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Study on battery pack consistency evolutions and equilibrium diagnosis for serial- connected lithium-ion batteries
Applied Energy xxx (2017) xxx–xxx
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Study on battery pack consistency evolutions and equilibrium diagnosis for serial- connected lithium-ion batteries Caiping Zhang ⇑, Yan Jiang, Jiuchun Jiang ⇑, Gong Cheng, Weiping Diao, Weige Zhang National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing 100044, China Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Jiaotong University, Beijing 100044, China
h i g h l i g h t s Impact sensitivity of parameters variety on energy utilization ratio is analyzed. The law of SOC consistency evolutions is deduced for parameter variations. The OCV consistency model of the battery pack is proposed. The mapping between OCV and SOC distribution of the battery pack is established.
a r t i c l e
i n f o
Article history: Received 16 February 2017 Received in revised form 23 May 2017 Accepted 26 May 2017 Available online xxxx Keywords: Lithium-ion battery Battery pack consistency Sensitivity analysis Modeling Equilibrium diagnosis
a b s t r a c t The consistency among lithium-ion battery pack is an important factor affecting their performance. The paper analyzes the impact sensitivity of parameters consistency including capacity, internal resistance and state of charge (SOC) on energy utilization efficiency of the battery pack. It turns out that SOC variations is the most significant influence on battery consistency, and hence is employed as evaluation index to characterize battery consistency level. Then the SOC evolution is explored under four scenarios, and the result reflects that the columbic efficiency is associated with prominent accumulative effect on SOC divergence of the battery group in use. The OCV consistency model is established based on longterm battery data of two trolley buses. It is observed that the fitted results match with the operational data very well. Finally, the mapping relation between the OCV distribution and the SOC distribution of the battery pack using dichotomy method is proposed. The calculation error of battery pack energy utilization efficiency by using such method is within 1.5%, which can be used for battery equilibrium diagnosis and prognosis at a certain aging state. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction To deal with the energy crisis and environmental pollution, vehicle electrification is one of the effective approaches. Since lithium-ion batteries possess high energy density, high terminal voltage, long life and none memory effect [1,2], they are widely used as power sources in electric vehicles (EVs). With the prevailing application of lithium-ion batteries, the durability and safety for battery packs receive more and more attentions from researchers and engineers since the improper usage of batteries will shorten the battery life [3–6] and even sometimes cause severe fire hazards [7,8]. ⇑ Corresponding authors at: National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing 100044, China. E-mail addresses: [email protected] (C. Zhang), [email protected] (J. Jiang).
In the practical applications, batteries are usually series connected to form a high voltage battery group in order to meet the power and energy requirements. The inconsistency among battery cells is a key factor influencing the performance of battery packs. Battery inconsistency, which is also called cell to cell variations, origins from two main factors. One happens in the procedure of battery production [9–11], such as coating, ingredients and unevenness of impure contents of batteries which give rise to the difference in the battery initial performance like original capacity, resistance, coulombic efficiency and self-discharge rate. Although such differences are minute, huge variations will occur during the operation of batteries. For example, researchers in Ref. [12] reported that 48 cells from a mass production were tested under equal conditions but a 10% capacity range showed after 1000 cycles due to the production inconsistency. Another occurs in usage. Inconsistency of battery original performance will cause difference in equivalent current rate, SOC usage range among cells. On
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Please cite this article in press as: Zhang C et al. Study on battery pack consistency evolutions and equilibrium diagnosis for serial- connected lithium-ion batteries. Appl Energy (2017), http://dx.doi.org/10.1016/j.apenergy.2017.05.176
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the other side, battery pack thermal distribution is always nonuniform, causing temperature variations among cells. The aforementioned factors will generate variations in SOC and aging (e.g. capacity fade, power fade) of the batteries [13]. The former will influence the utilization efficiency of the battery pack, the latter will impact the maximum available energy of the battery pack. In addition, both of them have an impact on the durability of battery packs. Experimental investigations on battery consistency have been implemented from various perspectives. In Ref. [14], Gogoana et al. reported that there is an additional 40% lifetime reduction when there is a 20% mismatch in the ohmic resistance for parallel connected batteries. During practical operation, the battery inconsistency becomes more and more severe because the temperature ingredient leads to different aging behaviors of batteries [5,6,15–17]. Su et al. [5] reported that batteries were more susceptible to temperature other than discharge current and discharge cutoff voltage. Susana et al. [15] compared the stability of the cathode material of lithium-ion batteries with blended cathode materials under 25 °C and 40 °C, and the experimental results showed that the elevated temperature is detrimental to the long term cycling stability of the cathode. Ganesan et al. developed an electrochemical-thermal coupled model for a battery pack to analyze the battery pack performance under various rates and temperatures. An additional 5% capacity loss of the battery pack was obtained when there is a temperature difference of 15 °C among the cells [18]. Before assembling the battery cells into a practical battery pack, batteries are usually connected in parallel in the first place. According to Ref. [19–21], either