# Second-Order Discrete-Time Sliding Mode Observer for State of Charge Determination Based on a Dynamic Resistance Li-Ion Battery Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Battery Modeling

_{oc}(Z) represents the relationship between the open circuit voltage (OCV) and the SOC, as shown in Figure 2. The OCV-SOC curve is obtained from the conditional discharge test data [16]. The nominal capacitance C

_{n}characterizes the capacity of the battery to store charge. The polarization capacitance C

_{p}describes the polarization effect of the battery. In addition, the terminal resistor R

_{t}, the diffusion resistor R

_{p}, and the propagation resistor R

_{b}represent the resistive components in the battery.

_{t}, the SOC Z, and the polarization voltage V

_{p}:

## 3. Second-Order DSMO for SOC Estimation

^{n}is the state variable vector; u ∈ ℜ is the input; y ∈ ℜ is the system output; and Δ represents the uncertainties caused by parameter variations and non-linearities. A, B, and H are the system nominal matrices, and (A, H) is assumed to be observable. Then, the second-order DSMO of the system [Equation (3)] has the form:

_{oc}is constant value α

_{k}over a sampling interval [k, k + 1]. This assumption is valid because the rate of change of Z over the interval is negligible, and errors caused by this assumption can be regarded as part of the uncertainty. Taking V

_{oc}(Z(t)) = α

_{k}, t ∈ [k, k + 1], Equation (1) is rewritten as follows:

_{i}(k), which is an element of the uncertainty vector Λ(k), is bounded by the known bound σ

_{i}:

_{i}is determined experimentally by comparing the modeling output with the actual battery output.

**Remark 1:**We note that the SOC at the sampling instant k + 1, i.e., Z

_{k+1}, is updated according to Equation (7), and the value of the voltage source is also reset as V

_{oc}(Z(t)) = V

_{oc}(Z

_{k+1}) = α

_{k+1}, t ∈ [k + 1, k + 2].

## 4. Experimental Results

- improvement of the battery modeling accuracy with the dynamic resistance varied with the operating conditions;
- the SOC estimation method using the second-order DSMO for the elimination of chattering.

#### 4.1. Parameter Extraction

_{p}is represented by its associated resistance:

_{int}is obtained using the direct current-internal resistance (DC-IR) method:

_{dis}represents the discharge current. Generally, it is assumed that R

_{b}and R

_{p}are taken to be 75% of R

_{int}and R

_{t}is equivalent to 25% of R

_{int}[10].

#### 4.2. Dynamic Resistance

_{int}curves for each discharge current. These results are obtained from 10 independent trials. As expected, the internal resistance has a higher value at higher discharge currents and at both ends of the SOC level. Therefore, allowing the resistance to vary with the discharge current and the SOC level is a reasonable way of more accurately representing the battery dynamics.

_{n}= 1.491 × 10

^{4}F, C

_{p}= 17.6735 F, R

_{b}= 0.165 Ω, R

_{p}= 0.165 Ω, and R

_{t}= 0.055 Ω. On the other hand, the resistance is updated using the SOC versus R

_{int}curves according to the operating conditions in the dynamic resistance model. Here, the capacitance is assumed to be constant, because its variation is negligibly small and the sampling period T is 0.01 s. The results show that the accuracy of both models is similar at 100% SOC. However, at 50% SOC, the dynamic resistance model shows better accuracy than the constant resistance model. Similarly, in other SOC levels, modeling errors are caused by differences between actual and constant resistance value. It represents that the proposed battery model is more suitable for the modeling of Li-ion batteries.

#### 4.3. Random Current Discharge Test

**Remark 2:**To compare the second-order DSMO with the first-order DSMO under the same conditions, the gain M is applied equally to each one and the sign function sign(·) is used for second-order DSMO [Equation (9)] instead of the saturation function sat(·).

**Figure 9.**One-cycle estimation results for the proposed method: (

**a**) output voltage estimation; and (

**b**) estimation errors.

## 5. Conclusions

_{int}curves for each discharge current can be obtained in advance through a simple experiment. Our experimental results show that the proposed battery model has a better modeling accuracy than the constant resistance model. In addition, elimination of chattering in both the SOC and output voltage estimates is achieved using the second-order DSMO.

