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A novel state of health estimation method of Li-ion battery using group method of data handling
Journal of Power Sources 327 (2016) 457e464
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Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
A novel state of health estimation method of Li-ion battery using group method of data handling Ji Wu, Yujie Wang, Xu Zhang, Zonghai Chen* Department of Automation, University of Science and Technology of China, Hefei, 230027, PR China
h i g h l i g h t s SoH estimation is treated and solved as a control theory issue. Philosophy of human health diagnosis is used as an analogy. Differential geometric is utilized to analyze battery terminal voltage curve. Group method of data handling is employed to estimate SoH. The method is applied to different types of battery showing universality.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 April 2016 Received in revised form 25 June 2016 Accepted 17 July 2016
In this paper, the control theory is applied to assist the estimation of state of health (SoH) which is a key parameter to battery management. Battery can be treated as a system, and the internal state, e.g. SoH, can be observed through certain system output data. Based on the philosophy of human health and athletic ability estimation, variables from a specific process, which is a constant current charge subprocess, are obtained to depict battery SoH. These variables are selected according to the differential geometric analysis of battery terminal voltage curves. Moreover, the relationship between the differential geometric properties and battery SoH is modelled by the group method of data handling (GMDH) polynomial neural network. Thus, battery SoH can be estimated by GMDH with inputs of voltage curve properties. Experiments have been conducted on different types of Li-ion battery, and the results show that the proposed method is valid for SoH estimation. © 2016 Elsevier B.V. All rights reserved.
Keywords: State of health Control theory Group method of data handling Differential geometry
1. Introduction Battery, especially lithium-ion battery, has been widely used in many renewable energy systems as an energy storage equipment because of its high energy density, stability and long service life [1e3]. However, battery would have a different performance when the health state changed. Furthermore, it will be unstable and risky when the battery is in a poor health state. Accordingly, a large amount of battery management systems (BMS) are developed to monitor battery health states, elevate battery safety and even enhance battery useful life for practical operation. The SoH is applied to describe how an aged battery differs from a fresh battery [4]. It is an internal parameter which cannot be measured directly, but just estimated. The same problem occurs in
* Corresponding author. E-mail address: [email protected] (Z. Chen). http://dx.doi.org/10.1016/j.jpowsour.2016.07.065 0378-7753/© 2016 Elsevier B.V. All rights reserved.
control theory where indirect effects of system internal state can only be inferred from the system outputs [5,6]. In this paper, the system is a Li-ion battery, whose internal state is the SoH, which cannot be measured directly. And the system outputs are the external features of the battery. Thus, the resolution of SoH estimation can be separated into 3 steps. The first step is to select suitable outputs (representative external features) of the battery, whose measurable quantities would be obtained by sensors in the BMS. The second step is to establish a model to describe the relationship between the internal state and external feature using mathematical functions. And the third step is to compute the internal state by using selected algorithms, e.g. nonlinear filters [7e10], machine learning methods [11e13]. Peculiarly, the second and the last step are usually combined into one in machine learning methods. Several different types of battery external features have been employed for SoH estimation. The features, e.g. the double layer
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neural network. Experiment results are presented and analyzed in Section 4. The conclusion is given in Section 5.
capacitances, the charge transfer resistances, the Warburg impedances and the electrolyte resistances, were extracted from electrochemical impedance spectroscopy (EIS) data and used as system output to describe SoH in Ref. [14]. A large amount of battery charge/discharge data of voltage at a low current (e.g. C/3) has been used by Feng et al. [15]. And, with the help of probability density function, the relationship between battery SoH and voltage data was revealed. The equivalent direct current resistances of Li-ion batteries under different health conditions during charging period have been obtained and utilized for SoH determination by Tsang et al. [16]. Currents, terminal voltages and state of charges (SoCs) were measured and computed in Ref. [17], and were then used to estimate battery SoH. Six critical factors of the battery, i.e. the peak power at 30% SoC, the capacity, the open circuit voltage, the voltage drop at 30% SoC with a C/3 pulse, the temperature rises at the end of discharge and charge at 1C, were required in Ref. [18] to describe SoH. Likewise, many valuable models and algorithms have been presented for SoH estimation in recent years. SoC and SoH were detected by a dual-sliding-mode observer based on the RC model which was presented using data of charge/discharge current, terminal voltage and temperature in Ref. [19]. Adaptive neural network and linear prediction error method were used by Rezvani et al. [20] for the SoH quantification and remaining useful life prediction of Li-ion battery cells. Hu et al. [21] presented a hybrid of coulomb counting and extended Kalman filter techniques to estimate battery capacity which indicates cell SoH. Chen et al. [22] established a battery model including the diffusion capacitance which was used to determine SoH. Then Genetic Algorithm (GA) was employed to identify model parameters and estimate SoH. Weng et al. [23] provided a quantitative relation between incremental capacity peaks and faded battery capacity using support vector regression (SVR), which was also developed to accomplish the task of SoH observation. In Ref. [24], a mixture of Gaussian process (MGP) was used to capture the time-varying degradation behavior of Li-ion battery, and particle filter was applied in the implementation of SoH monitor based on the MGP model. However, considering the limitation of the measurement devices in present BMS, many external features of the battery are hard or even impossible to be obtained in actual operation. Moreover, the applications of the above mentioned methods are also limited by the computational capability of real BMS. To address these issues, a suitable feature and an effective machine learning algorithm are utilized to estimate SoH in this paper. Specifically, terminal voltage curve at constant charge current is set as the feature to reflect SoH. A differential geometry based approach (DGA) is exploited to acquire featured parameters from thousands of sample points in the experimental data, followed by the application of GMDH polynomial neural network to establish the relation between voltage curves and SoH and the detection of SoH. Furthermore, experimental data from different Li-ion battery cells are applied to validate the SoH estimation method presented in the paper. The rest of this paper is organized as follows: In Section 2, battery external feature from CC charge subprocess is expressed by parameters using DGA. Section 3 introduces the GMDH polynomial
2. SoH reflection based on battery external features Battery external properties and features would change with the degradation of the battery. It is important to find a set of appropriate battery outputs to depict the variation of the SoH. 2.1. SoH definition A universal definition of the battery SoH is given in Eq. (1).