the internal resistance or the capacity mismatch would cause an unbalancing current distribution among parallel connected cells. These accelerate the capacity fade of a specific cell in the battery pack. Battery inconsistency directly influences the energy and capacity utilization efficiency of a battery pack. Different approaches have been developed to reduce inconsistency before battery pack assembling [22,23]. In Ref. [22], experimental charge/discharge voltage signals were analyzed by using discrete wavelet transform-based feature extraction method to select batteries with similar electrochemical characteristics. Although the initial performance of the battery pack was guaranteed, the above method failed to predict the consistency divergence along with usage. For a battery pack without regularly maintenance and balancing, the capacity and energy utilization ratio will be affected by internal resistance [14,19], capacity [24], coulombic efficiency [25–27] and SOC [28] variations of battery cells during the operation. Generally speaking, there are two ways to investigate the impact of the battery pack way on inconsistency. One way is based on all individual cells [9,14,22,23,27,29,30] and another way is to just choose some representative cells in a pack [31]. Jiang et al. [30] developed a novel energy utilization efficiency evaluation method based on the distribution of the capacity. This method took into account the influence of capacity variations but neglected the internal resistance and SOC variations. Moreover, the capacities of all batteries are different in a battery pack. Schuster et al. [9,29] studied the correlation among capacity, the impedance, the researched evolution of capacity and the impedance distribution of battery cells during operation. The aforementioned approach could predict the inconsistency divergence based on experimental results obtained from lab, however, the impedance data was not available on board. Ouyang et al. [31] presented a battery pack capacity estimation method by determining the remaining charging and discharging electric quantities of batteries which are the firstly reached to charge and discharge cutoff voltages respectively. A merit of this method consists in one only needs to pay attention to two specific cells, but the estimation error could be large if the battery group only operates in a small SOC range. In applications, the commonly used battery pack SOC estimation method is to
use current integration together with frequently calibration associated with OCV methods [24]. This would lead to a sudden drop or increase of the battery pack SOC when one specific battery voltage reaches beforehand to the cutoff conditions. The previous studies have achieved notable progresses, but there are also some limitations. One weakness of the existed literature is without considering the online application evaluation and the prediction of battery inconsistency. Most researches analyze the battery inconsistency and its influence on battery packs based on the battery capacity and the impedance. However, it is difficult to measure the impedance on board. Moreover, the online capacity estimation errors of battery cells can be large if battery packs operate in a small and almost fixed SOC range - such as the working conditions of electric buses. Additionally, there is little literature which clearly states and predicts the battery pack performance along with the inconsistency evolution of the battery capacity, the internal resistance, SOC and the coulombic efficiency. Since only the battery voltage, the current and the temperature data are measured on board, effective methods to evaluate the battery pack consistency and predict how the consistency will evolve are urgently needed. 1.1. Contributions of the paper The paper makes original contributions in the following aspects: (i) Sensitivity analysis of the impact of parameters consistency on energy utilization efficiency of the battery pack is performed. The SOC variation is the most significant influence on battery consistency, and hence is employed as evaluation index to characterize battery consistency level. (ii) The law of SOC consistency evolution is deduced for parameter variations of the capacity, the internal resistance, the columbic efficiency, and the initial SOC. Battery columbic efficiency shows prominent accumulative effect on SOC divergence of the battery group in use. (iii) Two OCV standard deviation models for describing the battery consistency are proposed based on the electric buses operating data, and L-M algorithms are used to identify the parameters. (iv) A mapping relationship between the OCV distribution and the SOC distribution of a battery pack is established by using dichotomy methods, and is applied for battery equilibrium diagnosis. 1.2. Organization of the paper The rest of the paper is organized as follows. Section 2 introduced the theoretical analysis method of battery consistency. Detailed sensitivity analysis of the impact of the parameters consistency on energy utilization efficiency of the battery pack is described, and one conclude that SOC consistency is the most significant factor on energy utilization efficiency. Battery SOC consistency evolutions under four scenarios are discussed, which shows that columbic efficiency will lead to prominently accumulative effect on SOC divergence. The OCV consistency model of the battery pack and identification algorithm are introduced in Section 3 and the corresponding accuracy is demonstrated. The mapping relationship between the OCV distribution and the SOC distribution of the battery pack using dichotomy method is presented in Section 4. The proposed method is validated by experimental data. Finally, Section 5 summarizes the main findings of this paper and related open issues. 2. Theoretical analysis of battery inconsistency 2.1. Sensitivity analysis of parameters consistency to energy utilization efficiency At a specified battery aging state, the distributions of parameters such as capacity, internal resistance and SOC will directly
Please cite this article in press as: Zhang C et al. Study on battery pack consistency evolutions and equilibrium diagnosis for serial- connected lithium-ion batteries. Appl Energy (2017), http://dx.doi.org/10.1016/j.apenergy.2017.05.176
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Fig. 1. Parameter distribution of the battery pack with 95 cells.
