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An open circuit voltage-based model for state-of-health estimation of lithium-ion batteries: Model development and validation
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An open circuit voltage-based model for state-of-health estimation of lithium-ion batteries: Model development and validation Xiaolei Bian a, *, Longcheng Liu a, Jinying Yan a, b, Zhi Zou a, Ruikai Zhao a a b
Department of Chemical Engineering, KTH-Royal Institute of Technology, S-100 44, Stockholm, Sweden Vattenfall AB, R&D, 169 92, Stockholm, Sweden
H I G H L I G H T S
� An open circuit voltage-based model is proposed to estimate the SOH of LIBs. � The accuracy of the model is validated by experimental results of an LFP cell. � The effect of size and location of voltage window on the model accuracy is studied. � ICA is used to explain the model accuracy change with different voltage window. � The versatility of the model to different chemistries is demonstrated. A R T I C L E I N F O
A B S T R A C T
Keywords: Lithium-ion battery State-of-health Open circuit voltage-based model Voltage window Incremental capacity analysis
An open circuit voltage-based model for state-of-health estimation of lithium-ion batteries is proposed and validated in this work. It describes the open circuit voltage as a function of the state-of-charge by a polynomial of high degree, with a lumped thermal model to account for the effect of temperature. When applied for practical use, the model requires a prior learning from the initial charging or discharging data for the sake of parameter identification, using e.g. a nonlinear least squares method, but it is undemanding to implement. The study shows that the model is able to estimate the state-of-health of a LiFePO4 cell cycled under conditions where the tem perature has fluctuated significantly with a relative error less than 0.45% at most. A short part of a constant current profile is enough for state-of-health estimation, and the effect of size and location of voltage window on the model’s accuracy is also studied. In particular, the reason of accuracy change with different voltage windows is explained by incremental capacity analysis. Additionally, the versatility and flexibility of the model to different chemistries and cell designs are demonstrated.
1. Introduction The lithium-ion batteries (LIBs) were first designed and developed by Asahi Kasei during the 1980s and commercialized by Sony and A & T Battery Corporation in 1991 and 1992, respectively [1]. A few years after commercialization, LIBs took over half of the portable electronic devices market [2], and the technology is now under rapid development for application in electric vehicles and stationary electrical energy storage [3,4]. High specific energy density, high efficiency, high nominal voltage, less maintenance requirement, and long cycling life are the main ad vantages of LIBs [5,6]. However, they inevitably degrade and deterio rate within the usage process, leading to a reduction of their abilities in
e.g. storing energy and powering performance [7]. In the worst case, safety issues and even catastrophic accidents may occur due to the failure of batteries [8]. In order to avoid the abuse and extend the life span by monitoring and controlling LIBs performance, battery man agement system (BMS) has already been developed and widely implemented [9]. Among many tasks, the most important one of BMS is to accurately estimate the state-of-charge (SOC) and state-of-health (SOH) of LIBs [10,11]. SOC is commonly defined as the percentage of the maximum possible charge that is now present inside a rechargeable battery [12]. It is equivalent to the remaining fuel gauge in internal combustion vehicles, signalling how long a battery can continue to work before it needs recharging. The definition of SOC varies slightly from different
* Corresponding author. E-mail address: [email protected] (X. Bian). https://doi.org/10.1016/j.jpowsour.2019.227401 Received 24 May 2019; Received in revised form 1 October 2019; Accepted 1 November 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Xiaolei Bian, Journal of Power Sources, https://doi.org/10.1016/j.jpowsour.2019.227401
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Fig. 1. Categorization of SOH estimation methods.
references [13,14]. In this study, SOC is defined as Q SOC ¼ SOC0 þ Qmax
voltage range [12]. By contrast, experimental techniques do not need data-learning and intensive computation, but remain mainly to be off-line, low accurate and high cost [14,19]. For a detailed discussion of the advantages and drawbacks of these available methods, the reader is referred to the review papers [16,18,20]. Noticeably, Guo et al. [24] recently developed an adaptive yet simple approach to estimate the SOH of LIBs. Instead of making an incremental capacity analysis (ICA), an equivalent circuit model was adopted by this approach to characterize the constant-current (CC) portion of the charging profiles, and a voltage transformation function with time-based parameters was employed. It was shown that, with only minor demand of learning from initial charging profiles, an error within 3% was achieved between the estimated and measured results about SOH for three kinds of commercial LIBs. This encouraging result has inspired us to apply the same method to estimate the SOH of a LiFePO4 (LFP) cell. However, it was failed with the usage of a simple function for learning, due to the very different chemical properties between the types of LIBs and also the essentiality of figuring out the correlation between the open circle voltage (OCV) and SOC from the learning process [25]. In reality, this inherent correlation is always contained in the voltage transformation function even though it is obviously absent. To avoid the use of a voltage transformation function with timebased parameters, we tried in this study to describe the OCV simply as a function of SOC by a polynomial of high degree, with a lumped thermal model to account for the effect of temperature. Although this approach also requires a prior learning from the initial charging or discharging profile at the CC step, it was shown to be able to evaluate the SOH of an LFP cell cycled under conditions where temperature fluctuated signifi cantly over a wide range with a relative error from 0.43% to 0.45% (the mean error is 0.18%). The effect of size and location of voltage window was also studied and it was found that with only short part of a CC profile, the model can still estimate SOH precisely as long as the window location was chosen properly. In addition, it was justified that a voltage window with four
(1)
where SOC0 is the SOC of the battery at t ¼ 0 when the charge or discharge commences, Q is the charged capacity and Qmax is the current maximum capacity of the battery. By contrast, SOH is a ‘measure’ that reflects the general condition of a battery and its ability to deliver the specified performance in com parison with a fresh battery [15,16], and it is commonly defined as [17–19], SOH ¼
Qmax � 100% Qnom
(2)
where Qnom is the nominal capacity of the battery. Accurate evaluation of the SOH of LIBs becomes, therefore, crucial in order to guarantee satisfactory performance and to replace the end-oflife battery when needed [20]. However, SOH cannot be directly measured; it can only be estimated according to measurable parameters, such as current, voltage and temperature [21]. As a result, extensive studies have been carried out over the last decades and different methodologies have been developed to estimate the SOH of LIBs, by accounting for e.g. different mechanisms of battery fading, which orig inates from many factors and interactions such as service type, working condition, and aging history, etc [22]. As summarized in Fig. 1, the methods of SOH estimation can roughly be divided into three groups: using either adaptive models, experimental techniques or differential approaches. All these methods have, however, distinct advantages and disadvantages [16,18]. Adaptive models often have strong physical meanings and the parameters involved generally have close relations with the underlying electrochemical processes in batteries, but usually need data-learning and intensive computation [17, 23]. Differential approaches also have advantages of battery degrada tion identification, but require static charging or discharging over a wide 2
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Therefore, the dynamic behaviour of the charging process of LIBs may be described by the Thevenin equivalent circuit [24,28], as shown in Fig. 3, to give OCVðSOC; TÞ ¼ V
(3)
V0
Vp
with 0 t
Vp ¼ Vp;0 e τ þ IRp @1
1 (4)
t
e τA
and (5)
V0 ¼ IR0
where V represents the terminal voltage, Vp is the electrochemical po larization voltage, V0 is the voltage drop between the ohmic resistance and OCV denotes the open circuit voltage, which is a function of SOC and temperature T. Vp,0 is the electrochemical polarization voltage at time t ¼ 0, τ is the time constant of the cell, I is the constant current (which is defined as positive for charging), Rp is the electrochemical polarization resistance, and R0 is the ohmic resistance. When Eqs. (3)–(5) are combined, we obtain � t � (6) OCVðSOC; TÞ ¼ V þ IRp Vp;0 e τ I R0 þ Rp
Fig. 2. Charging profiles of an LFP cell at different degradation statuses, ob tained from the constant-current step with CC ¼ 0.3 C.
or equivalently, OCVðSOC; TÞ ¼ V þ IRp
� Vp;0 e
Q Iτ
I R0 þ Rp
�
(7)
It follows, from Eqs. (1), (2) and (7), that an accurate description of the OCV(SOC) function, or its inverse function, has become critical to give a good estimation of Qmax and subsequently SOH. Unfortunately, it is widely known that the OCV(SOC) function is both temperature and age-dependent, making R0 and Rp apparently temperature-dependent and Q and τ mostly SOC-dependent in Eq. (7), and thus it may vary remarkably under different operating conditions [25,29–31]. As a result, most of the phenomenological OCV-SOC models that are built as a sum of simple functions [25,32] should be avoided for use in Eq. (7) because they do not well account for phase transition and staging phenomena associated with lithium-ion intercalation/deintercalation process of the active material and usually only show one or two peaks on the ICA curves [33]. This is, however, not the case if one uses a polynomial of degree around ten to describe the OCV(SOC) function [34], which enables to capture four ICA peaks for LFP batteries, in consistent with the results of ICA of electrochemical processes of LIBs [25]. For this reason, we pro pose in this study a polynomial-based, lumped thermal model that also accounts for the effect of temperature on the OCV(SOC) function by the Arrhenius law [35] to give � � �� m E 1 1 X OCVðSOC; TÞ ¼ exp ak;ref SOCk (8) R Tref T k¼0
Fig. 3. Schematic of the Thevenin equivalent circuit model.