SoH ¼
Cbat 100% C0
(1)
where Cbat is the present capacity, C0 is the capacity when the battery is fresh. Experimental data from NASA battery data set [25] is used in this paper. 3 18650-size Li-ion battery cells with same type are tested through 3 operational profiles (charge, discharge and impedance measurement) at room temperature of 25 C. The repetitive cycles of charge and discharge process are stopped when a certain cycle number is reached. The operating condition of NASA battery data set is shown in Table 1. Among them, battery # 6 has been over-discharged badly in every cycle. Moreover, SoH of the experimental battery cells from cycle 1 to cycle 168 are plotted in Fig. 1. As shown in Fig. 1, battery SoH changes as cycle number increases. Moreover, the curves in Fig. 1 also indicate that battery in different cycle number would have a same value of SoH, which means that SoH and cycle life are not equivalent for the batteries used by NASA. This phenomenon is easy to understand from the perspective of human health and kinesiology: health state and athletic ability may be similar for people in different ages. In the area of medicine and kinesiology, the health state and athletic ability of a person can be reflected by a set of indices whose parameters can be acquired through tests of specific process. For example, VO2max is used for evaluating the effects of aerobic exercise programs and classifying individuals for health risks. In Ref. [26], VO2max is predicted through the selected experimental data from a submaximal exercise test. Similarly, battery SoH can also be estimated based on the data of specific charging or discharging process. Thus, battery SoH is described and estimated based on the abovementioned philosophy in this paper. 2.2. External feature selection There are only a limited number of battery external features which can be captured by sensors in actual BMS to depict SoH. Some of them can be measured directly, e.g. terminal voltage, current, temperature, while some of them are obtained through calculation, e.g. open circuit voltage, incremental capacity and differential capacity. Moreover, to the best of our knowledge, charging process of the Li-ion battery during common actual application consists of 2 subprocesses, constant current (CC) charge and constant voltage (CV) charge. In practical operation, the value
Table 1 Test condition of NASA battery data set. Battery no.
Constant charge current (A)
Charge cut-off voltage (V)
Discharge current (A)
Discharge cut-off voltage (V)
Nominal capacity (Ah)
5 6 7
1.5 1.5 1.5
4.2 4.2 4.2
2.0 2.0 2.0
2.7 2.5 2.2
2.0 2.0 2.0
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of battery charging current is controlled by the BMS. Terminal voltage curves during CC charge subprocess of the battery at different SoH are shown in Fig. 2, whose shape varies from each other. For example, the line of the battery at 100% SoH has a bigger slope than the batteries at worse SoH before 300 s. Moreover, the slope of the terminal voltage curve changes more slowly while battery health deteriorates. Additionally, the values of initial voltage, mean voltage and final voltage are not equal in different curves. In summary, battery terminal voltage curve in specific charge/discharge process (CC charge subprocess) can reflect battery SoH. Thus it has been selected as the external feature for SoH detection. Considering the time cost of CC charge period of the battery with least SoH, terminal voltages are measured in the data collection interval (DCI), as shown in Fig. 2. The DCI is set from the charging start time to 1000 s later (from 0% SoC to about 60% SoC) to make it totally appropriate for batteries at different SoH. Fig. 1. SoH of the experimental batteries.
2.3. Curve description There are thousands of voltage points which may be measured in the DCI, which means that it is unpractical to utilize all of these voltage data to detect SoH in actual operation. Thus, typical features need to be extracted from certain number of the terminal voltage samples for curve representation. These features can be treated as a summary of the voltage curves and will be used as the inputs of the proposed machine learning algorithm in this paper. The properties of curve can be described by arclength, velocity, unit tangent, curvature, binormal, etc. in differential geometry [27,28]. Considering the characteristic of the discrete battery terminal voltage samples and the computation complexity in actual operation, value and velocity of selected voltage points are applied to describe the curve and act as the inputs of the GMDH in this paper. Velocity means the changing rate of voltage value with time. Thus, velocity, and arclength curvature are applied in DGA as the references for voltage sample selection from the DCI. The DGA is shown in Table 2. 3. GMDH based SoH estimation Fig. 2. Terminal voltages at CC charge subprocess of battery # 5.
Modeling the relation between battery outputs and internal
Table 2 Procedure of the DGA for voltage sample selection. Step 1 Step 2
Step 3
Initialization Confirm the number of target samples. In this paper, the number is equal to 4 in order to reduce the complexity of SoH estimation algorithm. Velocity calculation Calculate velocity of each sample by the equation below. fv ðkÞ ¼ Vkþ1 Vk
(2)
Arclength calculation Arclength in this paper is the length of DCI and can be calculated by accumulating the length of the curve in every unit interval of DCI [27]. The computation of approximate arclength of the DCI is given in Eq. (3). N 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X fa ðkÞ ¼ ðVkþ1 Vk Þ2 *h2 þ t 2 (3) k¼1
Step 4
where N is the number of voltage points, Vk is the terminal voltage at kth time step and t is the sample interval. h is the coefficient to transform x-axis and y-axis approximately and is equal to 1000/7 here. Curvature calculation Curvature is a measure of the bent degree of a curve at a certain point [28]. The curvature of each sample point in DCI is calculated by the function below. jV 2Vkþ1 þ Vk j fe ðkÞ ¼ h kþ2 (4) i3=2 1 þ ðVkþ1 Vk Þ2
Step 5
where Vk is the terminal voltage at kth time step. Sample selection 4 sample points are selected, such as the first sample with a velocity less than a certain value v (v is set to 0.005 in this paper), the last sample in the DCI, the point in the middle of the arc and the point with maximum curvature.
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state is fundamental for proper battery state estimation. Without an accurate battery model, it would be unrealistic to achieve a valid result for SoH estimation. In this section, GMDH polynomial neural network is employed to address the issue of modeling and estimation.
variables and single output can be established by the discrete form of functional Volterra series, which is Kolmogorov-Gabor polynomial, shown in Eq. (5).
