State of Charge Estimation of Lithium-Ion Batteries Using an Adaptive Cubature Kalman Filter

Abstract

Accurate state of charge (SOC) estimation is of great significance for a lithium-ion battery to ensure its safe operation and to prevent it from over-charging or over-discharging. However, it is difficult to get an accurate value of SOC since it is an inner sate of a battery cell, which cannot be directly measured. This paper presents an Adaptive Cubature Kalman filter (ACKF)-based SOC estimation algorithm for lithium-ion batteries in electric vehicles. Firstly, the lithium-ion battery is modeled using the second-order resistor-capacitor (RC) equivalent circuit and parameters of the battery model are determined by the forgetting factor least-squares method. Then, the Adaptive Cubature Kalman filter for battery SOC estimation is introduced and the estimated process is presented. Finally, two typical driving cycles, including the Dynamic Stress Test (DST) and New European Driving Cycle (NEDC) are applied to evaluate the performance of the proposed method by comparing with the traditional extended Kalman filter (EKF) and cubature Kalman filter (CKF) algorithms. Experimental results show that the ACKF algorithm has better performance in terms of SOC estimation accuracy, convergence to different initial SOC errors and robustness against voltage measurement noise as compared with the traditional EKF and CKF algorithms.

1. Introduction

As energy prices soar and environment pollution increases, electric vehicles (EVs) have become greatly considered in the past few years. Lithium-ion battery is currently considered to be the development trend of traction battery and has been widely used in EVs because of its high power and energy density, high voltage, pollution-free, no memory effect, long cycle life and low self-discharge. A battery management system (BMS) is essential for the lithium-ion battery to maximize its performance, ensure its safety and extend its life. Estimation for battery state of charge (SOC) is one of the most key techniques in the BMS, since SOC indicates the remaining capacity in the battery, which is helpful to dispel the diver’s range anxiety, predict the battery’s peak power capability and improve the battery’s safety by preventing it from possible over-charging or over-discharging. Nevertheless, it is difficult to accurately estimate SOC, because SOC is an inner state of each battery cell that cannot be directly measured and is greatly influenced by many factors, including ambient temperature, discharging current and battery aging. Therefore, the battery SOC has to be estimated with specific estimation techniques according to measured battery parameters, such as voltage, current and temperature.

Different approaches have been proposed to predict the battery SOC with the development of EVs. The existing SOC estimation algorithms can be divided into two categories, namely non-model-based type and model-based type. The former is typically based on Ampere-hour (Ah) or Coulomb counting [1,2], open-circuit voltage (OCV) [3,4,5], electrochemical impedance spectroscopy (EIS) [6,7], artificial neural networks (ANNs) [8,9,10,11] and fuzzy-logic (FL) [12,13]. The Ah counting method acquires the SOC by integrating the current over the time. This method is simple and can be easily implemented on-board, therefore it has been widely used in practice. However, as an open-loop estimation algorithm, Ah counting cannot deal with problems caused by measurement noise and inaccurate initial SOC. The OCV-based method obtains the SOC based on an OCV vs. SOC relationship. Unfortunately, this method is inappropriate for online applications since the battery has to be left in open circuit mode for a long time to reach the steady-state before measuring the OCV. Similarly, the EIS-based method estimates the SOC according to internal impedance of the battery and it is only suitable for offline applications. The ANNs- and FL-based methods predict the SOC according to the nonlinear relationship between the battery SOC and its influencing factors obtained by the trained black-box battery models. They do not require detailed knowledge of the battery systems so they can be applied to any battery type. Besides, these methods have excellent performance if the training data is sufficient to cover the total loading conditions. Nevertheless, it is time-consuming and nearly impossible to collect training data covering all of the battery loading conditions.

