Comparative Study of Online Open Circuit Voltage Estimation Techniques for State of Charge Estimation of Lithium-Ion Batteries

Abstract

Online estimation techniques are extensively used to determine the parameters of various uncertain dynamic systems. In this paper, online estimation of the open-circuit voltage ($OCV$) of lithium-ion batteries is proposed by two different adaptive filtering methods (i.e., recursive least square, $RLS$, and least mean square, $LMS$), along with an adaptive observer. The proposed techniques use the battery’s terminal voltage and current to estimate the $OCV$, which is correlated to the state of charge ($SOC$). Experimental results highlight the effectiveness of the proposed methods in online estimation at different charge/discharge conditions and temperatures. The comparative study illustrates the advantages and limitations of each online estimation method.

1. Introduction

Lithium-ion batteries have a higher energy and power density with respect to other chemistries like nickel cadmium (NiCad), nickel metal hydride (NiMH), and lead–acid [1,2]. Additionally, lithium-ion batteries have numerous advantages, such as compact size, low weight, high capacity, rapid charge capability, long cycle life, wide temperature operation range, low rate of self-discharge, no outgassing of hydrogen, and no memory effects [3]. These batteries have been widely used in real-time applications such as consumer electronics, automotive, and power tools. For these applications, the estimation of the state of charge ($SOC$) plays a vital role in their performance, since an inaccurate $SOC$ estimation would damage the battery and consequently reduce its lifetime and performance.

Traditional $SOC$ estimation techniques are used because of their simplicity. A basic real-time $SOC$ estimation is coulomb counting (also called ampere-hour counting method), which is an open-loop algorithm that uses the battery’s entering and leaving currents and integrates them through time. This method has several flaws, such as the accumulation of current sensor errors and difficulty in determining the initial value of $SOC$ [4]. Despite its flaws, it is preferred in real-time applications where high accuracy is not a requirement. The other method to estimate $SOC$ uses open-circuit voltage ($OCV$), which is related to the charge status of the battery [5]. However, this statement is true only when the battery is in steady-state. Therefore, a hybrid estimation technique combines the coulomb counting and $OCV$ methods. However, some applications need continuous operation and do not allow the battery to reach steady-state, which increases the need for online $SOC$ estimation techniques.

Numerous advanced estimation techniques are proposed at the expense of higher computation due to complex design. A simple battery model is implemented in [6] with a sliding mode observer to compensate for modeling uncertainties. In [7], a reduced observer technique is proposed to estimate $SOC$. However, this estimation technique requires the knowledge of the battery’s parameters, which leads to reduced accuracy with aging. This drawback has been overcome with the adaptive $SOC$ estimation strategy in [8]. Particle filter (PF) is a sequential Monte Carlo method which uses the samples of random weights (particles) for the estimation probability distribution function of nonlinear systems [8]. The Kalman filter (KF) has been widely used for the estimation of $OCV$ and other battery parameters which have a direct relationship with the $SOC$ [9]; this filter is a recursive algorithm that estimates the internal dynamic states of a system. To consider the nonlinear behavior of $OCV$, an extended Kalman filter (EKF) and unscented Kalman filter (UKF) are proposed in [10]. The fundamental principle of EKF is to linearize the nonlinear functions by using an expansion of a first-order Taylor series. Certain statistical assumptions and local linearization of state equations are the drawbacks of this well-known filter [11]. In [12], a different definition of $SOC$ is proposed for a pack of batteries connected in series by finding out the voltage at the battery’s terminals. Then, $SOC$ is estimated using EKF, which yields nearly half of the error compared to the two time constant (TTC) method. In [13], a comparison between EKF and square root unscented Kalman filter (SR-UKF) shows better performance of SR-UKF with respect to EKF. Another $SOC$ estimation technique with adaptive extended Kalman filter (AEKF) and wavelet transform matrix (WTM) is proposed in [14] to avoid the electromagnetic noise created in the measurement of voltage and current in electric vehicles (EVs). In addition, a fractional order Kalman Filter approach is introduced in [15] to estimate $SOC$ based on fractional order model. An $H_{\infty }$ observer is applied to estimate the $SOC$ of a battery depending on the equivalent circuit of the linear state space model used as an inspection robot on a power transmission line [16].

