⇦Chinese Physics Letters, 2016, Vol. 33, No. 7, Article code 078201 A Novel Real-Time State-of-Health and State-of-Charge Co-Estimation Method for LiFePO$_{4}$ Battery * Rong-Xue Qiao(乔荣学)1, Ming-Jian Zhang(张明建)1, Yi-Dong Liu(刘屹东)1, Wen-Ju Ren(任文举)1, Yuan Lin(林原)1,2**, Feng Pan(潘锋)1** Affiliations 1School of Advanced Materials, Peking University, Peking University Shenzhen Graduate School, Shenzhen 518055 2Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190 Received 27 March 2016 *Supported by the Guangdong Innovation Team Project under Grant No 2013N080, and the Peacock Plan of Shenzhen Science and Technology Research under Grant No KYPT20141016105435850.
**Corresponding author. Email: panfeng@pkusz.edu.cn; linyuan@iccas.ac.cn Citation Text: Qiao R X, Zhang M J, Liu Y D, Ren W J and Lin Y et al 2016 Chin. Phys. Lett. 33 078201 Abstract The state of charge (SOC) and state of health (SOH) are two of the most important parameters of Li-ion batteries in industrial production and in practical applications. The real-time estimation for these two parameters is crucial to realize a safe and reliable battery application. However, this is a great problem for LiFePO$_{4}$ batteries due to the large constant potential plateau in the charge/discharge process. Here we propose a combined SOC and SOH co-estimation method based on the experimental test under the simulating electric vehicle working condition. A first-order resistance-capacitance equivalent circuit is used to model the battery cell, and three parameter values, ohmic resistance ($R_{\rm s})$, parallel resistance ($R_{\rm p})$ and parallel capacity ($C_{\rm p})$, are identified from a real-time experimental test. Finally we find that $R_{\rm p}$ and $C_{\rm p}$ could be utilized to make a judgement on the SOH. More importantly, the linear relationship between $C_{\rm p}$ and the SOC is established to make the estimation of the SOC for the first time. DOI:10.1088/0256-307X/33/7/078201 PACS:82.47.Aa, 88.05.Hj, 88.05.Vx, 81.70.-q © 2016 Chinese Physics Society Article Text Due to the advantages of high voltage, light weight, long cycle life and high energy density, lithium ion batteries are widely used as a power source in the field of electric vehicles (EV), and shows very good application prospects.[1-3] The accurate estimation of battery state of charge (SOC) and state of health (SOH) is crucial to realize a safe and reliable battery application. The overcharge or deep discharge due to the inaccuracy of SOC estimation may bring irreversible damage to the battery cell, as well as the whole system. A battery in bad SOH which has not been replaced in time would lead to capacity loss, rapid warming, even a potential safety incident.[4-7] Battery SOC is the ratio between remaining available capacity and the maximum available capacity.[8] Battery SOH represents the battery health status, and can be defined as the ratio between the maximum available capacity and the rated capacity.[9] They are generally expressed in percentage (%). To date, a number of studies on the SOC and SOH estimation methods for lithium ion batteries have been carried out. Typically, there are several methods to derive the SOC. The Coulomb counting method[10] and the open circuit voltage (OCV) method[11-14] are widely used in battery management systems of EV due to the simple and direct characteristics. However, the uncertainty of initial SOC and accumulated errors seriously affect the accuracy of the Coulomb counting method. The OCV method requires the battery remaining unused for a long time to monitor terminal voltage, which limits the online usage. For LiFePO$_{4}$ (LFP) batteries, typical charge/discharge platform enlarges the measuring error of the OCV method.[15] The Kalman filter,[16-19] the sliding mode[20-22] and some intelligent methods, such as neural network,[23-25] fuzzy logic,[26,27] were introduced to enhance the precision of battery SOC estimation. The disadvantage of the former two is that they are computationally expensive, and that of the latter two is that they are too sensitive to the amount and quality of training data. Table 1 summarizes the advantages and disadvantages of these SOC estimation methods. Compared with the SOC estimation, less attention has been paid to the SOH estimation, and few results can be directly used for EV. According to the definition, SOH can be directly measured through a fully charge/discharge process at a low current rate. However, this method cannot be utilized in practical applications due to the fact that most batteries are operated in a range of 20–80% SOC to protect the battery from irreversible damage.[28] In this Letter, a discharge experiment at variable current rates is designed to simulate the working condition of LFP batteries in the EVs. By using an optimal first-order resistance-capacitance (RC) equivalent circuit model as the equivalent circuit model, three parameters are calculated, including ohmic resistance $R_{\rm s}$, parallel resistance $R_{\rm p}$, and parallel capacity $C_{\rm p}$. Finally, we find that $R_{\rm p}$ and $C_{\rm p}$ could be used to judge the SOH. Moreover, the linear relationship between $C_{\rm p}$ and battery SOC is established to evaluate the SOC. Accordingly, a real-time co-estimation method to evaluate the SOC and SOH is built. The method has a higher precision than the OCV method, and could be combined with other methods to reduce the complexity and to improve the accuracy of the estimation work.
