Energy 113 (2016) 64e75
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Energy journal homepage: www.elsevier.com/locate/energy
Temperature dependent power capability estimation of lithium-ion batteries for hybrid electric vehicles Fangdan Zheng a, b, c, Jiuchun Jiang a, b, *, Bingxiang Sun a, b, Weige Zhang a, b, Michael Pecht c a b c
National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing, 100044, China Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Jiaotong University, Beijing, 100044, China Center for Advanced Life Cycle Engineering (CALCE), University of Maryland, College Park, MD 20740, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 December 2015 Received in revised form 26 May 2016 Accepted 3 June 2016
The power capability of lithium-ion batteries affects the safety and reliability of hybrid electric vehicles and the estimate of power by battery management systems provides operating information for drivers. In this paper, lithium ion manganese oxide batteries are studied to illustrate the temperature dependency of power capability and an operating map of power capability is presented. Both parametric and nonparametric models are established in conditions of temperature, state of charge, and cell resistance to estimate the power capability. Six cells were tested and used for model development, training, and validation. Three samples underwent hybrid pulse power characterization tests at varied temperatures and were used for model parameter identiﬁcation and model training. The other three were used for model validation. By comparison, the mean absolute error of the parametric model is about 29 W, and that of the non-parametric model is around 20 W. The mean relative errors of two models are 0.076 and 0.397, respectively. The parametric model has a higher accuracy in low temperature and state of charge conditions, while the non-parametric model has better estimation result in high temperature and state of charge conditions. Thus, two models can be utilized together to achieve a higher accuracy of power capability estimation. © 2016 Published by Elsevier Ltd.
Keywords: Lithium-ion battery Hybrid electric vehicle Power capability estimation Temperature dependence Support vector machine Battery management system
1. Introduction Due to the shortage of oil energy and the raising of public environmental awareness , electric vehicles (EVs) are becoming more and more popular . Hybrid electric vehicles (HEVs) are claimed to be the most energy efﬁcient and to produce the lowest amount of greenhouse gas emissions compared to electric vehicles (EVs) and plug-in hybrid electric vehicles (PHEVs) . The hybrid architecture is exploited to achieve better fuel economy and lower exhaust emissions . HEVs improve on the traditional internal combustion engine (ICE) vehicle due to its ability to reduce the emissions of greenhouse gases . With lithium-ion batteries in the vehicle's power system, an HEV combines an ICE and an electric motor. To ensure that the HEV operates at the maximum efﬁciency, the ICE provides average power at constant speed area, while the
* Corresponding author. National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing, 100044, China. E-mail address: [email protected]
(J. Jiang). http://dx.doi.org/10.1016/j.energy.2016.06.010 0360-5442/© 2016 Published by Elsevier Ltd.
electric motor which is usually powered by batteries satisﬁes the high power demand by delivering short, high power discharges and charge current pulses during a vehicle's acceleration, gradient climbing and regenerative braking . As one of the critical components, battery performance determines the safety, reliability and efﬁciency of the vehicle system . Battery's power capability affects the vehicle's acceleration and maximum speed performance  as well as braking performance . Power capability is the ability of a battery to accept or deliver power at a given time . If the battery cannot deliver enough discharging power, the vehicle may fail to restart or inhibit acceleration. If the charging power during a vehicle's regenerative braking operation is beyond the acceptable range for the batteries, the converted energy from the vehicle's kinetic energy would be wasted. Moreover, things may get worse that the charging current may exceed the battery's design limit and result in high current if battery management systems (BMS) did not set a proper current limit. This may cause thermal issues due to rapid heat generation and temperature rise. It may reduce the battery's lifespan by damaging the internal chemical
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materials further. Since batteries' power capability directly affects the safety and reliability of HEV operation, an overview of usable power will be signiﬁcantly helpful. A power performance map which presents batteries' usable power in terms of the operation conditions (i.e. temperature, voltage, available capacity) will provide a guide to fully utilize the batteries to the extremes while maintaining safety. Moreover, accurate estimate of battery power capability provides a basis for the strategy of vehicle power management . Optimal power and energy management  for HEV and PHEV is able to utilize two sources, ICE and batteries, efﬁciently and thus achieve the best fuel economy . Therefore, estimating power capability accurately and generating a power performance map are crucial functions of the BMS. Guidelines for batteries used as an auxiliary propulsion source in HEVs have been established by the US Department of Energy (DOE) for use in the Partnership for a New Generation of Vehicle (PNGV) program . HEV duty cycle and battery requirements are described by Nelson . The power requirements for batteries in HEVs are speciﬁed in terms of a characteristic time instead of a maximum C-rate . To note, a C-rate is a measure of current