*2.2.1. Impedance spectroscopy test*

In an EIS test, an AC current is applied to the battery, and the voltage response to the current, namely, the amplitude and the phase, is recorded. By the complex division of AC voltage by AC current, the impedance is obtained. These procedures are repeated for different frequen‐ cies, and the properties of the battery in full range frequency is obtained [21]. The dynamic behavior of batteries could be activated and recorded by the EIS test, which gives a precise impedance measurement in a wide band of frequencies. The EIS test measures the nonlinear‐ ities, as well as very slow dynamics directly, thus it is considered as a unique tool to analyze battery dynamics.

The impedance spectra of a 50% SOC battery with 3 mHz to 2.1 kHz frequency range is shown in Figure 5. Three sections can be obtained for the impedance spectra, as shown in the figure, namely, the low-frequency section, the mid-frequency section, and the highfrequency section [21].

**Figure 5.** Impedance spectra of a Li-ion battery.

For the high-frequency section, where the frequency is larger than 144 Hz, the impedance spectra intersect with the real axis. According to the electrochemical theories (ET) and circuit theory (CT), this point could be explained by an ohmic resistance in the equivalent circuit.

When the frequency is smaller than 443 mHz, referred to as the low-frequency section, a straight line with a constant slope could be observed. This section could be expressed as a constant phase element (CPE) according to ET [18, 22, 23], which is often referred to as a Warburg element:

Battery Management System for Electric Drive Vehicles – Modeling, State Estimation and Balancing http://dx.doi.org/10.5772/61609 93

$$Z\_{\text{Murbung}}\text{(joo)} = \sqrt{\frac{R\_D}{j o C\_D}} \coth\left(\sqrt{R\_D j o C\_D}\right) \tag{6}$$

When in the mid-frequency section (443 mHz–144 Hz), the curve looks like a depressed semicircle. According to ET and CT, this depressed semicircle could be modeled by paralleling a CPE with a resistance in an equivalent circuit, which is called a ZARC element [23]. From the analysis above, the equivalent circuit is depicted as shown in Figure 6.

In Eq. 7, *Voc*denotes the open circuit voltage of the battery; *V*1, *V*2, and *V*<sup>3</sup> denote the voltage for *R*1, ZARC, and Warburg respectively; *Vo* is the voltage output of the battery that can be measured directly from the two terminals of the battery.

$$V\_o = V\_{oc} + V\_1 + V\_2 + V\_3 \tag{7}$$

**Figure 6.** Equivalent circuit for the impedance model.

*2.2.1. Impedance spectroscopy test*

92 New Applications of Electric Drives

battery dynamics.

frequency section [21].

**Figure 5.** Impedance spectra of a Li-ion battery.

Warburg element:

In an EIS test, an AC current is applied to the battery, and the voltage response to the current, namely, the amplitude and the phase, is recorded. By the complex division of AC voltage by AC current, the impedance is obtained. These procedures are repeated for different frequen‐ cies, and the properties of the battery in full range frequency is obtained [21]. The dynamic behavior of batteries could be activated and recorded by the EIS test, which gives a precise impedance measurement in a wide band of frequencies. The EIS test measures the nonlinear‐ ities, as well as very slow dynamics directly, thus it is considered as a unique tool to analyze

The impedance spectra of a 50% SOC battery with 3 mHz to 2.1 kHz frequency range is shown in Figure 5. Three sections can be obtained for the impedance spectra, as shown in the figure, namely, the low-frequency section, the mid-frequency section, and the high-

For the high-frequency section, where the frequency is larger than 144 Hz, the impedance spectra intersect with the real axis. According to the electrochemical theories (ET) and circuit theory (CT), this point could be explained by an ohmic resistance in the equivalent circuit. When the frequency is smaller than 443 mHz, referred to as the low-frequency section, a straight line with a constant slope could be observed. This section could be expressed as a constant phase element (CPE) according to ET [18, 22, 23], which is often referred to as a

Then comes the problem: how to use such an impedance model with CPE in it?

#### *2.2.2. Introduction to fractional order calculus*

Considered as a natural extension of the classical integral order calculus, fractional order calculus (FOC) is becoming more and more popular. According to previous studies, most phenomena, including damp, fluid, viscoelasticity, friction, chaos, dynamic backlash, me‐ chanical vibration, sound diffusion, etc., were considered to have fractional properties [24, 25]. Furthermore, the entire system would have fractional properties, as Machado pointed out, even if parts of it had integral properties [24, 26-28]. Based on FOC, more and more researches have been carried out to develop electrochemical models [22, 29], including lead-acid batteries, supercapacitors, li-ion batteries, fuel cells, and so on. In this section, to represent the impedance model in equations, a fractional modeling method is utilized. By this method, the impedance model can become more convenient to implement the SOC estimation of a battery.

### *2.2.3. Impedance model interpreted by FOC*

A FOC modeling method is introduced to interpret such an element. A fractional element is given in the following equation [20, 24]:

$$Z\_{fractional} \text{(joo)} = \frac{1}{\mathbb{Q}\left(joo\right)^{\prime}} \tag{8}$$

where *r* ∈ℝ (−1≤*r* ≤1) is the arbitrary order of the fractional element, which can be an integer or a fraction; *Q* ∈ℝ is the coefficient. When *r* =0, the fractional element is equivalent to a resistor; when *r* = −1, it is equivalent to an inductor; when *r* =1, it is equivalent to a capacitor.

Furthermore, the phase of the fractional element listed in Eq. (8) is *rπ* / 2, the magnitude is 20*r* dB dec-1, and the Nyquist plot of the fractional element is a straight line. The slope of the line is a constant value *rπ* / 2, which means the fractional element is a CPE. So, a CPE can be expressed by a fractional element, which is also proved in [24]. In this way, a Warburg element can be modeled by a fractional element Zwarburgf as shown in Eq. (9). For the relationship of fractional factor *α* and the slope of the curve in the low-frequency section, *α*can be determined by the curve slope. A Warburg element can be expressed in terms of a fractional element as follows:

$$Z\_{\text{Wurbung}}\left(jao\right) = \frac{1}{\mathcal{W}\left(jao\right)^{\alpha}}\tag{9}$$

where *α* ∈ℝ (−1≤*α* ≤1) is an arbitrary number, *W* ∈ℝis the coefficient. Similarly, assume the equation of *CPE*2 as follows:

$$Z\_{\rm CPE\_2} \left( joo \right) = \frac{1}{\mathcal{C}\_2 \left( joo \right)^{\rho}} \tag{10}$$

where *β* ∈ℝ (0≤*β* ≤1) is an arbitrary number, and *C*2∈ℝ is the coefficient, especially, when *β* =1,

$$\left. Z\_{\text{CPE}\_2} \left( jao \right) \right|\_{\beta=1} = \frac{1}{\mathcal{C}\_2 \left( jao \right)} \tag{11}$$

*CPE*2 is a capacitor and the capacitance is *C*2. The paralleling CPE and resistor will form a semicircle in the Nyquist plot.

Figure 7 depicts the Nyquist plot where Zwarburgf is used instead of Zwarburg and ZARC is represented by FOC. Figure 7 shows that the fractional method can fit the measured impedance spectra well, which means the FOC impedance model can characterize the battery based on the impedance spectra analysis.

**Figure 7.** Comparison between measured impedance spectra and the impedance model.