In order to improve the accuracy of SOC estimation, battery model-based and closed-loop estimation methods have been further developed. Despite requiring a higher computational cost than the Ah method, these methods have merits in terms of self-correcting, online computing and availability of the dynamic SOC estimation error range. Thus, they are increasingly popular and more suitable for real-time application than the other types of SOC estimation methods. A famous and widely used method is the Kalman filter (KF)-based [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] algorithm, which was originally developed to estimate state for linear systems. However, the lithium-ion battery is a strong nonlinear and time-variable system. Thus, modified KF algorithms have to be used in order to extend the application of KF in the nonlinear battery systems. Two commonly used types are extended Kalman filter (EKF) [15,16,17,18,19,20,21,22,23,24,25,26,27,28] and unscented Kalman filter (UKF) [29,30,31,32,33,34,35,36]. The EKF transforms a nonlinear system into a linear system by linearizing the nonlinear function on the basis of the first-order Taylor series expansion. However, the linearizing process inevitably causes large linearization error. Besides complicated computation, the Jacobian matrix may lead to the instability of the filter and inaccurate estimation for highly nonlinear battery systems. In addition, the case will be severer when more complicated OCV-SOC relationships are used and it increases the computational burden of the hardware. Unlike the EKF, the UKF introduces an unscented transformation to approximate the state distribution through a set of sample points, called sigma points, which capture the mean and covariance of the state distribution. It has been demonstrated that UKF has better performance than EKF in terms of accuracy and robustness [37,38]. Furthermore, UKF does not need to calculate the Jacobian matrix online. Unfortunately, EKF and UKF both suffer from divergence or the curse of dimensionality or both [39].

In 2009, a new nonlinear filter, called cubature Kalman filter (CKF), for high-dimensional state estimation was proposed by Arasaratnam and Haykin [39]. Based on the radial-spherical cubature rule, the CKF uses a set of 2n points, where n represents the state-vector dimension, to capture the mean and covariance of the states of a nonlinear system with additive Gaussian noise. It is considered to be more efficient and stable than the UKF [39,40,41]. The CKF has been successfully applied in many fields, such as mobile-station locating [42], moving-target tracking [43] and spacecraft attitude estimation [44].

This paper focuses on the application of the CKF in battery SOC estimation. An adaptive rule for the updating of the process and measurement noise covariance is presented to improve the algorithm performance. Two typical driving cycles, including the Dynamic Stress Test (DST) and New European Driving Cycle (NEDC) are applied to assess the performance of the proposed method by comparing with the standard CKF method. The assessment includes estimation accuracy and robustness against measurement noise.

The remainder of this paper is organized as follows. In Section 2, the experimental setup for data acquiring and processing is described. Section 3 derives the battery’s state–space equations based on the second-order RC equivalent circuit model, and determines parameters of the battery model using the least-square method. In Section 4, the principle of standard cubature Kalman filter and the adaptive rule for the updating of the process and measurement covariances are introduced in detail. In addition, the implementation of adaptive CKF-based SOC estimation algorithm is presented. The experimental results are discussed in Section 5, and Section 6 makes conclusions of the paper.

2. Experimental Setup

The schematic diagram of the battery test bench is shown in Figure 1. It consists of tested lithium-ion battery cells, a programmable power supply for cell charging (ITECH IT6952A, ITECH Electronics, Nanjing, China), a programmable electric load for cell discharging (ITECH IT8510, ITECH Electronics, Nanjing, China), a control board for data acquisition, and a host computer for monitoring and processing experimental data. The IT6952A power supply can charge the battery cell with a maximum current of 25 A at a maximum voltage of 60 V, while the IT8510 electric load is able to provide the maximum discharge current of 20 A with the maximum voltage of 120 V. The voltage setup accuracy of the IT6952A power supply is within 0.03% + 5 mV, while the error of current sensor of the IT8510 electric load is within 0.1%. Battery voltage and current are measured with a sampling rate of 10 Hz and transmitted to the host computer every second through RS485 ports. MATLAB R2010a software (MathWorks, Natick, MA, USA) installed in the host computer is used for data processing, such as battery parameters determination and SOC prediction. It is well known that the performance of lithium-ion battery is highly related with the cathode materials, such as LiFePO₄, LiCoO₂, LiMn₂O₄ and LiNi_xCo_yMn_zO₂, where x + y + z = 1. Compared with other materials, layered transition-metal oxide LiNi_xCo_yMn_zO₂ has the merits of high energy density, excellent consistency, mild thermal stability, low cost, and low toxicity [45]. Thus, the Samsung LiNi_xCo_yMn_zO₂ ICR18650-22F (Samsung SDI, Seoul, Korea) battery is used in the test. The ICR18650-22F lithium-ion battery has a nominal voltage of 3.6 V and a nominal capacity of 2.2 Ah.

Figure 1. Configuration of the battery test bench.

Figure 1. Configuration of the battery test bench.