Moreover, fuzzy logic and neural networks are widely used for robust approximation of systems subjected to uncertainties [17]. Various strategies are applied to estimate the $SOC$ and yield acceptable results [18,19]. Although neural networks provide satisfactory performance, they fail to incorporate any human expertise already acquired about the dynamics of the system at hand. This shortcoming has been overcome by adopting different models of fuzzy neural networks in [20,21], but at the cost of higher computation.

This paper presents a comparative study of different online $OCV$ estimation techniques for lithium-ion batteries. Here, two adaptive filtering methods— namely, least mean square ($LMS$) and recursive least square ($RLS$)—are designed and implemented. Moreover, an adaptive observer is also implemented for comparison purposes. It is important to note that all compared estimation strategies do not require any prior knowledge of the battery’s parameters. Numerous comparative studies have been presented for $SOC$ estimation [22,23,24,25,26,27,28,29,30,31]. While few studies investigate the impact of different battery circuit models on the $SOC$ estimation [22,23,24], others focus on the comparison of different estimation algorithms [25,26,27,28,29,30,31]. Among these, comparison of different kinds of Kalman filters, such as UKF, EKF, and AEKF, is popular [22,26,28]. However, the aforementioned comparative studies either use constant charge/discharge currents or do not consider the impact of temperature on the estimation. Temperature variations are known to introduce a drift in the battery’s parameters, which reduces the estimation accuracy. Unlike these studies, the proposed comparative analysis considers both temperature variations and highly nonlinear time-varying charge/discharge current profiles for validation. Design and implementation details of all three estimation algorithms are provided. Experimental results show the accuracy and convergence properties of all estimation methods. The rest of the paper is organized as follows. Section 2 depicts the lithium-ion battery circuit model along with its system dynamics. The proposed strategy of online parameter estimation is detailed in Section 3. Experimental results are reported and discussed in Section 4. A conclusion is drawn based on these results, and future studies in this field are suggested.

2. Modeling of Lithium-Ion Batteries

Similar to other types of batteries, the lithium-ion battery has four main components: positive and negative electrodes, electrolyte, and a separator. Its electric circuit model is shown in Figure 1. The voltage–current characteristic is modeled by an resistance-capacitance ($RC$) network, with $R_b$ being the internal battery’s resistance. On the other side, the $OCV$–$SOC$ characteristic is represented by a self-discharging resistance, a battery storage capacitor, and a current-controlled current source. To bridge these two networks, a voltage-controlled voltage source is used [32].

The voltage–current mathematical model can be represented by the following state equations:

$$\begin{array}{c}{\dot{V}}_p=\frac{1}{RC}{V}_p-\frac{1}{C}{I}_b\hfill \end{array}$$

(1)

$$\begin{array}{c}{V}_b=V_{oc}+V_p+R_b{I}_b\hfill \end{array}$$

(2)

where $V_b$ is the voltage at battery terminals, $I_b$ is the current at battery terminals, $V_{oc}$ is the open circuit voltage, $R_b$ is the internal resistance, R is the equivalent resistance, C is the equivalent capacitance, and $V_p$ is the voltage across the $RC$ network.

In this paper, we aim to estimate the $OCV$ (i.e., $V_{oc}$), as it is directly related to the battery’s $SOC$ by considering the parameters $R,C$, and $R_b$ as unknown. The battery system’s measurable parameters are the battery’s terminal voltage $V_b$ and current $I_b$. The current $I_b$ is assumed as positive in charge mode and negative in discharge. The battery’s terminal voltage $V_b$, current $I_b$, and their derivatives $\dot{V_b},\dot{I_b}$ are taken to be bounded and continuous in nature. The estimation algorithm sampling frequency is to be high enough such that the variation of the battery’s $OCV$ between two samples is negligible (i.e., ${\dot{V}}_{oc}\approx 0$). Finally, the $V_b$ and $I_b$ are continuously excited.