ICA peaks, or at least with the later three peaks, was necessary for the model to estimate SOH of an LFP battery reasonably well. To further verify the applicability and flexibility of the proposed model for SOH estimation, the datasets of LiNi0.8Co0.15Al0.05O2 (NCA) battery cells from the NASA Ames research center have also been uti lized [26,27]. It was shown that the estimated SOHs agree nicely with the measured results over the entire lifespan for the four NCA cells studied, with the relative errors ranging from 1.75% to 1.45%. This paper is organized as follows. In Section 2, the OCV-based model is formulated in detail, with specific analysis of the model characteris tics. In Section 3, the accuracy of the proposed model is verified and validated through five series of accelerated degradation tests, with a comparison of the estimated and measured SOH results of an LFP cell. In Section 4, the effect of size and location of the voltage window on SOH estimate is analysed and the number of ICA peaks should be covered by the voltage window is examined. In Section 5, the versatility and flexi bility of the model to different chemistries and cell designs are demon strated by the datasets from the NASA Ames research center. Some concluding remarks are, then, summarized in Section 6.
where E is the activation energy, R is the gas constant, Tref is the refer ence temperature at which the constants a0,ref, …, am,ref are evaluated for the OCV(SOC) function, and m is the degree of the polynomial in SOC. When Eq. (1) (7) and (8) are combined, we obtain a single-variable function V(Q) for the CC charging or discharging profile as � � � �k �� m � Q E 1 1 X Q V ¼ exp ak;ref SOC0 þ IRp Vp;0 e Iτ R Tref T Q max k¼0 � (9) þ I R0 þ Rp
2. Development of the SOH estimation model As discussed by Guo et al. [24], the charging (or discharging) profiles obtained from the CC step of a LIB during different degradation stages are similar to one another, albeit gradually shifting to a shorter time span and higher voltage range with aging. This is commonly true regardless of cell chemistry, as shown in Fig. 2 for a fresh LFP cell. More importantly, it is noted that all of the CC charging profiles of terminal voltage-charged capacity reproduce the main characteristics of the OCV-SOC curve [21,22] due to the predominant effect of the OCV in governing the profiles of the terminal voltage over the charging process with a constant current.
or simply m X
V ¼ c0 k¼1
3
� �k Q ak;ref SOC0 þ þ c1 ec2 Q þ c3 Qmax
(10)
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Fig. 5. Profile of the terminal voltage and current during the initial test.
similarity between the charging profiles at the CC step under different working conditions. It should, also, be noted that for LiCoO2/LiNixMnyCozO2-based LIBs, the rather smooth and uncomplicated dependence of the OCV on SOC, resulting from the approximate phase transformation voltages of LiCoO2 and LiNixMnyCozO2 [25], makes it possible to easily find a simple OCV (SOC) function with a polynomial of degree three [24]. For LFP batte ries, however, a polynomial of degree greater than 10 is required because only then a flat voltage plateau can be produced on the OCV-SOC curve with two-phase regions corresponding to FePO4 and LiFePO4, respectively [25]. As a result, we found that the curve-fitted charging profiles by the approach of Guo et al. [24] are very sensitive even to a slight change in the time-based parameters, making the process of parameter identifi cation often corrupt and ineffective. By contrast, our approach is much stable and fast in parameterizing the model for LFP batteries at any aging statuses of life. The main drawback of the polynomial-based OCV-SOC models is, perhaps, the limitation in its generality and adaptivity to different chemistries and cell designs. A logarithmic or exponential function might also be needed to include in the OCV(SOC) function in order to well represent different types of LIBs [24,25,32,33], especially when the degree m of the polynomial is small.
Fig. 4. Flowchart of the OCV-based model for SOH estimation.
where the introduced coefficients, c0, …, c3, can be regarded as con stants in a single CC charging or discharging process. Noticeably, the constants a1,ref, …, am,ref are only needed to deter mine once and they can readily be identified by fitting Eq. (10) to the initial charging or discharging profile when a battery is commenced for service with a nonlinear least squares method. If desired, the constant a0, ref can also be derived from Eq. (8) by specifying an OCV at which SOC would become zero. The result is, then, an estimate of the OCV(SOC) function at the reference temperature. The constants c0, …, c3, SOC0 and Qmax should, however, be updated all the time as battery ages. In particular, it should be noted that c0 represents actually the Arrhenius term in Eq. (8). It is the only constant that is apparently temperature-dependent and a certain constraint such as c0 2 ½0:5; 1:5� should be imposed. Likewise, Qmax 2 ½Qend ; Qmax;0 � had better be applied, where Qend is the charged capacity at the end of CC charging process and Qmax,0 is the initial maximum capacity of the battery. With initial training of the model to obtain parameters of a1,ref, …, am,ref, the evolution of Qmax with aging can be evaluated directly by fitting Eq. (10) to the charging/discharging profile of V-Q at later times. When use is made of Eq. (2), the SOH value can be readily calculated, as illustrated in Fig. 4 for the entire procedure of the model in estimating the SOH of batteries. Clearly, the methodology adopted in this study is quite different from that of Guo et al. [24]. Instead of using a transformation function, where the OCV(SOC) function was implicitly embedded in a base charging profile, we have now explicitly specified a polynomial form of the OCV (SOC) function, as given in Eq. (8), and used it to account for the
3. Validation and evaluation To evaluate and also validate the proposed model, SOH estimation of an LFP cell has been made in comparison with those obtained from five series of accelerated degradation tests. The cell used in the tests is an LFP battery with 60 Ah nominal capacity and 3.2 V nominal voltage. 3.1. Initial charge/discharge test A prior learning from the initial charging profile of a fresh battery is needed for training the proposed model. For this sake, an initial charge/ discharge experiment was made before commencing the accelerated degradation tests. This experiment contains 3.5 full charge-discharge cycles at an average temperature of 0 � C (varying between 5.8 � C and 5.5 � C). As illustrated in Fig. 5, the procedure of these cycles is: (1) rest for 5 min; (2) charged at a CC of 18 A (0.3 C) until an upper threshold voltage of 3.65 V was reached; (3) charged at a constant-voltage (CV) of 3.65 V until a cut-off current of 1.8 A (0.03 C) was achieved; (4) rest for another 5 min; (5) discharged at a constant current of 18 A (0.3 C) until a lower threshold voltage of 2.5 V was reached. The above process of rest-CC charge-CV charge-rest-CC discharge was repeated for three times, followed by steps of rest-CC charge-CV charge. After the initial test, the LFP cell was left in an open-circuit condition for around 2 days for depolarization and stabilization before 4
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Fig. 7. Illustration of the accuracy of the OCV-based model for comparison of experimental and curve-fitted V-Q results, including relative errors, (a) ob tained at the last full charge-discharge cycle before the aging test, with CC ¼ 0.3 C and T ¼ 0 � C, and (b) obtained at the end of the fifth series of the accelerated degradation tests, with CC ¼ 0.3 C and T ¼ 35– 40 � C.
Fig. 6. Profile of the accelerated degradation tests: (a) terminal voltage and current, and (b) temperature.