at which a battery is discharged relative to its maximum capacity. Idaho National Engineering & Environmental Laboratory (INEEL) proposed an evaluation method for determining a battery's power capability called the hybrid pulse power characterization (HPPC) test . Using this method, it is easy to determine a battery's power capability, which is represented by 10-s pulse discharge or charge peak power regarding different factors such as ambient temperature or state of charge (SOC) of the battery. SOC is a measure of the amount of charge stored in a battery at the present moment. In addition, the real driving conditions of HEVs can be simulated through changing the test control conditions based on conducting the HPPC test in laboratory environments. Subsequently, detailed information of a battery's power capability can be imported into a BMS to provide an optimal operation guideline for HEVs. Real-time power demands usually vary with the instantaneous working condition of the HEV. History of power consumption, speed changes and road information can be used to estimate the real-time state of power capability (SOP) of the HEV . To note, SOP is used to measure that the ICE and battery can meet the realtime power demand or not. A variety of studies have been conducted on the online prediction of SOP , which is commonly indicated by peak power . The deﬁnition of peak power as the maximum discharge or charge power that can be maintained constant for10 s within the operational design limits is proposed by Plett . Additionally, he presented a dynamic cell model for available power prediction of battery packs taking account current limit, voltage limit, and SOC limit. The difﬁculty is that this dynamic cell model is hard to simulate for on-board applications due to the low efﬁciency and high cost of complicated computation. To date, there are several approaches for online peak power prediction. Xiong et al.  proposed a dynamic electrochemical-polarization (EP) model based on multiple parameters. A data-driven adaptive SOC and SOP joint estimator was established to which the adaptive extended Kalman ﬁlter (AEKF) was subsequently applied for more convergent results . Efforts to improve the model, such as higher estimation accuracy as well as parameter updates requiring less computation, were made by the authors . Pei et al.  presented a training-free parameter and state estimator for online peak power estimation. An equivalent circuit model was used and a dual extended Kalman ﬁlter (DEKF) was applied for online parameter identiﬁcation. Jiang et al.  presented the testing methods for battery peak power with comparative analysis. In addition, experiments were designed to verify the accuracy of the peak power estimation results in this work. These studies focused
on real-time instantaneous power state prediction of batteries. However, the battery power supply can drop to zero almost instantly once its peak power exceeds the constraint boundary, such as current or voltage limitation, in accordance with the BMS control strategy. The ICE cannot take over providing power immediately since it takes time for the engine to respond. Thus the vehicle would stop moving, which is known as the “car frustration phenomenon” due to the instant disappearance of momentum. This phenomenon would degrade the user experience of driving and could also affect the braking performance of the vehicle, thus leading to trafﬁc accidents. Therefore, estimating exact power capability of an HEV battery and implement a power performance map into BMS in advance can present a view of real-time usable power while driving to guide optimal operation as well as to guarantee the safety and reliability of HEVs. The investigation of power capability was speciﬁed for lithiumion manganese oxide batteries (LiMn2O4), which are widely used in HEVS. As a bulk phase, LiMn2O4 possesses excellent rate capability . Moreover, it is highly favored as a positive electrode due to its merits such as lower cost, lower toxicity, and superior safety than V-, Co-, or Ni-based electrodes . In this study, a 10-s pulse discharge peak power was used to represent the power discharge capability of batteries. Similarly, corresponding application can be adopted for the charging case. Temperature dependency of power capability was investigated based on the experimental results of HPPC test. Both a parametric model and non-parametric model using data-driven approach were built to accurately estimate the power capability of LiMn2O4 batteries. A key advantage is that our models are only based on the data from several parameters whose real-time states are easily obtained while comparing with other model-based peak power estimators, which have more difﬁculty obtaining real-time model parameters accurately, such as the equivalent circuit model-based estimator or the electrochemistry model-based estimator. The performance of two proposed models were compared and evaluated via a list of statistical metrics under different temperatures and different SOCs. The reminder of the paper is arranged as follows. Section 2 demonstrates an experimental platform and tests under varied ambient temperatures for power capability determination. Section 3 analyzes the variation of test samples and illustrates the temperature dependency of power capability. An operating map of power in terms of temperature and SOC are presented as well. In section 4, both the parametric model and non-parametric model using data-driven approach are used for power estimation. The test data of three battery cells are used for model parameter identiﬁcation or model training. In section 5, the model validation results of two models are compared using statistical measures. Detailed discussion is presented including the applicability of two models. Finally, conclusions and suggestions for future work are given in section 6.
