. By assuming that the voltage of Ci is constant as *VCi* throughout a single switching cycle, the voltage of the

*V*

2

*<sup>V</sup> <sup>V</sup>*

*C*

*C*

 

3

*C*

*V*

*C*

4 3

*<sup>V</sup> V V*

$$I\_m = \frac{V\_{m-in} - V\_{m-P}}{|Z|} = \frac{V\_{m-in} - V\_{m-P}}{\sqrt{\left(\alpha L\_r - \frac{1}{\alpha C\_r}\right)^2}} \frac{V\_{m-in} - V\_{m-P}}{Z\_0 \sqrt{\left(\frac{\alpha}{\alpha\_r} - \frac{\alpha\_r}{\alpha}\right)^2}},\tag{25}$$

1

where *Z0* is the characteristic impedance of the resonant circuit given by

0

*Z L*

*r r*

**Figure 20.** Key waveforms and their fundamental components.

In order for the series-resonant inverter to transfer energies to the voltage multiplier connected to the secondary winding, *Vm-in* must be higher than *Vm-P*. Assuming that the number of series connections is four and *V1*–*V4* are balanced as *Vi*, the criterion of *N* is obtained from Eqs. (23) and (24), as

$$N < \frac{4V\_i}{V\_i + 2V\_D}.\tag{27}$$

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 171

variable resistor in order to emulate the voltage-balanced and -imbalanced conditions. With the tap Y selected, the current flow paths under the voltage-balanced condition are emulated, whereas those under the voltage-imbalanced condition of *V1* < *Vi* (*i* = 2…12) can be emulated by selecting the tap X. The input and the output of the equalizer were separated at the point Z to measure efficiencies. The efficiencies were measured by changing the ratio of *V1*/*Vin* between approximately 1/8 and 1/20. Cells were disconnected and only the smoothing capacitors (Cout1–

Cout8) were used to sustain the voltages of *V1*–*V8* in the efficiency measurement.

**Figure 22.** Experimental setup for efficiency measurement for the resonant equalizer.

components in the series-resonant inverter and the voltage multiplier.

The measured power conversion efficiencies and output power characteristics as a function of *V1* are shown in Fig. 23. As *V1* increased, the efficiencies significantly increased because the diode voltage drop accounted for a lesser portion of the output voltage (i.e., *V1*). The measured peak efficiencies under the voltage-balanced and -imbalanced conditions were 73% and 68%, respectively. The efficiency trends under the voltage-balanced condition were higher than those under the voltage-imbalanced condition. The lower efficiency trend under the voltage-balanced condition can be attributed to increased Joule losses in resistive

**N1 N2**

Z

**B4**

**B1**

**Rv ar** X Y

**D1 D2**

**C1**

**D8 D9 D10**

**C5**

**D7**

**B5**

**B8**

**D15 D16**

**C8**

**Lkg**

**Cr Lr**

**C4 Vext**

**Da**

**Qa**

**Qb**

**Db**

Measured waveforms of *iLr* and *vDSb* at *Vin* = 32 V and *V1* = 4 V under the voltage-balanced and -imbalanced conditions are shown in Figs. 24(a) and (b), respectively. The amplitude of *iLr* under the voltage-balanced condition was slightly greater than that under the voltageimbalanced condition. In the operation analysis made in Section 4.2.3, the voltage across Ci was assumed constant and the voltages of transformer windings were treated as square waves. However, in practice, the voltage across Ci varies as current flows, and transformer winding voltages are not ideal square waves. Under the voltage-imbalanced condition, as currents in the voltage multiplier concentrated to C1, *Vm-P* tended to increase because of an

Eqs. (23) and (25) indicate that the smaller *N* is, the larger *Im* will be, resulting in the larger power transfer from the resonant circuit to the voltage multiplier. Thus, with small *N*, an equalization speed can be accelerated, although it tends to cause increased losses in resistive components in the resonant circuit as well as in the voltage multiplier.

