*2.2. Li-Ion-RC Equivalent Circuit Model*

The 3RC ECM Li-ion battery model, as shown in Figure 6, consists of an OCV source in series with the internal Rin resistance of the battery and three parallel RC bias cells. RC cells are introduced into the circuit to capture the dynamic electrochemical behaviour of the battery and to increase the accuracy of the model. The first RC polarization cell captures the fast transient of the battery cell, and the last two RC cells capture the slow variation of the steady-state and increase the accuracy of the model. As the new technologies are largely dependent on batteries, it is important to develop accurate battery models that can be conveniently used with on-board power simulators and electronic on-board power systems, as mentioned in [14,17–23].

**Figure 6.** The third order 3RC electric circuit model (ECM)–Li-Ion battery represented in NI Multisim 14.1 editor (see [19]).

For simulation purpose, a specific setup for the 3RC ECM Li-ion battery model parameters, shown in Table 1 or directly on the electrical scheme from Figure 1, is available to prove the e ffectiveness of the proposed SOC estimation strategies. This setup is achieved from a generic ECM by changing only the values of the model parameters in state-space equations.

The Li-ion battery 3RC ECM model parameters are given in Table 3, and the OCV nonlinear model coe fficients are shown in Table 4.


**Table 3.** The RC ECM parameters [17–23].

**Table 4.** RC ECM Li-ion battery open-circuit controlled voltage (OCV) coefficients.


### 2.2.1. RC ECM Li-Ion Battery Model-Continuous Time State Space Representation

In a state-space representation, the continuous time 3RC ECM nonlinear model of SAFT Li-ion battery shown in Figure 6 is given by following Equations:

$$\frac{d\mathbf{x}\_1}{dt} = \frac{1}{R\_1 \mathbf{C}\_1} \mathbf{x}\_2 + \frac{1}{\mathbf{C}\_1} \boldsymbol{\mu}(t) \; \; \boldsymbol{\mu}(\tau) \ge 0 \tag{2}$$

$$\frac{d\mathbf{x}\_2}{dt} = \frac{1}{R\_2C\_2}\mathbf{x}\_2 + \frac{1}{C\_2}\boldsymbol{\mu}(t) \tag{3}$$

$$\frac{d\mathbf{x}\_3}{dt} = \frac{1}{R\_3 \mathbf{C}\_3} \mathbf{x}\_3 + \frac{1}{\mathbf{C}\_3} u(t) \tag{4}$$

$$\frac{d\mathbf{x}\_4}{dt} = -\frac{\eta\mu(t)}{\mathbb{C}\_{\text{non}}} \; , \; u(\tau) \ge 0 \tag{5}$$

*OCV*(*t*) = *K*0 − *K*2*x*4 − *K*1 *x*4 + *K*3 ln(*<sup>x</sup>*4) + *K*4 ln(1 − *<sup>x</sup>*4) (6)

$$y(t) = OCV(t) - x\_1 - x\_2 - x\_3 - R\_{in}u(t)\tag{7}$$

where the components of the state vector are: *x*4 = *SOC* is the state of charge of Li-ion battery, *x*1 = *V*1 is the voltage across first *R*1||*C*1 polarization cell, *x*2 = *V*2 denotes the voltage across the second *R*2||*C*2 polarization cell, *x*3 = *V*3 represents the third *R*3||*C*3 polarization cell *u*(*t*) = *<sup>i</sup>*(*t*) is the input discharging current (*u*(*t*) ≥ 0) or charging current (*u*(*t*) ≤ 0), *OCV*(*t*) represents the open-circuit voltage of Li-ion battery, and finally *y*(*t*) designates the terminal voltage of the battery. The open-circuit voltage of Li-ion battery *OCV*(*t*) given in (6) is a non-linear function of battery SOC, and contains a combination of the following three well-known generic battery models [17,19–21,25]:

(1) Shepherd model

$$y(t) = K\_0 - R\_{in}u(t) - \frac{K\_1}{x\_4} \tag{8}$$

(2) Unnewehr universal model

$$y(t) = K\_0 - R\_{in} R \mu(t) - K\_2 \mathbf{x}\_4 \tag{9}$$

(3) Nernst model

$$y(t) = \mathcal{K}\_0 - \mathcal{R}\_{\text{in}}u(t) + \mathcal{K}\_3 \ln(\mathbf{x}\_4) + \mathcal{K}\_4 \ln(1 - \mathbf{x}\_4) \tag{10}$$

The performance of the generic models in terms of voltage prediction and SOC estimation is analysed in [24], and the simulations result show that the Unnewehr and Nernst models compared to the Shepherd model, criticized in literature, increase significantly the accuracy of linear ECMs, more specifically, Nernst model "showed the best performance among the three mathematical models" due to its flexibility by using two parameters (correction factors instead of one). Last, the combination of all three mathematical models in (7) and their introduction in the terminal voltage relationship (8) increases considerable the Li-ion ECM accuracy. Also, the ECM combined model proved until now that it is "amongst the most accurate formulations seen in literature from EVs/HVs field" [17,21].

Since the parameters of 3 RC ECM Li-ion model strongly depend on temperature and SOC, the combined model is beneficial due to its simplicity, accuracy, and development of BMS SOC estimators for HEVs as a "proof concept" and fast real-time implementation.

It is important to underline that the values of coe fficients *K*0,*K*1,*K*2,*K*3, and *K*4 , provided in Table 2, are chosen to fit the Li-ion battery model accurately according to the manufacturers' data by using a least squares curve fitting estimation method, as is suggested in [17–23]. The values of the resistances *R*1, *R*2 and the capacitors *C*1, *C*2 , as well the value of the battery nominal capacity *Cnom* and its internal resistance are given in Table 1. The Simulink diagram of third order 3RC EMC–Li-Ion battery model that implements the Equations (3)–(8) is shown in Figure 7.

**Figure 7.** The Simulink diagram of third order 3RC ECM–Li-Ion battery model.

2.2.2. RC Electric Circuit Model (ECM)–Li-Ion Battery Model—Discrete Time State Space Representation

For the design and implementation of SOC estimators it is necessary to discretize over time the continuous model of the Li-ion battery. The discrete model of 3RC ECM Li-ion model is described in a compact state space matrix representation, as:

$$\mathbf{x}(k+1) = A\mathbf{x}(k) + bu(k) \text{ , } u(k) \ge 0 \tag{11}$$

*Batteries* **2020**, *6*, 42

$$\text{OCV}(k) = K\_0 - K\_2 \mathbf{x}\_4(k) - \frac{K\_1}{\mathbf{x}\_4(k)} + K\_3 \ln(\mathbf{x}\_4(k)) + K\_4 \ln(1 - \mathbf{x}\_4(k)) \tag{12}$$

$$y(k) = OCV(k) - \mathbf{x}\_1(k) - \mathbf{x}\_2(k) - \mathbf{x}\_3(k) - R\_{\text{in}}\boldsymbol{\mu}(k)\tag{13}$$

where *x*(*k*) = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *<sup>x</sup>*1(*k*)*<sup>x</sup>*2(*k*) *<sup>x</sup>*3(*k*) *<sup>x</sup>*4(*k*) ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ - *x*(*kTs*) is the state vector with 4 components *A* =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 1 − *Ts T*1 0 00 0 1 − *Ts T*2 0 0 0 01 − *Ts T*3 0 0 0 01 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ denotes the state matrix, *b* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *Ts C*1 *Ts C*2 *Ts C*3 − η *Cnom*⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a vector of the input coefficients

 (input excitation terms), *u*(*k*) - *u*(*kTs*) , *OCV*(*k*) - *OCV*(*kTs*), and *y*(*k*) - *y*(*kTs*) denotes the input, respectively the OCV and the terminal output voltage at discrete time instants *kTs*|*<sup>k</sup>*∈*Z*<sup>+</sup> . In our MATLAB simulations, the sampling time is set to *Ts* = 1 s, without any solver convergence problems.