3.1. GMDH polynomial neural network
y ¼ f ðXÞ
Due to the complexity of battery electrochemical reaction mechanisms, it is very difficult to describe the relation between battery SoH and battery terminal voltage curve, or the relation between SoH and the abovementioned geometric variables in Section 2, by using a certain equation. To address this issue, GMDH polynomial neural network is employed to establish the model of terminal voltage curve and battery SoH. Moreover, as mentioned in Section 1, this machine learning algorithm may also have the ability to estimate SoH. GMDH polynomial neural network is a self-organized deep learning algorithm originated by Prof. Alexey G. Ivakhnenko [29]. It has been used in a great variety of fields, e.g. data mining, knowledge discovery, forecasting and complex system modeling. For the principle of the GMDH, the relation between multiple input
¼ a0 þ
M X
ai xi þ
i¼1
M X M X i¼1 j¼1
aij xi xj þ
M X M X M X
aijk xi xj xk þ /
i¼1 j¼1 k¼1
(5) where y is the output, X is the input vector, xi, xj and xk are the input variables, ai, aj and ak the weights and M is the number of the input variables. The multiple input variables are the voltages and velocities of the selected samples, as diagrammatically shown in Fig. 3, and the single output is the SoH. Moreover, the structure of the GMDH of the paper is shown in Fig. 4. For the proposed GMDH, the output can be estimated by a series of partial quadratic polynomials which have just two input variables. The quadratic polynomial is shown in Eq. (6).
Fig. 3. (a) Feature voltage. (b) Feature velocity.
Fig. 4. Structure of the GMDH.
J. Wu et al. / Journal of Power Sources 327 (2016) 457e464
b y ¼ a0 þ a1 xi þ a2 xj þ a3 xi xj þ a4 x2i þ a5 x2j
detected by using a specific process, the CC charge subprocess.
(6)
where b y is the estimated value of each output. This quadratic polynomial is used to connect the hidden neurons and the inputs, the output and the hidden neurons, and two hidden neurons. For example, the SoH, which is the output of the proposed GMDH, can be estimated by two variables from the 4th hidden layers of GMDH using Eq. (7).
b ¼ f O xH4 ; xH4 h i j
2 2 O H4 O H4 O H4 H4 O H4 H4 þ aO ¼ aO 5 xj 0 þ a1 xi þ a2 xj þ a3 xi xj þ a4 xi (7)
where b h is the estimated SoH, f O ðÞ is a quadratic polynomial connecting neurons in the 4th hidden layer and output, aO i are the weights between the 4th hidden layer neurons and the output, and xH4 are the neurons in the 4th hidden layer. i Moreover, based on the combination of the partial quadratic polynomials, battery SoH can then be estimated by the input geometric properties. The expression is given in Eq. (8).
b ¼ f O xH4 ; xH4 h i j H3 H3 ; f H4 xH3 ¼ f O f H4 xH3 i ; xj k ; xl ¼ f O f H4 f H3 f H2 f H1 ðÞ; /
(8)
¼ f ðIÞ where f H4 ðÞ, f H3 ðÞ, f H2 ðÞ and f H1 ðÞ are the quadratic polynomials connecting different layers, I is the input vector, and xH3 is the i neuron in the 3rd hidden layer. After the construction of modeling, the weights in each polynomial are obtained by minimizing the mean square error (MSE) using least square (LS) method. The expression of MSE is given below. N 1 X 2 ðyn b ynÞ N n¼1
461
(9)
where N is the length of training data. 3.2. SoH estimation The procedure of battery SoH estimation based on the proposed method can be separated into several steps. Firstly, battery terminal voltages during CC charge subprocess at different SOH should be measured, where the sample area is the defined DCI. Secondly, properties are selected by DGA to describe battery charge curves which are dissimilar at different SoH. Then, the GMDH polynomial neural network is trained by the chosen variables and SoH. Battery voltages and velocities are the input of GMDH, and SoH is the output. In this paper, aiming at balancing the demand of computation and the accuracy of SoH estimation, the number of input is set to 8, the quantity of the hidden layers is 4, and the maximum number of hidden layer neurons is equal to 20. Finally, battery SoH can be detected by the established GMDH. This process is shown in Fig. 5. It is important to note that the abovementioned procedure for SoH estimation is consistent with the principle of control theory in Section 1. Battery internal state, e.g. SoH, is observed by utilizing the system outputs, and properties of battery terminal voltage curve. Moreover, similar to the estimation of human health state and athletic ability described in Section 2, battery SoH can also be
4. Experiment and analysis Experimental data from NASA battery data set is used to validate the accuracy of the SoH estimation method based on DGA and GMDH. Furthermore, to verify the universality of the presented method, experiments results of the IFP1865140 type LiFePO4 battery, which was developed by Hefei Guoxuan High-Tech Power Energy CO. LTD of China, and widely used in electric vehicles, hybrid electric vehicles and micro-grids, are also analyzed. Simulation of the experimental data is implemented by MATLAB in this paper.
