#### Abstract

Capacity degrading over repeated charge/discharge cycles is a main parameter for evaluating battery performance, which is commonly used for determining the state of health. However, it is difficult to measure the available capacity because it requires the normal operation to be terminated and a long time-consuming detection process. This study presents an online available-capacity estimation method by combining extended Kalman filter (EKF) with Gaussian process regression (GPR) for the daily partial charge data of lithium-ion batteries. First, GPR is used to establish an empirical model of the time-voltage curve in the constant current charge cases. Second, by analyzing the characteristics of the charge curve, the daily piecewise partially charge data are registered with the piecewise complete charge data to update GPR model and preestimate the equivalent complete charge time. On this basis, the equivalent complete charge time is refined by EKF. Furthermore, the available capacity estimation of the battery with constant current charge processes under different aging conditions is achieved. It is verified by experiments that the estimated error can be controlled within 5% when the actual available capacity is greater than 90% of the initial capacity.

#### 1. Introduction

Lithium-ion batteries are widely used as power sources for electrified vehicles due to their high energy density, high galvanic potential, and long lifetime. Battery capacity indicates the accumulation amount of electric energy under certain discharge conditions (e.g., temperature, discharge rate, termination voltage), which is a key parameter to evaluate battery’s performance over repeated charge/discharge cycles. The performance of a battery becomes unreliable when the battery’s capacity decreases below 80% of its initial capacity. Thus, accurate capacity estimation is critical to ensure reliable operation of the electric drive [1].

The most popular capacity test is conducted under the static condition approximately with the following processes: the battery is completely charged and then discharged with constant current to discharge cutoff voltage at certain temperature. Thus, the battery’s capacity can be obtained by integrating current value over discharge time [2]. The advantage of this method is the impressive accuracy of the measurement. However, in order to evaluate the battery capacity, complete charge and discharge processes always take huge time. In the practical application of power battery, the charge is generally conducted by the similar process of SOC from 20% to 80% or from 50% to 100%, although available capacity of the real-time battery cannot be estimated each time. Aimed at solving the problem, the method is proposed in the paper to assess the available capacity of power battery based on the practical charge data, providing reference for the electric vehicles and data mining of charging infrastructure’s big data platform. Although constant current charge is extensively used for the practical charge of electric vehicles, the maximum capacity from constant current charge is accepted as the actual capacity of power battery in this research. Sturlaugson and Sheppard [3] applied principal component analysis of data-driven Bayesian network to evaluate the battery capacity. Moreover, battery residual life prediction model was established in reference [4] with data-driven method and Nuhic et al. [5] especially focused on that predicting the development trend of battery health state would be data driven; In reference [6], the aging mechanism of battery life was analyzed and Gaussian process model was chosen based on the degenerating data of the battery cycle life; Sauer and Wenzl [7] calculated healthy life–predicting model using polynomial fitting, exponential fitting and a combination of both to assess health life of the battery. Furthermore, the LS-FDP [8] method was implemented for the residual life prediction, which was superior to the traditional fault diagnosis and prediction with higher accuracy and rapidity. Regarding references [9–11], various data-driven methods were introduced to evaluate the residual capacity or residual life. However, existing methods hardly deal with partial charge process in real applications.

#### 2. Materials and Methods

##### 2.1. Acquisition and Analysis of Experimental Data

The test platform comprises the tested batteries, tester to load the power battery, and an upper computer to store experimental data, which all come from a battery manufacturer in Shenzhen, China. A type of battery cell with a LiNi_{x}Co_{y}Mn_{z}O_{2} chemistry was tested [12]. The nominal voltage was 3.78 V, with the upper and lower cutoff voltages of 4.35 V and 2.75 V, respectively. The cell specifications are summarized in Table 1.

The cells were cycled in the same state of charge range under the same temperatures and current rates. At 25°C, the cells were fully charged with a constant current charge rate of 0.5 C to the upper cutoff voltage 4.35 V. And in turn, the test was paused for half an hour to allow the battery’s internal temperature to reach a steady state. Then, the cell was discharged to the lower cut-ff voltage under 0.5 C current rate.

The charge time until supper cutoff voltage is used to indicate the battery’s capacity because the capacity can be estimated by the product of current and time in constant current case.

Therefore, the relation between capacity and voltage can be converted into the curve between voltage and time. Then, under constant current charge conditions, the battery capacity estimation only needs to model the time-voltage curve. During the charging process, the voltage increases gradually, and the change of the curve’s gradient is influenced by different factors, including exterior and interior lithium-ion diffusion rates of positive electrode materials, phase change of electrode active materials, destruction of lattice structure and transition-metal ions migration into the electrolyte, etc.

In experiments, the battery was exposed to 1000 cycles. Under different aging degrees, the measured time-voltage curves are illustrated in Figure 1.