3. Battery Modeling and Parameters Identification

3.1. Battery Equivalent Circuit Model

The Kalman filter was developed based on state-space equations of the system and its accuracy is highly dependent on the accuracy of the system model. Thus, a battery model has to be constructed to estimate the SOC using Kalman filter-based approach. There are two basic requirements on a battery model for SOC estimation. Firstly, it can well simulate the dynamic behaviors of the battery. Secondly, the state-space equations can be easily derived according to the model. A commonly used model that well meets the above two requirements is the equivalent circuit model (ECM) with lumped parameters [46]. The most common ECM is comprised of resistor and parallel resistor-capacitor (RC) network(s) connected in series [47]. Although adding more RC networks is a benefit for improving the model accuracy, it leads to the increasing of computation complexity [48]. Accordingly, a tradeoff has to be made between the model accuracy and the computational complexity. Herein, the second-order RC equivalent circuit model is selected to meet the requirement of tradeoff. As shown in Figure 2, the second-order RC battery model consists of an open-circuit voltage U_oc (SOC), a resistor R_o, and two parallel RC networks connected in series. The resistor R_o indicates the ohmic resistance caused by the accumulation and dissipation of charge in the electrical double-layer, R_p1 and C_p1 are the activation polarization resistance and capacitance, respectively, while R_p2 and C_p2 separately are the concentration polarization resistance and capacitance.

Figure 2. Schematic diagram of the battery equivalent circuit model.

Figure 2. Schematic diagram of the battery equivalent circuit model.

3.2. State–Space Equations

The differential equations of the second-order RC equivalent circuit model shown in Figure 2 can be derived as:

$$\{\begin{array}{l}S\dot{O}C=-\frac{1}{Q_n}{I}_t+w_1\\ {\dot{U}}_{p1}=-\frac{1}{R_{p1}{C}_{p1}}{U}_{p1}+\frac{1}{C_{p1}}{I}_t+w_2\\ {\dot{U}}_{p2}=-\frac{1}{R_{p2}{C}_{p2}}{U}_{p2}+\frac{1}{C_{p2}}{I}_t+w_3\end{array}$$

(1)

$$U_t=U_{oc}(SOC)-U_{p1}-U_{p2}-R_o{I}_b+v$$

(2)

where Q_n is the battery nominal capacity; U_p1 and U_p2 are the terminal voltage of C_p1 and C_p2, respectively; U_t and I_t are the battery terminal voltage and current, respectively; U_oc represents the open circuit voltage (OCV), which is varied with change of SOC value; w₁, w₂ and w₃ are the process noise for SOC, U_p1 and U_p2, respectively; and v represents the measurement noise.

By selecting x = [SOC, U_p1, U_p2]^T as the state vector, and considering the current I_t and voltage U_t as the input and output variables respectively, the discrete-time state equations of the second-order RC battery model can be obtained as:

$$x_k=f(x_{k-1},u_k)+w_k$$

(3)

$$y_k=h(x_k,u_k)+v_k$$

(4)

where x_k represents the immeasurable state vector at time step k; u_k (=I_t,k) stands for the input vector; y_k (=U_t,k) is the observed output; v_k (=[v_1,k v_2,k v_3,k]^T) and w_k are separately the process and measurement noises, which are both uncorrelated zero-mean Gaussian white sequences; f (∙) and h (∙) indicate the process and measurement functions, respectively. Generally, f (∙) is linear while h (∙) is nonlinear due to the nonlinear relationship between the OCV and SOC, which will be illustrated in Section 3.3.

3.3. Parameters Identification with Forgetting Factor Least-Squares Algorithm

As shown in Figure 2, parameters needed to be determined include the OCV-SOC equation, R_o, R_p1, C_p1, R_p2 and C_p2. In order to acquire the data used to determine the OCV-SOC relationship, a sequence of discharging experiments were carried out. The measured data and fitted curves using the sixth-order polynomial, shown in Equation (5) [49], are shown in Figure 3. It is clear that the fitted equation can well simulate the nonlinear relationship between OCV and SOC.

$$\begin{array}{c}OCV=14.7958\times SO{C}^6-36.6148\times SO{C}^5+29.2355\times SO{C}^4-\\ 6.2817\times SO{C}^3-1.6476\times SO{C}^2+1.2866\times SOC+3.4049\end{array}$$

(5)

Figure 3. Measured and fitted OCV (open circuit voltage) vs. SOC (state of charge).