4.1. NASA battery data set Voltage samples from the DCI are obtained from NASA battery data set. Experimental results of battery No. 5, No. 6 and No. 7 in the data set are discussed in this paper. Each battery has been charged and discharged circularly during the experiment with a total cycle number of 168. Furthermore, variables selected by the DGA from 100 random cycle data is used to train the GMDH, and the rest of the experimental data is applied to test the estimation accuracy of the proposed method. The estimation results are plotted in Fig. 6. GMDH-DGA represents the proposed SoH estimation method based on GMDH using the DGA for input selection. And the true value of SoH is denoted by Real. As shown in Fig. 6 (a), 6 (b) and 6 (c), the estimation result matches the real SoH very well, which clearly indicates that the SoH can be accurately estimated by the GMDH using DGA as input selection approach. Moreover, according to Fig. 6 (d), the maximum estimation error of the presented method is 5%. In addition, a comparison with importance sampling (IS), which is used for battery terminal voltage sample selection and which was introduced in our previous study [13], is implemented to show the validity of the DGA. In order to ensure fairness, the same GMDH is used during the comparison. Moreover, an estimation method based on the random selected voltage samples is also presented in the comparison. The result of the comparison is shown in Fig. 7. The GMDH-RS in Fig 7 (a) means a GMDH method with random selected inputs, and GMDH-IS is the estimation method using GMDH and IS. It is obvious that GMDH-DGA has a better estimation performance than GMDH-RS. Fig 7 (b) shows the estimation errors of GMDH with different input selection approaches. Accordingly, it is undoubted that the terminal voltage curves during CC charge subprocess and GMDH algorithm are suitable for SoH detection. Furthermore, the numeric results of the comparison of all the three batteries are shown in Table 3. For battery No. 5, GMDH-DGA has a mean absolute error (MAE) of 0.4503, which is the least value among the three comparative methods. It is a 38.8% improvement of the GMDH-RS and about 15% of IS either. The same phenomenon occurs in the comparisons of battery No. 6 and MSE. As the table shows, GMDH-DGA and GMDHIS have similar estimation errors for battery No. 7, and still much better than GMDH-RS. These abovementioned comparisons indicate that the geometric properties of battery terminal voltage curves selected by DGA can be used to estimate SoH. Moreover, since SoH is estimated by GMDH-DGA, GMDH-IS and GMDH-RS based on the GMDH with same structure, these methods would have almost the same training complexity and estimation efficiency. Thus the method based on the geometric property analysis using GMDH is proven to be a more reliable performance for SoH estimation.
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Fig. 5. Procedure of SoH estimation method.
Fig. 6. (a) SoH estimation results of battery # 5. (b) SoH estimation results of battery # 6. (c) SoH estimation results of battery # 7. (d) SoH estimation errors.
4.2. IFP1865140 type LiFePO4 battery In this paper, more than 2000 cycles of the terminal voltages of the experimental LiFePO4 battery during charge and discharge process were measured. The battery testing platform is composed of a 63640-80-80 type DC electronic load and a programmable 62006P-30-80 type DC power supply model both develop by Chroma ATE Inc. The temperature of the experimental environment
is 25 C. A data set consists of 200 cycles of battery charge and discharge data was applied to validate the proposed SoH estimation method, where 120 cycles of data was used to train the GMDH, the other 80 cycles are the test data. To make the DCI consistent for different types of batteries, the DCI of the IFP1865140 type LiFePO4 battery is set from 0 s to 3600 s (from 0% SoC to about 60% SoC). Battery terminal voltage during one single cycle is plotted in
J. Wu et al. / Journal of Power Sources 327 (2016) 457e464
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Fig. 7. (a) SoH estimation results of the GMDH using different curve description methods. (b) Estimation errors of the test data.
Table 3 Numerical results of different GMDH based SoH estimation methods. GMDH with
a
MAE (%) MSE (%2) a
Battery # 5
Battery # 6
Battery # 7
RS
IS
DGA
RS
IS
DGA
RS
IS
DGA
0.7353 1.1507
0.5309 0.6213
0.4503 0.4258
1.0942 2.4347
0.5543 0.5343
0.4087 0.3600
0.6927 1.6739
0.4700 0.5953
0.4954 0.5640
MAE ¼ Mean absolute error. MSE ¼ Mean squared error.
Fig. 8. (a) Battery terminal voltage during one cycle. (b) Terminal voltages during CC charge subprocess at different SoH. (c) SoH estimation results. (d) SoH estimation errors.
Fig. 8 (a). During the cycle, battery is CC charged by 0.5 C until its terminal voltage reaches 3.65 V, and the discharge cut-off voltage is set to 2.0 V. The rest time between charge and discharge is 10 min. Battery terminal voltage curves at different SoH are plotted in Fig. 8 (b), and so does the DCI of this battery. The SoH estimation results of the test data are shown in Fig. 8 (c). And errors are plotted in
Table 4 Numerical results of SoH estimation of IFP1865140 type battery.
MAE (%) MSE (%2)
GMDH-RS
GMDH-IS
GMDH-DGA
1.3639 3.4864
0.7777 1.8117
0.8125 1.3119
Fig. 8 (d). Clearly, GMDH-DGA can achieve an accurate estimation result of SoH with a maximum error of about 5%. Numeric results of comparison among GMDH-RS, GMDH-IS and GMDH-DGA are shown in Table 4. The MAE of GMDH-DGA is 40.43% less than GMDH-RS. Moreover, the methods using DGA offer a minimum MSE among these estimation methods. Due to the voltage plateau of LiFePO4 battery under CC charge subprocess, GMDH-IS also showed a dependable performance of SoH detection in this comparison. Estimation results and statistical properties in Fig. 8 and Table 4 validate that the proposed GMDH-DGA can reach a good estimation result for the IFP1865140 type LiFePO4 battery as well as the Li-ion battery used for NASA data set. Hence, the universality of the developed SoH
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estimation method is verified by this comparison. 5. Conclusion In this study, Li-ion battery SoH is estimated by GMDH and DGA under the direction of control theory. The SoH is observed based on battery external features. Similar to the approach of human health and athletic ability description, geometric properties of battery terminal voltage curves during a specific process, which is a CC charge subprocess, are calculated by DGA to describe the SoH. The voltage of the selected samples is applied as the input of GMDH as well as their velocity. Relation between the geometric properties and SoH is established by GMDH, which is also applied to estimate the SoH. Experimental results show GMDH-DGA can detect battery SoH accurately with a margin of estimation error of 5%. Additionally, experiment has also been conducted on a LiFePO4 battery. The accurate estimation results verify the universality of the proposed method. In summary, the proposed method using GMDH and DGA can achieve an accurate and universal SoH estimation result for Liion batteries. Our future study will focus on SoH estimation considering the interference of voltage sampling and the inconsistency of batteries in the actual operation. Acknowledgments This work is supported by the National Natural Science Fund of China (Grant No. 61375079) and China Scholarship Council (No. 201606340099). The authors wish to thank Dr. Ya Zhang from the First Affiliated Hospital of Anhui Medical University and Dr. Xi Wang from the Institute of Intelligent Machines, Chinese Academy of Sciences for their valuable advices. Moreover, we would also like to express our sincere thanks to Mr. Jun Zhang for his useful suggestions. References [1] Y. He, X. Liu, C. Zhang, Z. Chen, Appl. Energy 101 (2013) 808e814.
