An Adaptive Estimation Scheme for Open-Circuit Voltage of Power Lithium-Ion Battery

Abstract

Open-circuit voltage (OCV) is one of the most important parameters in determining state of charge (SoC) of power battery. The direct measurement of it is costly and time consuming. This paper describes an adaptive scheme that can be used to derive OCV of the power battery. The scheme only uses the measurable input (terminal current) and the measurable output (terminal voltage) signals of the battery system and is simple enough to enable online implement. Firstly an equivalent circuit model is employed to describe the polarization characteristic and the dynamic behavior of the lithium-ion battery; the state-space representation of the electrical performance for the battery is obtained based on the equivalent circuit model. Then the implementation procedure of the adaptive scheme is given; also the asymptotic convergence of the observer error and the boundedness of all the parameter estimates are proven. Finally, experiments are carried out, and the effectiveness of the adaptive estimation scheme is validated by the experimental results.

1. Introduction

Electric vehicles (EVs) and hybrid electric vehicles (HEVs) have been getting more and more attention in recent years because they have the potential of improving the fuel efficiency and reducing the pollutant emissions [1]. The merits of the high energy density, high power density, no memory effect, and experience of low self-discharge when not in use compared to other batteries make the lithium-ion batteries the main contender for energy storage in EV/HEVs [2, 3].

Open-circuit voltage (OCV) is one of the most important parameters of a battery due to the intrinsic relationship between state of charge (SoC) and OCV [4, 5], which is defined as the measured terminal voltage when battery reaches steady-state. To the authors’ best knowledge there only exist two methods to get the value of OCV: experiment test and online estimation. The former method is very time consuming because the battery needs rest 10 h or more to reach the steady-state when charging or discharging at each measurement. Reference [6] has proposed a rapid test method, and the pause time at different SoC needs only one minute before the new experiment test is carried out. The voltage that reaches steady-state during the pauses at different SoC during discharging and charging is connected with line, and then the value of OCV takes the mean of the lines. This way can significantly save test time, but however the conventional experiment method or the new experiment method is suitable for measurement in laboratory, but not suitable for online measurement. The latter method is based on the equivalent circuit model to estimate OCV. The voltage of the electromotive force [7] or the voltage across the bulk capacitor [8, 9] is used to denote OCV. In this method, the parameter (capacitors and resistances) values of the equivalent circuit are generally considered to be constant, which contradict the fact that all parameter values usually change along with the SoC, temperature, and usage history change, and then the estimate precision will be degraded due to the inaccuracy of parameter values. Thus new theory and new method are demanded to enrich the estimate method and enhance the estimate accuracy. Adaptive technology is undoubtedly the best choice to solve such problem.

Battery is a complex electrochemical system; the accurate modeling for it is more difficult. As the adaptive scheme is implemented easily and the modeling error is within an acceptable range, the Thevenin battery model that includes one RC network is adopted here. The state-space formulation is obtained based on the equivalent circuit model, and an adaptive observer is designed according to the feature of the state-space. The designed adaptive observer together with the update laws gives accurate estimate of the true states but does not give the accurate estimate of the true parameters, which can only guarantee the boundedness of all the parameter estimates. Fortunately, the estimate of some parameters change in the same proportion and the estimate of OCV is the quotient of them, thus the accurate estimate of OCV can be obtained.

The remainder of this paper is organized as follows. Section 2 presents the main results of this paper. In Section 2.1, a dynamic equivalent circuit model is chosen, and the state-space formulation is obtained based on the equivalent circuit model; in Section 2.2, the adaptive observer and adaptive laws are designed. Section 3 provides the experimental results to demonstrate the correctness of the proposed adaptive estimate scheme. This paper ends with some conclusions.

2. Modeling, Adaptive Observer, and Adaptive Laws Design

In this section, an appropriate model of the battery is chosen firstly, and then the dynamic characteristic of the battery system is described by differential equations. The design procedure of the adaptive observer and adaptive laws will be shown step by step at last.