the accelerated degradation tests. 3.2. Accelerated degradation tests Following the initial test, five series of accelerated degradation tests were successively operated, with a rest interval of 2.5–4.75 days applied between two aging tests. The first four accelerated degradation tests were performed for twenty days, while the last one took about 19 days. At the end of each degradation test, 3.5 full charge-discharge cycles (in the same way as in the initial test) were also made in order to evaluate the capacity or SOH fade at a relative low C-rate. As exemplified in Fig. 6 (a), each of the degradation tests follows the periodic procedure: (1) rest for 5 min; (2) charged at a CC of 30 A (0.5 C) until an upper threshold voltage of 3.65 V was reached; (3) rest for another 5 min; (4) discharged at a CC of 30 A (0.5 C) until a lower threshold voltage of 2.5 V was reached. Compared with the initial test, the accelerated degradation tests consisted only of CC but no CV charging/discharging steps at a larger current, i.e., 30 A (0.5C). Consequently, the periodic time of the degradation tests is around 5 h, much shorter than the initial test of about 8 h. In addition, it is noted that these degradation tests were operated in an environment where temperature fluctuated in a certain range to mimic real working conditions. In the first series of the aging tests, as illustrated in Fig. 6 (b), the temperature fluctuated considerably, vary ing from 13.5 � C to 11.7 � C. By comparison, the temperature varied moderately but was high in the fifth series, ranging from 30 � C to 44.5 � C. The other three series of the aging tests also followed a similar pattern of temperature variations but ranged differently i.e. 4–27.6 � C
for second series, 22–35.5 � C for third, and 26–42 � C for forth. These significant changes in environmental temperature make the method of Guo et al. [24] less favorable to apply, because no constraints could reasonably be imposed on the model parameters. With our approach, however, the parameters in the OCV-based model can readily be identified by fitting the charging profiles of LIBs at any aging statuses, via a nonlinear least squares method, even if the working condition has changed severely over time. 3.3. Parameter identification In the LFP aging tests, the voltage window of the charging profile is 2.5–3.65 V, as shown in Fig. 6. Since it is unnecessary to use the whole voltage range for application of the OCV-based model, we chose a voltage window of 3.0–3.6 V (which is large enough to cover the dominant range of voltage changes), instead, in this section and also in Section 3.4 for both parameter identification and SOH estimation. As discussed in Section 2, the constants a1,ref, …, am,ref are deter mined only once by fitting Eq. (10) to the initial charging profile of V-Q and then kept unchanged in later usage. The other parameters, i.e. c0, …, c3, SOC0 and Qmax, evolve with time but can readily be identified by fitting Eq. (10) to the subsequent V-Q profiles, via the nonlinear least squares method. The result gives, then, a curve-fitted V (Q) function at 5
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Table 1 The eleven constants a1,ref, …, a11,ref determined from initial charging profile. a1,ref 20.31 a7,ref 226672.57
a2,ref
a3,ref
326.85
a4,ref
2942.28
16197.07
a8,ref
a9,ref
247039.31
a5,ref 57934.93 a10,ref
172411.88
69609.69
a6,ref 139162.86 a11,ref 12354.64
Table 2 The constants c0, …, c3 determined from initial charging profile. c0 1.00
c1 8.42 � 10
c2 5
9.99
c3 2.72
different aging status. As demonstrated in Fig. 7 for two cases, with m ¼ 11 and voltage ranging from 3.0 V to 3.6 V, the curve-fitted results of the OCV-based model show a good agreement to the charging profiles, with the rela tive error mainly falling in the range of �0.5% and mean error of 1:654 � 10 4 % and 9:776 � 10 5 %, respectively. As a consequence, it is deemed that the value of Qmax obtained in each of the cases can be used with great confidence to evaluate the capacity fade of the LFP cell with aging. The use of a polynomial of degree 11 in Eq. (10) is, therefore, appropriate for the LFP cell under study in describing the OCV(SOC) function in the OCV-based model, with the constants a1,ref, …, a11,ref, given in Table 1 and c0, …, c3 given in Table 2. As mentioned in Section 2, the constants a1,ref, …, a11,ref are only needed to determine once using the initial charging or discharging profile and can be applied directly to estimate SOH latter with Eqs. (10) and (2). It is also noteworthy to stress that the Arrhenius term c0 has become unity when fitting Eq. (10) to obtain these 11 constants by a nonlinear least squares method. 3.4. SOH estimation Qmax is the key parameter to be evaluated from the process of data fitting. As discussed already in Section 3.3, it can be identified by fitting Eq. (10) to the available V-Q profiles at different aging status after initial training of the OCV-based model. The result allows SOH to be estimated immediately from Eq. (2). To illustrate the accuracy of this approach, we now set the voltage window as 3.0 V–3.6 V and show in Fig. 8 (a) the estimated SOH for the entire period of accelerated degradation tests together with the available data measured at each series, and partial enlargement of the results in Fig. 8 (b). Clearly, the estimated SOH from the OCV-based model, SOHe, fluc tuates somehow in response to temperature variations in the tests. It well follows, however, the available data of measurement, SOHm, and shows a linear decline tendency as cell ages. The relative error between SOHm and SOHe, i.e. ðSOHm SOHe Þ=SOHm *100, was found to be in the range of 0.43%–0.45%, with a mean of 0.18%, as shown in Fig. 8 (c). This suggests that the proposed model might be used satisfactorily in practical applications to give a robust and accurate SOH estimate, even when the battery has been cycled under conditions where temperature fluctuated significantly. The premise is, however, that a voltage window on the charging or discharging profiles should be selected properly for parameter identification and SOH estimation. As a general rule of thumb, the larger the voltage windows ranges, the more accurate the estimated SOHs would be.
Fig. 8. The SOH estimation using the OCV-based model against experimental results (a) over the entire period of accelerated degradation tests, (b) partial enlargement, and (c) the relative error.
In the initial and accelerated degradation tests that have been dis cussed, the voltage ranges from 2.5 V to 3.65 V, allowing us to define a voltage window of 3.0–3.6 V for parameter identification and SOH estimation. In reality, however, it is seldom possible to have the CC charging or discharging profiles covering the whole-range voltage window. Therefore, it is important and of practical significance to esti mate SOH of LIBs with only part CC charging or discharging period.
4. Effect of the voltage window It is not hard to imagine that not only the size but also the location of the voltage window will have a considerable influence on the accuracy of the OCV-based model for SOH estimation. 6
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Fig. 9. The SOH estimation against experimental results (a) and the absolute value of the relative errors (b) over the entire period of accelerated degradation tests using the OCV-based model for different size of voltage window.
Fig. 10. The mean (a) and maximum (b) of the absolute value of the relative errors for the SOH estimated results shown in Fig. 8 over different cycles.
4.1. Effect of the size of voltage window
during 500 cycles. This is a significant contrast to the results obtained in the case of using 3.0–3.3 V as the voltage window, which gives a maximum of 8.08%. Additionally, Fig. 10 substantiates that both the mean and maximum of the errors well follow a downhill trend as the voltage-window size increases. The SOH estimation may, therefore, fail if the window size is too small, such as 3–3.3 V or 3–3.4 V in this study. In general, a larger voltage-window size will result in more accurate SOH estimate. How ever, the accuracy of the OCV-based model will not increase signifi cantly with increasing the window size once the voltage window has been defined large enough (the maximum of the errors was decreased by 5.38% when a voltage window of 3.0–3.5 V was used instead of 3.0–3.4 V, while it was decreased only by 0.50% when 3.0–3.6 V was considered instead of 3.0–3.5 V). The reason for this phenomenal effect of the voltage-window size on the SOH estimation can be explored in accordance with the ICA curves of the model results. As shown in Fig. 11, the ICA curves can well capture the fine features of the voltage plateaus of Q-V curves, and four ICA peaks associated with the phenomena of phase transition inside the cathode and anode are clearly identifiable over the voltage window of 3.0–3.5 V. The ICA peaks are asymmetric and, as battery ages, the changes in both the positions and the amplitudes of the ICA peaks are identifiable. The evolution of the ICA results may, therefore, be used to analyse the aging origins [36,37], such as the loss of active material, the
Using the same approach, as discussed in Section 3, but different size of voltage windows starting from 3.0 V, we re-estimated SOH of the LFP cell over the entire period of accelerated degradation tests with the OCVbased model. The results are shown in Fig. 9. It is seen clearly that, with the voltage window of 3.0–3.3 V or 3.0–3.4 V, the estimated SOHs are not only far away from the measured results but also not follow the changing trend. When the window size is increased to 0.45 V, however, the estimated SOH well follows the pattern of the measured SOHs yet with relatively low accuracy. As the window size continues to increase, the accuracy of the OCV-based model improves significantly; the estimated SOHs agree perfectly with the measured results when the window size has been increased to 0.6 V or 1.15 V. In particular, Fig. 9 (b) indicates that the relative errors of SOH estimation all become less than 1% with the voltage window of 3.0–3.5 V, 3.0–3.6 V, and 2.5–3.65 V. Compared to the results obtained with the full CC charging profiles, this simply implies that the whole size of voltage window, i.e. 2.5–3.65 V, is not a prerequisite to obtaining good SOH estimate; only part of it could still achieve satisfactory results. As also shown in Fig. 10 for the mean and maximum of the absolute value of the relative errors over different cycles, the voltage window of 3.0–3.6 V is sufficient in estimating the SOHs of the LFP cell under study. It gives a maximum of 0.45% in the absolute value of the relative errors 7
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Fig. 11. Illustration of the Q-V and ICA curves obtained at the initial test and the ends of the five series of accelerated degradation tests (a) over the voltage window of 3.0 V–3.55 V, and (b) part results to show the four ICA peaks.
loss of useable lithium-ion, and the increase of the internal resistance, etc. For our purpose, however, we note that the voltage window of 3.0–3.3 V covers only the first ICA peak, while the voltage window of 3.0–3.4 V covers three ICA peaks but not the last one. It seems, therefore, that only when all or most of the ICA peaks can be identified within the voltage window, an accurate SOH estimation could result. To see if this is the case and also figure out which ICA peaks are important in affecting the performance of the OCV-based model, we continue to study the ef fect of the location of voltage window.
