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State-of-health estimation of lithium-ion battery based on fractional impedance model and interval capacity
Electrical Power and Energy Systems 119 (2020) 105883
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State-of-health estimation of lithium-ion battery based on fractional impedance model and interval capacity
T
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Qingxia Yanga, Jun Xub, Xiuqing Lic, , Dan Xub, Binggang Caob a
Vehicle & Transportation Engineering Institute, Henan University of Science and Technology, Luoyang 471023, China School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, China c National Joint Engineering Research Center for Abrasion Control and Molding of Metal Materials, Henan University of Science and Technology, Luoyang 471023, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: State of health (SOH) Lithium-ion battery Fractional impedance model (FIM) Interval capacity
Lithium-ion batteries are being used in electric vehicles with very demanding duty schedules. The estimation of battery state of health is very important, so that it has become a research hotspot. This paper deals with the problem of lithium-ion battery state-of-health estimation based on a simplified fractional impedance model and the battery’s interval capacity. A simplified fractional impedance model based on the Grünwald-Letnikov definition is introduced, and the least-squares genetic algorithm is utilized to identify the model parameters with a voltage-tracing error rate less than 0.2%. In order to validate the battery ageing performance, a battery testbench has been established, and an accelerated ageing experiment has been carried out. Based on the identified model parameters and interval capacity combination with a voltage range from 3.95 V to 4.15 V, a back propagation neural network is introduced to estimate the battery state of health with an error margin of [−1.5%, 1.5%]. The effectiveness of the proposed method is verified through simulations and experiments.
1. Introduction In order to mitigate the global energy crisis, electric vehicles have witnessed a dramatic increase in attention. The traction battery has also been developed rapidly in recent years, for example, the lead-acid batteries, the Nickel-metal hydride batteries, the lithium-ion batteries, and so on. The lithium-ion batteries are more attractive for electric vehicle applications due to their superior characteristics: cost-effective, high-powered, long-lasting, high-energy density, and so on [1,2]. Unfortunately, unexpected battery system failures are usually coming with battery degradation, leading by environmental impacts, dynamic loading, and so on [3,4]. In order to prevent catastrophic failures, battery management systems (BMS) are developed with high efficiency and reliability. However, it is difficult to monitor battery states and to diagnose faults in real time due to the cell's sophisticated physical–chemical process, which has hindered the widespread applications of BMS. As the core function of the BMS designed for an energy storage system, the battery states, including the state of charge (SOC), the state of function (SOF), and the state of health (SOH), are required great effort to measure directly in the active application for electric vehicles. For the efficient use of a BMS, it is important to know the capacity available for the power system and the maximal capacity stored inside the battery. Different definitions of the SOH have been developed,
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including the calendar life, cycle life, capacity fading, and so on [5–8]. Considering to correlate the SOC with capacity fade, SOC should be defined as the percentage of the releasable capacity relative to the battery maximal releasable capacity [9,10]. However, as the battery maximal releasable capacity almost cannot be measured directly in engineering, the SOC is defined as function (1) in this study for the convenience of SOH analysis, which definition is also one of the commonly used SOC definitions [6]. The SOH is defined as function (2) [11]:
SOC = SOC_ini −
(∫ i·dt
)
Cn _rated × 100%
SOH = Cn _discharge Cn _rated × 100%
(1) (2)
where SOC_ini is the battery initial SOC. i is the battery current, the discharge current is positive, and the charge current is negative. Cn_rated is the battery rated capacity. Cn_discharge is the battery maximal releasable capacity. It is simpler to estimate these states for a fresh battery. Indeed, the estimation error would be considerable without knowing the battery capacity loss for an aged battery. The complicated ageing mechanisms for commercial lithium-ion batteries have been discussed in the literatures [12,13]. The capacity loss of LiNixCoyMn1-x-yO2 is mainly caused by: (1) changes in lattice volume during charge/discharge, (2) cation
Corresponding author. E-mail address: [email protected] (X. Li).
https://doi.org/10.1016/j.ijepes.2020.105883 Received 19 August 2019; Received in revised form 22 January 2020; Accepted 27 January 2020 0142-0615/ © 2020 Elsevier Ltd. All rights reserved.
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Table 1 Battery specifications and EIS test conditions. Battery specifications Type Item Voltage Temperature
EIS test conditions No. 1 2 3
Nickel Manganese Cobalt Oxide Lithium-ion Battery Panasonic NCR18650 2950 mAh 3.7 V (Nominal), 2.8 V ~ 4.2 V (Operation) Charging: 0 ~ 45℃;Discharging: −20 ~ 60℃
SOH (%) 82 90 97
SOC (%) 80 80 80
60 60 60
40 40 40
20 20 20
Fig. 1. EIS curves of lithium-ion batteries with different properties.
and SOC estimations under different conditions compared with the 1RC/2-RC models [26]. Xiao et al. compared a fractional order form of the Thevenin model with the integral order model, and they performed that fractional order form of the Thevenin model such that it can simulate the battery terminal voltage variation more precisely than the integral order model [27]. Some special methods have also been used for SOH estimations, such as the SOH estimated based on the changes of SOC and voltage [28]. However, the SOC should be estimated based on a special observer, and the SOH estimation errors based on this method accumulate with SOC estimation errors. A C_remaining (defined by the difference of the maximal releasable capacity and the actual discharge capacity) estimation method through differential voltage analysis was also proposed [29]. However, the accuracy of capacity estimations based on this method could only be guaranteed in a low-capacity range. Guo et al. [30] proposed an effective method to estimate the SOH based on charging curves, in which a typical constant current constant voltage charging method was designed for lithium-ion batteries, and the SOH of the battery could be estimated within 3% error range. Based on the above considerations, a fractional impedance model (FIM) and the interval capacity during the charging process are utilized to evaluate the SOH of lithium-ion batteries. The SOH estimation method was verified using a commercial lithium-ion battery, and the results are described in this paper. In section 2, a particular commercial lithium-ion battery is selected for the electrochemical impedance spectrum test at different states. Moreover, the FIM and the relevant
mixing, (3) transition metal ion dissolution, (4) side reactions between electrode and electrolyte, and so on [14,15]. Many studies have been conducted to estimate the SOH of batteries. Such methods mainly include electrochemical impedance spectroscopy technology [16], electromotive force analysis [17], sampling-based estimation methods [18], model-based estimation methods [19], and so on. As one of the most promising methods to characterize lithium-ion battery ageing effects, researchers indicated that many of the mechanisms responsible for battery ageing and degradation could be monitored and investigated non-destructively by the use of impedance spectroscopy [20]. Yet, this method requires special equipment for implementation, and the expensive cost of this equipment results in difficulties for BMS applications. A SOH estimation method based on an accurate battery model is one of the most popular solutions. Some attempts have been made to evaluate lithium-ion battery models to estimate states and parameters, such as equivalent circuit models, data based models and electrochemical models, and so on [21–24]. A model based on the Laplace transfer time for commercial lithium-ion batteries was proposed by Samadani et al. [25], in which the equivalent circuit model was constructed from the electrochemical impedance spectrum test results of a commercial battery at different temperatures. However, the constant phase element in the equivalent circuit model was approximated using a capacitor. Other models were derived from the models mentioned above. For example, the fractional-order model was proposed, which could maintain higher accuracy for voltage tracking
2
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Fig. 2. Simplified equivalent circuit based on the lithium-ion battery EIS and HPPC test.
