*2.1.1. The rint model*

The Rint model [8] takes advantage of an ideal voltage source *Voc* as the open circuit voltage (OCV) of the battery. Meanwhile, a resistor *R*1 is utilized to interpret the inner resistance of the battery. *Voc*and *R*1 are both functions of SOC, temperature, and so forth and will vary with these parameters. The equivalent circuit of the Rint model is as follows:

**Figure 1.** The equivalent circuit for Rint battery model.

In Figure 1, *V*1 is used to denote the voltage between terminals of *R*1, *V*o the voltage of the terminals of the battery, *I* the current of the battery. In this chapter, the current is assumed to be positive when the battery is charging, while negative when discharging.

According to the circuit theory, following equation could be obtained:

$$V\_o = V\_{oc} + R\_1 I \tag{1}$$

### *2.1.2. The RC model*

amount of battery cells in a battery string. This chapter will analyze the researches in this area according to literature review and a comparative analysis will be given. Several latest research results, including battery modeling, state estimation, and battery balancing, will also be

Firstly, battery modeling methods will be investigated, including the equivalent circuit methods, impedance methods, etc. The comparison between the proposed impedance model

Secondly, the state estimation method for the battery used in EDVs, including the state of charge (SOC) estimation, will be analyzed. And a novel adaptive proportional integral

Thirdly, the battery balancing method will be analyzed and compared. The positive balancing methods and active balancing methods will be introduced. A novel balancing topology will

Fourthly, the experimental validation will be introduced and the results will be analyzed.

As an important interpretation of an actual battery, a battery model could be used as an important reference when designing the vehicle controller and estimating the states of the battery. This section will study battery models describing the characteristics of batteries. Firstly, some existing battery models will be reviewed and analyzed. Then, an impedance battery model will be proposed according to the merits and demerits of the existing battery

The battery is an electrochemical system with strong nonlinearity. To model such a strong nonlinear system is very difficult. Some attempts have been made to evaluate the battery models for the state and parameter estimation of a li-ion battery, such as electrochemical models and equivalent circuit models. Electrochemical models take advantage of electro‐ chemical properties of the battery, and this method is relatively too complex for engineering, though it is relatively accurate. Thus, equivalent circuit models is very popular in practical applications, and several equivalent circuit models have been widely used, such as the Rint Model [8-10], the first order RC model [10-12], the second order RC model [10, 13], etc.

The Rint model [8] takes advantage of an ideal voltage source *Voc* as the open circuit voltage (OCV) of the battery. Meanwhile, a resistor *R*1 is utilized to interpret the inner resistance of the

and the measured impedance spectra will be shown and analyzed.

observer SOC estimation method will be proposed.

Finally, conclusions of this chapter will be given.

**2.1. Review of battery equivalent circuit models**

introduced in this chapter.

88 New Applications of Electric Drives

be proposed and analyzed.

**2. Battery modeling**

models.

*2.1.1. The rint model*

The RC model was proposed by the National Renewable Energy Laboratory (NREL), which has already been utilized in the automobile simulation software Advisor [8, 14]. The RC model consists of two capacitors and three resistors. The capacitance of *Cc* is relatively small to interpret the surface effects of the battery. Meanwhile, the capacitance of *C*<sup>b</sup> is relatively big to interpret the ampere capacity of the battery. The equivalent circuit for the RC model is shown as follows:

**Figure 2.** The equivalent circuit for the RC battery model.

Similarly, the mathematical relationship could be obtained as follows:

$$\begin{cases} \dot{V}\_b = \frac{-1}{\mathcal{C}\_b \left(\mathcal{R}\_c + \mathcal{R}\_c\right)} V\_b + \frac{1}{\mathcal{C}\_b \left(\mathcal{R}\_c + \mathcal{R}\_c\right)} V\_c + \frac{\mathcal{R}\_c}{\mathcal{C}\_b \left(\mathcal{R}\_c + \mathcal{R}\_c\right)} I \\ \dot{V}\_c = \frac{1}{\mathcal{C}\_c \left(\mathcal{R}\_c + \mathcal{R}\_c\right)} V\_b + \frac{-1}{\mathcal{C}\_c \left(\mathcal{R}\_c + \mathcal{R}\_c\right)} V\_c + \frac{\mathcal{R}\_c}{\mathcal{C}\_c \left(\mathcal{R}\_c + \mathcal{R}\_c\right)} I \end{cases} \tag{2}$$

$$V\_o = \frac{R\_c}{R\_c + R\_c} V\_b + \frac{R\_c}{R\_c + R\_c} V\_c + \left( R\_t + \frac{R\_c R\_c}{R\_c + R\_c} \right) I \tag{3}$$

where *V*˙ *<sup>b</sup>* and *V*˙ *<sup>c</sup>* means derivation of *Vb* and *Vc* respectively.