Fig. 12. The SOH estimation against experimental results (a) and the absolute value of the relative errors (b) over the entire period of accelerated degradation tests using the OCV-based model with a voltage window of a size of 0.3 V located in different positions.
accurately under, however, the condition that its location can be determined properly so as to cover all the ICA peaks associated with the phenomena of phase transition inside the LFP cell. In the worst case, i.e. when only the first ICA peak shown in Fig. 11 is contained in the voltage window of 3–3.3 V, the estimated SOHs from the model are entirely wrong compared with the measured results. This, together with the fact that good estimated SOHs can still be obtained with a voltage window of 3.3–3.6 V, indicates that the first ICA peak does not really affect the average model accuracy and that the first ICA peak might not be closely related to the aging origins of the LFP cell, in contrast to the other three ICA peaks [36,37]. To validate this assump tion, the evolution of ICA curves over the whole aging tests are plotted, as shown in Fig. 14. It is obvious that the attributes of peak 1 of the ICA curves only change slightly over the 500 cycles, in significantly contrast to those of the other three peaks. One may, therefore, deem that the first ICA peak might be unimportant in estimating the SOH of LFP batteries. It is deemed, therefore, that a voltage window covering all or at least the last three ICA peaks is a prerequisite to estimating SOH satisfactorily for an LFP cell using the OCV-based model. In addition, the model ac curacy would also depend on the size and location of the voltage window in the case when all the four ICA peaks are been contained. As discussed in Section 4.1 that a larger voltage-window size would give a more ac curate SOH estimation. If they cover all the peaks with same voltage size but are located at different positions, we believe that the difference in
4.2. Effect of the location of voltage window By moving the voltage window with a fixed size of 0.3 V from 3.0 V to 3.3 V, we show in Fig. 12 how the accuracy of the estimated SOHs would change. Clearly, only when the voltage window covers all the ICA peaks shown in Fig. 11, i.e. when it is located at 3.2 V or 3.25 V, good estimate of the SOHs by the model could be achieved. In the case of selecting 3.3–3.6 V as the voltage window, the estimated SOHs are still acceptable yet become a bit worse due to the fact that the first ICA peak located at ~3.28 V has been disappeared. This conclusion could also be drawn from Fig. 13, where we compared the mean and maximum of the absolute value of the relative SOH errors over different cycles when different voltage windows are selected. In particular, it is noted that the use of 3.25–3.55 V as the voltage window gives the best SOH estimate, with a mean of 0.36% and a maximum of 0.68% in the absolute value of the relative errors, respectively, over 500 cycles. This encouraging result suggests that a voltage window as small as 0.3 V is large enough to estimate SOH 8
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Fig. 15. Variation of the key parameters in the NASA tests: (a) terminal voltage and current of B0005, and (b) discharging profiles of B0005 at different degradation statuses.
short duration of the degradation tests encourages us to further explore its versatility and flexibility to different chemistries and cell designs. For this propose, the datasets from the NASA Ames research center [26] for the aging tests of four NCA cells with the nominal capacity of 2 Ah over nearly the whole useable lifetime could be utilized. As shown in Fig. 15 (a) for the cell labeled as B0005, all the aging tests of the NCA cells consisted of a series of CC-CV charge and CC discharge cycles at ambient temperature. The CC charge step was all performed at 1.5 A until a cut-off voltage of 4.2 V was reached. It was, then, immediately followed with the CV charge step until the current dropped to a cut-off value of 20 mA. Afterward, the CC discharge step was all performed at 2 A until, however, different cut-off voltages was approached, which were 2.7, 2.5, 2.2, and 2.5 V, respectively, for the B0005, B0006, B0007, and B0018 cells considered [26]. The results of the aging test of the four NCA cells allows us to obtain the CC charging or discharging profiles of terminal voltage-charged/ discharged capacity at different degradation statuses. In this examina tion, however, we chose to use the CC discharging profiles, as exem plified in Fig. 15 (b) for the B0005 cell, as the basis for SOH estimation to illustrate the universality of the OCV-based model. Similar to Figs. 2 and 15 (b) also shows that all the V-Q profiles obtained at the CC discharging steps are similar to each other, albeit gradually shifting to a shorter time span. We can, therefore, follow the same procedure as discussed in Section 3 to evaluate the parameters of the OCV-based model for the NCA cells, but with the degree of poly nomial m ¼ 8 in Eq. (10) instead of m ¼ 11. That is, we first use the initial CC discharging profile to train the model, obtaining the constants of a1, ref, …, a8,ref, and then evaluate Qmax for each of the succeeding V-Q profiles by a nonlinear least squares method. The use of Eq. (2) gives,
Fig. 13. The mean (a) and maximum (b) of the absolute value of the relative errors for the SOH estimated results shown in Fig. 12 over different cycles.
Fig. 14. The evolution of ICA curves over 500 cycles obtained from the OCVbased model, the initial cycle is coloured as green and the last cycle is red. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
model accuracy is not significant considering the results of voltage windows of 3.2–3.5 V and 3.25–3.55 V. 5. Applicability of the OCV-based model The success of the OCV-based model, with a proper selection of the voltage window, in estimating the SOH of an LFP cell over a relatively 9
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6. Conclusion An OCV-based model for SOH estimation of LIBs is developed and verified in this study. It uses a Thevenin equivalent circuit to charac terize the CC portion of the charging or discharging profiles, describing the OCV simply as a function of SOC by a polynomial of high degree, with a lumped thermal model to account for the effect of temperature. The model requires, however, prior learning from the initial charging or discharging profile when a battery is commenced for service but avoids the use of a transformation function [24]. As a result, it is much flexible and robust in parameterizing the model for SOH estimation of LFP batteries at any ages of life. To verify and validate the accuracy of the OCV-based model for SOH estimation, five series of accelerated degradation tests of an LFP cell was conducted. It was shown that the estimated SOH well follows the available data of measurement within a relative error from 0.43% to 0.45% (the mean error is 0.18%), even though the environmental tem perature has changed significantly. The effect of size and location of the voltage window on the per formance of the OCV-based model was also investigated. It was found that a voltage window covering all or at least the last three ICA peaks is necessary to estimate SOH satisfactorily for the LFP cell under study. In addition, the robustness and universality of the OCV-based model for SOH estimation of different LIB cells were also demonstrated by using the datasets from the NASA [26]. The results showed clearly that, for all the NCA cells under consideration, the estimated SOHs agree excellently with the measured results even if the SOH has been decayed to ~60%. It is deemed, therefore, that the OCV-based model can be used satisfactorily in practical applications for SOH estimation of different LIB cells over their service time, provided the CC charge or discharge is available. Declaration of competing interest We declare that there is no conflict of interest associated with this publication. Acknowledgments The authors are very grateful to China Electric Power Research Institute for doing the accelerated degradation tests and, in particular, to Dr. Chaoyong Hou for data collection and collation. The authors thank NASA for providing experimental datasets. Xiaolei Bian acknowledges the financial support from China Scholarship Council (CSC). References [1] M. Yoshio, R.J. Brodd, A. Kozawa, Lithium-ion Batteries: Science and Technologies, 2009. [2] G. Zhu, Y. Wang, Y. Xia, Ti-based compounds as anode materials for Li-ion batteries, Energy Environ. Sci. 5 (5) (2012) 6652. [3] M. Resch, J. Bühler, M. Klausen, A. Sumper, Impact of operation strategies of large scale battery systems on distribution grid planning in Germany, Renew. Sustain. Energy Rev. 74 (2017) 1042–1063. [4] A. Malhotra, B. Battke, M. Beuse, A. Stephan, T. Schmidt, Use cases for stationary battery technologies: a review of the literature and existing projects, Renew. Sustain. Energy Rev. 56 (2016) 705–721. [5] J. Cho, S. Jeong, Y. Kim, Commercial and research battery technologies for electrical energy storage applications, Prog. Energy Combust. Sci. 48 (2015) 84–101. [6] X. Lai, Y. Zheng, T. Sun, A comparative study of different equivalent circuit models for estimating state-of-charge of lithium-ion batteries, Electrochim. Acta 259 (2018) 566–577. [7] M.R. Palacin, A. de Guibert, Why do batteries fail? Science 351 (6273) (2016) 1253292. [8] K. Qian, B. Huang, A. Ran, Y. He, B. Li, F. Kang, State-of-health (SOH) evaluation on lithium-ion battery by simulating the voltage relaxation curves, Electrochim. Acta 303 (2019) 183–191. [9] Y. Shen, Adaptive extended Kalman filter based state of charge determination for lithium-ion batteries, Electrochim. Acta 283 (2018) 1432–1440. [10] N. Wassiliadis, J. Adermann, A. Frericks, M. Pak, C. Reiter, B. Lohmann, M. Lienkamp, Revisiting the dual extended Kalman filter for battery state-of-charge
Fig. 16. The SOH estimation against experimental results for (a) B0005 and B0006, and (b) B0007 and B0018 cells, with (c) the relative errors of all the NCA cells.