Fig. 3. Flow chart for parameter identification.
Fig. 4. Configuration of the battery test-bench.
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Fig. 5. Charging voltage and charging capacity of batteries with different SOH under ambient temperature.
Fig. 6. A hybrid pulse power characterization test of battery.
2. Fractional impedance model and parameter identification
parameter identification method based on a least-squares genetic algorithm (LSGA) are both introduced in this section. In section 3, a battery test-bench is established, and an accelerated ageing experiment for the lithium-ion battery is designed based on the results of a CC-CV charging test for the batteries at different SOH values. Results of the model parameter identification and the accelerated ageing experiment are both discussed in section 4. A novel back propagation neural network method is proposed to estimate the lithium-ion battery SOH based on identified internal parameters and experiment data, and the accuracy of SOH estimations is verified with an accelerated ageing experiment and simulation. Finally, some conclusions are given in section 5.
2.1. Fractional impedance model (FIM) The electrochemical impedance spectrum (EIS) is a powerful labbased diagnostic technique to characterize a battery’s dynamical performance [31]. In this paper, a commercially available lithium-ion battery cell based on Nickel Manganese Cobalt Oxide cathode material and graphitic anode designed for electric vehicle applications was selected. Table 1 presents the specifications of this battery cell. The lithium-ion battery EISs were measured at room temperature (25℃) by the Princeton electrochemical workstation at different SOH and SOC, as shown in Table 1. The EIS test results of batteries at different SOC and SOH are shown in Fig. 1. The different colors of curves represent the EIS of batteries at
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Fig. 7. Designed test cycle in the experiment.
The fractional impedance model is described as follows can be summed up as follows:
different SOH, and the EIS of batteries at different SOC are independently shown in the sub-graphs. It can be seen that the EIS curves are similar in shape. The total module of the impedance enhances as the battery ages, and the parameters of the equivalent circuit model could be extracted to characterize battery ageing. Whereas the SOC of batteries remained the same, the numerical variation tendencies of Zre and –Zim were rendered consistent in the same frequency section. The change tendencies of these curves are in well agreement with previous experimental results [16,32,33]. It can be seen from Fig. 1 that, there is a line going straight up, which is crossed with the axis of –Zim = 0, and there are two depressed semi-circles in each of the EIS curves. The intersection of this line and the axis of –Zim = 0 is regarded as the response of Ohmic resistance. The depressed semi-circle in mid-frequency section represents the concentration polarization due to a slow charge transfer reaction in electrodes, and the depressed semi-circle could be modeled by a parallel combination of a resistor and a constant phase element. The curve of the EIS in the low-frequency section is deemed as a part of depressed semi-circle with a large curvature radius, which can be utilized to represent the activation polarization response in electrodes. Based on the analysis above, a simplified equivalent circuit is established, as shown in Fig. 2. Here, Vser denotes the voltage for Ohmic resistance (Rser). The parallel combination of a resistor and a constant phase element is regarded as ZARC. Moreover, V1 denotes the voltage of ZARC1, which can be utilized to characterize the concentration polarization phenomenon inside the battery. Finally, V2 is regarded as the voltage of ZARC2, which represents the activation polarization voltage. Research shows that fractional order calculus can be utilized to establish a more accurate system model [34]. As discussed in [35,36], the FIM of lithium-ion batteries can be established based on the Grünwald-Letnikov definition. According to the circuit theory, from the equivalent circuit illustrated in Fig. 2 and the analysis above, the equations can be presented as follows:
Vo = Vocv + Vser + V1 + V2 ⎧ Vser = −Rser ·I ⎪ ⎪ − I = C1·Δα V1 + V1 R1 = C2·ΔβV2 + V2 R2 ⎨ Δα V1 = −I C1 − V1 R1 C1 ⎪ βV = −I C − V C R ⎪ Δ 2 2 2 2 2 ⎩
N ⎧ Δ x = A·x + B·I ⎨ ⎩ y = C·x + D·I
(4)
− 1 C1 ⎤ − 1 R1 C1 0 ⎤, B=⎡ , where A = ⎡ C = [1 1], ⎢ ⎢ 0 − 1 R2 C2 ⎥ ⎣− 1 C2 ⎥ ⎦ ⎣ ⎦ α V D = [−Rser ], N = ⎡ β ⎤, x = ⎡ 1 ⎤, y = [Vo − Vocv], x ∈ R2 ⎣ ⎦ ⎣V2 ⎦ The Grünwald-Letnikov definition is usually used to discretize the continuous fractional-order equations [37,38], and the Grünwald-Letnikov fractional order calculus is defined as: k G r a Δt f
(t ) = lim h−r h→0 kh = t − a
∑ (−1) j j=0
(rj) f (t − jh)
(5)
r aΔt
is the fractional order calculus operator, h is the sampling r j is the Newton binomial coefficient generalized to real numbers, which can be expressed as r r! j = j ! (j − r ) ! .
where
period, k is the amount of sampling, and
()
()
2.2. Battery parameter identification based on LSGA Many parameter identification methods have been proposed [36,39,40]. Due the battery fractional impedance model is strongly nonlinear, the least-squares genetic algorithm (LSGA), derived from the combination of the least squares method and the genetic algorithm, is selected to identify the internal parameters as discussed in the study [36]. They also developed an equivalent voltage tracking system for parameter identification, and the model is presented as: N ⎧ Δ x ̂ = A·x ̂ + B·I ⎨ ⎩ y ̂ = C·x ̂ + D·I
(6)
1 ⎤ V o − Vocv]. Here, x ̂ ∈ R2 is the state vector of the , y ̂ = [V where x ̂ = ⎡ ⎢V 2 ⎥ ⎣ ⎦ equivalent tracking system, and y ̂ is the estimated output voltage of the equivalent tracking system. The output voltage difference between the equivalent tracking system and battery system was defined as:
(3) 5
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Fig. 8. Results of the interval capacity test. (a) Changes to the interval capacity; (b) SOH and the interval capacities associated with the voltage region [3.95 V, 4.15 V].
e = ŷ− y
(7)
and humidity chamber was used to age the battery submitted to dynamic current cycles under controlled ambient temperature and relative humidity, and the battery internal parameters were identified via Matlab software. To charge the cells, the conventional constant current-constant voltage (CC-CV) protocol was adopted, where 0.3C rate (0.885 A) currents were used as the constant current, the voltage and current cutoff limits were set as 4.2 V and 0.06A. Batteries with different SOH were charged based on the CC-CV protocol, and the results are shown in Fig. 5. In Fig. 5, the charge voltage and capacity at different SOH are denoted by the symbols SOH_Vo and SOH-Cn, and ΔCni (i = 1, 2, 3) was utilized to represent the interval capacities associated with the same voltage difference. The results show that the interval capacities change as the battery ages. The characteristics of the hysteresis of OCV change completely during battery lifetime, and the hysteresis of the OCV can be detected through partial charge/discharge processes [41–43]. Taking into consideration that the battery OCV, in discharge process, is extremely difficult to be obtained while applying on the electric vehicle, the static OCV is selected for the battery model parameters identification. The static OCV can be obtained through long enough relaxed time, for example, 1 h relaxation time. A hybrid pulse power characterization test
The tracking target of the LSGA is represented through the following goal equation:
J (θ) =
t2
∫t1
(e T ·e )dt
(8)
where J (θ) is the objective function, t1 and t2 are the boundary of time. A parameter identification flow chart is presented in Fig. 3. The basic operations of LSGA include coding methods, individual fitness evaluation, genetic operators (such as selection, crossover, and mutation), and so on, as shown in Fig. 3. Here, I and Vocv are the input parameters of the equivalent tracking system. Further, Vo is the output o is voltage of battery system, and can also be measured directly. And V the output voltage of tracking system, and can be adjusted using the equivalent tracking model. Parameters are identified through the LSGA based on the voltage difference between the actual battery system and the equivalent tracking system. 3. Battery ageing experimental design A battery test-bench was established as shown in Fig. 4. The Neware battery testing system associated with a programmable temperature 6
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Fig. 9. Voltage tracing results of batteries with different SOH (98%, 92%, 87%, and 81%) based on the proposed FIM and LSGA.