## **4.3. Experimental**

#### *4.3.1. Prototype and its fundamental performance*

A 10-W prototype of the double-switch series-resonant equalizer was built for 8 cells connected in series, as shown in Fig. 21. Table 2 lists the component values used for the prototype. Cout1–Cout8 are smoothing capacitors connected to cells in parallel (not shown in Fig. 16 for the sake of simplicity). The transformer leakage inductance, *Lkg*, was used as the resonant inductor Lr, whereas the magnetizing inductance, *Lmg*, was designed to be large enough not to influence the series-resonant operation. The prototype equalizer was operated with a fixed *D* = 0.48 at a switching frequency of 220 kHz.

**Figure 21.** Photograph of a 10-W prototype of the double-switch series-resonant equalizer using a voltage multiplier for 8 cells connected in series.


**Table 2.** Component values used for the prototype of the series-resonant equalizer.

The experimental setup for the efficiency measurement for the resonant equalizer is shown in Fig. 22. The efficiency measurement was performed using the intermediate tap and the

obtained from Eqs. (23) and (24), as

**4.3. Experimental** 

number of series connections is four and *V1*–*V4* are balanced as *Vi*, the criterion of *N* is

4 2 *i i D V*

*V V*

Eqs. (23) and (25) indicate that the smaller *N* is, the larger *Im* will be, resulting in the larger power transfer from the resonant circuit to the voltage multiplier. Thus, with small *N*, an equalization speed can be accelerated, although it tends to cause increased losses in resistive

A 10-W prototype of the double-switch series-resonant equalizer was built for 8 cells connected in series, as shown in Fig. 21. Table 2 lists the component values used for the prototype. Cout1–Cout8 are smoothing capacitors connected to cells in parallel (not shown in Fig. 16 for the sake of simplicity). The transformer leakage inductance, *Lkg*, was used as the resonant inductor Lr, whereas the magnetizing inductance, *Lmg*, was designed to be large enough not to influence the series-resonant operation. The prototype equalizer was operated

**Figure 21.** Photograph of a 10-W prototype of the double-switch series-resonant equalizer using a

C1–C8 Tantalum Capacitor, 47 μF, 80 mΩ Cout1–Cout8 Ceramic Capacitor, 200 μF Cr Film Capacitor, 100 nF Qa, Qb N-Ch MOSFET, HAT2266H, Ron = 9.2 mΩ D1–D16 Schottky Diode, CRS08, *V <sup>D</sup>* = 0.36 V Transformer *N <sup>1</sup>* :*N <sup>2</sup>* = 30:5, *L kg* = 4.7 μH, *L mg* = 496 μH

Component Value

The experimental setup for the efficiency measurement for the resonant equalizer is shown in Fig. 22. The efficiency measurement was performed using the intermediate tap and the

**Table 2.** Component values used for the prototype of the series-resonant equalizer.

. (27)

*N*

components in the resonant circuit as well as in the voltage multiplier.

*4.3.1. Prototype and its fundamental performance* 

with a fixed *D* = 0.48 at a switching frequency of 220 kHz.

voltage multiplier for 8 cells connected in series.

variable resistor in order to emulate the voltage-balanced and -imbalanced conditions. With the tap Y selected, the current flow paths under the voltage-balanced condition are emulated, whereas those under the voltage-imbalanced condition of *V1* < *Vi* (*i* = 2…12) can be emulated by selecting the tap X. The input and the output of the equalizer were separated at the point Z to measure efficiencies. The efficiencies were measured by changing the ratio of *V1*/*Vin* between approximately 1/8 and 1/20. Cells were disconnected and only the smoothing capacitors (Cout1– Cout8) were used to sustain the voltages of *V1*–*V8* in the efficiency measurement.

**Figure 22.** Experimental setup for efficiency measurement for the resonant equalizer.

The measured power conversion efficiencies and output power characteristics as a function of *V1* are shown in Fig. 23. As *V1* increased, the efficiencies significantly increased because the diode voltage drop accounted for a lesser portion of the output voltage (i.e., *V1*). The measured peak efficiencies under the voltage-balanced and -imbalanced conditions were 73% and 68%, respectively. The efficiency trends under the voltage-balanced condition were higher than those under the voltage-imbalanced condition. The lower efficiency trend under the voltage-balanced condition can be attributed to increased Joule losses in resistive components in the series-resonant inverter and the voltage multiplier.

