*2.1.1 Li-ion battery electrical equivalent circuit model*

For simulation purposes and "proof concept," a starting point for developing new Li-ion battery alternative models might be a linear electrical circuit consisting of one

*Investigations of Using an Intelligent ANFIS Modeling Approach for a Li-Ion Battery… DOI: http://dx.doi.org/10.5772/intechopen.105529*

**Figure 4.** *MATLAB input-output simulation results.*

**Figure 5.**

*The input UDDS driving cycle current profile (battery charging and discharging periods).*

of the combinations of an open-circuit voltage (OCV) controlled source, known in the literature as Thevenin voltage source, connected in series with the internal resistance (Rint) of the battery, followed by one, two, or three parallel resistive and capacitive polarization cells (RC). These combinations lead to a simple electrical equivalent circuit models (ECMs) very spread in the literature field as is shown in **Figure 6** [7]. Until now, the ECMs proved that they are of the high simplicity and are the most suitable models to capture the battery's dynamic electrochemical behavior and increase the model's accuracy. Since in **Figure 6**, the ECM has three parallel RC bias

**Figure 6.** *Electrical schematic of third-order 3RC ECM battery selection (see [7]).*

polarization cells, it is known in the literature field as a three-order RC (3RC) ECM Li-ion battery model. The ECM schematic is built using the Multisim 14.1 software package provided by the well-known National Instruments (NI) company. The first R1pC1p polarization parallel cell captures the fast transient of the battery compared with the last two RC cells that capture only the slow steady state with a great impact in the increase of the battery model accuracy [7]. Since most HEV/EV technologies are very dependent on batteries nowadays, it is crucial for developing and implementing accurate Li-ion battery models. These models must suit better the BMS requirements to be easily deployed on-board power simulators and electronic on-board power systems. Moreover, the 3RC ECM accuracy performance is a baseline for all other alternative battery models developed in this research paper for comparison purposes. For MATLAB simulation's goal, a similar setup for the 3RC ECM Li-ion battery model parameters used in [7], shown in **Table 1** or directly on the electrical schematics from **Figure 6**, is considered to prove the effectiveness and the robustness of an adaptive extended Kalman filter SOC estimation strategy, similar to those used in [9] for a generic Li-ion cobalt battery and adapted to the 3RC ECM model, presented in Appendix A. This setup is achieved from a generic ECM by changing only the values of the model parameters in state-space equations.

#### *2.1.2 Li-ion battery 3RC ECM validation*

The Li-ion battery 3RC ECM model parameters and the OCV nonlinear model coefficients are given in **Tables 1** and **2**. The OCV shown in **Figure 7** is a nonlinear function of SOC that combines three additional well-known models, namely Shepherd, Unnewehr universal and Nernst (SUN-OCV) models, defined in [3, 5, 7, 9] with the coefficients set at same values as in [3, 7, 9].

According to the values of the parameters and coefficients set in the **Table 1** the Li-ion battery model dynamics is described by the following discrete-time Eqs. [7]:

$$
\omega\_1(k+1) = a\_{11}\omega\_1(k) + b\_1\omega(k) = V\_1(k) \tag{1}
$$

$$
\pi\_2(k+1) = a\_{22}\pi\_2(k) + b\_2\mu(k) = V\_2(k) \tag{2}
$$

$$
\omega\_3(k+1) = a\_{33}\omega\_3(k) + b\_3\omega(k) = V\_3(\mathbf{k})\tag{3}
$$

*Investigations of Using an Intelligent ANFIS Modeling Approach for a Li-Ion Battery… DOI: http://dx.doi.org/10.5772/intechopen.105529*


#### **Table 1.**

*The 3RC ECM parameters and OCV coefficients [3, 7, 9].*


#### **Table 2.**

*Performance of the Li-ion AEKF SOC ARX model compared with AEKF SOC estimator ANFIS model for UDDS driving cycle test [7].*

$$\varkappa\_4(k+1) = \varkappa\_4(k) + \frac{\eta T\_\* \mu(k)}{C\_{nom}},\\\varkappa\_4(k) = \text{SOC}(k) = \text{SOC}(kT\_s) \tag{4}$$

$$\text{OCV}(k) = k\_0 - k\_2 \mathbf{x}\_4(k) - \frac{k\_1}{\mathbf{x}\_4(k)} + k \ln \left( \mathbf{x}\_4(k) \right) + k \ln \left( \mathbf{1} - \mathbf{x}\_4(k) \right) \tag{5}$$

$$y(k) = \text{OCV}(k) - R\_{in}u(k) = V\_{bat}(k), u(k) = I\_{bat}(k) \tag{6}$$

where *Ts* ¼ 1 ½ �*s* is the sampling time, and the values of the equations' coefficients (1)–(6) are given by *<sup>a</sup>*<sup>11</sup> <sup>¼</sup> <sup>1</sup> � *Ts <sup>T</sup>*<sup>1</sup> , *<sup>a</sup>*<sup>22</sup> <sup>¼</sup> <sup>1</sup> � *Ts <sup>T</sup>*<sup>2</sup> , *<sup>a</sup>*<sup>33</sup> <sup>¼</sup> <sup>1</sup> � *Ts <sup>T</sup>*<sup>3</sup> , *<sup>a</sup>*<sup>44</sup> <sup>¼</sup> 1, *<sup>b</sup>*<sup>1</sup> <sup>¼</sup> *Ts Cp*<sup>1</sup> , *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> *Ts Cp*<sup>2</sup> , *<sup>b</sup>*<sup>3</sup> <sup>¼</sup> *Ts Cp*<sup>3</sup> , and *<sup>b</sup>*<sup>4</sup> ¼ � *<sup>η</sup>Ts* 3600*Qnom :* In the expression of the coefficient *b*4, *η* is the Coulombic efficiency, and *Qnom* represents the nominal capacity of the battery, set to the following values: *η* = 0.85, and *Qnom* = 6*Ah*. Also, the time constants of the polarization cells *T*1,*T*2, and *T*<sup>3</sup> are given by: *T*<sup>1</sup> ¼ *Rp*1*Cp*1, *T*<sup>2</sup> ¼ *Rp*2*Cp*2, and *T*<sup>3</sup> ¼ *Rp*3*Cp*3*:*

#### **Figure 7.**

*Battery terminal Rint-3RC ECM voltage versus ARX – ECM for the dynamic part of the battery.*

A Simulink model based on these previous equations is shown in **Figure 8a**, compact in compact form, and in **Figure 8b**, for a detailed form.

The MATLAB simulation results are shown in **Figure 9**. In **Figure 9a** and **b** are depicted the SOC of the battery 3RC model versus SOC estimated by the ADVISOR simulator. In **Figure 9c** and **d** are presented the OCV = f(SOC) curve and the battery SOC for a complete UDDS discharge cycle respectively. In **Figure 9e** is shown only the terminal voltage for a single UDDS cycle. The SOC residual represented in **Figure 9b** reveals a good SOC accuracy performance of the 3RC EMC battery model with respect to the estimated battery SOC on the ADVISOR simulator integrated with the MATLAB platform. This excellent result is a realistic argument that validates certainly the proposed 3RC ECM Li-ion battery attached to the generic Rint model of SAFT-type battery.