2. Experiments The experimental platform is shown in Fig. 1. It consisted of 5 parts: (1) six lithium-ion LiMn2O4 battery cells; (2) a temperature controlled chamber; (3) Digatron battery test system; (4) a data logger to record the battery data; and (5) a PC to give the orders and monitor data information. The test samples were composed of a graphite negative electrode and a lithium manganese oxide (LMO) positive electrode. Their basic speciﬁcations are given in Table 1. The open circuit voltageeSOC test and HPPC test at various ambient temperatures were conducted for the test samples. During the tests, data (current, voltage, and temperature of each cell) was measured and logged in1 second intervals.
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Fig. 1. The battery experimental platform.
Table 1 The speciﬁcations of the battery cells. Type
LiMn2O4 8 Ah
Upper/lower cut-off voltage
10 s pulse charging/discharging power (SOC ¼ 50%)
Maximum discharge current
Usage temperature range
4.2 V/3.0 V
783 W/725 W
e20 C, þ55 C
2.1. Open circuit voltage e SOC test Open circuit voltage (OCV) is the equilibrium potential of the battery which can be measured as terminal voltage when there are no polarization effects. It increases or decreases as the battery's SOC increases or decreases. In other words, OCV performance is a function of SOC:
OCV ¼ f ðSOCÞ
The relationship between OCV and SOC is commonly observed by the OCV e SOC test as follows. Firstly, the battery cell is fully charged via the constant current constant voltage (CCCV) method until the cell's terminal voltage reaches its upper cut-off voltage and the current is less than C/25. The constant current is 1C-rate, which means that the cell will be fully discharged from 100% SOC to 0% SOC in approximately 1 h. Secondly, the battery cell is rested for 2 h. Then the cell is discharged at a constant current of C/25 until the terminal voltage reaches its lower cut-off voltage and its SOC becomes 0%. Then the battery cell is rested for 2 h. Finally, the cell is charged to its upper cut-off voltage using a constant rate of C/25. The low discharge/charge current minimizes the effect of cell dynamics during charge and discharge. Therefore, the cell is at a closeto-equilibrium status and the corresponding terminal voltages are equilibrium potentials. The average value of discharge and charge equilibrium potentials is assumed to be the OCV, which reduces the effects of hysteresis and ohmic voltage. As shown in Fig. 2, the OCVSOC curve determined by this method at 20 C represents the OCVSOC relationship of our test samples.
Fig. 2. OCV-SOC curve by C/25 at 20 C.
2.2. Hybrid pulse power characterization test The HPPC test was proposed by the Idaho National Engineering and Environmental Laboratory (INEEL). It measures the power capability over the vehicle's useable voltage range, and the OCVSOC curve can be obtained as well . At ﬁrst, the battery is discharged to 90% SOC in relation to its rated capacity. Then it is kept in the open-circuit state for 1 h so that the OCV is observed and
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temperatures (20 C, 10 C, 0 C, 10 C, 20 C, 30 C, 40 C and 50 C). Due the nature of experiment and manufacturing variability, variation always exists. To ensure that the experimental data is independent from the variation, the analysis will be conducted in the following section. 3.1. Variations of battery cells
Fig. 3. Experiment time schedule and resistance calculation at 90% SOC.
recorded. After that, the test procedure includes 10-s pulse current discharging, a 40-s rest, and a10-s pulse current charging. Based on these two pulses, the discharge resistance and charge resistance are measured via Ohm's Law. The deﬁnition in equation (2) and Fig. 3 focuses on the discharge operation for the battery, but a corresponding deﬁnition can be used for the battery's charge operation. The discharge power capability is calculated based on the resistance, as shown in equation (3). Here, Vmin is the lower cut-off voltage of the test sample. It is notable that the positive value of the current denotes the discharging condition, and the negative value denotes the charging condition. This discharge pulsedrestdcharge pulse is conducted from 90% SOC to 10% SOC at intervals of 10% SOC, as shown in Fig. 4. Regarding the temperature dependence of the battery's power capability, the HPPC test was conducted from 20 C to 50 C at an interval of 10 C for six battery cells.