fully charged and discharged, with 2.95A charged current and 5.9 A discharged current, at 25 ℃ or −10℃. Such accelerated aging charge–discharge cycle is repeated for several times in each test cycle. The regular measurement of the battery’s capacity was mandatory alongside ageing campaigns in the capacity measurement phase.
was taken for the analysis of battery OCV, and results were shown in Fig. 6. It can be seen that the battery voltage error between relaxed 10 min and 1 h is less than 0.008 V, so that the battery voltage is regarded as the approximation of OCV. Ageing campaigns were launched to collect battery voltage and current data over time. The ageing experiment was intended to provoke changes to the internal parameters and interval capacities. Without a doubt, the capacity fading should be taken into consideration in the ageing protocol. Fig. 7 presents the current and voltage profiles in the designed test cycle, and the experiment protocol covered a number of such cycles. As shown in Fig. 7, the test cycle can be divided into four phases: the preparation phase, the step pulse test phase, the accelerated ageing phase, and the capacity measurement phase. One of the major tasks during the preparation phase was to ensure that the batteries were under the same state conditions. Thus, the lower cut-off voltage and the environmental conditions were enforced consistently at the end of the preparation phase. Internal parameters and interval capacities were taken into consideration during the step pulse test phase, as also shown in the subgraph in Fig. 7. Taking into account of the polarization and diffusion in charging process, the battery is relaxed 20 min to make sure that the battery is relatively stability, as shown in Fig. 7, when battery reaches the preset voltage values. Next, the step pulsing test is made, the battery is discharged 30 s with 5.9A current in this period. Then, the battery is relaxed 10 min, based on the analysis of OCV above, the battery voltage is regarded as the approximation of OCV at the end of this relaxation. In order to validate the battery’s fading performance, the number of the test cycles could be reduced with a different accelerated ageing method, as has been conducted in other works [44,45]. In this study, an accelerated ageing charge–discharge cycle was designed during the accelerated ageing phase. In the accelerated ageing phase, the battery was
4. Experimental results and SOH estimation 4.1. Results of the accelerated ageing experiment In the transient response test phase, the interval capacities associated with different voltage regions were measured. Many of the batteries based on the same cathode material were tested, and the experiment results of two batteries, distinguished by the symbols ‘_1’ and ‘_2’, are highly representative in Fig. 8. In Fig. 8(a), the x-axis represents the battery charging-discharging cycle times, and the y-axis represents the interval capacity. The interval capacities associated with different voltage regions are marked as Cn_375395_1, Cn_385405_1, Cn_395415_1, Cn_375395_2, Cn_385405_2, and Cn_395415_2. It can be seen that the interval capacities associated with different voltage regions change as the test cycle increases. The interval capacities associated with the voltage region [3.75 V, 3.95 V] change slightly near 815 mAh, and most of the interval capacities associated with the voltage region [3.85 V, 4.05 V] change slightly near 660 mAh. However, the interval capacity associated with voltage region [3.95 V, 4.15 V] consistently decreases as the test cycle increases. The SOH and the interval capacities associated with voltage region [3.95 V, 4.15 V] are shown in Fig. 8(b). The states of health are marked as SOH_1 and SOH_2. It can be seen that both the SOH and the interval capacities consistently decrease as the test times increases. During the transient response test stage, voltage step response 7
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Fig. 10. Internal parameter identification for batteries with different SOH. (a) Relationship between the SOH and the Ohmic resistances; (b) Relationship between the SOH and the concentration polarization fractional orders; (c) Relationship between the SOH and the activation polarization fractional orders.
batteries with different SOH (98%, 92%, 87%, and 81%) are shown in Fig. 8,after collecting and analyzing the battery voltage data. During the step pulse test phase, the battery was charged until it reached 3.95 V. Then, the battery was left unattended without charging or discharging it, and the end of this rest period was regarded as the
testing is also taken into consideration. The internal parameters of the tested batteries with different SOH were identified based on the FIM and LSGA parameter identification method. Battery voltage was traced via the identified internal parameters based on the equivalent tracking system described in section 2, and the voltage tracing results of 8
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Fig. 13. SOH estimation Results based on BPNN.
starting point shown in Fig. 9. Here, the battery reference voltage is presented by the black solid lines, the green dashed lines are used to describe the battery estimated voltage, and the red solid lines denote the voltage tracing error. It can be seen that the battery voltage tracing error could be well confined to an error region of [-5 mV, 5 mV], and the error rate was less than 0.2% for the lithium-ion battery. Based on the LSGA, result shows that the model parameters can be accurately identified as well as the method proposed in the study [39]. The relationships between the SOH and the internal parameters identified are shown in Fig. 10. In Fig. 10(a), the relationship between the SOH and the Ohmic resistances is shown, where the Ohmic resistances of batteries are represented as Rser_1 and Rser_2. The results show that the Ohmic resistances increase with the fading capacities, which means that the point, crossed with the axis of –Zim = 0 in the EIS, will move to the right with the battery aging at the same state, and this result is also well agreement with the EIS test result in Fig. 1. Fig. 10(b) shows the relationship between the battery SOH and the concentration polarization fractional orders, where the concentration polarization fractional orders are marked as α_1 and α_2. It can be seen that the concentration polarization fractional orders first decrease rapidly at first, and then increase when the aging state of the battery is 87.5%, the increasing rate gradually decreases and tends to be stable as the test cycle increases. The results are consistent with the changes of the concentration of lithium ion and SEI membrane inside the battery [46]. In Fig. 10(c), the relationship between the battery SOH and the activation polarization fractional orders is shown, where the activation polarization fractional orders are marked as β_1 and β_2. The results show that the activation polarization fractional orders increase rapidly at first, and then increase slowly. Results show that the variation regularity of the activation polarization fractional orders is agreement with the change of SEI membrane thickness inside the battery and the active substances in the electrolyte [46].
Fig. 11. The typical structure and the approximation system structure of BPNN. (a) The typical structure of BPNN; (b) The approximation system structure of BPNN.
