1. Introduction

2. Analysis of ECM Parameters in Frequency Domain

3. Least Squares Approach

3.2. Estimation of Diffusion Arc’s Gradient m

Considering the imaginary part of the measured impedance $z_i\left(k\right)$ and the real part of the measured impedance $z_r\left(k\right)$ in the Diffusion arc, it can be represented with a linear model [21]:

$$\begin{array}{c}\hfill z_i\left(k\right)=m{z}_r\left(k\right)+a\end{array}$$

(5)

Assuming the measurements are from $k_{DF1}$ to $k_{DF2}$ as shown in Figure 2, the following can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill z_i\left(k_{DF1}\right)=& m{z}_r\left(k_{DF1}\right)+a\hfill \\ \hfill z_i\left(k_{DF1+1}\right)=& m{z}_r\left(k_{DF1+1}\right)+a\hfill \\ \hfill & ⋮\hfill \\ \hfill z_i\left(k_{DF2}\right)=& m{z}_r\left(k_{DF2}\right)+a\hfill \end{array}\end{array}$$

(6)

Equation (6) can be written in the matrix form

$$\begin{array}{c}\hfill \underset{\mathbf{y}}{\underbrace{\left[\begin{array}{c}{z}_i\left(k_{DF1}\right)\\ z_i(k_{DF1}+1)\\ ⋮\\ z_i\left(k_{DF2}\right)\end{array}\right]}}=\underset{\mathbf{A}}{\underbrace{\left[\begin{array}{cc}{z}_r\left(k_{DF1}\right)& 1\\ z_r(k_{DF1}+1)& 1\\ ⋮\\ z_r\left(k_{DF2}\right)& 1\end{array}\right]}}\underset{\mathbf{k}}{\underbrace{\left[\begin{array}{c}m\\ a\end{array}\right]}}\end{array}$$

(7)

m and a can be estimated using the LS approach:

$$\begin{array}{c}\hfill \widehat{\mathbf{k}}={\left({\mathbf{A}}^T\mathbf{A}\right)}^{-1}\left({\mathbf{A}}^T\mathbf{y}\right)\end{array}$$

(8)

$$\widehat{m}=\widehat{\mathbf{k}}\left(1\right),\widehat{a}=\widehat{\mathbf{k}}\left(2\right)$$

(9)

Algorithm 1 estimates quantities (list) based on the following impedance values:

$$\begin{array}{cc}\hfill {\mathbf{z}}_r& =[z_r\left(1\right),z_r\left(2\right),\dots ,z_r\left(n\right)]\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\mathbf{z}}_i& =[z_i\left(1\right),z_i\left(2\right),\dots ,z_i\left(n\right)]\hfill \end{array}$$

(11)

In this paper, the uppermost bound of the DF arc is denoted as $k_{DF}^H$, the lowest bound of the CT arc is denoted as $k_{CT}^L$, and the lowest bound of the SEI arc is denoted as $k_{SEI}^L$, these boundaries can be identified by applying moving average filter (MAF) to process the measured impedance data via Algorithm 1, the filtered EIS data is shown in Figure 4a. The algorithms presented in this paper are written by utilizing MATLAB syntax. Algorithm 1 uses the following MATLAB commands: ’smooth’, ’length’, ’find’,’min’, ’break’, and ’continue’.

Figure 3. The procedure to obtain battery EIS [22].

Figure 3. The procedure to obtain battery EIS [22].

Algorithm 1 Boundary identification.

The gradient m of the Diffusion arc can be estimated by fitting the Diffusion arc with the linear model mentioned in (5) and searching out the best fit using Algorithm 2. The fitting process is also demonstrated in Figure 4b,c. Algorithm 2 uses the following MATLAB commands: ’mean’, ’find’, and ’max’.

Algorithm 2 Diffusion arc fitting.