Figure 3. Measured and fitted OCV (open circuit voltage) vs. SOC (state of charge).

Other parameters consisting of {R_o, R_p1, C_p1, R_p2, C_p2} can be determined using the least-square method with the following steps:

(i) Calculation of transfer function

By selecting x = I_t as the input and y = U_oc − U_t as the output, the transfer function of the battery model shown in Figure 1 can be obtianed as:

$$G(s)=\frac{Y(s)}{X(s)}=R_o+\frac{R_{p1}}{R_{p1}{C}_{p1}s}+\frac{R_{p2}}{R_{p2}{C}_{p2}s}$$

(6)

Assuming τ₁=R_p1C_p1 and τ₂=R_p2C_p2 yields

$$G(s)=\frac{R_o{s}^2+\frac{R_o({\tau }_1+{\tau }_2)+R_{p1}{\tau }_2+R_{p2}{\tau }_1}{{\tau }_1{\tau }_2}s+\frac{R_o+R_{p1}+R_{p2}}{{\tau }_1{\tau }_2}}{s^2+\frac{{\tau }_1+{\tau }_2}{{\tau }_1{\tau }_2}s+\frac{1}{{\tau }_1{\tau }_2}}$$

(7)

(ii) Discretization

Using the bilinear transform rule ($s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$, where T is the sample time) [50], the transfer function in Equation (7) can be discretized as:

$$G(z^{-1})=\frac{b_0+b_1{z}^{-1}+b_2{z}^{-2}}{1+a_1{z}^{-1}+a_2{z}^{-2}}$$

(8)

where b₀ b₁, b₂, a₁ and a₂ are undetermined coefficients.

Then, the time−domain difference equation of Equation can be expressed as:

$$y(k)=-a_{1}y(k-1)-a_{2}y(k-2)+b_{0}x(k)+b_{1}x(k-1)+b_{2}x(k-2)$$

(9)

(iii) Resolving

According to the forgetting factor least−squares algorithm, the parameter vector of the battery model can be resolved as follows:

$$\text{θ}={({\text{Ψ}}^T\text{Ψ})}^{-1}\text{Ψ}Y$$

(10)

where θ = [a₁, a₂, b₀, b₁, b₂]; Y = [λⁿ⁻³y(3), λⁿ⁻⁴y(4), …, y(n)]^T is the output sequence, where n is the number of sample data and λ (0<λ<1) is the forgetting factor [51]; Ψ = [Ψ₁, Ψ₂, …, Ψ_n−2] is the regress matrix in which Ψ_k = λ^n−2−k[−y(k+2), −y(k+1), x(k+2), x(k+1), x(k)]^T.

(iv) Parameters calculation

Using the inverse bilinear transform rule ($z^{-1}=\frac{2/T-s}{2/T+s}$) [50], the discrete transfer function in Equation (8) can be transformed as:

$$G(s)=\frac{\frac{b_0-b_1+b_2}{1-a_1+a_2}{s}^2+\frac{4(b_0-b_2)}{T(1-a_1+a_2)}s+\frac{4(b_0+b_1+b_2)}{T^2(1-a_1+a_2)}}{s^2+\frac{4(1-a_2)}{T(1-a_1+a_2)}s+\frac{4(1+a_1+a_2)}{T^2(1-a_1+a_2)}}$$

(11)

By comparing Equations (7) and (11), it can be derived that:

$$R_o=\frac{b_0-b_1+b_2}{1-a_1+a_2}$$

(12)

$${\tau }_1{\tau }_2=\frac{T^2(1-a_1+a_2)}{4(1+a_1+a_2)}$$

(13)

$${\tau }_1+{\tau }_2=\frac{T(1-a_2)}{1+a_1+a_2}$$

(14)

$$R_o+R_{p1}+R_{p2}=\frac{b_0+b_1+b_2}{1+a_1+a_2}$$

(15)

$$R_o({\tau }_1+{\tau }_2)+R_{p1}{\tau }_2+R_{p2}{\tau }_1=\frac{T(b_0-b_2)}{1+a_1+a_2}$$

(16)

By combining of Equations (12)–(16), τ₁ = R_p1C_p1 and τ₂ = R_p2C_p2, parameters {R_o, R_p1, C_p1, R_p2, C_p2} of the battery model can be determined. Herein, sample data collected from DST cycles are applied to identify the model parameters. The sample data are shown in Figure 4 and the identified parameters are shown in Table 1.