2.1. Modeling

The principle of selecting a battery model is as simple as possible but precise enough for the investigated problem. The model chosen here is the first-order Thevenin model (see Figure 1), which is simple yet accurate enough for the control-oriented purpose in hybrid electric vehicles [1]. The parallel 𝑅𝐶-branch, comprising 𝑅_𝑝 and 𝐶_𝑝, is used to model battery polarization effect; 𝑅 denotes the ohmic resistance, 𝑢_oc denotes the open circuit voltage, and 𝑢_𝑡 denotes the terminal voltage. 𝐼 denotes the terminal current and assumes it is positive when discharge otherwise is negative. Based on the Kirchhoff’s law, the electrical behavior of the circuit can be characterized as follows: 𝑢̇_𝑝1=−𝐶_𝑝𝑅_𝑝𝑢_𝑝+1𝐶_𝑝𝑢𝐼,_𝑡=𝑢_oc−𝑅𝐼−𝑢_𝑝. (1) From (1), the derivative of 𝑢_𝑡 can be obtained as 𝑢_𝑡̇=𝑢̇_oc−𝑅̇𝐼−𝑅𝐼̇−𝑢_𝑝̇≈−𝑅𝐼̇+1𝐶_𝑝𝑅_𝑝𝑢_𝑝−1𝐶_𝑝𝐼=−𝑅𝐼̇+1𝐶_𝑝𝑅_𝑝𝑢_oc−𝑅𝐼−𝑢_𝑡−1𝐶_𝑝𝐼1=−𝐶_𝑝𝑅_𝑝𝑢_𝑡−𝑅+𝑅_𝑝𝐶_𝑝𝑅_𝑝𝐼−𝑅𝐼̇+1𝐶_𝑝𝑅_𝑝𝑢_oc. (2)

Figure 1 

Thevenin equivalent model for lithium-ion battery.

Remark 1. The parameters 𝑅, 𝑅_𝑝, and 𝐶_𝑝 are functions of time, SoC, and temperature, but the partial differential of them is treated as zero due to the reason that the change rate is very slow with respect to the signals sampling period (the sampling period is 1 s). For the same reason, 𝑢̇_oc≈0.

2.2. Adaptive Observer and Adaptive Laws Design

This subsection is devoted to the constructive design of the adaptive observer and the update laws. To this end, choose 𝜃₁=1/𝐶_𝑝𝑅_𝑝, 𝜃₂=(𝑅+𝑅_𝑝)/𝐶_𝑝𝑅_𝑝, 𝜃₃=𝑅, 𝜃₄=(1/𝐶_𝑝𝑅_𝑝)𝑢_oc; then (2) can be rewritten as 𝑢̇_𝑡=−𝜃₁𝑢_𝑡−𝜃₂𝐼−𝜃₃𝐼̇+𝜃₄, (3) obviously 𝑢_oc=𝜃₄/𝜃₁.

Enlightened by [10], we design adaptive observer (4) for system (3) as well as for system (2) as follows: 𝑢̇_𝑡=−𝜃₁𝑢_𝑡−𝜃₂𝐼−𝜃₃𝐼̇+𝜃₄+𝐿𝑢_𝑡−𝑢_𝑡, (4) where 𝑢_𝑡 is the estimate of 𝑢_𝑡 and 𝐿 is the observer gain. The parameters 𝜃_𝑖(𝑖=1,…,4) are estimates of the parameters 𝜃_𝑖(𝑖=1,…,4), which will be adjusted adaptively such that 𝑢_𝑡→𝑢_𝑡 as 𝑡→∞.

Subtracting (4) from (3), the dynamic of observer error 𝑒=𝑢_𝑡−𝑢_𝑡 is as follows: 𝑒̇=−𝜃₁𝑢_𝑡−𝜃₂𝐼−𝜃₃𝐼̇+𝜃₄−𝐿𝑒, (5) where 𝜃_𝑖=𝜃_𝑖−𝜃_𝑖(𝑖=1,…,4).

The following Barbalat’s lemma that will be used in the development of our main results is given first.

Lemma 2 (see [11]). If the differentiable function 𝑓(𝑡) has a finite limit as 𝑡→∞ and if 𝑓̇ is uniformly continuous, then 𝑓(̇𝑡)→0 as 𝑡→∞.

The proof can be found in the reference and is omitted here.

To guarantee the asymptotical convergence of observer error system (5) and the boundedness of signals 𝜃_𝑖(𝑖=1,…,4), we have the following theorem.