3.4. Estimation of $R_{\mathrm{SEI}}$ and $C_{\mathrm{SEI}}$

As shown in Figure 2, to fit the SEI arc precisely, we select feature points that lie between $k_{SEI1}$ and $k_{SEI2}$. Let us denote the impedance measurements in the SEI arc as

$$\begin{array}{cc}\hfill & s_r\left(k\right)\triangleq z_r\left(k\right)\mathrm{s}.\mathrm{t}.k_{SEI1}\le k\le k_{SEI2}\hfill \\ \hfill & s_i\left(k\right)\triangleq z_i\left(k\right)\mathrm{s}.\mathrm{t}.k_{SEI2}\le k\le k_{SEI2}\hfill \end{array}$$

(16)

The estimation of $R_{\mathrm{SEI}}$ is to fit the SEI arc using a semicircle with its centre lying on the real axis; the coordinate of this semicircle’s center can be denoted as ($x_s$, 0); the radius of the semicircle can be denoted as $r_s$; thus, the measurements in (16) should satisfy the equation of the semicircle [20]:

$$\begin{array}{c}\hfill {\left(s_r\left(k\right)-x_s\right)}^2+{\left(s_i\left(k\right)-0\right)}^2={r_s}^2\end{array}$$

(17)

$$\begin{array}{c}\hfill {s_r\left(k\right)}^2-2x_s{s}_r\left(k\right)+x_s^2+{s_i\left(k\right)}^2={r_s}^2\end{array}$$

(18)

Let $c=-2x_s$ and $d=x_s^2-{r_s}^2$, thus

$$\begin{array}{c}\hfill {r_s}^2={\displaystyle \frac{c^2}{4}}-d\end{array}$$

(19)

$$\begin{array}{c}\hfill r_s=\sqrt{{\displaystyle \frac{c^2}{4}}-d}\end{array}$$

(20)

And (18) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {s_r\left(k\right)}^2+{s_i\left(k\right)}^2+c{s}_r\left(k\right)+d=0\end{array}\end{array}$$

(21)

In the matrix form, (21) can be written as

$$\begin{array}{cc}\hfill & \underset{\mathbf{z}}{\underbrace{\left[\begin{array}{c}-\left(s_r{\left(k_{SEI1}\right)}^2+s_i{\left(k_{SEI1}\right)}^2\right)\\ -\left(s_r{(k_{SEI1}+1)}^2+s_i{(k_{SEI1}+1)}^2\right)\\ -\left(s_r{(k_{SEI1}+2)}^2+s_i{(k_{SEI1}+2)}^2\right)\\ ⋮\\ -\left(s_r{\left(k_{SEI2}\right)}^2+s_i{\left(k_{SEI2}\right)}^2\right)\end{array}\right]}}\hfill \\ \hfill & =\underset{\mathbf{B}}{\underbrace{\left[\begin{array}{cc}{s}_r\left(k_{SEI1}\right)& 1\\ s_r(k_{SEI1}+1)& 1\\ s_r(k_{SEI1}+2)& 1\\ ⋮\\ s_r\left(k_{SEI2}\right)& 1\end{array}\right]}}\underset{{\mathbf{x}}_{\mathrm{SEI}}}{\underbrace{\left[\begin{array}{c}c\\ d\end{array}\right]}}+\underset{\mathbf{n}}{\underbrace{\left[\begin{array}{c}{n}_v\left(1\right)\\ n_v\left(2\right)\\ ⋮\\ n_v\left(n\right)\end{array}\right]}}\hfill \end{array}$$

(22)

Using LS algorithm, the estimate of ${\widehat{\mathbf{x}}}_{\mathrm{SEI}}$ will be given by

$$\begin{array}{c}\hfill {\widehat{\mathbf{x}}}_{\mathrm{SEI}}={\left({\mathbf{B}}^T\mathbf{B}\right)}^{-1}\left({\mathbf{B}}^T\mathbf{z}\right)\end{array}$$

(23)

The estimates of c and d are:

$$\widehat{c}={\widehat{\mathbf{x}}}_{\mathrm{SEI}}\left(1\right),\widehat{d}={\widehat{\mathbf{x}}}_{\mathrm{SEI}}\left(2\right)$$

(24)

From Figure 2, ${\mathrm{R}}_{\mathrm{SEI}}$ is the diameter of the SEI arc; thus, by substituting the values of c and d in (20), the estimate of $R_{\mathrm{SEI}}$ is

$$\begin{array}{c}\hfill {\widehat{R}}_{\mathrm{SEI}}=2{\widehat{r}}_s=2\sqrt{{\displaystyle \frac{{\widehat{c}}^2}{4}}-\widehat{d}}\end{array}$$