Table 1. Identified parameters of the battery model.

Parameters	Ro	Rp1	Cp1	Rp2	Cp2
Values	0.0380 Ω	0.0268 Ω	1125 F	0.0129 Ω	20701 F

Table 1. Identified parameters of the battery model.

Parameters	Ro	Rp1	Cp1	Rp2	Cp2
Values	0.0380 Ω	0.0268 Ω	1125 F	0.0129 Ω	20701 F

Figure 4. Sample data under DST (Dynamic Stress Test) cycles: (a) current; (b) voltage; (c) SOC.

Figure 4. Sample data under DST (Dynamic Stress Test) cycles: (a) current; (b) voltage; (c) SOC.

3.4. Model Validation

In order to assess the accuracy of the parameter identification, the measured and estimated battery voltages are compared in Figure 5. A locally enlarged portion of the first three driving cycles is shown in Figure 6a to show more details, and the corresponding voltage error is shown in Figure 6b. It can be seen that the maximum and mean relative errors are about 1.918% and 0.206%, respectively. Therefore, the battery model can well simulate the dynamic voltage behaviors of the battery. It is worth mentioning that online parameter estimation is helpful to improve the model accuracy since the fact that battery’s parameters are related to factors, such as the ambient temperature, operating current and cycling times. However, it is beyond the scope of this paper.

Figure 5. Measured and estimated voltage under DST cycles.

Figure 5. Measured and estimated voltage under DST cycles.

Figure 6. Voltage profiles in the first three DST cycles: (a) voltage; (b) voltage error.

Figure 6. Voltage profiles in the first three DST cycles: (a) voltage; (b) voltage error.

4. Adaptive Cubature Kalman Filter for SOC Estimation

The cubature Kalman filter (CKF) was first proposed by Arasaratnam and Haykin in 2009 [39]. It is based on the third-degree spherical-radial cubature rule and uses a set of points to approximate the mean and covariance of the states of a nonlinear system with additive Gaussian noise. The CKF is considered to be more accurate and stable in state estimation than the UKF. A comparison of the efficiency and complexity of the CKF-, EKF-, UKF- and particle filter (PF)-based SOC estimation algorithms for lithium-ion battery has been investigated in [52]. It is concluded that the CKF-based method performs better than both the UKF- and the EKF-based methods. Although the PF-based method has slightly better estimation accuracy compared to the CKF-based method, it is computationally more complex.

In standard CKF, process noise covariance and measurement noise covariance both are considered to be constant. Nevertheless, it is not the case for a practical battery system in electric vehicles due to random disturbance caused by sensor drift and parameter uncertainties caused by time-varying behaviors of the lithium-ion battery. Therefore, adaptively updating rules for covariance values of the process and measurement noise are required in order to improve performance of the algorithm. In this paper, the idea of covariance matching based on the residual sequence of battery model output voltage proposed in [31,53] is introduced to the CKF, and a residual-based Adaptive Cubature Kalman filter (ACKF) algorithm is developed accordingly. The process of the ACKF algorithm for battery SOC estimation on the basis of battery model is summarized as follows:

(i)

Initialization

• Initial posteriori error covariance: P₀;
• Initial process noise covariance: Q₀;
• Initial measurement noise covariance: R₀;
• Window size for covariance matching: L_w;
• Initial mean ${\overline{x}}_0$and covariance P₀ with a random state vector x₀ as follows

  $${\overline{x}}_0=E[x_0]$$

  (17)

  $$P_0=E[(x_0-{\overline{x}}_0){(x_0-{\overline{x}}_0)}^T]$$

  (18)

(ii)

Time update

• Factorize the error covariance

  $$S_{k-1}=chol(P_{k-1})$$

  (19)

  where chol(∙) represents a Cholesky decomposition of a matrix returning a lower triangular Cholesky factor. That’s to say:

  $$P_{k-1}=S_{k-1}{S}_{k-1}^T$$

  (20)
• Calculate the cubature points

  $$x_{k-1}^{(i)}=S_{k-1}{\xi }^{(i)}+{\widehat{x}}_{k-1}i=1,2,\cdots ,2n$$

  (21)

  where n is the number of state variables and ξ is the set of standard cubature points, which is given by

  $${\xi }^{(i)}=\{\begin{array}{l}\begin{array}{cc}\sqrt{n}{[1]}^{(i)}& i=1,2,\cdots n\end{array}\\ \begin{array}{cc}-\sqrt{n}{[1]}^{(i)}& i=n+1,n+2,\cdots 2n\end{array}\end{array}$$