Fig. 12. SOH estimation based on BPNN. Table 2 Parts of training data from one cell. Cn_discharge
Cn_395415
Rser
α
β
2790 2749 2724 … 2610 2593 2588 … 2458 2465 2467 2432
647.3 646.3 642.4 … 613.4 610.7 609 … 598 596.8 594.6 590.4
39.65 41.49 42.81 … 50.91 52.04 52.67 … 61.43 60.63 62 61.82
0.8733 0.8687 0.8645 … 0.8732 0.8778 0.8766 … 0.8924 0.8918 0.8924 0.8915
0.2107 0.2267 0.2513 … 0.3026 0.2976 0.3011 … 0.3327 0.319 0.3098 0.3182
4.2. SOH estimation based on BPNN Table 3 Parts of validation data from other cells. Cn_discharge
Cn_395415
Rser
α
β
2762 2662 … 2539 2765 2493 2751 2395
653.8 637.2 … 633.7 638.8 606.7 646.1 596.9
40.77 42.06 … 46.26 41.23 56.59 43.31 62.57
0.8721 0.8644 … 0.8568 0.8698 0.8813 0.8727 0.8897
0.2167 0.256 … 0.242 0.2482 0.3011 0.2308 0.3135
It can be seen from the results of the accelerated ageing experiment that the fractional impedance model could well characterize the battery’s performance, and the battery model-based internal parameters change as the battery’s capacity fades. Moreover, the interval capacities associated with the voltage region [3.95 V, 4.15 V] (Cn_395415) consistently decreases as the battery’s capacity fades. Due to the complex electrochemical process in the battery, it is difficult to describe the relationships between the SOH and these parameters with an exact function. As the back propagation neural network (BPNN) can be used to describe the strong nonlinear function without exact parameters, so the BPNN was introduced to depict the relationship between the 9
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genetic algorithm with a 0.005 V error margin. In order to validate the battery decaying performance, an accelerated ageing experiment was performed. The results of parameter identification and the accelerated ageing experiment show that the battery’s internal parameters and interval capacities change as the battery’s capacity fades. A back propagation neural network method was proposed based on the internal parameters and the interval capacities associated with the voltage region [3.95 V, 4.15 V], and was utilized to successfully estimate battery SOH with an error margin of merely [-1.5%, 1.5%] without a special state observer or filtering approach. It is worth noting that the SOH estimation method proposed in this study is still in the developmental stage, and needs further simplification and validation by a BMS designer.
internal parameters, the interval capacity and battery SOH. The typical structure BPNN and the approximation system structure of BPNN are shown in Fig. 11. In Fig. 11(a), the number of input neurons is m, the number of output neurons is n. The S-type function tansig is selected as the transfer function of the hidden layer, and the purelin liner function is used as the transfer function of the output layer. In Fig. 11(b), the error between the target system except values and the neural network system output values is fed back to the neural network system. According to the accuracy requirement for the nonlinear function approximation, the BPNN is designed with three layers in this study, as shown in Fig. 12: namely, the input layer, hidden layer, and output layer. The interval capacities associated with the voltage region [3.95 V, 4.15 V] (Cn_395415) and the internal parameters, including Ohmic resistance (Rser), the concentration polarization fractional order (α), and the activation polarization fractional order (β) were selected as the input parameters. The SOH estimation was designed as the goal of the network. The interval capacities associated with the voltage region [3.95 V, 4.15 V] (Cn_395415) can be measured directly during the charging process, and the internal parameters can be identified based on the FIM and the LSGA parameter identification methods mentioned in section 2. Data retrieved from one battery in the experiment were used as the training data, and the remaining were used for validation of the presented algorithm. Parts of training and validation data are given in Tables 2 and 3. The accuracy of the battery SOH estimation is shown in Fig. 13. The correlation between the estimated SOH (the blue line in Fig. 13 and the real SOH (the red line in Fig. 13 was quantified. It can be seen that the SOH error rate between estimated and real can be well confined to a small error region [−1.5%, 1.5%]. This result implies that the battery SOH can be precisely estimated via the internal parameters and the interval capacities based on the BPNN.
CRediT authorship contribution statement Qingxia Yang: Conceptualization, Methodology, Software, Writing - original draft, Data curation. Jun Xu: Visualization, Investigation. Xiuqing Li: Software, Writing - review & editing. Dan Xu: Writing review & editing. Binggang Cao: Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work has been funded by the Scientific Research Staring Foundation for Doctor of Henan University of Science and Technology, China (Grant no. 13480045), the National Natural Science Foundation of China, China (Grant no. 51405374), and the Key Research and Development Plan of China (Grant no. 2017YFB0103802) and National Science and Technology Ministry, China.
5. Future work Taking into account that the battery performances can be influence by many other factors, for example, the cell design, manufacture, boundary of working current and voltage, working conditions (including temperature, humidity), and so on. It is significant and necessary to design the aging test by considering the aforementioned impact factors in the future work, and the correlation between SOH and temperature will be further studied based on the proposed method. At moment, the single cell aging state are being studied. In realities, due to the lower voltage and capacity, the cells are often connected in series and/or parallel to form battery pack. Due to the in-homogeneous operating conditions, the inherent variations exist among the cells in a battery pack, and the cell-to-cell variation will become even larger with the cells aging. These variances result in SOH deviation among individual cells in the battery pack, therefore, the SOH estimation method for single cell may not be suitable for the battery pack. How to adapt the SOH estimation technique based on the proposed method from cell level to multicell module level remains an open question, which will be discussed in the following work. Moreover, the battery SOH was estimated based on the interval capacities, which were always obtained from a specific cycle, when the cell was fully discharged in this study. In future, the interval capacities can be obtained from other cycles, for example, the cell was discharged below preset voltage 3.9 V, or a much lower voltage level.