Rdis ¼ ðDVdis Þ=DIdis ¼ jV1 V0 =Idis j
Pdis ¼ ðVmin Þ ðOCVdis Vmin Þ=Rdis
Based on the tests conducted in section 2, OCV-SOC curve is determined as shown in Fig. 2. The resulting OCV-SOC curves for all six test samples are shown in Fig. 5. It can be seen that the six curves overlap during the entire SOC interval. In addition, the variations of OCVs among six cells at low (SOC ¼ 0.1), middle (SOC ¼ 0.5), and high (SOC ¼ 0.9) portions of SOC are shown in Fig. 6 using a boxplot. The mean OCV of six cells is 3.5175 V, 3.7105 V, and 4.0325 V at 10% SOC, 50% SOC, and 90% SOC, respectively. Furthermore, considering the average OCV at its corresponding SOC as the baseline, the voltage deviations for six samples are summarized as shown in Fig. 6 (d). The average deviation for OCV at low SOC is about 0.0005 V, whereas the average deviation for OCV is 0.0002 V and 0 V for middle SOC and high SOC, respectively. The variation of OCV-SOC for six cells is signiﬁcantly small, which means that one representative OCV-SOC curve is enough to indicate the relationship between OCV and SOC for all six cells of the same type. This OCV-SOC curve can be used as a look-up
3. Analysis of experimental data Six LiMn2O4 battery samples were tested with OCV-SOC test at 20 C and followed with HPPC test at eight different ambient
Fig. 4. HPPC test sequence .
Fig. 5. OCV-SOC relationship for six cells.
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Fig. 6. Variation of OCVs among the six cells (a) at low SOC level, (b) at middle SOC level, (a) at high SOC level, (d) relative voltage deviation for three SOC levels, measured from OCV e SOC test at 20 C.
table when calculating the power capability. 3.2. Temperature dependence of power capability As shown in Fig. 7, battery's power capability increases as its SOC gets larger. It can be seen that there is a nearly linear correlation between power and SOC even at different temperatures. Fig. 8 shows how power capability changes with temperature when a cell is at a given SOC. The power capability difference among varied SOCs grows larger as the temperature rises. The difference between 0.1 SOC and 0.9 SOC at high temperature (50 C) is two orders of magnitude larger than that at low temperature (20 C). The materials of lithium-ion batteries are affected by temperature, and the corresponding electrochemical model properties are temperaturedependent through the Arrhenius expression . A contour plot is presented in Fig. 9 for which the HPPC test data of six cells at 9 different SOCs and 8 different ambient temperatures are used. The x-axis denotes temperature, which ranges from 20 C to 50 C, and the y-axis denotes SOC, which ranges from 0.1 to 0.9. The varied color areas represent power capability levels throughout a range of temperatures and SOCs. It is worth noting that this ﬁgure can be taken as an operating map for indicating a battery's power
Fig. 8. Power capabilityetemperature curves at different SOC levels of Cell#1.
capability at varied temperatureeSOC conditions. Based on the temperature and SOC information given by the BMS, this operating map which provides a power capability range for users is helpful for optimal vehicle operation.
4. Power capability modeling In this paper, both a parametric model and a non-parametric model were conducted to estimate the power capability of LiMn2O4 batteries. For parametric model, temperature dependency of power capability was expressed by an exponential function. For non-parametric model, ambient temperature was considered as one input variable just like battery internal factors (i.e. SOC, resistance) of a data-driven model.
4.1. Parametric model As mentioned in section 2.1, OCV performances as a function of SOC. In statistics, several regression models are used to ﬁt the OCVSOC function. Based on the literature, four candidate models were selected to ﬁt our OCV-SOC data obtained from the test. According to the study of Plett , the OCV-SOC function is described by the following model. Model 1:
y ¼ a0
a2 s þ a3 lnðsÞ þ a4 lnð1 sÞ
Hu et al.  reported that a sixth-degree polynomial function depicts the OCV-SOC relationship most accurately. Model 2:
y ¼ b0 þ b1 s6 þ b2 s5 þ b3 s4 þ b4 s3 þ b5 s2 þ b6 s
Xiong et al.  proposed another OCV-SOC function. Model 3:
y ¼ g0 þ g1 s þ g2 s2 þ g3 s3 þ
Fig. 7. Power capability e SOC curves at varied temperatures of Cell#1.