(25)

The estimated centre of the semicircle can then be expressed as

$$({\widehat{x}}_s,0)=(-{\displaystyle \frac{\widehat{c}}{2}},0)$$

(26)

The fitting accuracy of the SEI arc can be evaluated as [23]

$$error=\sqrt{{\displaystyle \frac{{\sum }_{k=k_{SEI1}}^{k_{SEI2}}{d_k}^2}{k_{SEI2}-k_{SEI1}+1}}}$$

(27)

Where $d_k$ is the geometrical distance between the actual EIS data point and predicted EIS data point, which is defined as

$$d_k=\sqrt{{\left[s_r\left(k\right)-{\widehat{x}}_s\right]}^2+{\left[s_i\left(k\right)-0\right]}^2}-{\widehat{r}}_s$$

(28)

It can be shown in (1) that when the frequency is very high, the impedance in CT arc and Diffusion arc will be minimal so that it is negligible; thus, we assume the $Z_{CT\&DF}$ term will be zero, that is

$$\begin{array}{c}\hfill Z=Z_{\mathrm{RL}}+Z_{\mathrm{SEI}}+0\end{array}$$

(29)

Therefore, the impedance in SEI arc can be expressed as:

$$\begin{array}{cc}\hfill Z_{\mathrm{SEI}}& =Z-Z_{\mathrm{RL}}\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\displaystyle \frac{R_{\mathrm{SEI}}}{1+j\omega R_{\mathrm{SEI}}{C}_{\mathrm{SEI}}}}& =Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill 1+j\omega R_{\mathrm{SEI}}{C}_{\mathrm{SEI}}& ={\displaystyle \frac{R_{\mathrm{SEI}}}{Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}}}\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill j\omega R_{\mathrm{SEI}}{C}_{\mathrm{SEI}}& ={\displaystyle \frac{R_{\mathrm{SEI}}}{Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}}}-1\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill j\omega C_{\mathrm{SEI}}& ={\displaystyle \frac{1}{Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}}}-{\displaystyle \frac{1}{R_{\mathrm{SEI}}}}\hfill \end{array}$$

(34)

Take the imaginary part on both sides of the above equation,

$$\begin{array}{cc}\hfill C_{\mathrm{SEI}}& =\left({\displaystyle \frac{1}{\omega }}\right)\mathrm{Im}\left({\displaystyle \frac{1}{Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}}}-{\displaystyle \frac{1}{R_{\mathrm{SEI}}}}\right)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill C_{\mathrm{SEI}}& =\left({\displaystyle \frac{1}{\omega }}\right)\mathrm{Im}\left({\displaystyle \frac{1}{Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}}}\right)\hfill \end{array}$$

(36)

Substituting the expression for $R_{\mathsf{\Omega }}$ and L from (3) and (4), respectively, in (36) at $\omega ={\omega }_k$$(k_{SEI1}\le k\le k_{SEI2})$:

$$\begin{array}{cc}\hfill {\tilde{C}}_{\mathrm{SEI}}\left(k\right)& =\left({\displaystyle \frac{1}{{\omega }_k}}\right)\mathrm{Im}\left({\displaystyle \frac{1}{Z\left({\omega }_k\right)-j{\omega }_k\widehat{L}-{\widehat{R}}_{\mathsf{\Omega }}}}\right)\hfill \end{array}$$

(37)

Finally, average all the estimates ${\tilde{C}}_{\mathrm{SEI}}\left(k\right)$ to obtain the final estimate

$$\begin{array}{c}\hfill {\widehat{C}}_{\mathrm{SEI}}={\displaystyle \frac{1}{k_{SEI2}-k_{SEI1}+1}}\sum _{k=k_{SEI1}}^{k_{SEI2}}{\tilde{C}}_{\mathrm{SEI}}\left(k\right)\end{array}$$

(38)

Using the LS approach to identify $R_{SEI}$ and $C_{SEI}$ via the automatic selection of feature points are fully described in Algorithm 3. In addition, Figure 5a shows the RMSE of the fitted SEI arc in each iteration and Figure 5b shows the SEI arc, which is selected since it can reach the best fit. Algorithm 3 uses the following MATLAB commands: ’floor’, ’length’.