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Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
A novel state of health estimation method of Li-ion battery using group method of data handling Ji Wu, Yujie Wang, Xu Zhang, Zonghai Chen* Department of Automation, University of Science and Technology of China, Hefei, 230027, PR China
h i g h l i g h t s SoH estimation is treated and solved as a control theory issue. Philosophy of human health diagnosis is used as an analogy. Differential geometric is utilized to analyze battery terminal voltage curve. Group method of data handling is employed to estimate SoH. The method is applied to different types of battery showing universality.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 April 2016 Received in revised form 25 June 2016 Accepted 17 July 2016
In this paper, the control theory is applied to assist the estimation of state of health (SoH) which is a key parameter to battery management. Battery can be treated as a system, and the internal state, e.g. SoH, can be observed through certain system output data. Based on the philosophy of human health and athletic ability estimation, variables from a specific process, which is a constant current charge subprocess, are obtained to depict battery SoH. These variables are selected according to the differential geometric analysis of battery terminal voltage curves. Moreover, the relationship between the differential geometric properties and battery SoH is modelled by the group method of data handling (GMDH) polynomial neural network. Thus, battery SoH can be estimated by GMDH with inputs of voltage curve properties. Experiments have been conducted on different types of Li-ion battery, and the results show that the proposed method is valid for SoH estimation. © 2016 Elsevier B.V. All rights reserved.
Keywords: State of health Control theory Group method of data handling Differential geometry
1. Introduction Battery, especially lithium-ion battery, has been widely used in many renewable energy systems as an energy storage equipment because of its high energy density, stability and long service life [1e3]. However, battery would have a different performance when the health state changed. Furthermore, it will be unstable and risky when the battery is in a poor health state. Accordingly, a large amount of battery management systems (BMS) are developed to monitor battery health states, elevate battery safety and even enhance battery useful life for practical operation. The SoH is applied to describe how an aged battery differs from a fresh battery [4]. It is an internal parameter which cannot be measured directly, but just estimated. The same problem occurs in
* Corresponding author. E-mail address: [email protected] (Z. Chen). http://dx.doi.org/10.1016/j.jpowsour.2016.07.065 0378-7753/© 2016 Elsevier B.V. All rights reserved.
control theory where indirect effects of system internal state can only be inferred from the system outputs [5,6]. In this paper, the system is a Li-ion battery, whose internal state is the SoH, which cannot be measured directly. And the system outputs are the external features of the battery. Thus, the resolution of SoH estimation can be separated into 3 steps. The first step is to select suitable outputs (representative external features) of the battery, whose measurable quantities would be obtained by sensors in the BMS. The second step is to establish a model to describe the relationship between the internal state and external feature using mathematical functions. And the third step is to compute the internal state by using selected algorithms, e.g. nonlinear filters [7e10], machine learning methods [11e13]. Peculiarly, the second and the last step are usually combined into one in machine learning methods. Several different types of battery external features have been employed for SoH estimation. The features, e.g. the double layer
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neural network. Experiment results are presented and analyzed in Section 4. The conclusion is given in Section 5.
capacitances, the charge transfer resistances, the Warburg impedances and the electrolyte resistances, were extracted from electrochemical impedance spectroscopy (EIS) data and used as system output to describe SoH in Ref. [14]. A large amount of battery charge/discharge data of voltage at a low current (e.g. C/3) has been used by Feng et al. [15]. And, with the help of probability density function, the relationship between battery SoH and voltage data was revealed. The equivalent direct current resistances of Li-ion batteries under different health conditions during charging period have been obtained and utilized for SoH determination by Tsang et al. [16]. Currents, terminal voltages and state of charges (SoCs) were measured and computed in Ref. [17], and were then used to estimate battery SoH. Six critical factors of the battery, i.e. the peak power at 30% SoC, the capacity, the open circuit voltage, the voltage drop at 30% SoC with a C/3 pulse, the temperature rises at the end of discharge and charge at 1C, were required in Ref. [18] to describe SoH. Likewise, many valuable models and algorithms have been presented for SoH estimation in recent years. SoC and SoH were detected by a dual-sliding-mode observer based on the RC model which was presented using data of charge/discharge current, terminal voltage and temperature in Ref. [19]. Adaptive neural network and linear prediction error method were used by Rezvani et al. [20] for the SoH quantification and remaining useful life prediction of Li-ion battery cells. Hu et al. [21] presented a hybrid of coulomb counting and extended Kalman filter techniques to estimate battery capacity which indicates cell SoH. Chen et al. [22] established a battery model including the diffusion capacitance which was used to determine SoH. Then Genetic Algorithm (GA) was employed to identify model parameters and estimate SoH. Weng et al. [23] provided a quantitative relation between incremental capacity peaks and faded battery capacity using support vector regression (SVR), which was also developed to accomplish the task of SoH observation. In Ref. [24], a mixture of Gaussian process (MGP) was used to capture the time-varying degradation behavior of Li-ion battery, and particle filter was applied in the implementation of SoH monitor based on the MGP model. However, considering the limitation of the measurement devices in present BMS, many external features of the battery are hard or even impossible to be obtained in actual operation. Moreover, the applications of the above mentioned methods are also limited by the computational capability of real BMS. To address these issues, a suitable feature and an effective machine learning algorithm are utilized to estimate SoH in this paper. Specifically, terminal voltage curve at constant charge current is set as the feature to reflect SoH. A differential geometry based approach (DGA) is exploited to acquire featured parameters from thousands of sample points in the experimental data, followed by the application of GMDH polynomial neural network to establish the relation between voltage curves and SoH and the detection of SoH. Furthermore, experimental data from different Li-ion battery cells are applied to validate the SoH estimation method presented in the paper. The rest of this paper is organized as follows: In Section 2, battery external feature from CC charge subprocess is expressed by parameters using DGA. Section 3 introduces the GMDH polynomial
2. SoH reflection based on battery external features Battery external properties and features would change with the degradation of the battery. It is important to find a set of appropriate battery outputs to depict the variation of the SoH. 2.1. SoH definition A universal definition of the battery SoH is given in Eq. (1).