þ g5 lnðsÞ þ g6 lnð1 sÞ
Zhang et al.  presented an OCV-SOC function model that is claimed to be suitable for a lithium manganese-based cell. Based on their model, model 4 was put forward, as follows. Model 4:
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Table 2 Goodness-of-ﬁt of candidate models.
Model Model Model Model
1 2 3 4
0.9959 0.9985 0.9994 0.9879
0.0112 0.0068 0.0041 0.0192
relationship in this paper. Then Eq.(8) can be utilized to calculate the power capability based on Eq. (1) and Eq. (3), where Vmin is represented by 3.0 V.
Pdis ¼ 3:0 ½f ðSOCÞ 3:0=Rdis
Fig. 9. Operating map of power capability throughout varied SOC and temperature ranges.
y ¼ a bðlnðsÞÞd1 þ cs þ ded2 ðs1Þ
where y is OCV, s is the SOC, and a0 ~ 4, b0 ~ 6, g0 ~ 6, and a, b, c, d, d1 ~ 2 are the model parameters to be determined in Eqs. (4) to (7). OCV-SOC data from the test in Fig. 2 is used to reﬁt the model parameters via least squares optimization. The ﬁtting results of the 4 candidate models for LiMn2O4 battery OCV-SOC relationship at 20 Care shown in Fig. 10. To compare four models quantitatively, two indicators, the adjusted R-square (R2adj ) and the root mean square error (RMSE), are used to evaluate the goodness-of-ﬁt of these candidate models, as shown in Table 2. R2adj modiﬁes the phenomenon of R-square as well as compensates for the effects of the outliers or extra values in a model relative to the data points . A larger R2adj denotes a better ﬁtting performance of the model, and 1 is a perfect ﬁt. RMSE is the square root of the mean square error, which shows how close a ﬁtted line is to the data points. A smaller RMSE value represents a better ﬁt of a model, and 0 is the best. Model 1, model 2, and model 3 show slightly better performance in R2adj than model 4. Among the ﬁrst three models, model 3 has the largest R2adj value, which is closer to a perfect ﬁt, 1. At the same time, model 3 has smallest RMSE than others. Therefore, model 3 is selected to characterize the OCV-SOC
Fig. 10. OCV-SOC ﬁtting results of 4 candidate models at 20 C.
Battery's OCV-SOC performance varies at different temperatures . Thus, its power capability changes with ambient temperatures. By introducing a function of temperature C(T) which facilitates the reduction of offset caused by the environmental conditions, the accuracy of power capability estimation is improved. The improved power capability estimator is shown below:
Pdis ¼ 3:0 ½f ðSOCÞ 3:0=Rdis þ CðTÞ
HPPC test data sets of Cell#1, Cell#2 and Cell#3were used to ﬁgure out the C values in terms of the ambient temperature. The calculated Pdis value based on Eq.(3) is assumed to be the true value, since both OCV and Rdis were measured from the experiments. The power capability observed by SOC as shown in Eq.(9) is considered to be the estimated value. Fig. 11 shows the true and estimated power capability for three cells at low (10 C), ordinary (20 C), and high (40 C) temperatures. As shown in Fig. 11 (a), the x-axis denotes the number of data sets and the y-axis denotes the power capability. The black solid line represents the true value and the red dashed line represents the estimated value of power capability. Data of seven SOCs (0.1, 0.2, 0.4, 0.5, 0.6, 0.8 and 0.9, respectively) were recorded including power values for each cell at 10 C. The ﬁrst seven data points represent power capabilities for Cell#1; the second and last seven data points are for Cell#2 and Cell#3. Overall, the estimated power capability follows the true value well. In addition, Fig. 11 (b) and (c) show estimated results at 20 C and 40 C, respectively. It can be seen that the power capability estimator works better for Cell#2 than the other two cells from the ﬁgures. This is probably because the OCV-SOC model characterizes the OCV-SOC relationship of Cell#2 best. There are slight differences among test samples, even though they are from the same batch of cells and were supposed to have identical characteristics. The estimated errors and the ﬁtted C values at varied ambient temperatures are shown in Table 3. Battery's power capability increased as its ambient temperature rose. The true power capability is one order of magnitude larger at 50 C (around 1300 W) than that at 20 C (around 130 W), taking 90% SOC as an example. Therefore, the mean absolute error at 50 C accordingly is larger than at 20 C. The mean absolute error is less than 2% of the maximum power capability at 20 C and that at 50 C is around 5%. The maximum absolute error and minimum absolute error are listed in Table 3 as well. The C value-temperature relationship can be modeled by an exponential function (see Fig. 12) on temperature . All the C values and their corresponding ﬁtted curve for 8 temperatures fall into the 95% prediction bounds. The adjusted R-square equals to 0.9914 which indicates that the function ﬁts the C(T) data reasonably well. Furthermore, C values at other temperatures are predictably available. Thus, a parametric model (Model A) for power capability estimation at varied ambient temperatures is established
Fig. 11. The true and estimated power capability for three cells at (a) low temperature (10 C), (b) ordinary temperature (20 C), and (c) high temperature (40 C).