Algorithm 3 Estimate $R_{SEI}$ and $C_{SEI}$ via automatic feature detection.

3.5. Estimation of $R_{\mathrm{CT}}$ and $C_{\mathrm{DL}}$

It can be observed in Figure 2 that to fit the CT arc using a semicircle precisely, we need to select feature points that lie between $k_{CT1}$ to $k_{CT2}$; therefore, the impedance measurements in the CT arc can be denoted as:

$$\begin{array}{cc}\hfill & c_r\left(k\right)\triangleq z_r\left(k\right)\mathrm{s}.\mathrm{t}.k_{CT1}\le k\le k_{CT2}\hfill \\ \hfill & c_i\left(k\right)\triangleq z_i\left(k\right)\mathrm{s}.\mathrm{t}.k_{CT1}\le k\le k_{CT2}\hfill \end{array}$$

(39)

Assuming that the centre of the semicircle lies on the real axis, which is noted as ($x_c$, 0), the radius of the semicircle can be noted as $r_c$; thus, the measurements in (39) should satisfy the equation of the semicircle [20]:

$$\begin{array}{c}\hfill {\left(c_r\left(k\right)-x_c\right)}^2+{\left(c_i\left(k\right)-0\right)}^2={r_c}^2\end{array}$$

(40)

$$\begin{array}{c}\hfill {c_r\left(k\right)}^2-2x_c{c}_r\left(k\right)+x_c^2+{c_i\left(k\right)}^2={r_c}^2\end{array}$$

(41)

Let $g=-2x_c$ and $h=x_c^2-{r_c}^2$, thus

$$\begin{array}{c}\hfill {r_c}^2={\displaystyle \frac{g^2}{4}}-h\end{array}$$

(42)

$$\begin{array}{c}\hfill r_c=\sqrt{{\displaystyle \frac{g^2}{4}}-h}\end{array}$$

(43)

And (41) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {c_r\left(k\right)}^2+{c_i\left(k\right)}^2+g{c}_r\left(k\right)+h=0\end{array}\end{array}$$

(44)

In the matrix form, (44) can be written as

$$\begin{array}{cc}\hfill & \underset{\mathbf{p}}{\underbrace{\left[\begin{array}{c}-\left(c_r{\left(k_{CT1}\right)}^2+c_i{\left(k_{CT1}\right)}^2\right)\\ -\left(c_r{(k_{CT1}+1)}^2+c_i{(k_{CT1}+1)}^2\right)\\ -\left(c_r{(k_{CT1}+2)}^2+c_i{(k_{CT1}+2)}^2\right)\\ ⋮\\ -\left(c_r{\left(k_{CT2}\right)}^2+c_i{\left(k_{CT2}\right)}^2\right)\end{array}\right]}}\hfill \\ \hfill & =\underset{\mathbf{C}}{\underbrace{\left[\begin{array}{cc}{c}_r\left(k_{CT1}\right)& 1\\ c_r(k_{CT1}+1)& 1\\ c_r(k_{CT1}+2)& 1\\ ⋮\\ c_r\left(k_{CT2}\right)& 1\end{array}\right]}}\underset{{\mathbf{x}}_{\mathrm{CT}}}{\underbrace{\left[\begin{array}{c}g\\ h\end{array}\right]}}+\underset{\mathbf{n}}{\underbrace{\left[\begin{array}{c}{n}_v\left(1\right)\\ n_v\left(2\right)\\ ⋮\\ n_v\left(n\right)\end{array}\right]}}\hfill \end{array}$$

(45)

From (45), ${\mathbf{x}}_{\mathrm{CT}}$ can be estimated using LS algorithm

$${\widehat{\mathbf{x}}}_{\mathrm{CT}}={\left({\mathbf{C}}^T\mathbf{C}\right)}^{-1}\left({\mathbf{C}}^T\mathbf{p}\right)$$

(46)

Thus, the estimates of a and b are:

$$\widehat{g}={\widehat{\mathbf{x}}}_{\mathrm{CT}}\left(1\right),\widehat{h}={\widehat{\mathbf{x}}}_{\mathrm{CT}}\left(2\right)$$

(47)