3. Online Parameter Estimation

In this section, $LMS$, $RLS$, and adaptive observer algorithms are designed to estimate the online battery’s $OCV$. Rearranging Equation (2) leads to:

$$V_p=V_b-V_{oc}-R_b{I}_b$$

Substituting $V_p$ in (1) and considering the aforementioned assumptions gives:

$$\dot{V_b}-\frac{1}{RC}{V}_b-R_b\dot{I_b}+\frac{R_b}{RC}{I}_b+\frac{1}{C}{I}_b+\frac{1}{RC}{V}_{oc}=0$$

(3)

Multiplying with $RC$ yields:

$$V_b=RC\dot{V_b}-R_{b}RC\dot{I_b}+(R+R_b)I_b+V_{oc}$$

(4)

Therefore, the battery’s dynamics (4) can be expressed using the following regression model:

$$RC{\dot{V}}_b-R_{b}RC{\dot{I}}_b+(R+R_b)I_b+V_{oc}={\mathsf{\Psi }}^{T}W$$

(5)

where $\mathsf{\Psi }\in {\mathbb{R}}^4=[\dot{V_b}\dot{I_b}{I}_{b}1]$ is a vector of known functions (regressor), and $W\in {\mathbb{R}}^4$ is a vector of parameters:

$$\begin{array}{cc}\hfill W_1& =RC\hfill \\ \hfill W_2& =-R_{b}RC\hfill \\ \hfill W_3& =R+R_b\hfill \\ \hfill W_4& =V_{oc}\hfill \end{array}$$

(6)

Hence, precise estimation of parameter $W_4$ eventually leads to an accurate $SOC$ estimation. Thus, the output of an adaptive filter can be expressed as:

$${\widehat{V}}_b={\mathsf{\Psi }}^T\widehat{W}$$

(7)

where $\widehat{W}$ is the filter’s parameter estimate vector. Therefore, the estimation error can be defined as:

$$e_b=V_b-{\widehat{V}}_b$$

(8)

To achieve accurate $OCV$ estimation, three different algorithms (i.e., $LMS$, $RLS$, and adaptive observer) are designed to drive the estimation error $e_b$ to zero. $LMS$ algorithm is a widely used algorithm in adaptive filters, and is known for a low computational complexity. In addition, $RLS$ algorithm is another adaptive filter where the coefficients are calculated recursively, which minimizes the weighted linear least squares cost function related to input signals. Unlike the other algorithms, $RLS$ input signals are considered as deterministic, whereas in $LMS$ they are considered as stochastic. In contrast to the other algorithms, $RLS$ converges extremely quickly, but at the cost of higher computational complexity. Similar to $RLS$, adaptive observers offer fast convergence with the simplicity of $LMS$ [8]. Moreover, The adaptive observer estimator’s stability is guaranteed by Lyapunov’s direct method, unlike the aforementioned methods. Next, the proposed methods for the online $OCV$ estimation approach are explained.

3.1. Least Mean Square Filter

The $LMS$ algorithm emerged as a simple yet effective method for the operation of adaptive finite impulse response (FIR) filters. $LMS$-based algorithms are model-independent because no statistical knowledge about the system in hand is needed in deriving them. Rather, it is a stochastic gradient algorithm that uses simple computational terms to iterate the weights in the direction of the gradient of the squared magnitude of the error signal, as follows:

$$\Delta \widehat{W}(k)=\mathsf{\mu }\mathsf{\Psi }(k)e_b(k)$$

(9)

where k is the discrete-time index and $\mathsf{\mu }$ is a step-size or adaptation constant rate. Here, the step-size $\mathsf{\mu }$ influences the filter’s coefficients or weights since a large value would lead to high fluctuations in filter weights estimation. On the other hand, if the chosen $\mathsf{\mu }$ is too small, time taken for convergence to optimal weights would be too long. Therefore, optimal selection of $\mathsf{\mu }$ is necessary, and a series of tests are usually performed to find a reasonable step-size. However, it is better to have a slower convergence rate and a more accurate output than fluctuating output because of high step-size. Algorithm 1 shows the step-by-step process used by the $LMS$ filter.

Algorithm 1 $LMS$ filter.

3.2. Recursive Least Square Filter

The $RLS$ algorithm uses the inverse correlation matrix of the input data, leading to a higher performance at the expense of an increase in computational complexity with respect to $LMS$. $RLS$-based algorithms are model-dependent, since their derivatives assume the use of a multivariate Gaussian model. The $RLS$ filter recursively computes the filter coefficients update by minimizing a weighted least-squared cost function as follows:

$$\Delta \widehat{W}(k)=G(k)e_b(k)$$

(10)

where $G(k)$ is the gain vector, which is updated as:

$$G(k+1)=\frac{P(k)\mathsf{\Psi }(k+1)}{\mathsf{\lambda }+{\mathsf{\Psi }}^T(k+1)P(k)\mathsf{\Psi }(k+1)}$$