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6. Conclusions This paper presented a fractional impedance model, and proposed a SOH estimation method based on a battery’s internal parameters and interval capacity. A simplified fractional impedance model was introduced to characterize the battery’s performance, and the internal parameters of the impedance model were identified via a least-squares 10
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identification. J Power Sources 2014;251:38–54. [15] Huang YY, Chen JT, Ni JF, Zhou HH, Zhang XX. A modified ZrO2-coating process to improve electrochemical performance of Li(Ni1/3Co1/3Mn1/3)O2. J Power Sources 2009;188:538–45. [16] Galeotti M, Cinà L, Giammanco C, Cordiner S, Di Carlo A. Performance analysis and SOH (state of health) evaluation of lithium polymer batteries through electrochemical impedance spectroscopy. Energy 2015;89:678–86. [17] Farmann A, Sauer DU. A comprehensive review of on-board state-of-availablepower prediction techniques for lithium-ion batteries in electric vehicles. J Power Sources 2016;329:123–37. [18] Camci F, Ozkurt C, Toker O, Atamuradov V. Sampling based State of Health estimation methodology for Li-ion batteries. J Power Sources 2015;278:668–74. [19] Chen Z, Xiong R, Tian J, Shang X, Lu J. Model-based fault diagnosis approach on external short circuit of lithium-ion battery used in electric vehicles. Appl Energ 2016;184:365–74. [20] Klett M, Zavalis TG, Kjell MH, Lindstrom RW, Behm M, Lindbergh G. Altered electrode degradation with temperature in LiFePO4/mesocarbon microbead graphite cells diagnosed with impedance spectroscopy. Electrochim Acta 2014;141:173–81. [21] Jiang Y, Zhao X, Valibeygi A, De Callafon RA. Dynamic prediction of power storage and delivery by data-based fractional differential models of a lithium iron phosphate battery. Energies 2016;9:590. [22] He Z, Yang G, Lu L. A parameter identification method for dynamics of lithium iron phosphate batteries based on step-change current curves and constant current curves. Energies 2016;9:444. [23] Hu X, Li SE, Yang Y. Advanced machine learning approach for lithium-ion battery state estimation in electric vehicles. IEEE Trans Transp Electrif 2016;2(2):140–9. [24] Rahman MA, Anwar S, Izadian A. Electrochemical model parameter identification of a lithium-ion battery using particle swarm optimization method. J Power Sources 2016;307:86–97. [25] Samadani E, Farhad S, Scott W, Mastali M, Gimenez LE, Fowler M, et al. Empirical modeling of lithium-ion batteries based on electrochemical impedance spectroscopy tests. Electrochim Acta 2015;160:169–77. [26] J.M. Francisco, J. Sabatier, L. Lavigne, F. Guillemard, Lithium-ion battery state of charge estimation using a fractional battery model, International Conference onFractional Differentiation and its Applications (ICFDA), 2014, 1-6. [27] Xiao R, Shen J, Li X, Yan W, Pan E, Chen Z. Comparisons of modeling and state of charge estimation for lithium-ion battery based on fractional order and integral order methods. Energies 2016;9:184. [28] Wang L, Pan C, Liu L, Cheng Y, Zhao X. On-board state of health estimation of LiFePO4 battery pack through differential voltage analysis. Appl Energy 2016;168:465–72. [29] Liu GM, Ouyang MG, Lu LG, Li JQ, Han XB. Online estimation of lithium-ion battery remaining discharge capacity through differential voltage analysis. J Power Sources 2015;274:971–89. [30] Guo Z, Qiu XP, Hou GD, Liaw BY, Zhang CS. State of health estimation for lithium
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
State-of-health estimation of lithium-ion battery based on fractional impedance model and interval capacity
T
⁎
Qingxia Yanga, Jun Xub, Xiuqing Lic, , Dan Xub, Binggang Caob a
Vehicle & Transportation Engineering Institute, Henan University of Science and Technology, Luoyang 471023, China School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, China c National Joint Engineering Research Center for Abrasion Control and Molding of Metal Materials, Henan University of Science and Technology, Luoyang 471023, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: State of health (SOH) Lithium-ion battery Fractional impedance model (FIM) Interval capacity
Lithium-ion batteries are being used in electric vehicles with very demanding duty schedules. The estimation of battery state of health is very important, so that it has become a research hotspot. This paper deals with the problem of lithium-ion battery state-of-health estimation based on a simplified fractional impedance model and the battery’s interval capacity. A simplified fractional impedance model based on the Grünwald-Letnikov definition is introduced, and the least-squares genetic algorithm is utilized to identify the model parameters with a voltage-tracing error rate less than 0.2%. In order to validate the battery ageing performance, a battery testbench has been established, and an accelerated ageing experiment has been carried out. Based on the identified model parameters and interval capacity combination with a voltage range from 3.95 V to 4.15 V, a back propagation neural network is introduced to estimate the battery state of health with an error margin of [−1.5%, 1.5%]. The effectiveness of the proposed method is verified through simulations and experiments.
1. Introduction In order to mitigate the global energy crisis, electric vehicles have witnessed a dramatic increase in attention. The traction battery has also been developed rapidly in recent years, for example, the lead-acid batteries, the Nickel-metal hydride batteries, the lithium-ion batteries, and so on. The lithium-ion batteries are more attractive for electric vehicle applications due to their superior characteristics: cost-effective, high-powered, long-lasting, high-energy density, and so on [1,2]. Unfortunately, unexpected battery system failures are usually coming with battery degradation, leading by environmental impacts, dynamic loading, and so on [3,4]. In order to prevent catastrophic failures, battery management systems (BMS) are developed with high efficiency and reliability. However, it is difficult to monitor battery states and to diagnose faults in real time due to the cell's sophisticated physical–chemical process, which has hindered the widespread applications of BMS. As the core function of the BMS designed for an energy storage system, the battery states, including the state of charge (SOC), the state of function (SOF), and the state of health (SOH), are required great effort to measure directly in the active application for electric vehicles. For the efficient use of a BMS, it is important to know the capacity available for the power system and the maximal capacity stored inside the battery. Different definitions of the SOH have been developed,
⁎
including the calendar life, cycle life, capacity fading, and so on [5–8]. Considering to correlate the SOC with capacity fade, SOC should be defined as the percentage of the releasable capacity relative to the battery maximal releasable capacity [9,10]. However, as the battery maximal releasable capacity almost cannot be measured directly in engineering, the SOC is defined as function (1) in this study for the convenience of SOH analysis, which definition is also one of the commonly used SOC definitions [6]. The SOH is defined as function (2) [11]:
SOC = SOC_ini −
(∫ i·dt
)
Cn _rated × 100%
SOH = Cn _discharge Cn _rated × 100%
(1) (2)
where SOC_ini is the battery initial SOC. i is the battery current, the discharge current is positive, and the charge current is negative. Cn_rated is the battery rated capacity. Cn_discharge is the battery maximal releasable capacity. It is simpler to estimate these states for a fresh battery. Indeed, the estimation error would be considerable without knowing the battery capacity loss for an aged battery. The complicated ageing mechanisms for commercial lithium-ion batteries have been discussed in the literatures [12,13]. The capacity loss of LiNixCoyMn1-x-yO2 is mainly caused by: (1) changes in lattice volume during charge/discharge, (2) cation
Corresponding author. E-mail address: [email protected] (X. Li).
https://doi.org/10.1016/j.ijepes.2020.105883 Received 19 August 2019; Received in revised form 22 January 2020; Accepted 27 January 2020 0142-0615/ © 2020 Elsevier Ltd. All rights reserved.
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Table 1 Battery specifications and EIS test conditions. Battery specifications Type Item Voltage Temperature
EIS test conditions No. 1 2 3
Nickel Manganese Cobalt Oxide Lithium-ion Battery Panasonic NCR18650 2950 mAh 3.7 V (Nominal), 2.8 V ~ 4.2 V (Operation) Charging: 0 ~ 45℃;Discharging: −20 ~ 60℃
SOH (%) 82 90 97
SOC (%) 80 80 80
60 60 60
40 40 40
20 20 20
Fig. 1. EIS curves of lithium-ion batteries with different properties.