As shown in Figure 2, $R_{\mathrm{CT}}$ is the diameter of the CT arc; thus, by substituting the values of a and b in (43), the estimate of $R_{\mathrm{CT}}$ is

$$\begin{array}{c}\hfill {\widehat{R}}_{\mathrm{CT}}=2{\widehat{r}}_c=2\sqrt{{\displaystyle \frac{{\widehat{g}}^2}{4}}-\widehat{h}}\end{array}$$

(48)

The estimated centre of the semicircle can then be expressed as

$$({\widehat{x}}_c,0)=(-{\displaystyle \frac{\widehat{g}}{2}},0)$$

(49)

The fitting accuracy of the CT arc can be evaluated as [23]

$$error=\sqrt{{\displaystyle \frac{{\sum }_{k=k_{CT1}}^{k_{CT2}}{d_k}^2}{k_{CT2}-k_{CT1}+1}}}$$

(50)

Where $d_k$ is the geometrical distance between the actual EIS data point and predicted EIS data point, which is defined as [23]

$$d_k=\sqrt{{\left[c_r\left(k\right)-{\widehat{x}}_c\right]}^2+{\left[c_i\left(k\right)-0\right]}^2}-{\widehat{r}}_c$$

(51)

Based on (1),

$$\begin{array}{c}\hfill Z=Z_{\mathrm{RL}}+Z_{\mathrm{SEI}}+Z_{\mathrm{CT}\&\mathrm{DF}}\end{array}$$

(52)

Therefore, the impedance in CT arc and DF arc can be expressed as:

$$\begin{array}{cc}\hfill Z_{\mathrm{CT}\&\mathrm{DF}}& =Z-Z_{\mathrm{RL}}-Z_{\mathrm{SEI}}\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{R_{\mathrm{CT}}+Z_{\mathrm{w}}\left(j\omega \right)}{1+j\omega \left(R_{\mathrm{CT}}+Z_{\mathrm{w}}\left(j\omega \right)\right)C_{\mathrm{DL}}}}\hfill \\ & =Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}-{\displaystyle \frac{R_{\mathrm{SEI}}}{1+j\omega R_{\mathrm{SEI}}{C}_{\mathrm{SEI}}}}\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill & j\omega \left(R_{\mathrm{CT}}+Z_{\mathrm{w}}\left(j\omega \right)\right)C_{\mathrm{DL}}\hfill \\ \hfill & ={\displaystyle \frac{R_{\mathrm{CT}}+Z_{\mathrm{w}}\left(j\omega \right)}{Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}-\frac{R_{\mathrm{SEI}}}{1+j\omega R_{\mathrm{SEI}}{C}_{\mathrm{SEI}}}}}-1\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill j\omega C_{\mathrm{DL}}& ={\displaystyle \frac{1}{Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}-\frac{R_{\mathrm{SEI}}}{1+j\omega R_{\mathrm{SEI}}{C}_{\mathrm{SEI}}}}}\hfill \\ \hfill & -{\displaystyle \frac{1}{R_{\mathrm{CT}}+Z_{\mathrm{w}}\left(j\omega \right)}}\hfill \end{array}$$

(56)

Take the imaginary part on both sides of (56), and substitute $Z_w\left(jw\right)$ with the expression given in (13), we obtain

$$\begin{array}{cc}\hfill & C_{\mathrm{DL}}\hfill \\ \hfill & =\left({\displaystyle \frac{1}{\omega }}\right)\mathrm{Im}\left({\displaystyle \frac{1}{Z\left(\omega \right)-j\omega L-R_{\mathsf{\Omega }}-\frac{R_{\mathrm{SEI}}}{1+j\omega R_{\mathrm{SEI}}{C}_{\mathrm{SEI}}}}}\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{R_{\mathrm{CT}}+(1-jm)\frac{\sigma }{\sqrt{\omega }}}}\right)\hfill \end{array}$$

(57)