SoH ¼
Cbat 100% C0
(1)
where Cbat is the present capacity, C0 is the capacity when the battery is fresh. Experimental data from NASA battery data set [25] is used in this paper. 3 18650-size Li-ion battery cells with same type are tested through 3 operational profiles (charge, discharge and impedance measurement) at room temperature of 25 C. The repetitive cycles of charge and discharge process are stopped when a certain cycle number is reached. The operating condition of NASA battery data set is shown in Table 1. Among them, battery # 6 has been over-discharged badly in every cycle. Moreover, SoH of the experimental battery cells from cycle 1 to cycle 168 are plotted in Fig. 1. As shown in Fig. 1, battery SoH changes as cycle number increases. Moreover, the curves in Fig. 1 also indicate that battery in different cycle number would have a same value of SoH, which means that SoH and cycle life are not equivalent for the batteries used by NASA. This phenomenon is easy to understand from the perspective of human health and kinesiology: health state and athletic ability may be similar for people in different ages. In the area of medicine and kinesiology, the health state and athletic ability of a person can be reflected by a set of indices whose parameters can be acquired through tests of specific process. For example, VO2max is used for evaluating the effects of aerobic exercise programs and classifying individuals for health risks. In Ref. [26], VO2max is predicted through the selected experimental data from a submaximal exercise test. Similarly, battery SoH can also be estimated based on the data of specific charging or discharging process. Thus, battery SoH is described and estimated based on the abovementioned philosophy in this paper. 2.2. External feature selection There are only a limited number of battery external features which can be captured by sensors in actual BMS to depict SoH. Some of them can be measured directly, e.g. terminal voltage, current, temperature, while some of them are obtained through calculation, e.g. open circuit voltage, incremental capacity and differential capacity. Moreover, to the best of our knowledge, charging process of the Li-ion battery during common actual application consists of 2 subprocesses, constant current (CC) charge and constant voltage (CV) charge. In practical operation, the value
Table 1 Test condition of NASA battery data set. Battery no.
Constant charge current (A)
Charge cut-off voltage (V)
Discharge current (A)
Discharge cut-off voltage (V)
Nominal capacity (Ah)
5 6 7
1.5 1.5 1.5
4.2 4.2 4.2
2.0 2.0 2.0
2.7 2.5 2.2
2.0 2.0 2.0
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of battery charging current is controlled by the BMS. Terminal voltage curves during CC charge subprocess of the battery at different SoH are shown in Fig. 2, whose shape varies from each other. For example, the line of the battery at 100% SoH has a bigger slope than the batteries at worse SoH before 300 s. Moreover, the slope of the terminal voltage curve changes more slowly while battery health deteriorates. Additionally, the values of initial voltage, mean voltage and final voltage are not equal in different curves. In summary, battery terminal voltage curve in specific charge/discharge process (CC charge subprocess) can reflect battery SoH. Thus it has been selected as the external feature for SoH detection. Considering the time cost of CC charge period of the battery with least SoH, terminal voltages are measured in the data collection interval (DCI), as shown in Fig. 2. The DCI is set from the charging start time to 1000 s later (from 0% SoC to about 60% SoC) to make it totally appropriate for batteries at different SoH. Fig. 1. SoH of the experimental batteries.
2.3. Curve description There are thousands of voltage points which may be measured in the DCI, which means that it is unpractical to utilize all of these voltage data to detect SoH in actual operation. Thus, typical features need to be extracted from certain number of the terminal voltage samples for curve representation. These features can be treated as a summary of the voltage curves and will be used as the inputs of the proposed machine learning algorithm in this paper. The properties of curve can be described by arclength, velocity, unit tangent, curvature, binormal, etc. in differential geometry [27,28]. Considering the characteristic of the discrete battery terminal voltage samples and the computation complexity in actual operation, value and velocity of selected voltage points are applied to describe the curve and act as the inputs of the GMDH in this paper. Velocity means the changing rate of voltage value with time. Thus, velocity, and arclength curvature are applied in DGA as the references for voltage sample selection from the DCI. The DGA is shown in Table 2. 3. GMDH based SoH estimation Fig. 2. Terminal voltages at CC charge subprocess of battery # 5.
Modeling the relation between battery outputs and internal
Table 2 Procedure of the DGA for voltage sample selection. Step 1 Step 2
Step 3
Initialization Confirm the number of target samples. In this paper, the number is equal to 4 in order to reduce the complexity of SoH estimation algorithm. Velocity calculation Calculate velocity of each sample by the equation below. fv ðkÞ ¼ Vkþ1 Vk
(2)
Arclength calculation Arclength in this paper is the length of DCI and can be calculated by accumulating the length of the curve in every unit interval of DCI [27]. The computation of approximate arclength of the DCI is given in Eq. (3). N 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X fa ðkÞ ¼ ðVkþ1 Vk Þ2 *h2 þ t 2 (3) k¼1
Step 4
where N is the number of voltage points, Vk is the terminal voltage at kth time step and t is the sample interval. h is the coefficient to transform x-axis and y-axis approximately and is equal to 1000/7 here. Curvature calculation Curvature is a measure of the bent degree of a curve at a certain point [28]. The curvature of each sample point in DCI is calculated by the function below. jV 2Vkþ1 þ Vk j fe ðkÞ ¼ h kþ2 (4) i3=2 1 þ ðVkþ1 Vk Þ2
Step 5
where Vk is the terminal voltage at kth time step. Sample selection 4 sample points are selected, such as the first sample with a velocity less than a certain value v (v is set to 0.005 in this paper), the last sample in the DCI, the point in the middle of the arc and the point with maximum curvature.
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state is fundamental for proper battery state estimation. Without an accurate battery model, it would be unrealistic to achieve a valid result for SoH estimation. In this section, GMDH polynomial neural network is employed to address the issue of modeling and estimation.
variables and single output can be established by the discrete form of functional Volterra series, which is Kolmogorov-Gabor polynomial, shown in Eq. (5).