subsequently, the SOH of the NCA cells at different aging statuses, as shown in Fig. 16 (a) and (b). Clearly, the estimated SOHs agree exceedingly well with the measured results over the entire lifetime of the four NCA cells. The relative SOH errors are found to be well in the range from 1.75% to 1.45%, as indicated in Fig. 16 (c). This substantiates strongly that the OCV-based model is capable of giving reliable and efficient SOH esti mates for different LIB cells over their lifespan. 10
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Contents lists available at ScienceDirect
Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
An open circuit voltage-based model for state-of-health estimation of lithium-ion batteries: Model development and validation Xiaolei Bian a, *, Longcheng Liu a, Jinying Yan a, b, Zhi Zou a, Ruikai Zhao a a b
Department of Chemical Engineering, KTH-Royal Institute of Technology, S-100 44, Stockholm, Sweden Vattenfall AB, R&D, 169 92, Stockholm, Sweden
H I G H L I G H T S
� An open circuit voltage-based model is proposed to estimate the SOH of LIBs. � The accuracy of the model is validated by experimental results of an LFP cell. � The effect of size and location of voltage window on the model accuracy is studied. � ICA is used to explain the model accuracy change with different voltage window. � The versatility of the model to different chemistries is demonstrated. A R T I C L E I N F O
A B S T R A C T
Keywords: Lithium-ion battery State-of-health Open circuit voltage-based model Voltage window Incremental capacity analysis
An open circuit voltage-based model for state-of-health estimation of lithium-ion batteries is proposed and validated in this work. It describes the open circuit voltage as a function of the state-of-charge by a polynomial of high degree, with a lumped thermal model to account for the effect of temperature. When applied for practical use, the model requires a prior learning from the initial charging or discharging data for the sake of parameter identification, using e.g. a nonlinear least squares method, but it is undemanding to implement. The study shows that the model is able to estimate the state-of-health of a LiFePO4 cell cycled under conditions where the tem perature has fluctuated significantly with a relative error less than 0.45% at most. A short part of a constant current profile is enough for state-of-health estimation, and the effect of size and location of voltage window on the model’s accuracy is also studied. In particular, the reason of accuracy change with different voltage windows is explained by incremental capacity analysis. Additionally, the versatility and flexibility of the model to different chemistries and cell designs are demonstrated.
1. Introduction The lithium-ion batteries (LIBs) were first designed and developed by Asahi Kasei during the 1980s and commercialized by Sony and A & T Battery Corporation in 1991 and 1992, respectively [1]. A few years after commercialization, LIBs took over half of the portable electronic devices market [2], and the technology is now under rapid development for application in electric vehicles and stationary electrical energy storage [3,4]. High specific energy density, high efficiency, high nominal voltage, less maintenance requirement, and long cycling life are the main ad vantages of LIBs [5,6]. However, they inevitably degrade and deterio rate within the usage process, leading to a reduction of their abilities in
e.g. storing energy and powering performance [7]. In the worst case, safety issues and even catastrophic accidents may occur due to the failure of batteries [8]. In order to avoid the abuse and extend the life span by monitoring and controlling LIBs performance, battery man agement system (BMS) has already been developed and widely implemented [9]. Among many tasks, the most important one of BMS is to accurately estimate the state-of-charge (SOC) and state-of-health (SOH) of LIBs [10,11]. SOC is commonly defined as the percentage of the maximum possible charge that is now present inside a rechargeable battery [12]. It is equivalent to the remaining fuel gauge in internal combustion vehicles, signalling how long a battery can continue to work before it needs recharging. The definition of SOC varies slightly from different
* Corresponding author. E-mail address: [email protected] (X. Bian). https://doi.org/10.1016/j.jpowsour.2019.227401 Received 24 May 2019; Received in revised form 1 October 2019; Accepted 1 November 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Xiaolei Bian, Journal of Power Sources, https://doi.org/10.1016/j.jpowsour.2019.227401
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Fig. 1. Categorization of SOH estimation methods.
references [13,14]. In this study, SOC is defined as Q SOC ¼ SOC0 þ Qmax
voltage range [12]. By contrast, experimental techniques do not need data-learning and intensive computation, but remain mainly to be off-line, low accurate and high cost [14,19]. For a detailed discussion of the advantages and drawbacks of these available methods, the reader is referred to the review papers [16,18,20]. Noticeably, Guo et al. [24] recently developed an adaptive yet simple approach to estimate the SOH of LIBs. Instead of making an incremental capacity analysis (ICA), an equivalent circuit model was adopted by this approach to characterize the constant-current (CC) portion of the charging profiles, and a voltage transformation function with time-based parameters was employed. It was shown that, with only minor demand of learning from initial charging profiles, an error within 3% was achieved between the estimated and measured results about SOH for three kinds of commercial LIBs. This encouraging result has inspired us to apply the same method to estimate the SOH of a LiFePO4 (LFP) cell. However, it was failed with the usage of a simple function for learning, due to the very different chemical properties between the types of LIBs and also the essentiality of figuring out the correlation between the open circle voltage (OCV) and SOC from the learning process [25]. In reality, this inherent correlation is always contained in the voltage transformation function even though it is obviously absent. To avoid the use of a voltage transformation function with timebased parameters, we tried in this study to describe the OCV simply as a function of SOC by a polynomial of high degree, with a lumped thermal model to account for the effect of temperature. Although this approach also requires a prior learning from the initial charging or discharging profile at the CC step, it was shown to be able to evaluate the SOH of an LFP cell cycled under conditions where temperature fluctuated signifi cantly over a wide range with a relative error from 0.43% to 0.45% (the mean error is 0.18%). The effect of size and location of voltage window was also studied and it was found that with only short part of a CC profile, the model can still estimate SOH precisely as long as the window location was chosen properly. In addition, it was justified that a voltage window with four
(1)
where SOC0 is the SOC of the battery at t ¼ 0 when the charge or discharge commences, Q is the charged capacity and Qmax is the current maximum capacity of the battery. By contrast, SOH is a ‘measure’ that reflects the general condition of a battery and its ability to deliver the specified performance in com parison with a fresh battery [15,16], and it is commonly defined as [17–19], SOH ¼
Qmax � 100% Qnom
(2)
where Qnom is the nominal capacity of the battery. Accurate evaluation of the SOH of LIBs becomes, therefore, crucial in order to guarantee satisfactory performance and to replace the end-oflife battery when needed [20]. However, SOH cannot be directly measured; it can only be estimated according to measurable parameters, such as current, voltage and temperature [21]. As a result, extensive studies have been carried out over the last decades and different methodologies have been developed to estimate the SOH of LIBs, by accounting for e.g. different mechanisms of battery fading, which orig inates from many factors and interactions such as service type, working condition, and aging history, etc [22]. As summarized in Fig. 1, the methods of SOH estimation can roughly be divided into three groups: using either adaptive models, experimental techniques or differential approaches. All these methods have, however, distinct advantages and disadvantages [16,18]. Adaptive models often have strong physical meanings and the parameters involved generally have close relations with the underlying electrochemical processes in batteries, but usually need data-learning and intensive computation [17, 23]. Differential approaches also have advantages of battery degrada tion identification, but require static charging or discharging over a wide 2
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Therefore, the dynamic behaviour of the charging process of LIBs may be described by the Thevenin equivalent circuit [24,28], as shown in Fig. 3, to give OCVðSOC; TÞ ¼ V
(3)
V0
Vp
with 0 t
Vp ¼ Vp;0 e τ þ IRp @1
1 (4)
t
e τA
and (5)
V0 ¼ IR0
where V represents the terminal voltage, Vp is the electrochemical po larization voltage, V0 is the voltage drop between the ohmic resistance and OCV denotes the open circuit voltage, which is a function of SOC and temperature T. Vp,0 is the electrochemical polarization voltage at time t ¼ 0, τ is the time constant of the cell, I is the constant current (which is defined as positive for charging), Rp is the electrochemical polarization resistance, and R0 is the ohmic resistance. When Eqs. (3)–(5) are combined, we obtain � t � (6) OCVðSOC; TÞ ¼ V þ IRp Vp;0 e τ I R0 þ Rp
Fig. 2. Charging profiles of an LFP cell at different degradation statuses, ob tained from the constant-current step with CC ¼ 0.3 C.
or equivalently, OCVðSOC; TÞ ¼ V þ IRp
� Vp;0 e
Q Iτ
I R0 þ Rp
�
(7)
It follows, from Eqs. (1), (2) and (7), that an accurate description of the OCV(SOC) function, or its inverse function, has become critical to give a good estimation of Qmax and subsequently SOH. Unfortunately, it is widely known that the OCV(SOC) function is both temperature and age-dependent, making R0 and Rp apparently temperature-dependent and Q and τ mostly SOC-dependent in Eq. (7), and thus it may vary remarkably under different operating conditions [25,29–31]. As a result, most of the phenomenological OCV-SOC models that are built as a sum of simple functions [25,32] should be avoided for use in Eq. (7) because they do not well account for phase transition and staging phenomena associated with lithium-ion intercalation/deintercalation process of the active material and usually only show one or two peaks on the ICA curves [33]. This is, however, not the case if one uses a polynomial of degree around ten to describe the OCV(SOC) function [34], which enables to capture four ICA peaks for LFP batteries, in consistent with the results of ICA of electrochemical processes of LIBs [25]. For this reason, we pro pose in this study a polynomial-based, lumped thermal model that also accounts for the effect of temperature on the OCV(SOC) function by the Arrhenius law [35] to give � � �� m E 1 1 X OCVðSOC; TÞ ¼ exp ak;ref SOCk (8) R Tref T k¼0
Fig. 3. Schematic of the Thevenin equivalent circuit model.