Measured waveforms of *iLr* and *vDSb* at *Vin* = 32 V and *V1* = 4 V under the voltage-balanced and -imbalanced conditions are shown in Figs. 24(a) and (b), respectively. The amplitude of *iLr* under the voltage-balanced condition was slightly greater than that under the voltageimbalanced condition. In the operation analysis made in Section 4.2.3, the voltage across Ci was assumed constant and the voltages of transformer windings were treated as square waves. However, in practice, the voltage across Ci varies as current flows, and transformer winding voltages are not ideal square waves. Under the voltage-imbalanced condition, as currents in the voltage multiplier concentrated to C1, *Vm-P* tended to increase because of an increased voltage variation of Ci. Consequently, the amplitude of *iLr*, *Im*, decreased as *Vm-P* increased, as can be understood from Eq. (25).

Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 173

the cell voltages eventually decreased down to approximately 4 mV at the end of the equalization test. The cell voltages were almost completely balanced, and hence, the

**Figure 25.** Experimental profiles of 8 series-connected (a) SCs and (b) lithium-ion cells equalized by the

A similar experimental equalization was performed for 8-series lithium-ion cells with a capacity of 2200 mAh at a rated charge voltage of 4.2 V. The initial SOCs of the cells were imbalanced between 0%–100%. The measured equalization profiles are shown in Fig. 25(b). Despite the nonlinear characteristics of lithium-ion cells, the standard deviation of the cell voltages gradually decreased, and the voltage imbalance was successfully eliminated.

The presented single- and double-switch equalizers are compared with conventional equalizers in terms of the number of required power components in Table 3, where *n* is the number of series connections of cells. Obviously, the passive dissipative equalizers using resistors or diodes are the simplest topology, although they are neither efficient nor practical as mentioned in Section 2.1. Except for the equalizers using a multi-winding transformer, the required number of switches in conventional nondissipative equalizers is proportional to the number of series connections, and therefore, the circuit complexity tends to significantly increase for applications requiring a large number of series connections. Although the equalizers using a multi-winding transformer need only one or two switches, the need of a multi-winding transformer is considered as their major drawback because of the requirement for strict parameter matching among multiple secondary windings,

series-resonant equalizer.

**5. Comparison with conventional equalizers** 

resulting in design difficulty and poor modularity.

equalization performance of the series-resonant equalizer was demonstrated.

**Figure 23.** Measured power conversion efficiencies and output powers of the series-resonant equalizer as a function of *V1* under (a) voltage-balanced and (b) -imbalanced conditions.

**Figure 24.** Measured waveforms under (a) voltage-balanced and (b) -imbalanced conditions.

#### *4.3.2. Equalization*

The experimental equalization test using the prototype of the series-resonant equalizer was performed for 8-series SCs with a capacitance of 500 F at a rated charge voltage of 2.5 V. The initial voltages of SCs were imbalanced in the range 1.8–2.5 V. The results of the equalization test are shown in Fig. 25(a). The cell voltages with a high initial voltage decreased, while those with a low initial voltage increased by the energy redistribution mechanism. The voltage imbalance gradually disappeared, and the standard deviation of the cell voltages eventually decreased down to approximately 4 mV at the end of the equalization test. The cell voltages were almost completely balanced, and hence, the equalization performance of the series-resonant equalizer was demonstrated.

**Figure 25.** Experimental profiles of 8 series-connected (a) SCs and (b) lithium-ion cells equalized by the series-resonant equalizer.

A similar experimental equalization was performed for 8-series lithium-ion cells with a capacity of 2200 mAh at a rated charge voltage of 4.2 V. The initial SOCs of the cells were imbalanced between 0%–100%. The measured equalization profiles are shown in Fig. 25(b). Despite the nonlinear characteristics of lithium-ion cells, the standard deviation of the cell voltages gradually decreased, and the voltage imbalance was successfully eliminated.

