2017 International Conference on Energy Development and Environmental Protection (EDEP 2017) ISBN: 978-1-60595-482-0
A Novel Online SOC Estimation Method for the Power Lithium Battery Pack Based on the Unscented Kalman Filter Shun-Li WANG1,a,*, Li-Ping SHANG1, Zhan-Feng LI1, Wei XIE2 and Hui-Fang YUAN1 1
Southwest University of Science and Technology, Mianyang 621010, China 2
Sichuan Huatai Electric Co., Ltd., Suining 629000, China a
[email protected] *Corresponding author
Keywords: lithium battery pack; SOC estimation; unscented Kalman filter; equivalent circuit model; state of balance.
Abstract. The SOC (State Of Charge) estimation is a core aspect for the associated BMS (Battery Management System) equipment of the lithium battery pack, in which the KF (Kalman Filter) -based methods have been extensively used but suffers from the linearization accuracy drawbacks. A novel UKF (Unscented Kalman Filter) estimation method battery model is proposed for the SOC estimation of the lithium battery pack, in which the linearization treatment is not required and fewer Sigma data points are used, reducing the computational requirement of the SOC estimation. The UKF method improves the SOC covariance properties for the lithium battery pack, the estimation performance of which has been validated by the experimental results. The proposed SOC estimation method has a RMSE (Root Mean Square Error) value of 1.42%, playing an important role in the popularization and application of the lithium battery pack. 1. Introduction The lithium battery gained its popularity in many applications ranging from portable devices, EVs (Electric Vehicles), renewable energy systems to the antenna-powered applications energy source. Due to the high energy density, low self-discharge rate and no memory effect, the lithium battery has been playing an important role in the high-power energy supply applications. Comparing with other type batteries, the lithium battery has a higher energy density, lower self-discharge rate, and long cycle life. However, the lithium battery over-charge or discharge will make it reduce irreversible damage to its performance and cell life because of the influence of its internal structure. In order to maintain security and performance of the LIB, reliable and accurate methods of estimating the SOC value are desirable in the attached BMS of the lithium battery pack. The particle-filtering-based estimation is conducted by Burgos-Mellado et al. (2016) to evaluate the maximum available power state in LIBs. The implementation of discharge and charge current sensor-less SOC estimator is conducted by Chun et al. (2016) in reflecting cell-to-cell variations in lithium battery packs. A mixed SOC estimation algorithm is proposed by Lim et al. (2016) with high accuracy in various driving patterns of EVs. The lithium battery security guaranteeing method based on the SOC estimation is proposed by Wang et al. (2015). In order to allow the controllability of sigma point distribution better, a reduced order treatment is
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introduced in this paper, the distribution of which is independent of the number of sigma points as the standard UKF method. 2. Mathematical Analysis The UKF-based SOC estimation method is a nonlinear estimation method that does not need the linearization treatment towards the nonlinear state-equation function of f(*) and the observation-equation function of h(*), which is different from the traditional EKF-based SOC estimation method. It just uses the unscented transform treatment process to find out the detection data point around the SOC estimation data point for the power lithium battery. After that, the Gaussian probability density of these SOC sample data points of the lithium battery can be applied to the state-space probability density function of the nonlinear lithium battery power supply system. The unscented transformation treatment process can be realized by using the following steps. Firstly, some detection data points for the working state of lithium battery can be obtained by selecting from the original state distribution of the SOC according to a certain data chosen formula. Then, these selected target detection data points can be substituted into the nonlinear state and observation functions for the SOC estimation of the lithium battery. After that, the nonlinear functional data point set can be obtained, the mean and covariance value of which can be obtained though the analysis of these data points at the same time. According to this calculation process, the accuracy of the mean and covariance without the linearization treatment can reach two-order, which is higher than the EKF method realized with the Taylor series expending process. Eventually, the accuracy can reach three-order, if the nonlinear SOC estimation process of the lithium battery is adapted to the Gaussian distribution. The realization process for the sampling data point selection is based on the proposed unscented transform treatment together with the relevant columns of the prior mean and prior covariance matrices, the nonlinear transform principle of which can be described as shown in Fig. 1.