#### *2.1.3. The first order model*

The first order model [9, 15] is realized by adding a parallel RC network to the Rint model, that's why it is called first order model. The added parallel RC network is used to interpret the dynamic response of the battery. The equivalent circuit of the first order model is shown in Figure 3.

**Figure 3.** The equivalent circuit for the first order RC battery model.

Four parts could be included in the first order model: the OCV *Voc*, the inner resistance *R*1, the polarization resistance *R*2, and the polarization capacitance *C*2. *V*2is used to denote the voltage over the parallel RC network. The mathematical relationship is as follows:

$$\begin{cases} \dot{V}\_2 = -\frac{1}{R\_2 C\_2} V\_2 + \frac{I}{C\_2} \\ V\_o = V\_{oc} + V\_2 + R\_1 I \end{cases} \tag{4}$$

## *2.1.4. The second order model*

Similarly, the mathematical relationship could be obtained as follows:

1 1

*bbc*

<sup>ì</sup> - <sup>ï</sup> =++

&

&

*2.1.3. The first order model*

90 New Applications of Electric Drives

Figure 3.

1 1

*o b ct*

where *V*˙ *<sup>b</sup>* and *V*˙ *<sup>c</sup>* means derivation of *Vb* and *Vc* respectively.

**Figure 3.** The equivalent circuit for the first order RC battery model.

*c bc*

( ) ( ) ( )

*be c be c be c*

*<sup>R</sup> VVVI CR R CR R CR R <sup>R</sup> VVVI CR R CR R CR R*

+++ ï

<sup>í</sup> - <sup>ï</sup> =++ <sup>ï</sup> +++ <sup>î</sup>

*c*

*e*

æ ö

(2)

(3)

(4)

( ) ( ) ( )

*ce c ce c ce c*

*c e e c*

*ec ec e c R R R R V V VR I RR RR RR*

The first order model [9, 15] is realized by adding a parallel RC network to the Rint model, that's why it is called first order model. The added parallel RC network is used to interpret the dynamic response of the battery. The equivalent circuit of the first order model is shown in

Four parts could be included in the first order model: the OCV *Voc*, the inner resistance *R*1, the polarization resistance *R*2, and the polarization capacitance *C*2. *V*2is used to denote the voltage

> 22 2 2 1

over the parallel RC network. The mathematical relationship is as follows:

ì

&

í

2 2

ï =- +

<sup>ï</sup> = ++ <sup>î</sup>

*o oc*

1

*<sup>I</sup> V V RC C*

*V V V RI*

= + ++ ç ÷ ++ + è ø Another parallel RC network added to the first order model forms the second order model. The resistance of the RC network *R*<sup>3</sup> is used to describe the concentration polarization, while the capacitance *C*3 is used to describe the electrochemical polarization. The Second order model is shown in Figure 4 and the mathematical expression is shown in Eq. (5):

**Figure 4.** The equivalent circuit for the second order RC battery model.

$$\begin{cases} \dot{V}\_2 = -\frac{1}{R\_2 C\_2} V\_2 + \frac{I}{C\_2} \\ \dot{V}\_3 = -\frac{1}{R\_3 C\_3} V\_3 + \frac{I}{C\_3} \\ V\_o = V\_{oc} + V\_2 + V\_3 + R\_1 I \end{cases} \tag{5}$$

#### **2.2. The battery impedance model**

Normally, a more accurate battery model could lead to more accurate state estimation. The models analyzed above have been widely used to estimate the states of the battery, but the problem is that they are sometimes not accurate enough to get satisfying estimation results. The electrochemical impedance spectroscopy (EIS) method is an experimental method to characterize electrochemical systems, and it is considered as one of the most accurate methods to model electrochemical systems, including li-ion batteries and supercapacitors [16-18]. However, studies have shown that the EIS method is too complex to be implemented in real time applications [19]. Besides, the impedance spectra vary with temperature, which adds difficulty to derive SOC directly from the impedance spectra. To solve this problem, an impedance model is proposed in the section [20].