Table 1. Advantages and disadvantages of SOC estimation methods.
Method Advantage Disadvantage
Coulomb counting[10] Simple and fast Uncertain to initial SOC, affected
heavily by the accumulated error
OCV method[11-14] Simple and fast Unsuitable for LFP batteries with flat
OCV-SOC curve and requires a
sufficient rest period, unsuitable for
real-time SOC estimation
Kalman filter method[16-19] Accuracy, closed-loop Computationally expensive and highly
Sliding mode method[20-22] depend on the model accuracy
Neural network method[23,25] Good nonlinearity, mapping approximation Sensitive to the amount and quality
Fuzzy logic method[26,27] of training data, slow to converge
LFP-32650 batteries (cathode: LiFePO$_{4}$, anode: graphene, electrolyte: a mixture of EC and DMC with 1 M LiPF$_{6}$ dissolved, size: diameter 32 mm$\times$height 65 mm, rated capacity: 5 Ah) were adopted in our test. According to their SOH, batteries are divided into four categories A, B, C and D, respectively. Each category contains four cells. Table 2 lists the SOH of these batteries. Here the SOH is defined from the aspect of capacity fading, that is, $$\begin{align} {\rm SOH}=C_{\rm aged}/C_{\rm new},~~ \tag {1} \end{align} $$ where $C_{\rm aged}$ is the capacity of the battery after certain cycles, and $C_{\rm new}$ is the original capacity of the brand new battery. When the SOH of a battery is less than 80%, we call it a 'bad battery', and it should be replaced.[29]
Table 2. The SOH of four categories of batteries under test.
Battery categories Charge/discharge SOH
cycles in the past
Healthy batteries A 500 $>90\% $
B 1000 $>90\% $
Bad batteries C 400 $ < 80\% $
D 1200 $ < 80\% $
The working condition test is programmed according to the standard stipulated in the IEC 61982–3–61982. The output power of each cell is converted into the output current, the maximum is 19.5 A (about 4 C). As shown in Fig. 1(a), each discharge stage contains three sections of working condition to be simulated according to the real applications in city driving and one of working condition to be simulated according to the real applications in country driving. A braking energy recovery process (rechargeable) is also included. A fully discharge process includes about 18 discharge stages. Thus we could use the SOC to represent the whole discharge procedure. A general battery model to describe the electrical behavior is usually derived from an equivalent circuit. As shown in Fig. 1(b), $R_{\rm s}$ represents the bulk resistance of the battery, which reflects the electrical conductivity of the electrolyte, the separator, and the electrodes. The parallel connection of $R_{\rm p}$ and $C_{\rm p}$ is used to describe the effects caused by activation polarization and Li-ion diffusion while the battery is charged or discharged. Here $U_{\rm p}$ denotes the terminal voltage of the parallel $R_{\rm p}$ and $C_{\rm p}$. The parameterization values of the resistances and capacitances greatly depend on the particular operating point of the battery and the cell degradation. Therefore, they will change with time, temperature, power demand, the SOC and SOH.[30] All tests in this work are performed under a constant ambient temperature. In the whole test process, we monitored the voltage and current data of each battery by using a battery charge/discharge instrument (Maccor MC16) with the interval time of 1 s. Based on the ohm law, the value of $R_{\rm s}$ could be estimated by Eq. (2). Simultaneously, the values of $R_{\rm p}$ and $C_{\rm p}$ could be calculated by fitting the polarization voltage curve with Eq. (3). Three parameters were calculated by the data of section 1 marked in Fig. 1(a). Section 1 represents a current step from 0 C to 0.5 C, and its zoom-in graph was presented in Fig. 1(b) $$\begin{align} R_{\rm S}=\,&\frac{U_1 -U_2 }{I_2 -I_1},~~ \tag {2} \end{align} $$ $$\begin{align} U_{\rm P}=\,&I\cdot R_{\rm P} \cdot (1-e^{\frac{-t}{R_{\rm P} \cdot C_{\rm P} }}),~~ \tag {3} \end{align} $$ As shown in Fig. 2(a), all the values of $R_{\rm s}$ are calculated according to the above analysis. The calculated $R_{\rm s}$ values for four types of LFP batteries (A, B, C, D) mainly fall in the ranges of 17.9–25.3, 23.5–27.8, 24.2–28.0 and 24.6–29.9 m$\Omega$, respectively. According to the statistical result shown in Fig. 2(a), the $R_{\rm s}$ values of healthy and bad batteries seriously overlap, although the $R_{\rm s}$ values of healthy batteries (A and B) are slightly lower than those of bad batteries (C and D). Thus we could not judge the SOH of LFP batteries according to $R_{\rm s}$. In addition, for healthy batteries, $R_{\rm s}$ of battery B (1000 cycles) do have a slight increase compared with that of battery A (500 cycles). The similar phenomenon is observed in the bad batteries, indicating that $R_{\rm s}$ will increase with the cycle numbers. This is consistent with the previous report.[31] Moreover, $R_{\rm s}$ almost remains unchanged with the decrease of SOC except in the early stage of the discharge for four types of LFP batteries, which makes no sense for the SOC estimation. Consequently, $R_{\rm s}$ could not be used to estimate the SOC and SOH well and accurately.