It can be seen from Figure 1 that charge curves according to different cycles have a similar trend, which indicates that these curves are approximately parallel to each other. The key point is to exactly estimate complete charge time until reaching the upper cutoff voltages. In this study, Gaussian process regression is used to learn the complete charge experience model of the lithium-ion battery.

##### 2.2. A Complete Charge Model Based on Gaussian Process Regression

Gaussian process regression (GPR) models are nonparametric kernel-based probabilistic models used to describe the distribution of random variables in the framework of Bayesian inference. Consider a set of random variables , its corresponding states obey joint Gaussian process defined by its mean function and covariance function , that is,

Then, .

Given a training set and its output , now consider the following regression model:where . GPR model aims to train nonlinear mapping from input to output and predicts the value of a response variable to the new input .

For a new testing sample , and its predicted value , is a joint Gaussian distribution, that is,where stands for the covariance matrices between the training set and the testing sample , and is the variance of the testing sample.

Based on Bayesian regression method, the posterior probability of subjects to Gaussian distribution with mean function and covariance function , that is,where

Gaussian processes usually have different covariance functions. A common covariance function is the squared exponent covariance function:where is the covariance of with the scale . The hyperparameters can be obtained by maximizing the logarithmic marginal likelihood function of the training set.

At present, there is no unified theoretical support for the selection of kernel functions in Gaussian process. For a system with high complexity, it is difficult to accurately represent the data by the GPR model based on a single kernel function which usually only describes one aspect of the data. Therefore, the combined kernel function could be used to characterize the heterogeneous information of the data and enhance the capability in representing the nonlinear mapping. Suppose kernel functions and are independent of each other, define

It is easily shown that and are also kernel functions, respectively, called additive and multiplicative kernel functions. According to the characteristics of the time-voltage curve in constant current charge process of the battery, the local neural network kernel function is selected as the kernel function to describe nonlinearity of the data while the periodic kernel function is chosen as the global kernel function to describe periodic characteristics of the data.

For comparison, this study conducted tests on synthetic datasets and evaluated the influence of two kinds of single kernel functions and combined kernel functions on the predictive distribution to determine an appropriate kernel function. The 64 data points are produced by a step function with Gaussian noise with standard deviation which is totally same as the example taken in [13]. Figure 2 shows the means and 95% confidence intervals for the noisy signal in grey. Specially, Figure 2(a) expresses the single neural network kernel function, Figure 2(b) illustrates the periodic kernel function.

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New kernels can be generated with these single kernel functions based on sum and product given as equations (9) and (10). Figures 2(c) and 2(d) show the results of regression by combined kernel functions. Obviously, the combined kernel functions are more flexible than a single kernel function. It turns out that the multiplicative function can be viewed as a suitable choice shown as Figure 2(e). Here, can be taken as a modification of .

In this study, the charge voltage and the charge time are taken as the input and output variables, respectively. Combined kernel-based GPR is used to obtain the correspondence between the charge voltage and the complete charge time by maximum likelihood estimation.

##### 2.3. Online Available Capacity Estimation of the Battery for Partial Charge Process

The daily charge processes of lithium-ion batteries are usually partial, so the actual available charge capacity cannot be directly obtained based on charge time. This study proposed an online capacity evaluation method for the daily charge data of the battery. The specific steps are as follows:

(1)Analyze the initial complete charge data (for about 10 cycles), charge end-off voltage, and the initial complete charge model can be acquired based on GPR. Note that lithium-ion membrane formed within first several charge cycles for a new battery where about 18% of energy was irreversibly consumed in generating ionic membrane. Thus, the initial complete charge data were obtained by complete charge curve after several cycles.(2)All inflection points of the initial complete charge curve are extracted and the fully charge curve is split. Screen the partial charge data segment during the daily use of the battery to be estimated. Then inflection points are also used to divide the charge curve of estimated daily partial charge data into multiple segments.(3)The slopes of corresponding segments in different cycles are basically unchanged. In addition, the charge curves are parallel in general. Thus, the daily charge data can be compared and registered with the initial complete charge curve. The parameters in the model are adjusted to accommodate the current charge process and a preestimate of the equivalent complete charge time for the daily partial charge process.(4)The preestimated equivalent of the complete charge time is taken as the initial value of adjusted model, and the extended Kalman filter is used to describe the current charge process and refined equivalent complete charge time . The available capacity is estimated as . The procedure of the proposed algorithm is shown in Figure 3.

##### 2.4. Feature-Based Charge Curve Subsection and Equivalent Complete Charge Time Preestimation for Partial Charge Process

According to the characteristics of the voltage-time curve of the constant current charge process, the feature points are provided with the following properties: the feature points often exist at inflection points of the curve, and the curve between feature points can be approximated by a straight line. Then, the slope of these lines is calculated. Feature points can effectively reflect the charge process of the battery and improve the estimation accuracy of the battery capacity.