ICA peaks, or at least with the later three peaks, was necessary for the model to estimate SOH of an LFP battery reasonably well. To further verify the applicability and flexibility of the proposed model for SOH estimation, the datasets of LiNi0.8Co0.15Al0.05O2 (NCA) battery cells from the NASA Ames research center have also been uti lized [26,27]. It was shown that the estimated SOHs agree nicely with the measured results over the entire lifespan for the four NCA cells studied, with the relative errors ranging from 1.75% to 1.45%. This paper is organized as follows. In Section 2, the OCV-based model is formulated in detail, with specific analysis of the model characteris tics. In Section 3, the accuracy of the proposed model is verified and validated through five series of accelerated degradation tests, with a comparison of the estimated and measured SOH results of an LFP cell. In Section 4, the effect of size and location of the voltage window on SOH estimate is analysed and the number of ICA peaks should be covered by the voltage window is examined. In Section 5, the versatility and flexi bility of the model to different chemistries and cell designs are demon strated by the datasets from the NASA Ames research center. Some concluding remarks are, then, summarized in Section 6.
where E is the activation energy, R is the gas constant, Tref is the refer ence temperature at which the constants a0,ref, …, am,ref are evaluated for the OCV(SOC) function, and m is the degree of the polynomial in SOC. When Eq. (1) (7) and (8) are combined, we obtain a single-variable function V(Q) for the CC charging or discharging profile as � � � �k �� m � Q E 1 1 X Q V ¼ exp ak;ref SOC0 þ IRp Vp;0 e Iτ R Tref T Q max k¼0 � (9) þ I R0 þ Rp
2. Development of the SOH estimation model As discussed by Guo et al. [24], the charging (or discharging) profiles obtained from the CC step of a LIB during different degradation stages are similar to one another, albeit gradually shifting to a shorter time span and higher voltage range with aging. This is commonly true regardless of cell chemistry, as shown in Fig. 2 for a fresh LFP cell. More importantly, it is noted that all of the CC charging profiles of terminal voltage-charged capacity reproduce the main characteristics of the OCV-SOC curve [21,22] due to the predominant effect of the OCV in governing the profiles of the terminal voltage over the charging process with a constant current.
or simply m X
V ¼ c0 k¼1
3
� �k Q ak;ref SOC0 þ þ c1 ec2 Q þ c3 Qmax
(10)
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Fig. 5. Profile of the terminal voltage and current during the initial test.
similarity between the charging profiles at the CC step under different working conditions. It should, also, be noted that for LiCoO2/LiNixMnyCozO2-based LIBs, the rather smooth and uncomplicated dependence of the OCV on SOC, resulting from the approximate phase transformation voltages of LiCoO2 and LiNixMnyCozO2 [25], makes it possible to easily find a simple OCV (SOC) function with a polynomial of degree three [24]. For LFP batte ries, however, a polynomial of degree greater than 10 is required because only then a flat voltage plateau can be produced on the OCV-SOC curve with two-phase regions corresponding to FePO4 and LiFePO4, respectively [25]. As a result, we found that the curve-fitted charging profiles by the approach of Guo et al. [24] are very sensitive even to a slight change in the time-based parameters, making the process of parameter identifi cation often corrupt and ineffective. By contrast, our approach is much stable and fast in parameterizing the model for LFP batteries at any aging statuses of life. The main drawback of the polynomial-based OCV-SOC models is, perhaps, the limitation in its generality and adaptivity to different chemistries and cell designs. A logarithmic or exponential function might also be needed to include in the OCV(SOC) function in order to well represent different types of LIBs [24,25,32,33], especially when the degree m of the polynomial is small.
Fig. 4. Flowchart of the OCV-based model for SOH estimation.
where the introduced coefficients, c0, …, c3, can be regarded as con stants in a single CC charging or discharging process. Noticeably, the constants a1,ref, …, am,ref are only needed to deter mine once and they can readily be identified by fitting Eq. (10) to the initial charging or discharging profile when a battery is commenced for service with a nonlinear least squares method. If desired, the constant a0, ref can also be derived from Eq. (8) by specifying an OCV at which SOC would become zero. The result is, then, an estimate of the OCV(SOC) function at the reference temperature. The constants c0, …, c3, SOC0 and Qmax should, however, be updated all the time as battery ages. In particular, it should be noted that c0 represents actually the Arrhenius term in Eq. (8). It is the only constant that is apparently temperature-dependent and a certain constraint such as c0 2 ½0:5; 1:5� should be imposed. Likewise, Qmax 2 ½Qend ; Qmax;0 � had better be applied, where Qend is the charged capacity at the end of CC charging process and Qmax,0 is the initial maximum capacity of the battery. With initial training of the model to obtain parameters of a1,ref, …, am,ref, the evolution of Qmax with aging can be evaluated directly by fitting Eq. (10) to the charging/discharging profile of V-Q at later times. When use is made of Eq. (2), the SOH value can be readily calculated, as illustrated in Fig. 4 for the entire procedure of the model in estimating the SOH of batteries. Clearly, the methodology adopted in this study is quite different from that of Guo et al. [24]. Instead of using a transformation function, where the OCV(SOC) function was implicitly embedded in a base charging profile, we have now explicitly specified a polynomial form of the OCV (SOC) function, as given in Eq. (8), and used it to account for the
3. Validation and evaluation To evaluate and also validate the proposed model, SOH estimation of an LFP cell has been made in comparison with those obtained from five series of accelerated degradation tests. The cell used in the tests is an LFP battery with 60 Ah nominal capacity and 3.2 V nominal voltage. 3.1. Initial charge/discharge test A prior learning from the initial charging profile of a fresh battery is needed for training the proposed model. For this sake, an initial charge/ discharge experiment was made before commencing the accelerated degradation tests. This experiment contains 3.5 full charge-discharge cycles at an average temperature of 0 � C (varying between 5.8 � C and 5.5 � C). As illustrated in Fig. 5, the procedure of these cycles is: (1) rest for 5 min; (2) charged at a CC of 18 A (0.3 C) until an upper threshold voltage of 3.65 V was reached; (3) charged at a constant-voltage (CV) of 3.65 V until a cut-off current of 1.8 A (0.03 C) was achieved; (4) rest for another 5 min; (5) discharged at a constant current of 18 A (0.3 C) until a lower threshold voltage of 2.5 V was reached. The above process of rest-CC charge-CV charge-rest-CC discharge was repeated for three times, followed by steps of rest-CC charge-CV charge. After the initial test, the LFP cell was left in an open-circuit condition for around 2 days for depolarization and stabilization before 4
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Fig. 7. Illustration of the accuracy of the OCV-based model for comparison of experimental and curve-fitted V-Q results, including relative errors, (a) ob tained at the last full charge-discharge cycle before the aging test, with CC ¼ 0.3 C and T ¼ 0 � C, and (b) obtained at the end of the fifth series of the accelerated degradation tests, with CC ¼ 0.3 C and T ¼ 35– 40 � C.
Fig. 6. Profile of the accelerated degradation tests: (a) terminal voltage and current, and (b) temperature.