Figure 1. The nonlinear transform principle of the unscented transform treatment process. The state-space description of the nonlinear lithium battery pack power supply system for the SOC estimation can be expressed by using the nonlinear transform function y=f(x), in which the state vector x is a n dimensional random variable and its mean value and variance value are known previously. The 2n+1 dimensional Sigma point set X and its weight coefficients can be obtained by using the following
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unscented transformation treatment, according to the process of which the statistical characteristics of the SOC state for the lithium battery pack can be calculated and obtained. Firstly, the 2n+1 Sigma point data set can be calculated that is also named as detection point set as shown in equation 1, in which the parameter n indicates the state dimension of the data set and the parameter i indicates the i-th column of the data set and its variance root matrices. X (i ) = X , i = 0 X (i ) = X + ( n + λ ) P , i = 1, , n i (i ) X = X − ( n + λ ) P i , i = n + 1, , 2n T P P =P
( (
) )
(1)
( )( )
Then, the weight coefficients of this sampling point set can be calculated according to the equations as shown in equation 2. Wherein, the subscript m indicates the mean value of the sampling data point set and the subscript c indicates the covariance of the sampling data point set. The superscript i indicates the serial number of the sampling data points. The factor λ is a scaling parameter, which is used to reduce the total SOC estimation error by changing it adaptively. The selection of the parameter α decides the state distribution of the sampling data point set. The parameter κ is a parameter waiting to be selected, the specific value of which has no limitation but normally ensuring the matrix (n+λ)*P to be a positive semi-definite matrix. The waiting selected parameter β is a non-negative weight coefficient, which can merge the high-order dynamic error of the state-space function, making the high order influence to be included in the unscented transform treatment process.
λ λ ( 0) ( 0) 2 2 ωm = n + λ , ωc = n + λ + (1 − α + β ) , λ =α ( n + κ ) − n λ ωm( i ) = ωc( i ) = , i = 1, , 2n 2(n + λ )
(2)
It can be known from the above calculation process that the Sigma point data set obtained from the transform process has following characteristics. Firstly, as the Sigma data point set has the same weight coefficient around the expected mean symmetric data point, the sample mean of the Sigma set is the same as that of the random variable parameter X. Secondly, the variance of the sampling Sigma point data set is the same with the variance of the standard random vector. Thirdly, the Sigma point set of any normal distribution is obtained by a transformation treatment of the Sigma set of the standard normal distribution. Several Sigma data point set transformation methods can be used in the transform process, including unscented transform, simplex transform, and spherical transform and so on. Because the computation complexity of UKF-based SOC estimation is positively correlated with the number of Sigma data points, it is beneficial for the integrated application to have fewer Sigma points in the unscented transformation process. The proposed unscented transform treatment process requires 2n+1 Sigma data point selection, in which the parameter n is the dimension of the nonlinear SOC estimation system for the lithium battery pack. The simplex transform requires only n+1 Sigma points, but it suffers from numerical stability issues because these obtained Sigma points are based on the radius sphere of 2n/2. As a result, it does not considered in the unscented transform treatment process.
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The SOC estimation of lithium battery is conducted by Meng et al. (2016) based on adaptive UKF and support vector machine. The least-squares based coulomb counting method is studied by Zhao et al.(2016) which is applied for SOC estimation in EVs. The spherical transform considered in this unscented transform treatment process for the SOC estimation of the lithium battery pack only requires n+2 Sigma points. However, its numerical stability is improved by reducing the sphere radius. In the spherical transform process of the n dimensional nonlinear SOC estimation system for the lithium battery pack, the initial weight coefficient W0 is set firstly and the choice of the parameter W0 affects only the fourth or higher order moments of the Sigma point data set. By the analysis and calculation treatment of the parameters W0 and n, the rest of the weight coefficients from W1 to Wn are obtained and selected. The three-element vector of the parameter X from 0 to 2 for the first column can be generated by using the weight coefficient W1. In order to generate the required n+2 set of Sigma point vectors with n dimensional characteristics, this three-element vector are calculated with recursive expanded treatment in the SOC estimation process of the lithium battery pack as shown in equation 3. 1 − W0 0 ≤ W0 ≤ 1;Wi = n + 1 1 1 1 1 1 , X2 = X 0 = [ 0] , X 1 = − 2W1 2W1 X i j −1 j X j = X j −1 , i = 0; X j = − , i = 1, , j; X i j = ,i = j +1 i 0 i j ( j + 1) W1 j ( j + 1) W1 X ij n / (1 − W0 )
(3)
There are several data stability problems that are remaining in the SP-UKF calculation process, such as the negative ECM model parameters and so on for the SOC estimation process of the lithium battery pack. The reason for this phenomenon is that the sphere radius for Sigma point distribution depends on the size of estimated SOC state-vector. Aiming to ensure the sphere radius is independent with the estimated state-vector size and the parameter ζ always falls within the expected variance range of the function f(ζ), all Sigma data points are normalized with respect to the sphere radius. As a result, all the sigma points should be guaranteed to be projected within a unit hyper sphere. The realization process of the unscented transform for the SOC estimation of the lithium battery pack can be summarized and descript as follows. Firstly, the initial weight coefficient parameter W0 should be chosen previously. Secondly, the rest of the weight coefficient parameters can be computed by using the initial coefficient parameter W0 and the numerical parameter n. And thirdly, the three-element vectors should be initialized by using the coefficient parameter W1. Fourthly, the rest of the three-element vectors should be calculated by the recursively expanding, for j=2,…,n. Finally, the vectors should be arranged in a unit hyper sphere. According to calculation process described above, the unscented transform treatment can be realized in the linearization process for the SOC estimation of the lithium battery pack.