(11)

where $P(k)$ is the inverse correlation matrix of the input vector. The standard $RLS$ algorithm uses the following equation to update the inverse correlation matrix:

$$P(k+1)=\frac{1}{\mathsf{\lambda }}\left(P(k)-\frac{P(k)\mathsf{\Psi }(k+1)\mathsf{\Psi }{(k+1)}^{T}P(k)}{\mathsf{\lambda }+{\mathsf{\Psi }}^T(k+1)P(k)\mathsf{\Psi }(k+1)}\right)$$

(12)

Here, $\mathsf{\lambda }$ is the forgetting factor required as an exponential factor to give less weight to past errors. If $\mathsf{\lambda }=0$, the algorithm has no memory. Conversely, the algorithm has an infinite memory with $\mathsf{\lambda }=1$. Therefore, this factor is usually a constant that lies between 0 and 1 in conventional $RLS$ algorithms. Algorithm 2 shows the step-by-step process used by the $RLS$ filter.

Algorithm 2 $RLS$ filter.

3.3. Adaptive Observer

Similar to adaptive filters, adaptive observers can also track parameters online as they vary over time. In here, the design and implementation details of an adaptive observer are laid out and explained [8]. The battery’s voltage estimation law is defined as:

$${\widehat{V}}_b=\widehat{R}\widehat{C}{\dot{V}}_r-{\widehat{R}}_b\widehat{R}\widehat{C}{\dot{I}}_b+(\widehat{R}+{\widehat{R}}_b)I_b+{\widehat{V}}_{oc}$$

(13)

with:

$${\dot{V}}_r={\widehat{\dot{V}}}_b-K_p{e}_b-K_i\int e_b$$

(14)

where $K_p$ and $K_i$ are the proportional and integral gains, respectively. Substituting (14) into (13) gives:

$${\widehat{V}}_b=\widehat{R}\widehat{C}{\widehat{\dot{V}}}_b-\widehat{R}\widehat{C}{K}_p{e}_b-\widehat{R}\widehat{C}{K}_i\int e_b-{\widehat{R}}_b\widehat{R}\widehat{C}{\dot{I}}_b+(\widehat{R}+{\widehat{R}}_b)I_b+{\widehat{V}}_{oc}$$

(15)

Adding and subtracting $\widehat{R}\widehat{C}{\dot{V}}_b$ on the right side of the above equation gives:

$${\widehat{V}}_b=-\widehat{R}\widehat{C}{\dot{e}}_b-\widehat{R}\widehat{C}{K}_p{e}_b-\widehat{R}\widehat{C}{K}_i\int e_b-{\widehat{R}}_b\widehat{R}\widehat{C}{\dot{I}}_b+(\widehat{R}+{\widehat{R}}_b)I_b+\widehat{R}\widehat{c}{\dot{V}}_b+{\widehat{V}}_{oc}$$

(16)

Subtracting (5) from (16) and using linear regression gives:

$${\dot{e}}_b+(K_p+\widehat{\mathsf{\beta }})e_b+K_i\int e_b=\widehat{\mathsf{\beta }}{\mathsf{\Psi }}^T\tilde{W}$$

(17)

where $\tilde{W}=W-\widehat{W}$ and $\mathsf{\beta }=(1/RC)$. So, the estimation law (13) gives the closed-loop dynamics as follows:

$${\dot{e}}_b+(K_p+\widehat{\mathsf{\beta }})e_b+K_i\int e_b=0$$

(18)

The state-space form of the above equation can be written as:

$$\dot{X}=AX+BU$$

(19)

where $X\in {\mathbb{R}}^2={[\int e_b{e}_b]}^T$ is the state vector and $U\in \mathbb{R}=\widehat{\mathsf{\beta }}{\mathsf{\Psi }}^T\tilde{W}$ is the state-space input. $A\in {\mathbb{R}}^{2x2}$ and $B\in {\mathbb{R}}^2$ are given by:

$$A=\left[\begin{array}{cc}0& 1\\ -K_i& -(K_p+\widehat{\mathsf{\beta }})\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right]$$

Hence, the estimator’s gains $K_p$ and $K_i$ can be chosen to place the closed loop poles at desired places using a pole placement technique. For the given nonlinear system, the adaptive estimator’s stability and error’s convergence to zero can be guaranteed with the following adaptation law:

$$\widehat{\dot{W}}=-\mathrm{\Gamma }\mathsf{\Psi }{B}^{T}PX$$

(20)

where $\mathrm{\Gamma }=[{\mathsf{\gamma }}_1,{\mathsf{\gamma }}_2]$ and ${\mathsf{\gamma }}_i$ is a positive constant gain and P is a symmetric positive definite matrix which is chosen to satisfy the Lyapunov equation:

$$A^{T}P+PA=-Q$$

(21)

where Q is a positive definite matrix. Algorithm 3 shows the step-by-step process of the adaptive observer.