the accelerated degradation tests. 3.2. Accelerated degradation tests Following the initial test, five series of accelerated degradation tests were successively operated, with a rest interval of 2.5–4.75 days applied between two aging tests. The first four accelerated degradation tests were performed for twenty days, while the last one took about 19 days. At the end of each degradation test, 3.5 full charge-discharge cycles (in the same way as in the initial test) were also made in order to evaluate the capacity or SOH fade at a relative low C-rate. As exemplified in Fig. 6 (a), each of the degradation tests follows the periodic procedure: (1) rest for 5 min; (2) charged at a CC of 30 A (0.5 C) until an upper threshold voltage of 3.65 V was reached; (3) rest for another 5 min; (4) discharged at a CC of 30 A (0.5 C) until a lower threshold voltage of 2.5 V was reached. Compared with the initial test, the accelerated degradation tests consisted only of CC but no CV charging/discharging steps at a larger current, i.e., 30 A (0.5C). Consequently, the periodic time of the degradation tests is around 5 h, much shorter than the initial test of about 8 h. In addition, it is noted that these degradation tests were operated in an environment where temperature fluctuated in a certain range to mimic real working conditions. In the first series of the aging tests, as illustrated in Fig. 6 (b), the temperature fluctuated considerably, vary ing from 13.5 � C to 11.7 � C. By comparison, the temperature varied moderately but was high in the fifth series, ranging from 30 � C to 44.5 � C. The other three series of the aging tests also followed a similar pattern of temperature variations but ranged differently i.e. 4–27.6 � C
for second series, 22–35.5 � C for third, and 26–42 � C for forth. These significant changes in environmental temperature make the method of Guo et al. [24] less favorable to apply, because no constraints could reasonably be imposed on the model parameters. With our approach, however, the parameters in the OCV-based model can readily be identified by fitting the charging profiles of LIBs at any aging statuses, via a nonlinear least squares method, even if the working condition has changed severely over time. 3.3. Parameter identification In the LFP aging tests, the voltage window of the charging profile is 2.5–3.65 V, as shown in Fig. 6. Since it is unnecessary to use the whole voltage range for application of the OCV-based model, we chose a voltage window of 3.0–3.6 V (which is large enough to cover the dominant range of voltage changes), instead, in this section and also in Section 3.4 for both parameter identification and SOH estimation. As discussed in Section 2, the constants a1,ref, …, am,ref are deter mined only once by fitting Eq. (10) to the initial charging profile of V-Q and then kept unchanged in later usage. The other parameters, i.e. c0, …, c3, SOC0 and Qmax, evolve with time but can readily be identified by fitting Eq. (10) to the subsequent V-Q profiles, via the nonlinear least squares method. The result gives, then, a curve-fitted V (Q) function at 5
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Table 1 The eleven constants a1,ref, …, a11,ref determined from initial charging profile. a1,ref 20.31 a7,ref 226672.57
a2,ref
a3,ref
326.85
a4,ref
2942.28
16197.07
a8,ref
a9,ref
247039.31
a5,ref 57934.93 a10,ref
172411.88
69609.69
a6,ref 139162.86 a11,ref 12354.64
Table 2 The constants c0, …, c3 determined from initial charging profile. c0 1.00
c1 8.42 � 10
c2 5
9.99
c3 2.72
different aging status. As demonstrated in Fig. 7 for two cases, with m ¼ 11 and voltage ranging from 3.0 V to 3.6 V, the curve-fitted results of the OCV-based model show a good agreement to the charging profiles, with the rela tive error mainly falling in the range of �0.5% and mean error of 1:654 � 10 4 % and 9:776 � 10 5 %, respectively. As a consequence, it is deemed that the value of Qmax obtained in each of the cases can be used with great confidence to evaluate the capacity fade of the LFP cell with aging. The use of a polynomial of degree 11 in Eq. (10) is, therefore, appropriate for the LFP cell under study in describing the OCV(SOC) function in the OCV-based model, with the constants a1,ref, …, a11,ref, given in Table 1 and c0, …, c3 given in Table 2. As mentioned in Section 2, the constants a1,ref, …, a11,ref are only needed to determine once using the initial charging or discharging profile and can be applied directly to estimate SOH latter with Eqs. (10) and (2). It is also noteworthy to stress that the Arrhenius term c0 has become unity when fitting Eq. (10) to obtain these 11 constants by a nonlinear least squares method. 3.4. SOH estimation Qmax is the key parameter to be evaluated from the process of data fitting. As discussed already in Section 3.3, it can be identified by fitting Eq. (10) to the available V-Q profiles at different aging status after initial training of the OCV-based model. The result allows SOH to be estimated immediately from Eq. (2). To illustrate the accuracy of this approach, we now set the voltage window as 3.0 V–3.6 V and show in Fig. 8 (a) the estimated SOH for the entire period of accelerated degradation tests together with the available data measured at each series, and partial enlargement of the results in Fig. 8 (b). Clearly, the estimated SOH from the OCV-based model, SOHe, fluc tuates somehow in response to temperature variations in the tests. It well follows, however, the available data of measurement, SOHm, and shows a linear decline tendency as cell ages. The relative error between SOHm and SOHe, i.e. ðSOHm SOHe Þ=SOHm *100, was found to be in the range of 0.43%–0.45%, with a mean of 0.18%, as shown in Fig. 8 (c). This suggests that the proposed model might be used satisfactorily in practical applications to give a robust and accurate SOH estimate, even when the battery has been cycled under conditions where temperature fluctuated significantly. The premise is, however, that a voltage window on the charging or discharging profiles should be selected properly for parameter identification and SOH estimation. As a general rule of thumb, the larger the voltage windows ranges, the more accurate the estimated SOHs would be.
Fig. 8. The SOH estimation using the OCV-based model against experimental results (a) over the entire period of accelerated degradation tests, (b) partial enlargement, and (c) the relative error.
In the initial and accelerated degradation tests that have been dis cussed, the voltage ranges from 2.5 V to 3.65 V, allowing us to define a voltage window of 3.0–3.6 V for parameter identification and SOH estimation. In reality, however, it is seldom possible to have the CC charging or discharging profiles covering the whole-range voltage window. Therefore, it is important and of practical significance to esti mate SOH of LIBs with only part CC charging or discharging period.
4. Effect of the voltage window It is not hard to imagine that not only the size but also the location of the voltage window will have a considerable influence on the accuracy of the OCV-based model for SOH estimation. 6
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Fig. 9. The SOH estimation against experimental results (a) and the absolute value of the relative errors (b) over the entire period of accelerated degradation tests using the OCV-based model for different size of voltage window.
Fig. 10. The mean (a) and maximum (b) of the absolute value of the relative errors for the SOH estimated results shown in Fig. 8 over different cycles.
4.1. Effect of the size of voltage window
during 500 cycles. This is a significant contrast to the results obtained in the case of using 3.0–3.3 V as the voltage window, which gives a maximum of 8.08%. Additionally, Fig. 10 substantiates that both the mean and maximum of the errors well follow a downhill trend as the voltage-window size increases. The SOH estimation may, therefore, fail if the window size is too small, such as 3–3.3 V or 3–3.4 V in this study. In general, a larger voltage-window size will result in more accurate SOH estimate. How ever, the accuracy of the OCV-based model will not increase signifi cantly with increasing the window size once the voltage window has been defined large enough (the maximum of the errors was decreased by 5.38% when a voltage window of 3.0–3.5 V was used instead of 3.0–3.4 V, while it was decreased only by 0.50% when 3.0–3.6 V was considered instead of 3.0–3.5 V). The reason for this phenomenal effect of the voltage-window size on the SOH estimation can be explored in accordance with the ICA curves of the model results. As shown in Fig. 11, the ICA curves can well capture the fine features of the voltage plateaus of Q-V curves, and four ICA peaks associated with the phenomena of phase transition inside the cathode and anode are clearly identifiable over the voltage window of 3.0–3.5 V. The ICA peaks are asymmetric and, as battery ages, the changes in both the positions and the amplitudes of the ICA peaks are identifiable. The evolution of the ICA results may, therefore, be used to analyse the aging origins [36,37], such as the loss of active material, the
Using the same approach, as discussed in Section 3, but different size of voltage windows starting from 3.0 V, we re-estimated SOH of the LFP cell over the entire period of accelerated degradation tests with the OCVbased model. The results are shown in Fig. 9. It is seen clearly that, with the voltage window of 3.0–3.3 V or 3.0–3.4 V, the estimated SOHs are not only far away from the measured results but also not follow the changing trend. When the window size is increased to 0.45 V, however, the estimated SOH well follows the pattern of the measured SOHs yet with relatively low accuracy. As the window size continues to increase, the accuracy of the OCV-based model improves significantly; the estimated SOHs agree perfectly with the measured results when the window size has been increased to 0.6 V or 1.15 V. In particular, Fig. 9 (b) indicates that the relative errors of SOH estimation all become less than 1% with the voltage window of 3.0–3.5 V, 3.0–3.6 V, and 2.5–3.65 V. Compared to the results obtained with the full CC charging profiles, this simply implies that the whole size of voltage window, i.e. 2.5–3.65 V, is not a prerequisite to obtaining good SOH estimate; only part of it could still achieve satisfactory results. As also shown in Fig. 10 for the mean and maximum of the absolute value of the relative errors over different cycles, the voltage window of 3.0–3.6 V is sufficient in estimating the SOHs of the LFP cell under study. It gives a maximum of 0.45% in the absolute value of the relative errors 7
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Fig. 11. Illustration of the Q-V and ICA curves obtained at the initial test and the ends of the five series of accelerated degradation tests (a) over the voltage window of 3.0 V–3.55 V, and (b) part results to show the four ICA peaks.
loss of useable lithium-ion, and the increase of the internal resistance, etc. For our purpose, however, we note that the voltage window of 3.0–3.3 V covers only the first ICA peak, while the voltage window of 3.0–3.4 V covers three ICA peaks but not the last one. It seems, therefore, that only when all or most of the ICA peaks can be identified within the voltage window, an accurate SOH estimation could result. To see if this is the case and also figure out which ICA peaks are important in affecting the performance of the OCV-based model, we continue to study the ef fect of the location of voltage window.