Theorem 3. Consider battery system (2) as well as its parametric form (3), design the observer (4), if one chooses the parameter update laws for 𝜃_𝑖(𝑖=1,…,4) as ̇𝜃₁=−𝑐₁𝑒𝑢_𝑡,̇𝜃₂=−𝑐₂̇𝜃𝑒𝐼,₃=−𝑐₃𝑒̇̇𝐼,𝜃₄=𝑐₄𝑐𝑒,_𝑖>0,𝑖=1,…,4 (6) and the observer gain 𝐿>0, then the signal 𝑒 is convergent to zero as 𝑡→∞ and all the signals 𝜃_𝑖(𝑖=1,…,4) are uniformly bounded.

Proof. Define a candidate Lyapunov function 1𝑉=2𝑒²+12⁴_𝑖=11𝑐_𝑖𝜃²_𝑖,𝑐_𝑖>0,𝑖=1,…,4; (7) from this together with (5) and (6), we have 𝑉̇=𝑒𝑒̇−⁴_𝑖=11𝑐_𝑖𝜃_𝑖̇𝜃_𝑖=−𝑒𝜃₁𝑢_𝑡−𝑒𝜃₂𝐼−𝑒𝜃₃𝐼̇+𝑒𝜃₄−𝑒𝐿𝑢_𝑡−𝑢_𝑡−1𝑐₁𝜃₁̇𝜃₁−1𝑐₂𝜃₂̇𝜃₂−1𝑐₃𝜃₃̇𝜃₃−1𝑐₄𝜃₄̇𝜃₄=−𝜃₁𝑒𝑢_𝑡+1𝑐₁̇𝜃₁−𝜃₂1𝑒𝐼+𝑐₂̇𝜃₂−𝜃₃𝑒̇1𝐼+𝑐₃̇𝜃₃−𝜃₄1𝑒−𝑐₄̇𝜃₄−𝑒𝐿𝑢_𝑡−𝑢_𝑡,=−𝐿𝑒²≤0. (8) Thus 𝑉(𝑡) is a nonincreasing function of time; that is, sup_𝑡≥0𝑉(𝑡)≤𝑉(0). (9) This proves that 𝑒 and 𝜃_𝑖(𝑖=1,…,4) remain bounded for all 𝑡≥0, namely, lim_𝑡→∞‖𝑒(𝑡)‖<∞,lim_𝑡→∞𝜃_𝑖(𝑡)<∞,𝑖=1,…,4. (10) The derivative of 𝑉̇ is 𝑉̈(𝑡)=2𝐿𝑒(𝜃₁𝑢_𝑡+𝜃₂𝐼+𝜃₃𝐼̇−𝜃₄+𝐿(𝑢_𝑡−𝑢_𝑡)); this shows that 𝑉̈(𝑡) is bounded since 𝑒 and 𝜃_𝑖(𝑖=1,…,4) are bounded. Hence, 𝑉̇(𝑡) is uniformly continuous. After the application of Barbalat’s lemma we obtain the conclusion that 𝑒→0 as 𝑡→∞.
From (10) and the definition 𝜃_𝑖=𝜃_𝑖−𝜃_𝑖, (𝑖=1,…,4), we know that 𝜃_𝑖(𝑖=1,…,4) are bounded.
Thus, we obtain the conclusion of the asymptotical convergence to zero of the signal 𝑒 and the uniform boundedness of the signals 𝜃_𝑖(𝑖=1,…,4).

3. Experimental Results

There are three purposes for the experiment: the first one is to verify the adaptive observer which can give an accurate estimate of the system states; the second one is to verify the estimates of the parameters 𝜃₁ and 𝜃₄ which change with the same scaling proportion. The fact that OCV can be estimated by 𝜃₄/𝜃₁ will be verified at last.

3.1. Parameter Extraction

The extraction experiment that can obtain the parameters value is firstly carried out. The experimental sample is a large size lithium iron phosphate battery. The nominal capacity and voltage of the selected battery is 50 Ah and 51.2 V, respectively. The tests are carried out on an AVL e-Storage Tester. The tester is capable of charging and discharging the battery at a maximum rate of 285 A. It can operate in several modes such as alternating current and constant current. It comprises power, current, voltage, and temperature measurements. The battery tester and the data logger are connected to a PC through a bus system. The host computer with the installed software can program the experiment procedure. Several limits for the battery can be defined, which are necessary in order to avoid damage, for example, caused by overcharging.