  (22)

  where [1] represents the identity matrix and [1]⁽ⁱ⁾ denotes its i-th column vector. For example, assuming n = 3 yields:

  $$[1]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

  (23)
• Propagate the cubature points and calculate the predicted state

  $${\chi }_{k|k-1}^{(i)}=f(x_{k-1}^{(i)},u_{k-1})$$

  (24)

  $${\overline{x}}_{k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{\chi }_{k|k-1}^{(i)}}$$

  (25)
• Calculate the propagated covariance

  $$P_{k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}({\chi }_{k|k-1}^{(i)}-{\overline{x}}_{k|k-1}}){({\chi }_{k|k-1}^{(i)}-{\overline{x}}_{k|k-1})}^T+Q_{k-1}$$

  (26)

  where Q_k-1 is the process noise covariance matrix at time step k−1.

(iii)

Measurement update

• Factorize the error covariance

  $$S_{k|k-1}=chol(P_{k|k-1})$$

  (27)
• Recalculate the cubature points

  $$x_{k|k-1}^{(i)}=S_{k|k-1}{\xi }^{(i)}+{\widehat{x}}_{k|k-1}i=1,2,\cdots ,2n$$

  (28)
• Propagate the cubature points and calculate the predicted measurement

  $$y_{k|k-1}^{(i)}=h(x_{k|k-1}^{(i)},u_k)$$

  (29)

  $${\overline{y}}_{k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{y}_{k|k-1}^{(i)}}$$

  (30)
• Calculate the estimated covariance

  $$P_{k|k-1}^y=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}(y_{k|k-1}^{(i)}}-{\overline{y}}_{k|k-1}){(y_{k|k-1}^{(i)}-{\overline{y}}_{k|k-1})}^T+R_{k-1}$$

  (31)

  $$P_{k|k-1}^{xy}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}(x_{k|k-1}^i}-{\overline{x}}_{k|k-1}){(y_{k|k-1}^i-{\overline{y}}_{k|k-1})}^T$$

  (32)

  where R_k-1 is the measurement noise covariance matrix at time step k-1.
• Calculate the Kalman gain

  $$K_k=P_{k|k-1}^{xy}{(P_{k|k-1}^y)}^{-1}$$

  (33)
• Update the predicted state

  $${\widehat{x}}_k={\overline{x}}_{k|k-1}+K_k(y_k-{\overline{y}}_{k|k-1})$$

  (34)

  where y_k is the measured output at time step k.
• Update the error covariance

  $$P_k=P_{k|k-1}-K_k{P}_{k|k-1}^y{K}_k^T$$

  (35)

(iv)

Adjustment of Q_k and R_k

In this step, the process noise covariance Q and measurement noise covariance R are adaptively estimated according to the output voltage residual sequence of the battery model. Thus, Q and R can be iteratively updated as:

$$Q_k=K_k{F}_k{K}_k^T$$

(36)

$$R_k=F_k+{\displaystyle \sum _{i=0}^{2n}{W}_n^{(i)}\left(y_{k|k-1}^{(i)}-y_k\right){\left(y_{k|k-1}^{(i)}-y_k\right)}^T}$$

(37)

where F_k is an approximation to the covariance of the voltage residual at time step k and is defined as:

$$F_k={\displaystyle \sum _{i=k-L_w+1}^k{e}_i{e}_i^T}$$

(38)

where e_i is the voltage residual of the battery model at time step i, and L_w is window size for covariance matching.

Figure 7. Schematic of the ACKF (adaptive cubature Kalman filter)-based SOC estimation algorithm.

Figure 7. Schematic of the ACKF (adaptive cubature Kalman filter)-based SOC estimation algorithm.

The schematic diagram of the proposed ACKF-based SOC estimation algorithm is shown in Figure 7. After initialization, the estimated state vector ${\widehat{x}}_k$ and the estimation error covariance P_k can be firstly achieved according to the prediction and update processes. The voltage residual error e_k is computed on the basis of the measurement equation of the battery model. Afterwards, the process noise covariance Q_k and the measurement noise covariance R_k are obtained though the voltage residual-based updating law. Then, ${\widehat{x}}_k$, P_k, Q_k and R_k are used for the next prediction and update processes. The battery SOC can be recursively estimated by repeating the above procedure.