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Table 3 Estimated model parameter and statistics of power capability modeling. T ( C)
Mean absolute error (W)
Maximum absolute error (W)
Minimum absolute error (W)
20 10 0 10 20 30 40 50
3.5315 8.8987 3.5425 33.4591 46.305 72.3353 187.3785 342.7275
2.5869 3.4092 9.0104 16.5291 21.0391 37.4973 49.0354 70.8996
5.5655 8.1968 21.6390 66.3984 47.5794 84.2579 103.3211 191.8510
0.2075 0.5735 1.5622 0.0928 0.7337 4.4325 3.8891 1.4943
Fig. 13. Scatter plot of resistance and power capability.
Fig. 12. The exponential curve ﬁtted result for the C(T) relationship.
as shown in Eq.(9), where C(T) is an exponential function. Model parameters are: battery state of charge (SOC), battery resistance (Rdis) and ambient temperature (T). 4.2. Non-parametric model As mentioned in section 3.1, the characteristics of all six battery samples are assumed to be the same. Cell#1 is taken as an example to study the correlation between SOC and power capability at different ambient temperatures, as shown in Fig. 9. Pearson correlation coefﬁcient is introduced as a measure of linear correlation between the two variables. The coefﬁcient ranges from 1 to þ1; negative value denotes a negative correlation, and vice versa. The closer its absolute value to 1, the stronger the correlation is. The value “0” indicates no correlation between the two variables. A strong linear correlation between SOC and power capability is shown in Table 4. A battery's power capability has a strong dependence on its ambient temperature, as discussed in section 3.2. Moreover, temperature affects cell resistance as well. Cell resistance decreases as temperature rises. A negative correlation between resistance and power capability is shown in Fig. 13. Scatter plot is a type of mathematical diagram using Cartesian coordinates to display
values for a set of data as a collection of points . It can be seen that power capability drops rapidly as resistance increases from 0 to 10 mU, and then power capability tends to drop slowly for large resistance (10 mU) regions. Obviously, correlation between resistance and power capability is nonlinear and cell resistance can be considered as a dependent factor that affects the battery's power capability. Based on what have been discussed, it can be seen that a battery's power capability is subject to the combined actions of different factors. Taking into account the ambient temperature, SOC and battery resistance (Rdis) as factors of power capability, a datadriven approach can be used to build a non-parametric model without investigating the details of the temperature dependency of battery characteristics. In our paper, support vector machine (SVM) was utilized. SVM is suitable for complex system modeling and solving nonlinear problem. It is comparatively efﬁcient since detailed physical knowledge of the system is not necessarily required. It is a nonlinear generalization of the Generalized Portrait algorithm proposed by Vapnik  in 1963 for both classiﬁcation and regression problems. Support vector machines, also called support vector networks, are supervised learning models which analyze data and recognize patterns . The SVM method implements the structural risk minimization (SRM) principle , which aims to ﬁnd the best compromise solution that balances model complexity and learning ability . Unlike the usual empirical risk
Table 4 Pearson correlation coefﬁcient between SOC and power capability. Temperature ( C) Pearson Correlation Coefﬁcient
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minimization, structural risk minimization increases the generalization ability of the model by not only minimizing the ﬁtting error, but also reducing the upper bound of the generalization error . This helps the SVM model to acquire the best promotion ability even with limited sample information . The input variables for a task in original ﬁnite space may not be linearly separable, and thus leads to difﬁculty for people to deal with this task. However, data can be mapped into a high- or inﬁnite-dimensional space where input variables are separated by a hyperplane or set of hyperplanes and thereby the original problem in ﬁnite space can be addressed. The hyperplanes are constructed by SVMs according to statistical theory . SVMs used for regression problems is called support vector regression (SVR), which consists of linear regression and nonlinear regression . Nonlinear SVR is developed from the linear SVR and introduces the kernel function, which operates in the original ﬁnite space but enables mapped data in high-dimensional to become separable; thus the task is addressed . Notably, SVR as a non-linear estimator is more robust than a least-squares estimator because it is insensitive to small changes. To date, Three kernel functions are commonly used : 1) Polynomial function:
q K xi ; xj ¼ xi ; xj þ a
and thus affect the regression results. Generally, any symmetric functions that satisfy Mercer conditions can be chosen as a kernel function according to the Mercer Theorem . In this paper, RBF is selected as model kernel function due to its simple form and better estimation performance than others. Therefore, a non-parametric model based on SVM (Model B) for power capability estimation is established. The input variables are ambient temperature, SOC, and Rdis; the output is the power capability. Similar to parametric model A, Cell#1, Cell#2, and Cell#3 were used for SVR data training, whereas Cell#4, Cell#5, and Cell#6were used for model validation. To give an example, inputs and output for Cell#1are shown in Fig. 14. Here, the x-axis denotes the number of data sets, the y-axis on the left side represents ambient temperature and SOC, respectively, and the y-axis on the right side denotes the power capability. Notably, the black solid line indicates the model's output, power capability. The model training results based on three cells are shown in Fig. 15.It is can be seen from the ﬁgure that the estimated value captures the evolution of power capability for all SOCs and temperatures. The squared correlation coefﬁcient is utilized to evaluate the regression performance of the model. The value closer to 1 indicates a better ﬁt between regression result and the input data. In the training process, model regression squared correlation coefﬁcient is 0.990 which means the regression result ﬁt the data reasonably well.
(10) 5. Results and discussion
2) Radial basis function (RBF):
K xi ; xj ¼ exp
! xi xj 2
3) Sigmoid function:
K xi ; xj ¼ tanh g xi ; xj þ r
wherexi is the input variable and xj is the support vector. In addition, a, q, s, g and r are parameters that deﬁne the kernel's behavior
As mentioned in previous sections, the experimental data of Cell#4, Cell#5 and Cell#6 are used to validate parametric model (Model A) and non-parametric model (Model B). Fig. 16 shows the power capability estimation results of two models. The green circle represents the true power capability, while the black asterisk and blue asterisk represent the estimated value of Model A and Model B, respectively. In the low power capability range (<300 W), the estimation of Model A is closer to the true value than Model B. More intuitive comparisons are shown in Fig. 17, which introduces the mean absolute error (MAE), and in Fig. 18, which introduces the relative error to evaluate the performance of two models. Moreover, mean absolute percentage error (MAPE) is involved as an
Fig. 14. Input variables of Cell#1 in the training process of Model B.
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Fig. 15. Output in the training process of Model B. Fig. 18. Relative error of Model A and Model B.
additional metrics for model comparison. The MAE, relative error and MAPE are given by the following equations, respectively.
n n 1X 1X jfi yi j ¼ je j n i¼1 n i¼1 i
f yi jei j ¼ ; hi ¼ i yi jyi j MAPE ¼
Fig. 16. Power capability estimation results for model validation.
n 1X fi yi *100% n i¼1 yi
where fi is the forecast value, yi is the true value, ei is the modeling error, and hi is the relative error of data set i. MAE is derived from the absolute value of each difference, and it is an index of the average model performance . MAE describes how close the forecasts are to the true value; therefore, it can easily be compared for two models. From the ﬁgures it can be seen that Model A has a higher MAE but a lower MAPE than Model B, and the values are given in Table 5. The relative error is equal to the absolute error divided by the magnitude of the true value. Model A has a larger MAE but a smaller mean relative error than Model B. In fact, the battery characterizes a high power capability (>700 W) with large SOC under high temperature and a low power capability (<300 W) with small SOC under low temperature. A small error leads to a large relative error in the low power region but a small relative error in the high power region. Model B captures the true power capability better in the high power range but worse in the low power range than Model A. Therefore, Model B has a lower MAE and Model A has a lower MAPE. Generally, Model A considers the temperature dependency of a battery's power
Table 5 Comparison between parametric model and non-parametric model.