Algorithm 3 Adaptive Observer [8].

4. Experimental Results

4.1. Setup

To evaluate the performance of the proposed techniques, a 15 Ah lithium-ion battery was subjected to predefined charge/discharge current profiles at various temperatures (0 ${}^{\circ }$C, 10 ${}^{\circ }$C, 25 ${}^{\circ }$C, and 40 ${}^{\circ }$C). For each charge/discharge cycle, the battery’s terminal voltage and current were measured as is illustrated in Figure 2. The collected experimental data at different temperatures was used with a sampling time of 1 s to validate the effectiveness of the adaptive filters. The estimated voltage of each technique is compared against the measured terminal voltage. For a better comparison, the relative $OCV$ estimation error of between both $LMS$ and $RLS$ algorithms is calculated with the following relation:

$$\mathrm{Relative}\mathrm{error}=V_{oc}(LMS)-V_{oc}(RLS)$$

In addition, the estimated battery’s voltage and $OCV$ obtained with the adaptive observer presented in [5] are shown as a benchmark.

4.2. Results

The lithium-ion battery was charged using the battery’s current profile shown in Figure 3. Experimental results under this condition are depicted in Figure 4, Figure 5 and Figure 6. As it is shown in Figure 4, both adaptive filters demonstrated a good ability in providing smooth $OCV$ estimation at transient and steady-state conditions for different temperatures (0 ${}^{\circ }$C, 10 ${}^{\circ }$C, 25 ${}^{\circ }$C, and 40 ${}^{\circ }$C). As expected, the $RLS$ algorithm showed a faster convergence. The battery’s voltage estimation (shown in Figure 5) displayed good tracking for all methods. On the other hand, spikes were observed in the battery’s voltage due to discontinuous charging current profile (Figure 3). Despite these nonlinearities, the adaptive approaches were able to provide smooth $OCV$ estimation (Figure 4). The relative $OCV$ estimation error depicted in Figure 6 shows that although $LMS$ had a slower convergence compared to $RLS$, it was able to gradually drive its relative estimation error to $RLS$ to zero. Moreover, the initial error for all temperatures was kept in an acceptable range ($\pm 0.05$ V). It is noteworthy that even if the estimation of parameters requires persistent excitation in several adaptive systems, the fact that $OCV$ can be estimated at equilibrium state makes the estimator independent of this requirement.

Then, the battery was discharged using the current profile shown in Figure 7. Experimental results for discharge mode are illustrated in Figure 8, Figure 9 and Figure 10. In this case, the deviation of the $OCV$ estimates from the expected value was larger, which resulted in higher errors at startup and few oscillations. This was expected, since the other estimated parameters were not yet converged to the expected values. As illustrated in Figure 8, the $OCV$ gradually converged to its desired values for all temperatures (0 ${}^{\circ }$C, 10 ${}^{\circ }$C, 25 ${}^{\circ }$C, and 40 ${}^{\circ }$ C). Again, $RLS$ showed a faster convergence. In addition, the abrupt change in current again caused spikes in voltage estimation, as shown in Figure 9. The relative error for discharge mode is depicted in Figure 10. Again, smooth error convergence to zero was observed in the presence of current nonlinearities.

5. Conclusions

In this paper, online $OCV$ estimation was achieved for lithium-ion batteries at different temperatures. The proposed strategy makes use of three different algorithms to provide smooth $OCV$ estimation using the battery’s terminal voltage and current. Experimental results highlight the proposed estimator’s performance. Both adaptive filters show high performance in estimating the battery’s $OCV$ for charge/discharge modes at various temperatures. $RLS$ showed a faster convergence, which confirms the credentials for this technique in providing better transient performance at the expense of a higher computation. However, the $LMS$ algorithm gradually eliminates the gap, which yields the same estimation as its $RLS$ counterpart. Experimental results for an adaptive observer are presented as a benchmark. All methods demonstrate different estimation capabilities at various operating conditions.