and SOC estimations under different conditions compared with the 1RC/2-RC models [26]. Xiao et al. compared a fractional order form of the Thevenin model with the integral order model, and they performed that fractional order form of the Thevenin model such that it can simulate the battery terminal voltage variation more precisely than the integral order model [27]. Some special methods have also been used for SOH estimations, such as the SOH estimated based on the changes of SOC and voltage [28]. However, the SOC should be estimated based on a special observer, and the SOH estimation errors based on this method accumulate with SOC estimation errors. A C_remaining (defined by the difference of the maximal releasable capacity and the actual discharge capacity) estimation method through differential voltage analysis was also proposed [29]. However, the accuracy of capacity estimations based on this method could only be guaranteed in a low-capacity range. Guo et al. [30] proposed an effective method to estimate the SOH based on charging curves, in which a typical constant current constant voltage charging method was designed for lithium-ion batteries, and the SOH of the battery could be estimated within 3% error range. Based on the above considerations, a fractional impedance model (FIM) and the interval capacity during the charging process are utilized to evaluate the SOH of lithium-ion batteries. The SOH estimation method was verified using a commercial lithium-ion battery, and the results are described in this paper. In section 2, a particular commercial lithium-ion battery is selected for the electrochemical impedance spectrum test at different states. Moreover, the FIM and the relevant
mixing, (3) transition metal ion dissolution, (4) side reactions between electrode and electrolyte, and so on [14,15]. Many studies have been conducted to estimate the SOH of batteries. Such methods mainly include electrochemical impedance spectroscopy technology [16], electromotive force analysis [17], sampling-based estimation methods [18], model-based estimation methods [19], and so on. As one of the most promising methods to characterize lithium-ion battery ageing effects, researchers indicated that many of the mechanisms responsible for battery ageing and degradation could be monitored and investigated non-destructively by the use of impedance spectroscopy [20]. Yet, this method requires special equipment for implementation, and the expensive cost of this equipment results in difficulties for BMS applications. A SOH estimation method based on an accurate battery model is one of the most popular solutions. Some attempts have been made to evaluate lithium-ion battery models to estimate states and parameters, such as equivalent circuit models, data based models and electrochemical models, and so on [21–24]. A model based on the Laplace transfer time for commercial lithium-ion batteries was proposed by Samadani et al. [25], in which the equivalent circuit model was constructed from the electrochemical impedance spectrum test results of a commercial battery at different temperatures. However, the constant phase element in the equivalent circuit model was approximated using a capacitor. Other models were derived from the models mentioned above. For example, the fractional-order model was proposed, which could maintain higher accuracy for voltage tracking
2
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Fig. 2. Simplified equivalent circuit based on the lithium-ion battery EIS and HPPC test.
Fig. 3. Flow chart for parameter identification.
Fig. 4. Configuration of the battery test-bench.
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Fig. 5. Charging voltage and charging capacity of batteries with different SOH under ambient temperature.
Fig. 6. A hybrid pulse power characterization test of battery.
2. Fractional impedance model and parameter identification
parameter identification method based on a least-squares genetic algorithm (LSGA) are both introduced in this section. In section 3, a battery test-bench is established, and an accelerated ageing experiment for the lithium-ion battery is designed based on the results of a CC-CV charging test for the batteries at different SOH values. Results of the model parameter identification and the accelerated ageing experiment are both discussed in section 4. A novel back propagation neural network method is proposed to estimate the lithium-ion battery SOH based on identified internal parameters and experiment data, and the accuracy of SOH estimations is verified with an accelerated ageing experiment and simulation. Finally, some conclusions are given in section 5.
2.1. Fractional impedance model (FIM) The electrochemical impedance spectrum (EIS) is a powerful labbased diagnostic technique to characterize a battery’s dynamical performance [31]. In this paper, a commercially available lithium-ion battery cell based on Nickel Manganese Cobalt Oxide cathode material and graphitic anode designed for electric vehicle applications was selected. Table 1 presents the specifications of this battery cell. The lithium-ion battery EISs were measured at room temperature (25℃) by the Princeton electrochemical workstation at different SOH and SOC, as shown in Table 1. The EIS test results of batteries at different SOC and SOH are shown in Fig. 1. The different colors of curves represent the EIS of batteries at
4
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Fig. 7. Designed test cycle in the experiment.
The fractional impedance model is described as follows can be summed up as follows:
different SOH, and the EIS of batteries at different SOC are independently shown in the sub-graphs. It can be seen that the EIS curves are similar in shape. The total module of the impedance enhances as the battery ages, and the parameters of the equivalent circuit model could be extracted to characterize battery ageing. Whereas the SOC of batteries remained the same, the numerical variation tendencies of Zre and –Zim were rendered consistent in the same frequency section. The change tendencies of these curves are in well agreement with previous experimental results [16,32,33]. It can be seen from Fig. 1 that, there is a line going straight up, which is crossed with the axis of –Zim = 0, and there are two depressed semi-circles in each of the EIS curves. The intersection of this line and the axis of –Zim = 0 is regarded as the response of Ohmic resistance. The depressed semi-circle in mid-frequency section represents the concentration polarization due to a slow charge transfer reaction in electrodes, and the depressed semi-circle could be modeled by a parallel combination of a resistor and a constant phase element. The curve of the EIS in the low-frequency section is deemed as a part of depressed semi-circle with a large curvature radius, which can be utilized to represent the activation polarization response in electrodes. Based on the analysis above, a simplified equivalent circuit is established, as shown in Fig. 2. Here, Vser denotes the voltage for Ohmic resistance (Rser). The parallel combination of a resistor and a constant phase element is regarded as ZARC. Moreover, V1 denotes the voltage of ZARC1, which can be utilized to characterize the concentration polarization phenomenon inside the battery. Finally, V2 is regarded as the voltage of ZARC2, which represents the activation polarization voltage. Research shows that fractional order calculus can be utilized to establish a more accurate system model [34]. As discussed in [35,36], the FIM of lithium-ion batteries can be established based on the Grünwald-Letnikov definition. According to the circuit theory, from the equivalent circuit illustrated in Fig. 2 and the analysis above, the equations can be presented as follows:
Vo = Vocv + Vser + V1 + V2 ⎧ Vser = −Rser ·I ⎪ ⎪ − I = C1·Δα V1 + V1 R1 = C2·ΔβV2 + V2 R2 ⎨ Δα V1 = −I C1 − V1 R1 C1 ⎪ βV = −I C − V C R ⎪ Δ 2 2 2 2 2 ⎩
N ⎧ Δ x = A·x + B·I ⎨ ⎩ y = C·x + D·I
(4)
− 1 C1 ⎤ − 1 R1 C1 0 ⎤, B=⎡ , where A = ⎡ C = [1 1], ⎢ ⎢ 0 − 1 R2 C2 ⎥ ⎣− 1 C2 ⎥ ⎦ ⎣ ⎦ α V D = [−Rser ], N = ⎡ β ⎤, x = ⎡ 1 ⎤, y = [Vo − Vocv], x ∈ R2 ⎣ ⎦ ⎣V2 ⎦ The Grünwald-Letnikov definition is usually used to discretize the continuous fractional-order equations [37,38], and the Grünwald-Letnikov fractional order calculus is defined as: k G r a Δt f
(t ) = lim h−r h→0 kh = t − a
∑ (−1) j j=0
(rj) f (t − jh)
(5)
r aΔt
is the fractional order calculus operator, h is the sampling r j is the Newton binomial coefficient generalized to real numbers, which can be expressed as r r! j = j ! (j − r ) ! .