#### **5. Comparison with conventional equalizers**

172 Energy Storage – Technologies and Applications

*4.3.2. Equalization* 

increased, as can be understood from Eq. (25).

increased voltage variation of Ci. Consequently, the amplitude of *iLr*, *Im*, decreased as *Vm-P*

**Figure 23.** Measured power conversion efficiencies and output powers of the series-resonant equalizer

**Figure 24.** Measured waveforms under (a) voltage-balanced and (b) -imbalanced conditions.

The experimental equalization test using the prototype of the series-resonant equalizer was performed for 8-series SCs with a capacitance of 500 F at a rated charge voltage of 2.5 V. The initial voltages of SCs were imbalanced in the range 1.8–2.5 V. The results of the equalization test are shown in Fig. 25(a). The cell voltages with a high initial voltage decreased, while those with a low initial voltage increased by the energy redistribution mechanism. The voltage imbalance gradually disappeared, and the standard deviation of

as a function of *V1* under (a) voltage-balanced and (b) -imbalanced conditions.

The presented single- and double-switch equalizers are compared with conventional equalizers in terms of the number of required power components in Table 3, where *n* is the number of series connections of cells. Obviously, the passive dissipative equalizers using resistors or diodes are the simplest topology, although they are neither efficient nor practical as mentioned in Section 2.1. Except for the equalizers using a multi-winding transformer, the required number of switches in conventional nondissipative equalizers is proportional to the number of series connections, and therefore, the circuit complexity tends to significantly increase for applications requiring a large number of series connections. Although the equalizers using a multi-winding transformer need only one or two switches, the need of a multi-winding transformer is considered as their major drawback because of the requirement for strict parameter matching among multiple secondary windings, resulting in design difficulty and poor modularity.


(Smoothing Capacitors are excluded) \* Inductor is not necessary when leakage inductance of the transformer is used as a resonant inductor Single- and Double-Switch Cell Voltage Equalizers for Series-Connected Lithium-Ion Cells and Supercapacitors 175

The single-switch equalizer using multi-stacked buck–boost converters can be derived by multi-stacking any of the traditional buck–boost converters: SEPIC, Zeta, or Ćuk converters. In addition to the single-switch configuration, a multi-winding transformer is not necessary, and therefore, the circuit complexity can be significantly reduced as well as improving the modularity when compared with conventional equalizers, which require multiple switches and/or a multi-winding transformer. The detailed operation analysis was mathematically

The double-switch equalizer using a resonant inverter and voltage multiplier can be synthesized by, namely, combining a resonant inverter and a voltage multiplier. Although two switches are necessary, the required number of switches is sufficiently small to achieve a reduced circuit complexity. Since the number of required magnetic components is only one (i.e., a transformer), the size and cost of the equalizer are considered to be minimal when compared with equalizers requiring multiple magnetic components. The seriesresonant inverter was used as a resonant inverter, and a detailed operation analysis was

The prototypes of the single- and double-switch equalizers were built for series-connected cells, and experimental equalization tests were performed for series-connected SCs and lithium-ion cells from initially-voltage-imbalanced conditions. The energies of cells with a high initial voltage are redistributed to the cells with a low initial voltage, and eventually, voltage imbalance of SCs and lithium-ion cells were almost perfectly eliminated by the

Cao, J., Schofield, N. & Emadi, A. (2008). Battery Balancing Methods: A Comprehensive Review, *Proceedings of IEEE Vehicle Power and Propulsion Conference*, ISBN 978-1-4244-

Guo, K. Z., Bo, Z. C., Gui, L. R. & Kang, C. S. (2006). Comparison and Evaluation of Charge Equalization Technique for Series Connected Batteries, *Proceedings of IEEE Applied Power Electronics Conference and Exposition*, ISBN 0-7803-9716-9, Jeju, South Korea, June 18-22,

Isaacson, M. J., Hollandsworth, R. P., Giampaoli, P. J., Linkowaky, F. A., Salim, A. & Teofilo, V. L. (2000). Advanced Lithium Ion Battery Charger, *Proceedings of Battery Conference on Applications and Advances*, ISBN 0-7803-5924-0, Long Beach, California, USA, January 11-

Kim, C. H., Kim. M. Y., Kim. Y. D. & Moon, G. W. (2011). A Modularized Charge Equalizer Using Battery Monitoring IC for Series Connected Li-Ion Battery String in an Electric

separately made for the voltage multiplier and the series-resonant inverter.

made for the isolated Ćuk converter-based topology.

equalizers after sufficient time elapsed.