The pulsed-discharge experiments are carried out at different SoC, and the terminal voltage is monitored simultaneously (see Figure 2). The battery parameters can be extracted through the transient response.

Figure 2 

Terminal voltage response under step-current discharge.

The terminal voltage response at different SoC can be fitted as in the following form: 𝑢_𝑡(𝑡)=𝑉₂+𝑐₁1−𝑒^−𝑐2𝑡, (11) where 𝑐₁ and 𝑐₂ are the least-squares curve fitting coefficients. 𝑉₂ is as shown in Figure 2. Then the parameters value of battery model (see Figure 1) can be derived from 𝑉𝑅=²−𝑉₁𝐼,𝑅_𝑝=𝑐₁𝐼,𝐶_𝑝=1𝑐₂𝑅_𝑝. (12) The parameter values identified for the battery module at different SoC are given in Table 1. From Table 1, it can be seen that the parameters 𝑅, 𝑅_𝑝, and 𝐶_𝑝 are all varying with the SoC changes, which further confirms that the system state estimation needs adaptive scheme.

Table 1 

Model parameter values list identified for the battery module.

3.2. Experiment Validation

This subsection is to validate the correctness of the theory of Section 2.2. The urban dynamometer driving schedule (UDDS) driving cycle of Advisor is adopted to generate terminal current (see Figure 3(a)). According to the parameter values at SoC = 1.0 (see Table 1), set the initial conditions of system (3) to be 𝑢_𝑡(0)=54.108, 𝜃₁=0.0863, 𝜃₂=0.00412, 𝜃₃=0.0438, and 𝜃₄=4.67, and set the initial conditions of observer system (4) to be 𝑢_𝑡=51.459, 𝜃₁(0)=0.0833, 𝜃₂(0)=0.0035, 𝜃₃(0)=0.043, and 𝜃₄(0)=4. Set coefficients of update laws (6) to be 𝑐₁=0.1, 𝑐₂=0.1, 𝑐₃=0.1, and 𝑐₄=0.1, and set the observer gain to be 𝐿=2. The experiment results are shown in Figures 3–5.

(a)

(b)

(c)

Figure 3 

Comparison of the true and estimated battery terminal voltage.

(a)

(b)

(c)

(d)

(e)

Figure 4 

Comparison of the true and estimated parameters value for the equivalent circuit model.

(a)

(b)

Figure 5 

Comparison of the true and estimated battery open-circuit voltage.

Figure 3(a) is the system input, namely, battery terminal current. Figure 3(b) is the original and observer system outputs. The observer error is shown in Figure 3(c), from which we can see that the observer error converges to zero within twelve seconds although the parameter values of original system and observer system are different.

Figures 4(a)–4(d) are the true and estimated values of the parameters 𝜃_𝑖(𝑖=1,…,4), from which we can see that the estimated values of the parameters are bounded. Assume that the scaling of the estimated values of 𝜃₁ and 𝜃₄ is 𝑠₁=𝜃₁/𝜃₁ and 𝑠₄=𝜃₄/𝜃₄, accordingly, Figure 4(e) shows that the difference of 𝑠₁ and 𝑠₄ is approximately zero. In other words 𝑠₁≈𝑠₄; so the estimated value of 𝑢_oc can be obtained from 𝜃₄/𝜃₁.

Figures 5(a) and 5(b) are the true and estimated battery OCV and the estimated error. It can be seen that the adaptive observer designed here can estimate the OCV accurately.

Remark 4. Battery discharging at a constant current value 15 A from SoC=1.0 to SoC=0.1 takes 10800 s (which does not include the standing time of the discharge experiment). 𝑢_oc changes from the maximum 54.108 V to the minimum 51.459 V; so 𝑢_oc is considered to be a constant when discharging and charging for tens of seconds.

4. Conclusions

In this paper, an adaptive observer is designed to estimate OCV of a lithium-ion battery pack. The parameter values of the battery equivalent circuit model are extracted through nonlinear fitting. The performance of the proposed estimator is confirmed by a series of experiment results. The validity results show that the estimator has a good performance under the condition that it has no requirement for knowing the exact parameter values.

Acknowledgments

The authors are grateful to the editor and reviewers for their comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grants numbers 61034007, 61273097, 61104034, and 61104087) and Research Fund for the Doctoral Programme of Higher Education of China (Grant number 20120131130007).