Substituting L, $R_{\mathsf{\Omega }}$, $R_{\mathrm{SEI}}$, $C_{\mathrm{SEI}}$, $R_{\mathrm{CT}}$ and $\sigma$ with the estimations given in (4), (3), (25), (38), (48) and (15) respectively, in the above equation at $\omega ={\omega }_k$$(k_{CT1}\le k\le k_{CT2})$:

$$\begin{array}{cc}\hfill & {\tilde{C}}_{\mathrm{DL}}\left(k\right)\hfill \\ \hfill & =\left({\displaystyle \frac{1}{{\omega }_k}}\right)\mathrm{Im}\left({\displaystyle \frac{1}{Z\left({\omega }_k\right)-j{\omega }_k\widehat{L}-{\widehat{R}}_{\mathsf{\Omega }}-\frac{{\widehat{R}}_{\mathrm{SEI}}}{1+j{\omega }_k{\widehat{R}}_{\mathrm{SEI}}{\widehat{C}}_{\mathrm{SEI}}}}}\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{{\widehat{R}}_{\mathrm{CT}}+(1-j\widehat{m})\frac{\widehat{\sigma }}{\sqrt{{\omega }_k}}}}\right)\hfill \end{array}$$

(58)

Finally, average all the estimates ${\tilde{C}}_{\mathrm{DL}}\left(k\right)$ to obtain the final estimate

$$\begin{array}{c}\hfill {\widehat{C}}_{\mathrm{DL}}={\displaystyle \frac{1}{k_{CT2}-k_{CT1}+1}}\sum _{k=k_{CT1}}^{k_{CT2}}{\tilde{C}}_{\mathrm{DL}}\left(k\right)\end{array}$$

(59)

Using the LS approach to identify $R_{CT}$ and $C_{DL}$ via the automatic selection of feature points is shown in Algorithm 4. Furthermore, Figure 6a shows the RMSE of the fitted CT arc in each iteration, and Figure 6b shows the CT arc selected since it can reach the best fit. Algorithm 4 uses the following MATLAB commands: ’floor’ and ’round’.

Algorithm 4 Estimate $R_{CT}$ and $C_{DL}$ via automatic feature detection.

4. Exhaustive Search Approach

4.2. Specify the Range of Parameters for Exhaustive Search

As presented in Section 3, the rough estimations of ECM parameters are given; therefore, we can assign the lower and upper bound for each parameter such that the exhaustive search approach can identify the most suitable parameter between the boundary. As shown in Algorithm 5, the range of each ECM parameter is assigned based on the empirical coefficient. In this paper, the size of possible values in each ECM parameter is restricted to 20 such that the computational time is within the acceptable range. Algorithm 5 uses the MATLAB command ’linspace’.

Algorithm 5 Set the range of ECM parameters estimated by LS approach.

4.3. Implement Exhaustive Search

Algorithm 6 describes the ES approach that can be applied to precisely identify the ECM parameters; where, the input ${\widehat{R}}_{\mathsf{\Omega }}$ is estimated via (3), and $\widehat{L}$ is estimated via (4). Figure 9 shows the MAE evaluated from the initial iteration throughout the end of the exhaustive search process; by finding the lowest MAE in this figure, one can identify the best estimation of ECM parameters via the ES approach.

Algorithm 6 Exhaustive Search Approach.

5. Nonlinear Least Squares Approach

5.3. Algorithm Switch

To reach the best fit, the NLS approach will apply ’trust-region-reflective’ algorithm [24,25] and ’levenberg-margquardt’ algorithm [26,27,28] to fit the EIS data with different initial guess, after that, the algorithm that can achieve the lowest mean absolute error (MAE) will be select as the approach that can reach the best fit. The detailed approach is shown in Algorithm 7. This algorithm uses the following MATLAB commands: ’abs’ and ’randn’.

Algorithm 7 NLS Approach.

6. Implementation

6.1. Simulate EIS Data

The simulated EIS data was generated using Algorithm 8, where the frequency ranges from 0.01 Hz to 10 kHz, and the number of EIS measurements is 121; this is to keep the conformity with the sampling size we set in the EIS experiment [20]. In addition, Table 1 shows the true ECM parameters for simulating EIS data. To generate various noise levels for the simulated EIS measurements, we implement Gaussian noise with zero-mean, standard deviation ${\sigma }_{noise}$, and independent outcome.

Algorithm 8 uses the MATLAB command ’linspace’, ’real’, ’imag’, and ’randn’.

Algorithm 8 EIS simulation.

7. Results

8. Conclusions and Discussions

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