cpl-33-7-078201-fig1.png
Fig. 1. (a) Current and voltage profiles of LFP batteries in the test dataset; (b) the first-order RC equivalent circuit model for LFP batteries (inset), and a zoom-in current and voltage stage in the test dataset.
cpl-33-7-078201-fig2.png
Fig. 2. (a) The calculated $R_{\rm s}$ as a function of the SOC for four types of LFP batteries (A, B, C and D); (b) the calculated $R_{\rm p}$ as a function of the SOC for four types of batteries: A, B, C, D.
The $R_{\rm p}$ values are also calculated and plotted versus the SOC in Fig. 2(b). As shown in the inset of Fig. 2(b), the $R_{\rm p}$ values of healthy batteries (A and B) are relatively dispersive and slightly higher than that of bad batteries (C and D) in the main part (10–95%) of the whole SOC window. For example, when SOC is 50%, $R_{\rm p}$ of the healthy batteries (A and B) fall in the range of 6.8–8.3 m$\Omega$, and the values of the bad batteries (C and D) fall in the range of 6.0–6.3 m$\Omega$. They are not overlapped. Thus the $R_{\rm p}$ value could be acted on as a criterion to distinguish healthy batteries from bad batteries. According to the variation trend of $R_{\rm p}$ versus the SOC, the inset could be divided into two regions, regions I and II. In region I (50–95%), $R_{\rm p}$ almost remains unchanged, while in region II (10–50%), the $R_{\rm p}$ value increases slowly with the discharge depth. The SOC in the range of 10–50% can be estimated by the $R_{\rm p}$ value. Here $R_{\rm p}$ steeply rises in the end of the discharge period. Based on the above analysis, it is concluded that the SOH could be partly judged by the $R_{\rm p}$ value, which is not an ideal criterion to estimate the SOC. As shown in Fig. 3(a), the $C_{\rm p}$ values are calculated and plotted versus the SOC. The $C_{\rm p}$ values of healthy batteries (A and B) are significantly lower than that of bad ones within the whole SOC range. The $C_{\rm p}$ curves of healthy batteries and bad batteries also do not overlap. In this light, $C_{\rm p}$ can act as an outstanding SOH estimator. Notably, the $C_{\rm p}$ value decreases linearly with the decrease of the SOC for all the batteries when the SOC is less than 60%. This result is obviously different from the plateau characteristic presented in $R_{\rm p}$ and the OCV for LFP batteries. When the SOC is larger than 60%, $C_{\rm p}$ changes slowly without any obvious rising or reducing tendency. Based on the above analysis, $C_{\rm p}$ can be used to fully estimate both the SOH and the SOC concurrently.
cpl-33-7-078201-fig3.png
Fig. 3. (a) The calculated $C_{\rm p}$ as a function of the SOC for four types of batteries (A, B, C and D); (b) and (c) comparison of curves for $C_{\rm p}$ versus SOC between A4 and B4 at different current rates.
According to the above discussion of three parts, we could conclude that $R_{\rm p}$ and $C_{\rm p}$ could be utilized in a two-dimensional way to judge the SOH, and $C_{\rm p}$ could be an ideal parameter to estimate the SOC for LFP batteries. To further confirm the reliability of this method, parameter values at another two current rates (1 C and 4 C marked as 2 and 3 in Fig. 1(b)) are calculated. Particularly, we plot the $C_{\rm p}$ values versus the SOC of healthy batteries (A4 and B4) in Fig. 3(b) and Fig. 3(c). The $C_{\rm p}$ values at these current rates all show excellent linear relationship with the SOC in the range of 0–60%. There seemingly is consistent change rule in the values from 0.5 C to higher current rates, which shows that $C_{\rm p}$ value identified from higher current rates could also be used to judge the SOH. Linear regression models are established to verify the accuracy of this SOC estimation method for healthy batteries and bad batteries, respectively. The significance test of regression equation ($F$), correlation coefficient ($R$) and 95% confidence interval are calculated and listed in Table 3. The values of $F_{\rm c}$ and $R_{\rm c}$ are obtained by look-up table.