*Japan Aerospace Exploration Agency, Japan* 

1848-0, Harbin, China, September 3-5, 2008

**Author details** 

Masatoshi Uno

**7. References** 

2006

14, 2000

**Table 3.** Comparison in terms of required number of power components.

On the other hand, the single-switch equalizer using multi-stacked buck–boost converters (isolated Ćuk converter-based) can operate with a single switch, and therefore, the circuit complexity can be significantly reduced when compared with conventional equalizers that require many switches in proportion to the number of series connections. In addition, since a multi-winding transformer is not necessary and the number of series connections can be readily arbitrary extended by stacking a circuit consisting of L, C, and D. A drawback of this single-switch equalizer is the need of multiple inductors, with which the equalizer is prone to be bulky and costly as the number of the series connections increases.

The double-switch resonant equalizer using a voltage multiplier is able to operate with two switches and a single transformer (in the case that the leakage inductance of the transformer is used as a resonant inductor). In addition to the reduced number of switches, the required number of magnetic components is only one, and hence, the resonant equalizer achieves simplified circuitry coupled with a reduction in size and cost when compared with those requiring multiple magnetic components. The modularity of the resonant equalizer is also good; by adding C and D, the number of series connections can be arbitrarily extended.

## **6. Conclusions**

Cell voltage equalizers are necessary in order to ensure years of safe operation of energy storage cells, such as SCs and lithium-ion cells, as well as to maximize available energies of cells. Although various kinds of equalization techniques have been proposed, demonstrated, and implemented, the requirement of multiple switches and/or a multi-winding transformer in conventional equalizers is not desirable; the circuit complexity tends to significantly increase with the number of switches, and the strict parameter matching among multiple secondary windings of a multi-winding transformer is a serious issue resulting in design difficulty and poor modularity. Two novel equalizers, (a) the single-switch equalizer using multi-stacked buck–boost converters and (b) the double-switch equalizer using a resonant inverter and voltage multiplier are presented in this chapter in order to address the above issues.

The single-switch equalizer using multi-stacked buck–boost converters can be derived by multi-stacking any of the traditional buck–boost converters: SEPIC, Zeta, or Ćuk converters. In addition to the single-switch configuration, a multi-winding transformer is not necessary, and therefore, the circuit complexity can be significantly reduced as well as improving the modularity when compared with conventional equalizers, which require multiple switches and/or a multi-winding transformer. The detailed operation analysis was mathematically made for the isolated Ćuk converter-based topology.

The double-switch equalizer using a resonant inverter and voltage multiplier can be synthesized by, namely, combining a resonant inverter and a voltage multiplier. Although two switches are necessary, the required number of switches is sufficiently small to achieve a reduced circuit complexity. Since the number of required magnetic components is only one (i.e., a transformer), the size and cost of the equalizer are considered to be minimal when compared with equalizers requiring multiple magnetic components. The seriesresonant inverter was used as a resonant inverter, and a detailed operation analysis was separately made for the voltage multiplier and the series-resonant inverter.

The prototypes of the single- and double-switch equalizers were built for series-connected cells, and experimental equalization tests were performed for series-connected SCs and lithium-ion cells from initially-voltage-imbalanced conditions. The energies of cells with a high initial voltage are redistributed to the cells with a low initial voltage, and eventually, voltage imbalance of SCs and lithium-ion cells were almost perfectly eliminated by the equalizers after sufficient time elapsed.

## **Author details**

174 Energy Storage – Technologies and Applications

Topology

Single-Switch Equalizer (Isolated Ćuk-Based) Double-Switch Series-Resonant Equalizer

Multi-Winding Transformer-Based

Single Converter with Selection Switches

Induvidual Cell Equalizer

Dissipative Equalizer

**6. Conclusions** 

**Table 3.** Comparison in terms of required number of power components.

to be bulky and costly as the number of the series connections increases.