3. Experimental analysis In order to validate the accuracy and the reliability of the proposed SOC estimation method based on the UKF algorithm for the lithium battery pack and obtain the ECM battery model parameters, some experimental studies are conducted based on the associated BMS equipment that is designed and
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applied for the energy management of the lithium battery packs, which is used to perform the aviation environmental conditions for subsequent experiments and shown in Fig. 2.
Figure 2. The BMS equipment system structure of the lithium battery pack. This BMS equipment is used to perform the airborne working conditions. As shown in the above figure, a DC power supply is sued to simulate the output power and a DC electronic load is used to simulate the loadings of the subsystems. A data acquisition system is used to record the battery terminal voltage, terminal current and temperature of the referenced SOC calculation. The referenced SOC is obtained using the calibrated Ah counting via the high precision current sensors from the power supply and the DC electronic load with the sensor accuracy of 0.20% and 0.10% respectively. A C# program has been written to control all the hardware equipments. A thermal chamber is used to maintain the battery temperature between 5.00 to 25.00 ℃ aiming to emulate the battery heater. A microcontroller is used to process the acquired data as well as the real-time experimental SOC for the comparison. In addition, a portable fuel gauge with an expected accuracy of 3.00% to 5.00% is used for the further benchmarking. The OCV value of the lithium battery cell, named as UOC, has a nonlinear relationship with the SOC value of the lithium battery cell. In order to obtain this nonlinear function relationship, the OCV test is conducted using the lithium battery cell with 45Ah capacity as an experimental sample. In this experimental study, the hysteresis effect is ignored. If an additional voltage source is placed in parallel to the parameter UOC at the expense of increased complexity, the hysteresis effect can be included. The lithium battery cell is first fully charged by using the CC-CV method and then rest for an hour, allowing it to reach the steady-state voltage before the UOC value is measured. At different SOC levels measured for the follow-up UOC, the lithium battery cell is discharged at 4.50A (0.10C5A) for half an hour, and rested for another hour to reach the static state before another test is carried out. The SOC-OCV diagram obtained from the experiment is shown in Fig. 3.
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Figure 3. The OCV versus SOC diagram of the lithium battery cell. In the figure, the horizontal axis represents the SOC value of the lithium battery cell and the vertical axis represents the obtained OCV value. By means of intermittent discharge maintenance, the discrete point OCV-SOC value of the lithium battery cell can be obtained, and the overall variation acquisition can be realized by using the curve fitting method. In order to obtain a good estimation effect of the SOC value for the LIB, as can be seen from the above figure, the relationship curve can be divided into three phases of abrupt - slow - steep zones, which can be described in subsection. In the estimation experiment, the SOC estimation performance using the proposed UKF estimation method with unknown initial SOC value is performed. The true initial SOC value is set as 0.40 and the initial estimated SOC value is set as 0.32. The estimated SOC value converges to the true SOC value within 10s in the discharging process when the estimated SOC error is lower than 5%, which is shown in Fig. 4.
Figure 4. The SOC estimation with unknown initial state.
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As can be seen from the above experimental results, the output voltage detection accuracy also affects the SOC estimation process of the lithium battery pack in certain content. Because the terminal voltage of the lithium battery pack increases nonlinearly when it is being charged or discharged. The experimental results show that the proposed method has good SOC estimation performance in the discharging and charging process, which has small SOC estimation error for the lithium battery pack.
4. Conclusion A novel SOC estimation method named as UKF is proposed for the lithium battery pack. The proposed UKF method has Jacobian-free advantages and requires fewer Sigma data points than the regular UKF algorithm. The unscented transform treatment in the SOC estimation process with UKF algorithm improves the SOC covariance properties. The experimental results show that the proposed SOC estimation method performs well with unknown initial SOC values. The SOC estimation performance of the proposed method has been validated with experimental results, which indicates that the UKF method has the low absolute mean error, absolute maximum error and RMSE values and it is applicable for the SOC estimation of the lithium battery pack and plays an important role in its popularization and application.
Acknowledgments The work was supported by Sichuan Science and Technology Support Program (No. 2017FZ0013), Scientific Research Fund of Sichuan Provincial Education Department (No. 17ZB0453) and Mianyang Science and Technology Project (No. 15G-03-3). We would like to thank the sponsors.
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