## Appendix: Stability Analysis

**Case 1**: suppose that $\left|{\tilde{V}}_{t}\left(k\right)\right|\ge \varphi $.In this case, we have:$$\begin{array}{c}\hfill \Lambda \left(k\right)-\Lambda (k-1)-Msat\left(\frac{{\tilde{V}}_{t}\left(k\right)}{\varphi}\right)=-|M\pm (\Lambda \left(k\right)-\Lambda (k-1))|sign\left({\tilde{V}}_{t}\left(k\right)\right)\end{array}$$$$\begin{array}{c}\hfill -F\left(k\right)|{\tilde{V}}_{t}\left(k\right)|=\Lambda \left(k\right)-\Lambda (k-1)-Msat\left(\frac{{\tilde{V}}_{t}\left(k\right)}{\varphi}\right)\end{array}$$$$\begin{array}{c}\hfill 0\le F\left(k\right)\triangleq \frac{|M\pm (\Lambda \left(k\right)-\Lambda (k-1))|sign\left(\tilde{y}\left(k\right)\right)}{\tilde{y}\left(k\right)}\le \frac{\sigma +M}{\varphi}\end{array}$$$$\begin{array}{c}\hfill \tilde{x}(k+1)-(A-LH+I+FH)\tilde{x}\left(k\right)+(A-LH)\tilde{x}(k-1)=0\end{array}$$$$\begin{array}{c}\hfill Z(k+1)=\mathbf{A}Z\left(k\right)\end{array}$$$$\begin{array}{c}Z\left(k\right)=\left[\begin{array}{c}\tilde{x}(k-1)\\ \tilde{x}\left(k\right)\end{array}\right],\phantom{\rule{3.33333pt}{0ex}}\mathbf{A}=\left[\begin{array}{cc}0& I\\ -(A-LH)& A-LH+I+FH\end{array}\right]\end{array}$$**A**are smaller than one. Therefore, the convergence of the estimation error is also guaranteed.**Case 2**: $\left|{\tilde{V}}_{t}\left(k\right)\right|<\varphi $.In this case, Equation (18) can be expressed as:$$\begin{array}{c}\hfill \tilde{x}(k+1)-\left(A-LH+I-\frac{MH}{\varphi}\right)\tilde{x}\left(k\right)+(A-LH)\tilde{x}(k-1)-\Lambda \left(k\right)+\Lambda (k-1)=0\end{array}$$$$\begin{array}{c}\hfill {Z}^{\prime}(k+1)={\mathbf{A}}^{\prime}{Z}^{\prime}\left(k\right)+\left[\begin{array}{c}0\\ \Lambda \left(k\right)-\Lambda (k-1)\end{array}\right]\end{array}$$$$\begin{array}{c}{Z}^{\prime}\left(k\right)=\left[\begin{array}{c}\tilde{x}(k-1)\\ \tilde{x}\left(k\right)\end{array}\right],\phantom{\rule{3.33333pt}{0ex}}{\mathbf{A}}^{\prime}=\left[\begin{array}{cc}0& I\\ -(A-LH)& A-LH+I+FH\end{array}\right]\end{array}$$**A**’ are smaller than one, Z(k)′ is bound. Also, the estimation error $\tilde{x}\left(k\right)$ is bounded.

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kim, D.; Koo, K.; Jeong, J.J.; Goh, T.; Kim, S.W. Second-Order Discrete-Time Sliding Mode Observer for State of Charge Determination Based on a Dynamic Resistance Li-Ion Battery Model. *Energies* **2013**, *6*, 5538-5551.
https://doi.org/10.3390/en6105538

**AMA Style**

Kim D, Koo K, Jeong JJ, Goh T, Kim SW. Second-Order Discrete-Time Sliding Mode Observer for State of Charge Determination Based on a Dynamic Resistance Li-Ion Battery Model. *Energies*. 2013; 6(10):5538-5551.
https://doi.org/10.3390/en6105538

**Chicago/Turabian Style**

Kim, Daehyun, Keunhwi Koo, Jae Jin Jeong, Taedong Goh, and Sang Woo Kim. 2013. "Second-Order Discrete-Time Sliding Mode Observer for State of Charge Determination Based on a Dynamic Resistance Li-Ion Battery Model" *Energies* 6, no. 10: 5538-5551.
https://doi.org/10.3390/en6105538