On the other hand, the single-switch equalizer using multi-stacked buck–boost converters (isolated Ćuk converter-based) can operate with a single switch, and therefore, the circuit complexity can be significantly reduced when compared with conventional equalizers that require many switches in proportion to the number of series connections. In addition, since a multi-winding transformer is not necessary and the number of series connections can be readily arbitrary extended by stacking a circuit consisting of L, C, and D. A drawback of this single-switch equalizer is the need of multiple inductors, with which the equalizer is prone

Switch Resistor Inductor Capacitor Diode Transformer

1 - *n* + 1 *n* + 1 *n* 1 2 - (1)\* *n* + 1 2*n* 1

(Smoothing Capacitors are excluded)

Resistor - *n* --- - Zener Diode - - - - *n* - Active Shunting *n n* --- - Switched Capacitor 2*n* - - *n* - 1 - - Buck-Boost Converter 2(*n* - 1) - *n* - 1 - - -

Flyback Converter 1 - - - *n* 1 (*n +* 1 windings) Forward Converter 2 - - - *n* + 2 1 (*n +* 1 windings) Flying Capacitor 2*n* --1- - Flyback Converter 2*n* + 1 - - 1 1 1

\* Inductor is not necessary when leakage inductance of the transformer is used as a resonant inductor

The double-switch resonant equalizer using a voltage multiplier is able to operate with two switches and a single transformer (in the case that the leakage inductance of the transformer is used as a resonant inductor). In addition to the reduced number of switches, the required number of magnetic components is only one, and hence, the resonant equalizer achieves simplified circuitry coupled with a reduction in size and cost when compared with those requiring multiple magnetic components. The modularity of the resonant equalizer is also good; by adding C and D, the number of series connections can be arbitrarily extended.

Cell voltage equalizers are necessary in order to ensure years of safe operation of energy storage cells, such as SCs and lithium-ion cells, as well as to maximize available energies of cells. Although various kinds of equalization techniques have been proposed, demonstrated, and implemented, the requirement of multiple switches and/or a multi-winding transformer in conventional equalizers is not desirable; the circuit complexity tends to significantly increase with the number of switches, and the strict parameter matching among multiple secondary windings of a multi-winding transformer is a serious issue resulting in design difficulty and poor modularity. Two novel equalizers, (a) the single-switch equalizer using multi-stacked buck–boost converters and (b) the double-switch equalizer using a resonant inverter and

voltage multiplier are presented in this chapter in order to address the above issues.

Masatoshi Uno *Japan Aerospace Exploration Agency, Japan* 

## **7. References**


Vehicle, *Proceedings of IEEE International Power Electronics Conference*, ISBN 978-1-61284- 956-0, Jeju, South Korea, May 30-June 3, 2011

**Chapter 8** 

© 2013 Zhang and Liu, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Zhang and Liu, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(1)

**Hybrid Energy Storage and Applications Based** 

In the fields of electrical discipline, power electronics and pulsed power technology, the common used modes of energy transferring and energy storage include mechanical energy storage (MES), chemical energy storage (CHES), capacitive energy storage (CES), inductive energy storage (IES) and the hybrid energy storage (HES) [1-3]. The MES and CHES are important ways for energy storage employed by people since the early times. The MES transfers mechanical energy to pulse electromagnetic energy, and the typical MES devices include the generator for electricity. The CHES devices, such as batteries, transfer the chemical energy to electrical energy. The energy storage modes aforementioned usually combine with each other to form an HES mode. In our daily life, the MES and CHES usually need the help of other modes to deliver or transfer energy to drive the terminal loads. As a result, CES, IES and HES become the most important common used energy storage modes for users. So, these three

The CES is an energy storage mode employing capacitors to store electrical energy [3-5]. As Fig. 1(a) shows, *C*0 is the energy storage component in CES, and the load of *C*0 can be inductors, capacitors and resistors respectively. Define the permittivity of dielectric in capacitor *C*0 as *ε*, the electric field intensity of the stored electrical energy in *C*0 as *E*. The

> <sup>1</sup> <sup>2</sup> . <sup>2</sup> *W E <sup>E</sup>*

Usually, *W*E which is restricted to *ε* and the breakdown electric field intensity of *C*0 is about

energy storage modes are analyzed in detail as the central topics in this chapter.

104~105 *J*/*m*3. The traditional Marx generators are in the CES mode [4-5].

**on High Power Pulse Transformer Charging** 

Yu Zhang and Jinliang Liu

http://dx.doi.org/10.5772/52217

energy density *W*E of CES is as

**1. Introduction** 

Additional information is available at the end of the chapter

**1.1. HES based on pulse transformer charging** 