3.1. GMDH polynomial neural network
y ¼ f ðXÞ
Due to the complexity of battery electrochemical reaction mechanisms, it is very difficult to describe the relation between battery SoH and battery terminal voltage curve, or the relation between SoH and the abovementioned geometric variables in Section 2, by using a certain equation. To address this issue, GMDH polynomial neural network is employed to establish the model of terminal voltage curve and battery SoH. Moreover, as mentioned in Section 1, this machine learning algorithm may also have the ability to estimate SoH. GMDH polynomial neural network is a self-organized deep learning algorithm originated by Prof. Alexey G. Ivakhnenko [29]. It has been used in a great variety of fields, e.g. data mining, knowledge discovery, forecasting and complex system modeling. For the principle of the GMDH, the relation between multiple input
¼ a0 þ
M X
ai xi þ
i¼1
M X M X i¼1 j¼1
aij xi xj þ
M X M X M X
aijk xi xj xk þ /
i¼1 j¼1 k¼1
(5) where y is the output, X is the input vector, xi, xj and xk are the input variables, ai, aj and ak the weights and M is the number of the input variables. The multiple input variables are the voltages and velocities of the selected samples, as diagrammatically shown in Fig. 3, and the single output is the SoH. Moreover, the structure of the GMDH of the paper is shown in Fig. 4. For the proposed GMDH, the output can be estimated by a series of partial quadratic polynomials which have just two input variables. The quadratic polynomial is shown in Eq. (6).
Fig. 3. (a) Feature voltage. (b) Feature velocity.
Fig. 4. Structure of the GMDH.
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b y ¼ a0 þ a1 xi þ a2 xj þ a3 xi xj þ a4 x2i þ a5 x2j
detected by using a specific process, the CC charge subprocess.
(6)
where b y is the estimated value of each output. This quadratic polynomial is used to connect the hidden neurons and the inputs, the output and the hidden neurons, and two hidden neurons. For example, the SoH, which is the output of the proposed GMDH, can be estimated by two variables from the 4th hidden layers of GMDH using Eq. (7).
b ¼ f O xH4 ; xH4 h i j
2 2 O H4 O H4 O H4 H4 O H4 H4 þ aO ¼ aO 5 xj 0 þ a1 xi þ a2 xj þ a3 xi xj þ a4 xi (7)
where b h is the estimated SoH, f O ðÞ is a quadratic polynomial connecting neurons in the 4th hidden layer and output, aO i are the weights between the 4th hidden layer neurons and the output, and xH4 are the neurons in the 4th hidden layer. i Moreover, based on the combination of the partial quadratic polynomials, battery SoH can then be estimated by the input geometric properties. The expression is given in Eq. (8).
b ¼ f O xH4 ; xH4 h i j H3 H3 ; f H4 xH3 ¼ f O f H4 xH3 i ; xj k ; xl ¼ f O f H4 f H3 f H2 f H1 ðÞ; /
(8)
¼ f ðIÞ where f H4 ðÞ, f H3 ðÞ, f H2 ðÞ and f H1 ðÞ are the quadratic polynomials connecting different layers, I is the input vector, and xH3 is the i neuron in the 3rd hidden layer. After the construction of modeling, the weights in each polynomial are obtained by minimizing the mean square error (MSE) using least square (LS) method. The expression of MSE is given below. N 1 X 2 ðyn b ynÞ N n¼1
461
(9)
where N is the length of training data. 3.2. SoH estimation The procedure of battery SoH estimation based on the proposed method can be separated into several steps. Firstly, battery terminal voltages during CC charge subprocess at different SOH should be measured, where the sample area is the defined DCI. Secondly, properties are selected by DGA to describe battery charge curves which are dissimilar at different SoH. Then, the GMDH polynomial neural network is trained by the chosen variables and SoH. Battery voltages and velocities are the input of GMDH, and SoH is the output. In this paper, aiming at balancing the demand of computation and the accuracy of SoH estimation, the number of input is set to 8, the quantity of the hidden layers is 4, and the maximum number of hidden layer neurons is equal to 20. Finally, battery SoH can be detected by the established GMDH. This process is shown in Fig. 5. It is important to note that the abovementioned procedure for SoH estimation is consistent with the principle of control theory in Section 1. Battery internal state, e.g. SoH, is observed by utilizing the system outputs, and properties of battery terminal voltage curve. Moreover, similar to the estimation of human health state and athletic ability described in Section 2, battery SoH can also be
4. Experiment and analysis Experimental data from NASA battery data set is used to validate the accuracy of the SoH estimation method based on DGA and GMDH. Furthermore, to verify the universality of the presented method, experiments results of the IFP1865140 type LiFePO4 battery, which was developed by Hefei Guoxuan High-Tech Power Energy CO. LTD of China, and widely used in electric vehicles, hybrid electric vehicles and micro-grids, are also analyzed. Simulation of the experimental data is implemented by MATLAB in this paper.