where
period, k is the amount of sampling, and
()
()
2.2. Battery parameter identification based on LSGA Many parameter identification methods have been proposed [36,39,40]. Due the battery fractional impedance model is strongly nonlinear, the least-squares genetic algorithm (LSGA), derived from the combination of the least squares method and the genetic algorithm, is selected to identify the internal parameters as discussed in the study [36]. They also developed an equivalent voltage tracking system for parameter identification, and the model is presented as: N ⎧ Δ x ̂ = A·x ̂ + B·I ⎨ ⎩ y ̂ = C·x ̂ + D·I
(6)
1 ⎤ V o − Vocv]. Here, x ̂ ∈ R2 is the state vector of the , y ̂ = [V where x ̂ = ⎡ ⎢V 2 ⎥ ⎣ ⎦ equivalent tracking system, and y ̂ is the estimated output voltage of the equivalent tracking system. The output voltage difference between the equivalent tracking system and battery system was defined as:
(3) 5
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Fig. 8. Results of the interval capacity test. (a) Changes to the interval capacity; (b) SOH and the interval capacities associated with the voltage region [3.95 V, 4.15 V].
e = ŷ− y
(7)
and humidity chamber was used to age the battery submitted to dynamic current cycles under controlled ambient temperature and relative humidity, and the battery internal parameters were identified via Matlab software. To charge the cells, the conventional constant current-constant voltage (CC-CV) protocol was adopted, where 0.3C rate (0.885 A) currents were used as the constant current, the voltage and current cutoff limits were set as 4.2 V and 0.06A. Batteries with different SOH were charged based on the CC-CV protocol, and the results are shown in Fig. 5. In Fig. 5, the charge voltage and capacity at different SOH are denoted by the symbols SOH_Vo and SOH-Cn, and ΔCni (i = 1, 2, 3) was utilized to represent the interval capacities associated with the same voltage difference. The results show that the interval capacities change as the battery ages. The characteristics of the hysteresis of OCV change completely during battery lifetime, and the hysteresis of the OCV can be detected through partial charge/discharge processes [41–43]. Taking into consideration that the battery OCV, in discharge process, is extremely difficult to be obtained while applying on the electric vehicle, the static OCV is selected for the battery model parameters identification. The static OCV can be obtained through long enough relaxed time, for example, 1 h relaxation time. A hybrid pulse power characterization test
The tracking target of the LSGA is represented through the following goal equation:
J (θ) =
t2
∫t1
(e T ·e )dt
(8)
where J (θ) is the objective function, t1 and t2 are the boundary of time. A parameter identification flow chart is presented in Fig. 3. The basic operations of LSGA include coding methods, individual fitness evaluation, genetic operators (such as selection, crossover, and mutation), and so on, as shown in Fig. 3. Here, I and Vocv are the input parameters of the equivalent tracking system. Further, Vo is the output o is voltage of battery system, and can also be measured directly. And V the output voltage of tracking system, and can be adjusted using the equivalent tracking model. Parameters are identified through the LSGA based on the voltage difference between the actual battery system and the equivalent tracking system. 3. Battery ageing experimental design A battery test-bench was established as shown in Fig. 4. The Neware battery testing system associated with a programmable temperature 6
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Fig. 9. Voltage tracing results of batteries with different SOH (98%, 92%, 87%, and 81%) based on the proposed FIM and LSGA.
fully charged and discharged, with 2.95A charged current and 5.9 A discharged current, at 25 ℃ or −10℃. Such accelerated aging charge–discharge cycle is repeated for several times in each test cycle. The regular measurement of the battery’s capacity was mandatory alongside ageing campaigns in the capacity measurement phase.
was taken for the analysis of battery OCV, and results were shown in Fig. 6. It can be seen that the battery voltage error between relaxed 10 min and 1 h is less than 0.008 V, so that the battery voltage is regarded as the approximation of OCV. Ageing campaigns were launched to collect battery voltage and current data over time. The ageing experiment was intended to provoke changes to the internal parameters and interval capacities. Without a doubt, the capacity fading should be taken into consideration in the ageing protocol. Fig. 7 presents the current and voltage profiles in the designed test cycle, and the experiment protocol covered a number of such cycles. As shown in Fig. 7, the test cycle can be divided into four phases: the preparation phase, the step pulse test phase, the accelerated ageing phase, and the capacity measurement phase. One of the major tasks during the preparation phase was to ensure that the batteries were under the same state conditions. Thus, the lower cut-off voltage and the environmental conditions were enforced consistently at the end of the preparation phase. Internal parameters and interval capacities were taken into consideration during the step pulse test phase, as also shown in the subgraph in Fig. 7. Taking into account of the polarization and diffusion in charging process, the battery is relaxed 20 min to make sure that the battery is relatively stability, as shown in Fig. 7, when battery reaches the preset voltage values. Next, the step pulsing test is made, the battery is discharged 30 s with 5.9A current in this period. Then, the battery is relaxed 10 min, based on the analysis of OCV above, the battery voltage is regarded as the approximation of OCV at the end of this relaxation. In order to validate the battery’s fading performance, the number of the test cycles could be reduced with a different accelerated ageing method, as has been conducted in other works [44,45]. In this study, an accelerated ageing charge–discharge cycle was designed during the accelerated ageing phase. In the accelerated ageing phase, the battery was
4. Experimental results and SOH estimation 4.1. Results of the accelerated ageing experiment In the transient response test phase, the interval capacities associated with different voltage regions were measured. Many of the batteries based on the same cathode material were tested, and the experiment results of two batteries, distinguished by the symbols ‘_1’ and ‘_2’, are highly representative in Fig. 8. In Fig. 8(a), the x-axis represents the battery charging-discharging cycle times, and the y-axis represents the interval capacity. The interval capacities associated with different voltage regions are marked as Cn_375395_1, Cn_385405_1, Cn_395415_1, Cn_375395_2, Cn_385405_2, and Cn_395415_2. It can be seen that the interval capacities associated with different voltage regions change as the test cycle increases. The interval capacities associated with the voltage region [3.75 V, 3.95 V] change slightly near 815 mAh, and most of the interval capacities associated with the voltage region [3.85 V, 4.05 V] change slightly near 660 mAh. However, the interval capacity associated with voltage region [3.95 V, 4.15 V] consistently decreases as the test cycle increases. The SOH and the interval capacities associated with voltage region [3.95 V, 4.15 V] are shown in Fig. 8(b). The states of health are marked as SOH_1 and SOH_2. It can be seen that both the SOH and the interval capacities consistently decrease as the test times increases. During the transient response test stage, voltage step response 7
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Fig. 10. Internal parameter identification for batteries with different SOH. (a) Relationship between the SOH and the Ohmic resistances; (b) Relationship between the SOH and the concentration polarization fractional orders; (c) Relationship between the SOH and the activation polarization fractional orders.
batteries with different SOH (98%, 92%, 87%, and 81%) are shown in Fig. 8,after collecting and analyzing the battery voltage data. During the step pulse test phase, the battery was charged until it reached 3.95 V. Then, the battery was left unattended without charging or discharging it, and the end of this rest period was regarded as the
testing is also taken into consideration. The internal parameters of the tested batteries with different SOH were identified based on the FIM and LSGA parameter identification method. Battery voltage was traced via the identified internal parameters based on the equivalent tracking system described in section 2, and the voltage tracing results of 8
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Fig. 13. SOH estimation Results based on BPNN.