Fig. 17. Estimated error of Model A and Model B.
Mean absolute error (W) Maximum absolute error (W) Mean absolute percentage error Maximum relative error
28.931 251.903 7.626% 0.579
20.043 298.138 39.732% 16.117
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capability and thus results in a more accurate overall estimation, with a 7.626% mean relative error for all SOCs and temperatures. However, it is notable that Model B can provide a better estimation result than Model A when the cell's power capability is higher than 1000 W. Furthermore, two models can be combined for power capability estimation application to acquire a more accurate power value for battery cells at every data point of temperature and SOC.
The authors would like to thank the members of Battery Management and Application Group at National Active Distribution Network Technology Research Center (NANTEC) of Beijing Jiaotong University and the Battery Group at Center for Advanced Life Cycle Engineering (CALCE) of the University of Maryland.
Rdis DIdis DVdis Pdis Vmin OCVdis R2adj C(T) xi xj fi yi ei
As the main power source for hybrid electric vehicles, LiMn2O4 battery's power capability directly affects the safety and reliability of vehicle operation. In other words, power capability is a signiﬁcant indicator for battery performance evaluation. Ambient temperature and battery SOC affect battery's power capability. This paper presents a detailed analysis of power capability and its related factors. Based on the experimental results of HPPC tests under different temperatures and different SOCs, a strong temperature dependency of power capability is illustrated. The correlation between power capability and temperature is useful for power modeling of lithium-ion batteries. Two models are proposed in this study: a parametric model (Model A) and a non-parametric model using data-driven approach (Model B). For Model A, four SOC-OCV functions were assessed with statistical model selection metrics, and the temperature dependency of power capability was modeled by exponential function. For Model B, the power estimation model was established without detailed physical knowledge based on the SVM approach. The inputs were temperature, SOC, and cell resistance, and the output was the power. The data collected from the HPPC tests under varied temperatures of three cells (Cell#1, Cell#2, and Cell#3) among six test samples were used to estimate the parameters of Model A or to train the support vectors of Model B. The data from the other three cells (Cell#4, Cell#5, and Cell#6) were used to validate both two models. The MAEs of the two models were 28.931 W and 20.043 W, while the MAPEs were 7.626% and 39.732%, respectively. Model A estimated power more accurately than Model B in low temperature and low SOC regimes. Model B estimated power more accurately in high temperature and SOC regimes. In other words, Model A is more accurate for low power capability (<300 W) estimation while Model B is more accurate for high power capability (>700 W). Both Model A and Model B can ensure high estimation accuracy when the power capability falls within the range of middle region (300 We700 W). Therefore, the two models can be implemented in combination in a BMS to improve the accuracy of power capability estimation for on-board application. Furthermore, the proposed approach can be applied to complicated operation situations, for instance, varied SOC conditions, different temperature environments, and diverse aging levels. Besides, the operating map of power capability as a function of temperature and SOC can provide an operation guideline for HEV drivers. The study presented in this paper primarily focused on the power capability modeling and estimation for a single battery cell. In future work, our research will focus on improving developing the power estimation for battery packs, which is more complex than a single cell due to the issue of inconsistencies among cells. In addition, real-time model implementation and readable graphical user interface display will be considered which is absolutely attractive for practical HEV applications. Acknowledgements This work is supported by the National Key Technology Support Program (Grant No. 2013BAG24B02) and the Fundamental Research Funds for the Central Universities (Grant No. 2016YJS144).
hi EV HEV PHEV ICE BMS HPPC SOC SOP OCV C-rate RMSE SVM MAE MAPE
Discharge resistance Current change during discharging Voltage change during discharging Discharge power capability Lower cut-off voltage Open circuit voltage during discharging Adjusted R-square Function of temperature Input variable Support vector Forecast value True value Modeling error Relative error Electric vehicle Hybrid electric vehicle Plug-in hybrid electric vehicle Internal combustion engine Battery management system Hybrid pulse power characterization State of charge State of power Open circuit voltage Capacity rate Root mean square error Support vector machine Mean absolute error Mean absolute percentage error
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