4.1. NASA battery data set Voltage samples from the DCI are obtained from NASA battery data set. Experimental results of battery No. 5, No. 6 and No. 7 in the data set are discussed in this paper. Each battery has been charged and discharged circularly during the experiment with a total cycle number of 168. Furthermore, variables selected by the DGA from 100 random cycle data is used to train the GMDH, and the rest of the experimental data is applied to test the estimation accuracy of the proposed method. The estimation results are plotted in Fig. 6. GMDH-DGA represents the proposed SoH estimation method based on GMDH using the DGA for input selection. And the true value of SoH is denoted by Real. As shown in Fig. 6 (a), 6 (b) and 6 (c), the estimation result matches the real SoH very well, which clearly indicates that the SoH can be accurately estimated by the GMDH using DGA as input selection approach. Moreover, according to Fig. 6 (d), the maximum estimation error of the presented method is 5%. In addition, a comparison with importance sampling (IS), which is used for battery terminal voltage sample selection and which was introduced in our previous study [13], is implemented to show the validity of the DGA. In order to ensure fairness, the same GMDH is used during the comparison. Moreover, an estimation method based on the random selected voltage samples is also presented in the comparison. The result of the comparison is shown in Fig. 7. The GMDH-RS in Fig 7 (a) means a GMDH method with random selected inputs, and GMDH-IS is the estimation method using GMDH and IS. It is obvious that GMDH-DGA has a better estimation performance than GMDH-RS. Fig 7 (b) shows the estimation errors of GMDH with different input selection approaches. Accordingly, it is undoubted that the terminal voltage curves during CC charge subprocess and GMDH algorithm are suitable for SoH detection. Furthermore, the numeric results of the comparison of all the three batteries are shown in Table 3. For battery No. 5, GMDH-DGA has a mean absolute error (MAE) of 0.4503, which is the least value among the three comparative methods. It is a 38.8% improvement of the GMDH-RS and about 15% of IS either. The same phenomenon occurs in the comparisons of battery No. 6 and MSE. As the table shows, GMDH-DGA and GMDHIS have similar estimation errors for battery No. 7, and still much better than GMDH-RS. These abovementioned comparisons indicate that the geometric properties of battery terminal voltage curves selected by DGA can be used to estimate SoH. Moreover, since SoH is estimated by GMDH-DGA, GMDH-IS and GMDH-RS based on the GMDH with same structure, these methods would have almost the same training complexity and estimation efficiency. Thus the method based on the geometric property analysis using GMDH is proven to be a more reliable performance for SoH estimation.
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Fig. 5. Procedure of SoH estimation method.
Fig. 6. (a) SoH estimation results of battery # 5. (b) SoH estimation results of battery # 6. (c) SoH estimation results of battery # 7. (d) SoH estimation errors.
4.2. IFP1865140 type LiFePO4 battery In this paper, more than 2000 cycles of the terminal voltages of the experimental LiFePO4 battery during charge and discharge process were measured. The battery testing platform is composed of a 63640-80-80 type DC electronic load and a programmable 62006P-30-80 type DC power supply model both develop by Chroma ATE Inc. The temperature of the experimental environment
is 25 C. A data set consists of 200 cycles of battery charge and discharge data was applied to validate the proposed SoH estimation method, where 120 cycles of data was used to train the GMDH, the other 80 cycles are the test data. To make the DCI consistent for different types of batteries, the DCI of the IFP1865140 type LiFePO4 battery is set from 0 s to 3600 s (from 0% SoC to about 60% SoC). Battery terminal voltage during one single cycle is plotted in
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Fig. 7. (a) SoH estimation results of the GMDH using different curve description methods. (b) Estimation errors of the test data.
Table 3 Numerical results of different GMDH based SoH estimation methods. GMDH with
a
MAE (%) MSE (%2) a
Battery # 5
Battery # 6
Battery # 7
RS
IS
DGA
RS
IS
DGA
RS
IS
DGA
0.7353 1.1507
0.5309 0.6213
0.4503 0.4258
1.0942 2.4347
0.5543 0.5343
0.4087 0.3600
0.6927 1.6739
0.4700 0.5953
0.4954 0.5640
MAE ¼ Mean absolute error. MSE ¼ Mean squared error.
Fig. 8. (a) Battery terminal voltage during one cycle. (b) Terminal voltages during CC charge subprocess at different SoH. (c) SoH estimation results. (d) SoH estimation errors.
Fig. 8 (a). During the cycle, battery is CC charged by 0.5 C until its terminal voltage reaches 3.65 V, and the discharge cut-off voltage is set to 2.0 V. The rest time between charge and discharge is 10 min. Battery terminal voltage curves at different SoH are plotted in Fig. 8 (b), and so does the DCI of this battery. The SoH estimation results of the test data are shown in Fig. 8 (c). And errors are plotted in
Table 4 Numerical results of SoH estimation of IFP1865140 type battery.
MAE (%) MSE (%2)
GMDH-RS
GMDH-IS
GMDH-DGA
1.3639 3.4864
0.7777 1.8117
0.8125 1.3119
Fig. 8 (d). Clearly, GMDH-DGA can achieve an accurate estimation result of SoH with a maximum error of about 5%. Numeric results of comparison among GMDH-RS, GMDH-IS and GMDH-DGA are shown in Table 4. The MAE of GMDH-DGA is 40.43% less than GMDH-RS. Moreover, the methods using DGA offer a minimum MSE among these estimation methods. Due to the voltage plateau of LiFePO4 battery under CC charge subprocess, GMDH-IS also showed a dependable performance of SoH detection in this comparison. Estimation results and statistical properties in Fig. 8 and Table 4 validate that the proposed GMDH-DGA can reach a good estimation result for the IFP1865140 type LiFePO4 battery as well as the Li-ion battery used for NASA data set. Hence, the universality of the developed SoH
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estimation method is verified by this comparison. 5. Conclusion In this study, Li-ion battery SoH is estimated by GMDH and DGA under the direction of control theory. The SoH is observed based on battery external features. Similar to the approach of human health and athletic ability description, geometric properties of battery terminal voltage curves during a specific process, which is a CC charge subprocess, are calculated by DGA to describe the SoH. The voltage of the selected samples is applied as the input of GMDH as well as their velocity. Relation between the geometric properties and SoH is established by GMDH, which is also applied to estimate the SoH. Experimental results show GMDH-DGA can detect battery SoH accurately with a margin of estimation error of 5%. Additionally, experiment has also been conducted on a LiFePO4 battery. The accurate estimation results verify the universality of the proposed method. In summary, the proposed method using GMDH and DGA can achieve an accurate and universal SoH estimation result for Liion batteries. Our future study will focus on SoH estimation considering the interference of voltage sampling and the inconsistency of batteries in the actual operation. Acknowledgments This work is supported by the National Natural Science Fund of China (Grant No. 61375079) and China Scholarship Council (No. 201606340099). The authors wish to thank Dr. Ya Zhang from the First Affiliated Hospital of Anhui Medical University and Dr. Xi Wang from the Institute of Intelligent Machines, Chinese Academy of Sciences for their valuable advices. Moreover, we would also like to express our sincere thanks to Mr. Jun Zhang for his useful suggestions. References [1] Y. He, X. Liu, C. Zhang, Z. Chen, Appl. Energy 101 (2013) 808e814.
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