starting point shown in Fig. 9. Here, the battery reference voltage is presented by the black solid lines, the green dashed lines are used to describe the battery estimated voltage, and the red solid lines denote the voltage tracing error. It can be seen that the battery voltage tracing error could be well confined to an error region of [-5 mV, 5 mV], and the error rate was less than 0.2% for the lithium-ion battery. Based on the LSGA, result shows that the model parameters can be accurately identified as well as the method proposed in the study [39]. The relationships between the SOH and the internal parameters identified are shown in Fig. 10. In Fig. 10(a), the relationship between the SOH and the Ohmic resistances is shown, where the Ohmic resistances of batteries are represented as Rser_1 and Rser_2. The results show that the Ohmic resistances increase with the fading capacities, which means that the point, crossed with the axis of –Zim = 0 in the EIS, will move to the right with the battery aging at the same state, and this result is also well agreement with the EIS test result in Fig. 1. Fig. 10(b) shows the relationship between the battery SOH and the concentration polarization fractional orders, where the concentration polarization fractional orders are marked as α_1 and α_2. It can be seen that the concentration polarization fractional orders first decrease rapidly at first, and then increase when the aging state of the battery is 87.5%, the increasing rate gradually decreases and tends to be stable as the test cycle increases. The results are consistent with the changes of the concentration of lithium ion and SEI membrane inside the battery [46]. In Fig. 10(c), the relationship between the battery SOH and the activation polarization fractional orders is shown, where the activation polarization fractional orders are marked as β_1 and β_2. The results show that the activation polarization fractional orders increase rapidly at first, and then increase slowly. Results show that the variation regularity of the activation polarization fractional orders is agreement with the change of SEI membrane thickness inside the battery and the active substances in the electrolyte [46].
Fig. 11. The typical structure and the approximation system structure of BPNN. (a) The typical structure of BPNN; (b) The approximation system structure of BPNN.
Fig. 12. SOH estimation based on BPNN. Table 2 Parts of training data from one cell. Cn_discharge
Cn_395415
Rser
α
β
2790 2749 2724 … 2610 2593 2588 … 2458 2465 2467 2432
647.3 646.3 642.4 … 613.4 610.7 609 … 598 596.8 594.6 590.4
39.65 41.49 42.81 … 50.91 52.04 52.67 … 61.43 60.63 62 61.82
0.8733 0.8687 0.8645 … 0.8732 0.8778 0.8766 … 0.8924 0.8918 0.8924 0.8915
0.2107 0.2267 0.2513 … 0.3026 0.2976 0.3011 … 0.3327 0.319 0.3098 0.3182
4.2. SOH estimation based on BPNN Table 3 Parts of validation data from other cells. Cn_discharge
Cn_395415
Rser
α
β
2762 2662 … 2539 2765 2493 2751 2395
653.8 637.2 … 633.7 638.8 606.7 646.1 596.9
40.77 42.06 … 46.26 41.23 56.59 43.31 62.57
0.8721 0.8644 … 0.8568 0.8698 0.8813 0.8727 0.8897
0.2167 0.256 … 0.242 0.2482 0.3011 0.2308 0.3135
It can be seen from the results of the accelerated ageing experiment that the fractional impedance model could well characterize the battery’s performance, and the battery model-based internal parameters change as the battery’s capacity fades. Moreover, the interval capacities associated with the voltage region [3.95 V, 4.15 V] (Cn_395415) consistently decreases as the battery’s capacity fades. Due to the complex electrochemical process in the battery, it is difficult to describe the relationships between the SOH and these parameters with an exact function. As the back propagation neural network (BPNN) can be used to describe the strong nonlinear function without exact parameters, so the BPNN was introduced to depict the relationship between the 9
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genetic algorithm with a 0.005 V error margin. In order to validate the battery decaying performance, an accelerated ageing experiment was performed. The results of parameter identification and the accelerated ageing experiment show that the battery’s internal parameters and interval capacities change as the battery’s capacity fades. A back propagation neural network method was proposed based on the internal parameters and the interval capacities associated with the voltage region [3.95 V, 4.15 V], and was utilized to successfully estimate battery SOH with an error margin of merely [-1.5%, 1.5%] without a special state observer or filtering approach. It is worth noting that the SOH estimation method proposed in this study is still in the developmental stage, and needs further simplification and validation by a BMS designer.
internal parameters, the interval capacity and battery SOH. The typical structure BPNN and the approximation system structure of BPNN are shown in Fig. 11. In Fig. 11(a), the number of input neurons is m, the number of output neurons is n. The S-type function tansig is selected as the transfer function of the hidden layer, and the purelin liner function is used as the transfer function of the output layer. In Fig. 11(b), the error between the target system except values and the neural network system output values is fed back to the neural network system. According to the accuracy requirement for the nonlinear function approximation, the BPNN is designed with three layers in this study, as shown in Fig. 12: namely, the input layer, hidden layer, and output layer. The interval capacities associated with the voltage region [3.95 V, 4.15 V] (Cn_395415) and the internal parameters, including Ohmic resistance (Rser), the concentration polarization fractional order (α), and the activation polarization fractional order (β) were selected as the input parameters. The SOH estimation was designed as the goal of the network. The interval capacities associated with the voltage region [3.95 V, 4.15 V] (Cn_395415) can be measured directly during the charging process, and the internal parameters can be identified based on the FIM and the LSGA parameter identification methods mentioned in section 2. Data retrieved from one battery in the experiment were used as the training data, and the remaining were used for validation of the presented algorithm. Parts of training and validation data are given in Tables 2 and 3. The accuracy of the battery SOH estimation is shown in Fig. 13. The correlation between the estimated SOH (the blue line in Fig. 13 and the real SOH (the red line in Fig. 13 was quantified. It can be seen that the SOH error rate between estimated and real can be well confined to a small error region [−1.5%, 1.5%]. This result implies that the battery SOH can be precisely estimated via the internal parameters and the interval capacities based on the BPNN.
CRediT authorship contribution statement Qingxia Yang: Conceptualization, Methodology, Software, Writing - original draft, Data curation. Jun Xu: Visualization, Investigation. Xiuqing Li: Software, Writing - review & editing. Dan Xu: Writing review & editing. Binggang Cao: Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work has been funded by the Scientific Research Staring Foundation for Doctor of Henan University of Science and Technology, China (Grant no. 13480045), the National Natural Science Foundation of China, China (Grant no. 51405374), and the Key Research and Development Plan of China (Grant no. 2017YFB0103802) and National Science and Technology Ministry, China.
5. Future work Taking into account that the battery performances can be influence by many other factors, for example, the cell design, manufacture, boundary of working current and voltage, working conditions (including temperature, humidity), and so on. It is significant and necessary to design the aging test by considering the aforementioned impact factors in the future work, and the correlation between SOH and temperature will be further studied based on the proposed method. At moment, the single cell aging state are being studied. In realities, due to the lower voltage and capacity, the cells are often connected in series and/or parallel to form battery pack. Due to the in-homogeneous operating conditions, the inherent variations exist among the cells in a battery pack, and the cell-to-cell variation will become even larger with the cells aging. These variances result in SOH deviation among individual cells in the battery pack, therefore, the SOH estimation method for single cell may not be suitable for the battery pack. How to adapt the SOH estimation technique based on the proposed method from cell level to multicell module level remains an open question, which will be discussed in the following work. Moreover, the battery SOH was estimated based on the interval capacities, which were always obtained from a specific cycle, when the cell was fully discharged in this study. In future, the interval capacities can be obtained from other cycles, for example, the cell was discharged below preset voltage 3.9 V, or a much lower voltage level.
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