Cyclic charge test was conducted at room temperature with charge rate of 0.3 C and sampling interval of 1 s, and the experimental data were analyzed, as shown in Figure 4. The initial complete charge curve was combined with the partial data to form a more complete charge curve, the error of the third section compared with the original curve is about 0.8% and with errors of other sections are in the range of 0.5%. According to the error analysis, charge data segment was considered as a part of the complete charge curve. Thus, the complete charge curve can be substituted by the piecewise curve.

Let and be the feature points of time-voltage curve in initial complete charge process and current partial charge processes, respectively. For the convenience of calculation and explanation, () are denoted as the beginning and ending times of the charge process here. Thus, () make up the endpoints of piecewise charge data. Let and stand for the last time intervals of initial and current charge processes, respectively. Then the pre-estimation of the equivalent complete charge time of the current partial charge process is calculated as where is original complete charge time.

##### 2.5. Equivalent Complete Charge Time Modification and Available Capacity Estimation Based on EKF-GPR

As mentioned above, the precision available capacity of lithium-ion battery cannot be directly obtained in daily use because complete charge or discharge process is hard to achieve. In this study, the Gaussian process regression (GPR) is integrated into the extended Kalman filter (EKF) [14] to establish the EKF-GPR model. The piecewise data of the initial complete charge process and the fragmentation data of the daily partial charge process are translated, registered, and spliced to reconstruct the current partial charge data. Accordingly, using the idea in Section 2.2, the Gaussian process regression is used to update the charge curve model which is taken as the system model in the EKF including the state model, the measurement model and the corresponding noise covariance matrices. Then the pre-estimation of the equivalent complete charge time obtained above is taken as the initial state of the EKF to effectively solve the capacity estimation problem with the unknown or inaccurate battery model.

Nonparametric EKF-GPR model is established to estimate the equivalent complete charge time as follows:where and stand for sampling time and corresponding voltage, is sampling interval and is the measurement noise. GP is the Gaussian process regression employed to learn measurement model of EKF. The initial state is set as .

It is assumed here that the measurement noise follows the Gaussian distribution with covariance . The combined kernel function defined as equations (9)–(11) is used in GPR for prediction. Based on EKF model, construct the following recursive formulas:

Finally, the equivalent complete charge time of the daily partial charge data is predicted as

Then, the current available capacity in constant current charge process is estimated as

The online state of health estimation is calculated bywhere stands for initial available capacity.

#### 3. Results and Discussion

In this section, an analysis of real degradation datasets obtained as Section 2.1 was conducted to validate the developed approach. To show the effectiveness of the proposed technique, we tested capacity degradation trajectories of five batteries. The battery charge capacity is reduced from 100% to 80% after 1000 charge/discharge cycle with constant current rate. For instance, Figure 5 shows the comparisons between actual and estimated voltage-tine curves of 100, 500, and 1000 charge cycles, respectively.

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Based on charge time estimation, Figure 6 and Table 2 also show the capacity estimation errors of different batteries under different cycles based on proposed method.

Through 1000 charge/discharge cycles, the capacity estimation accuracy achieves good performance, where error is less than 3.5% within the first 800 s of the cycles. However, when the capacity of the battery with hops or changes in the opposite direction during cycles, the estimation algorithm features relative high error and the robustness needs to be improved.

Note that charge and discharge for long cycles were conducted in the test by using Neware Battery Test System. During the estimation, the true value was adopted from the last complete charge curve. It had a high accuracy of 0.1% and the curves from two consecutive cycles overlapped completely. Besides, charge data segment was a part of the complete charge curve so that the segment could be arbitrarily chosen from the complete charge curve. The observed data corresponding with the actual data were from the same battery without differences. Consequently, estimation results were reliable for the research.

#### 4. Conclusions

In this study, we developed an online battery capacity estimation method for partial charge process. According to the characteristics of the charge curve, feature points are introduced for piecewise-linearized processing procedure of the data. Then, the daily partial charge data are registered with the complete charge data to learn GPR model and preestimate the equivalent complete charge time. With the proposed method, a joint EKF-GPR model is constructed to exploit a quantitative correlation between equivalent charge time and battery voltage. For partial charge data, when the available capacity exceeded more than 80 percent of the initial capacity, the obtained estimations of equivalent charge time are accurate and precise with 7% estimation error on average. The battery capacity can thereby be estimated from partial charge data with flexible initial available capacity. The results demonstrate that the online capacity estimation method possess high precision and good potential for practical applications.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Key Research and Development Program of China under grant no. 2016YFF0201204 and the Natural Scientific Research Foundation of Heilongjiang Province under grant no, F2016026.