Table 3. Analysis results of regression model and estimation error.
Battery categories $F$ $F_{\rm c}$ $R$ $R_{\rm c}$ 95% confidence interval
Healthy batteries 744.13 3.92 0.94 0.21 [SOC$_{0}$-6.33%, SOC$_{0}$+6.33%]
Bad batteries 544.30 3.92 0.92 0.21 [SOC$_{0}$-7.20%, SOC$_{0}$+7.20%]
Here SOC$_{0}$ represents the estimation value of SOC, $F>F_{\rm c}$, indicating that the regression models established are effective, $R$ is close to 1, and $R>R_{\rm c}$, showing a highly linear correlation between SOC and $C_{\rm p}$. The absolute errors of the estimated SOC values for healthy and bad batteries are 6.33% and 7.20%, respectively, further confirming the accuracy of this estimation method. In summary, we have designed and performed the discharge experiments of healthy and bad LFP batteries to simulate various working conditions of EV. Based on a first-order RC equivalent circuit, we systematically investigate the relationship of three parameters ($R_{\rm s}$, $R_{\rm p}$ and $C_{\rm p}$) and the SOC, SOH. Lastly, we find that $R_{\rm p}$ and $C_{\rm p}$ could be used to judge the SOH. Moreover, $C_{\rm p}$ is found to be linearly correlated with the SOC for the first time. Thus the novel method based on $R_{\rm p}$ and $C_{\rm p}$ to co-estimate the SOC and SOH can be herein established, which is more precise than the traditional OCV method. The internal mechanism of this method and the physical meanings of three parameters ($R_{\rm s}$, $R_{\rm p}$ and $C_{\rm p}$) would be further studied based on the finding. We could predict that the combination with this method and other reported methods, such as the Kalman filter, neural network and fuzzy logic, would greatly reduce complexity and improve the accuracy of the estimation work. The LFP-32650 batteries were provided by OptimumNano Energy Co., Ltd. References From single cell model to battery pack simulation for Li-ion batteriesSearch for an Optimal Rapid-Charging Pattern for Li-Ion Batteries Using the Taguchi ApproachBrief overview of electrochemical potential in lithium ion batteriesThe testing of batteries linked to supercapacitors with electrochemical impedance spectroscopy: A comparison between Li-ion and valve regulated lead acid batteriesA practical framework of electrical based online state-of-charge estimation of lithium ion batteriesExtended Kalman filter method for state of charge estimation of vanadium redox flow battery using thermal-dependent electrical modelSequential Monte Carlo filter for state estimation of LiFePO4 batteries based on an online updated modelState-of-Charge (SOC)-Balancing Control of a Battery Energy Storage System Based on a Cascade PWM ConverterSimulation of capacity fade in lithium-ion batteriesEnhanced coulomb counting method for estimating state-of-charge and state-of-health of lithium-ion batteriesCapacity loss in rechargeable lithium cells during cycle life testing: The importance of determining state-of-chargeModeling Battery Behavior for Accurate State-of-Charge IndicationState-of-the-art of battery state-of-charge determinationBattery open-circuit voltage estimation by a method of statistical analysisDynamic electric behavior and open-circuit-voltage modeling of LiFePO4-based lithium ion secondary batteriesExtended Kalman filtering for battery management systems of LiPB-based HEV battery packsState-of-Charge Estimation for Lithium-Ion Batteries Using Neural Networks and EKFLi-ion battery SOC estimation method based on the reduced order extended Kalman filteringAdaptive Kalman filter based state of charge estimation algorithm for lithium-ion batteryNonlinear State of Charge Estimator for Hybrid Electric Vehicle BatteryA Technique for Estimating the State of Health of Lithium Batteries Through a Dual-Sliding-Mode ObserverThe novel state of charge estimation method for lithium battery using sliding mode observerThe available capacity computation model based on artificial neural network for lead–acid batteries in electric vehiclesState of charge estimation of Li-ion batteries in an electric vehicle based on a radial-basis-function neural networkA new battery capacity indicator for lithium-ion battery powered electric vehicles using adaptive neuro-fuzzy inference systemDesign and implementation of a fuzzy logic-based state-of-charge meter for Li-ion batteries used in portable defibrillatorsA Novel Battery State-of-Health Estimation Method for Hybrid Electric VehiclesOnline battery state of health estimation based on Genetic Algorithm for electric and hybrid vehicle applicationsState-of-health monitoring of lithium-ion batteries in electric vehicles by on-board internal resistance estimationBehavior and state-of-health monitoring of Li-ion batteries using impedance spectroscopy and recurrent neural networks
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