5. Results and Discussion

5.1. Estimation Results without Measurement Noise

In this section, experimental data collected from two typical driving cycles, including the 360 s Dynamic Stress Test (DST) cycle and the 1184 s New European Driving Cycle (NEDC) are applied to evaluate the performance of SOC estimation algorithms under the typical loading conditions of EVs. The profiles under the DST cycles are shown in Figure 4 in Section 3.3, and that under the NEDC cycles are illustrated in Figure 8.

Figure 8. Current and SOC under NEDC (New European Driving Cycle) test: (a) current; (b) SOC.

Figure 8. Current and SOC under NEDC (New European Driving Cycle) test: (a) current; (b) SOC.

The proposed method was compared with the widely used EKF and the standard CKF algorithms in terms of estimation accuracy, convergence rate and computational complexity. The comparison results, including root mean square error (RMSE), convergence rate to 5% SOC error from different initial SOCs and execution time are summarized in Table 2. As an example, the SOC estimation results at 80% initial SOC under DST and NEDC cycles are separately shown in Figure 9 and Figure 10, where the black solid-line presents the reference SOC computed using Coulomb counting method with accurate current values, the green chain-line represents the estimated value using the EKF algorithm, the blue dotted-line indicates the estimated value using the traditional CKF algorithm, while the red dashed-line describes the estimated value using the proposed adaptive CKF (ACKF) algorithm. It can be found that although the ACKF takes more computational cost, it can improve both the estimation accuracy and convergence rate in comparison with the EKF and CKF algorithms. For example, the RMSE is reduced from 1.2% to 0.6% under DST cycles with accurate initial SOC compared with CKF, and from 1.2% to 0.5% under NEDC cycles. The average convergence rate can be improved about 20% compared with EKF, and about 45% compared with CKF. In addition, the proposed ACKF can reduce the fluctuation of SOC estimation compared with both the EKF and CKF methods.

Figure 9. SOC estimation results under DST test: (a) SOC; (b) SOC error; (c) zoom figure for (b).

Figure 9. SOC estimation results under DST test: (a) SOC; (b) SOC error; (c) zoom figure for (b).

Table 2. Comparison of SOC estimation without measurement noise.

Methods	Initial SOC	Execution time	DST			NEDC
			Maximum error	RMSE	Convergence rate	Maximum error	RMSE	Convergence rate
EKF	100%	0.76 s	4.0%	0.8%	1 step	4.3%	0.7%	1 step
	80%			1.3%	108 step		1.2%	105 step
	70%			1.8%	205 step		1.7%	203 step
	60%			2.3%	310 step		2.2%	270 step
CKF	100%	1.36 s	3.8%	1.2%	1 step	3.8%	1.2%	1 step
	80%			1.6%	160 step		1.6%	155 step
	70%			2.0%	350 step		2.0%	300 step
	60%			2.4%	405 step		2.4%	390 step
ACKF	100%	1.89 s	3.8%	0.6%	1 step	3.8%	0.5%	1 step
	80%			1.2%	88 step		1.1%	90 step
	70%			1.6%	160 step		1.5%	155 step
	60%			2.1%	255 step		2.0%	250 step

Table 2. Comparison of SOC estimation without measurement noise.

Methods	Initial SOC	Execution time	DST			NEDC
			Maximum error	RMSE	Convergence rate	Maximum error	RMSE	Convergence rate
EKF	100%	0.76 s	4.0%	0.8%	1 step	4.3%	0.7%	1 step
	80%			1.3%	108 step		1.2%	105 step
	70%			1.8%	205 step		1.7%	203 step
	60%			2.3%	310 step		2.2%	270 step
CKF	100%	1.36 s	3.8%	1.2%	1 step	3.8%	1.2%	1 step
	80%			1.6%	160 step		1.6%	155 step
	70%			2.0%	350 step		2.0%	300 step
	60%			2.4%	405 step		2.4%	390 step
ACKF	100%	1.89 s	3.8%	0.6%	1 step	3.8%	0.5%	1 step
	80%			1.2%	88 step		1.1%	90 step
	70%			1.6%	160 step		1.5%	155 step
	60%			2.1%	255 step		2.0%	250 step

Figure 10. SOC estimation results under NEDC test: (a) SOC; (b) SOC error; (c) zoom figure for (b).