Fig. 12. The SOH estimation against experimental results (a) and the absolute value of the relative errors (b) over the entire period of accelerated degradation tests using the OCV-based model with a voltage window of a size of 0.3 V located in different positions.
accurately under, however, the condition that its location can be determined properly so as to cover all the ICA peaks associated with the phenomena of phase transition inside the LFP cell. In the worst case, i.e. when only the first ICA peak shown in Fig. 11 is contained in the voltage window of 3–3.3 V, the estimated SOHs from the model are entirely wrong compared with the measured results. This, together with the fact that good estimated SOHs can still be obtained with a voltage window of 3.3–3.6 V, indicates that the first ICA peak does not really affect the average model accuracy and that the first ICA peak might not be closely related to the aging origins of the LFP cell, in contrast to the other three ICA peaks [36,37]. To validate this assump tion, the evolution of ICA curves over the whole aging tests are plotted, as shown in Fig. 14. It is obvious that the attributes of peak 1 of the ICA curves only change slightly over the 500 cycles, in significantly contrast to those of the other three peaks. One may, therefore, deem that the first ICA peak might be unimportant in estimating the SOH of LFP batteries. It is deemed, therefore, that a voltage window covering all or at least the last three ICA peaks is a prerequisite to estimating SOH satisfactorily for an LFP cell using the OCV-based model. In addition, the model ac curacy would also depend on the size and location of the voltage window in the case when all the four ICA peaks are been contained. As discussed in Section 4.1 that a larger voltage-window size would give a more ac curate SOH estimation. If they cover all the peaks with same voltage size but are located at different positions, we believe that the difference in
4.2. Effect of the location of voltage window By moving the voltage window with a fixed size of 0.3 V from 3.0 V to 3.3 V, we show in Fig. 12 how the accuracy of the estimated SOHs would change. Clearly, only when the voltage window covers all the ICA peaks shown in Fig. 11, i.e. when it is located at 3.2 V or 3.25 V, good estimate of the SOHs by the model could be achieved. In the case of selecting 3.3–3.6 V as the voltage window, the estimated SOHs are still acceptable yet become a bit worse due to the fact that the first ICA peak located at ~3.28 V has been disappeared. This conclusion could also be drawn from Fig. 13, where we compared the mean and maximum of the absolute value of the relative SOH errors over different cycles when different voltage windows are selected. In particular, it is noted that the use of 3.25–3.55 V as the voltage window gives the best SOH estimate, with a mean of 0.36% and a maximum of 0.68% in the absolute value of the relative errors, respectively, over 500 cycles. This encouraging result suggests that a voltage window as small as 0.3 V is large enough to estimate SOH 8
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Fig. 15. Variation of the key parameters in the NASA tests: (a) terminal voltage and current of B0005, and (b) discharging profiles of B0005 at different degradation statuses.
short duration of the degradation tests encourages us to further explore its versatility and flexibility to different chemistries and cell designs. For this propose, the datasets from the NASA Ames research center [26] for the aging tests of four NCA cells with the nominal capacity of 2 Ah over nearly the whole useable lifetime could be utilized. As shown in Fig. 15 (a) for the cell labeled as B0005, all the aging tests of the NCA cells consisted of a series of CC-CV charge and CC discharge cycles at ambient temperature. The CC charge step was all performed at 1.5 A until a cut-off voltage of 4.2 V was reached. It was, then, immediately followed with the CV charge step until the current dropped to a cut-off value of 20 mA. Afterward, the CC discharge step was all performed at 2 A until, however, different cut-off voltages was approached, which were 2.7, 2.5, 2.2, and 2.5 V, respectively, for the B0005, B0006, B0007, and B0018 cells considered [26]. The results of the aging test of the four NCA cells allows us to obtain the CC charging or discharging profiles of terminal voltage-charged/ discharged capacity at different degradation statuses. In this examina tion, however, we chose to use the CC discharging profiles, as exem plified in Fig. 15 (b) for the B0005 cell, as the basis for SOH estimation to illustrate the universality of the OCV-based model. Similar to Figs. 2 and 15 (b) also shows that all the V-Q profiles obtained at the CC discharging steps are similar to each other, albeit gradually shifting to a shorter time span. We can, therefore, follow the same procedure as discussed in Section 3 to evaluate the parameters of the OCV-based model for the NCA cells, but with the degree of poly nomial m ¼ 8 in Eq. (10) instead of m ¼ 11. That is, we first use the initial CC discharging profile to train the model, obtaining the constants of a1, ref, …, a8,ref, and then evaluate Qmax for each of the succeeding V-Q profiles by a nonlinear least squares method. The use of Eq. (2) gives,
Fig. 13. The mean (a) and maximum (b) of the absolute value of the relative errors for the SOH estimated results shown in Fig. 12 over different cycles.
Fig. 14. The evolution of ICA curves over 500 cycles obtained from the OCVbased model, the initial cycle is coloured as green and the last cycle is red. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
model accuracy is not significant considering the results of voltage windows of 3.2–3.5 V and 3.25–3.55 V. 5. Applicability of the OCV-based model The success of the OCV-based model, with a proper selection of the voltage window, in estimating the SOH of an LFP cell over a relatively 9
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6. Conclusion An OCV-based model for SOH estimation of LIBs is developed and verified in this study. It uses a Thevenin equivalent circuit to charac terize the CC portion of the charging or discharging profiles, describing the OCV simply as a function of SOC by a polynomial of high degree, with a lumped thermal model to account for the effect of temperature. The model requires, however, prior learning from the initial charging or discharging profile when a battery is commenced for service but avoids the use of a transformation function [24]. As a result, it is much flexible and robust in parameterizing the model for SOH estimation of LFP batteries at any ages of life. To verify and validate the accuracy of the OCV-based model for SOH estimation, five series of accelerated degradation tests of an LFP cell was conducted. It was shown that the estimated SOH well follows the available data of measurement within a relative error from 0.43% to 0.45% (the mean error is 0.18%), even though the environmental tem perature has changed significantly. The effect of size and location of the voltage window on the per formance of the OCV-based model was also investigated. It was found that a voltage window covering all or at least the last three ICA peaks is necessary to estimate SOH satisfactorily for the LFP cell under study. In addition, the robustness and universality of the OCV-based model for SOH estimation of different LIB cells were also demonstrated by using the datasets from the NASA [26]. The results showed clearly that, for all the NCA cells under consideration, the estimated SOHs agree excellently with the measured results even if the SOH has been decayed to ~60%. It is deemed, therefore, that the OCV-based model can be used satisfactorily in practical applications for SOH estimation of different LIB cells over their service time, provided the CC charge or discharge is available. Declaration of competing interest We declare that there is no conflict of interest associated with this publication. Acknowledgments The authors are very grateful to China Electric Power Research Institute for doing the accelerated degradation tests and, in particular, to Dr. Chaoyong Hou for data collection and collation. The authors thank NASA for providing experimental datasets. Xiaolei Bian acknowledges the financial support from China Scholarship Council (CSC). References [1] M. Yoshio, R.J. Brodd, A. Kozawa, Lithium-ion Batteries: Science and Technologies, 2009. [2] G. Zhu, Y. Wang, Y. Xia, Ti-based compounds as anode materials for Li-ion batteries, Energy Environ. Sci. 5 (5) (2012) 6652. [3] M. Resch, J. Bühler, M. Klausen, A. Sumper, Impact of operation strategies of large scale battery systems on distribution grid planning in Germany, Renew. Sustain. Energy Rev. 74 (2017) 1042–1063. [4] A. Malhotra, B. Battke, M. Beuse, A. Stephan, T. Schmidt, Use cases for stationary battery technologies: a review of the literature and existing projects, Renew. Sustain. Energy Rev. 56 (2016) 705–721. [5] J. Cho, S. Jeong, Y. Kim, Commercial and research battery technologies for electrical energy storage applications, Prog. Energy Combust. Sci. 48 (2015) 84–101. [6] X. Lai, Y. Zheng, T. Sun, A comparative study of different equivalent circuit models for estimating state-of-charge of lithium-ion batteries, Electrochim. Acta 259 (2018) 566–577. [7] M.R. Palacin, A. de Guibert, Why do batteries fail? Science 351 (6273) (2016) 1253292. [8] K. Qian, B. Huang, A. Ran, Y. He, B. Li, F. Kang, State-of-health (SOH) evaluation on lithium-ion battery by simulating the voltage relaxation curves, Electrochim. Acta 303 (2019) 183–191. [9] Y. Shen, Adaptive extended Kalman filter based state of charge determination for lithium-ion batteries, Electrochim. Acta 283 (2018) 1432–1440. [10] N. Wassiliadis, J. Adermann, A. Frericks, M. Pak, C. Reiter, B. Lohmann, M. Lienkamp, Revisiting the dual extended Kalman filter for battery state-of-charge
Fig. 16. The SOH estimation against experimental results for (a) B0005 and B0006, and (b) B0007 and B0018 cells, with (c) the relative errors of all the NCA cells.
subsequently, the SOH of the NCA cells at different aging statuses, as shown in Fig. 16 (a) and (b). Clearly, the estimated SOHs agree exceedingly well with the measured results over the entire lifetime of the four NCA cells. The relative SOH errors are found to be well in the range from 1.75% to 1.45%, as indicated in Fig. 16 (c). This substantiates strongly that the OCV-based model is capable of giving reliable and efficient SOH esti mates for different LIB cells over their lifespan. 10
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