Figure 10. SOC estimation results under NEDC test: (a) SOC; (b) SOC error; (c) zoom figure for (b).

5.2. Estimation Results with Measurement Noise

In practice, it is difficult to always get accurate measurement values from an online battery system due to noise caused by factors, such as the electromagnetic interference (EMI) generated by electronic equipment on EVs and temperature drift of sensors. To further evaluate the robustness of the proposed ACKF algorithm against measurement noise, a sequence of voltage error shown in Figure 11 is added to the measured voltage. The total voltage error profile vs. time is shown in Figure 11a, and a locally enlarged portion is shown Figure 11b to illustrate more details. The SOC estimation results with voltage measurement noise under DST and NEDC cycles are shown in Figure 12 and Figure 13, respectively. In both Figure 12 and Figure 13, the black solid-line presents the reference SOC values, the green chain-line represents the estimated SOC using the EKF algorithm, the blue dotted-line indicates the estimated SOC using the traditional CKF algorithm, and the red dashed-line describes the estimated SOC using the ACKF algorithm. The corresponding maximum error and RMSE of SOC estimation are summarized in Table 3.

With all the results shown in Table 3 and Figure 11, Figure 12 and Figure 13, it is clear that SOC estimation errors using the EKF approach are highly increased due to the introduction of the voltage noise, while those using CKF and ACKF are slightly increased. For example, the maximum error using EKF increases from 4.0% to 8.7%, and the RMSE increases from 0.8% to 3.6% under DST cycles. However, it is not the case for the CKF and ACKF algorithms. As for the ACKF, the maximum error is slightly increased, while the RMSE is even slightly reduced due to the adaptively updating of the voltage covariance. It is accordingly demonstrated that the proposed ACKF algorithm is greatly robust to measurement noise compared with the EKF and CKF algorithms.

Figure 11. Voltage noise: (a) voltage noise vs. time; (b) zoom figure for (a).

Figure 11. Voltage noise: (a) voltage noise vs. time; (b) zoom figure for (a).

Table 3. Comparison of SOC estimation with measurement noise.

Methods	DST		NEDC
	Maximum error	RMSE	Maximum error	RMSE
EKF	8.7%	3.6%	9.1%	3.7%
CKF	4.8%	1.5%	4.8%	1.5%
ACKF	4.3%	0.5%	4.3%	0.4%

Table 3. Comparison of SOC estimation with measurement noise.

Methods	DST		NEDC
	Maximum error	RMSE	Maximum error	RMSE
EKF	8.7%	3.6%	9.1%	3.7%
CKF	4.8%	1.5%	4.8%	1.5%
ACKF	4.3%	0.5%	4.3%	0.4%

Figure 12. SOC estimation results with voltage noise under DST test: (a) SOC; (b) SOC error.

Figure 12. SOC estimation results with voltage noise under DST test: (a) SOC; (b) SOC error.

Figure 13. SOC estimation results with voltage noise under NEDC test: (a) SOC; (b) SOC error.

Figure 13. SOC estimation results with voltage noise under NEDC test: (a) SOC; (b) SOC error.

6. Conclusions

In this paper, an Adaptive Cubature Kalman filter (ACKF) algorithm is presented to accurately estimate SOC of the lithium-ion batteries in electric vehicles. The commonly used second-order RC equivalent circuit is applied to simulate the nonlinear behaviors of the lithium-ion battery and establish the battery state-space equations. The OCV-SOC relationship is fitted using the sixth-order polynomial and the other RC parameters of the battery model are determined by the forgetting factor least-squares algorithm. The principle of adaptive cubature Kalman filter for battery SOC estimation is introduced and the estimated process is presented in detail. Two typical driving cycles, including the Dynamic Stress Test and New European Driving Cycle, are applied to assess the performance of the proposed method by comparing with the traditional EKF and CKF algorithms. Experimental results indicate that, although the proposed ACKF algorithm takes more computational time compared with EKF and CKF, it is helpful to improve the SOC estimation accuracy and convergence to different initial SOC error. Furthermore, it is more robust against voltage measurement noise than EKF and CKF.