Q j ¼ DSOCðjÞ C j ¼ C j ½SOC I ðjÞ f
1
ðU cutoff ðdisÞ þ I Rj Þ
ð1Þ
where {C j ; j ¼ 1; . . . ; N} and {Rj ; j ¼ 1; . . . ; N} are the current nominal capacities and internal resistances of the N battery cells, respectively, U cutoff ðdisÞ is the discharge cut-off voltage, SOC I ðjÞ is the initial 1
SOC of the jth battery cell, f ðU OCV Þ is addressed as a function of SOC. Thus the available capacity of the battery pack Q A discharged at the discharge current I is given by:
Q A ¼ min Q j
ð2Þ
16j6N
The SOC of the jth battery cell when the discharge process of the pack is cut off, is given by:
SOC 0 ðjÞ ¼ SOC I ðjÞ
QA CðjÞ
ð3Þ
Thus, in this case, the available SOC range of the jth battery cell is SOC 0 ðjÞ SOC I ðjÞ. The available energy of the jth battery cell being discharged in a battery pack is described by:
Z Ea ðjÞ ¼
Z U OCVj ðtÞIdt
Rj I2 dt jSOCðjÞ¼½SOC 0 ðjÞ;SOC I ðjÞ
ð4Þ
The energy utilization efficiency of a battery pack is defined as in Eq. (5).
PN EUE ¼
j¼1 Ea ðjÞ
EM
100%
ð5Þ
where EM is the total energy that can be stored in a battery pack. To investigate the most significant factor for the energy utilization efficiency, four cases are studied based on the battery data illustrated in Fig. 1. Case 1 represents the original distribution of
80
Energy Utilization Efficiency(%)
affect the energy utilization efficiency of a battery pack. Analyzing the sensitivity of the distributions of the three parameters on energy utilization efficiency of the battery pack is necessary to determine a battery consistency evaluation index. A group of battery data is collected from a company’s EV operated for three years is used in the study. The inconsistency of the battery capacity, the internal resistance and the initial SOC of discharge is illustrated in Fig. 1. In this battery pack, the ratios of the range to mean value for the capacity, the internal resistance and the initial SOC of discharge are 16%, 21%, 34%, respectively. Each of the parameter variations has an impact on the energy utilization efficiency of the battery pack. To investigate the corresponding effect, a calculation method has been introduced [32]. The available capacity of each battery cell discharged individually at the discharge current I is Qj {j = 1, . . . , N}, which is given by Eq. (1).
70 60 50 40 30 20 10 0
Case 1
Case 2
Case 3
Case 4
Fig. 2. Comparisons of energy utilization efficiency at various parameters distributions.
batteries including the inconsistency of the internal resistance, capacity and SOC. We assume that the inconsistency of the internal resistance, capacity and SOC is eliminated in Case 2, Case 3 and Case 4, respectively, and the other two parameter distributions are the same as the original distribution shown in Fig. 1. As shown in Fig. 2, the energy utilization efficiency of Case 1 is almost the same as that of Case 2. In case 3, the energy utilization efficiency is increased by 3.72% compared with Case 1. Furthermore, when the inconsistency of the initial SOCs of discharge (Case 4) is eliminated, the energy utilization efficiency is increased by 16.01% compared with Case1. Comparing the contributions of each parameter’s inconsistency to the energy utilization efficiency, it is obvious that the SOC variation has the most impact while the internal resistance has the least influence on the energy utilization efficiency of the battery pack. 2.2. Deduction of SOC consistency evolutions for a battery group As discussed in Section 2.1, the SOC variations has the most significant effect on the energy utilization efficiency of a battery pack. The evolution of SOC variations of the battery group are therefore deduced as battery usage to investigate the impact of various parameters divergence on battery inconsistency. Three batteries connected in series are used in the following scenarios, and their capacity, resistance, columbic efficiency are symbolized as Q , R, and g, respectively. Scenario I: Suppose the available discharged capacity Q , resistance R and columbic efficiency of three batteries are the same, but their initial SOC are different. Let SOC 1 ¼ s, SOC 2 ¼ s þ a, SOC 3 ¼ s þ b, where a and b are constant, and b > a > 0, g ¼ 1, and let the discharge and charge current be I1 and I2 , respectively.
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At the end of the first discharge, battery #1 is fully discharged, and the SOC for each battery can be given by:
I1 t 1 ¼0 Q
SOC 11st
dch
¼s
SOC 21st
dch
¼sþa
I1 t 1 ¼a Q
ð6bÞ
SOC 31st
dch
¼sþb
I1 t 1 ¼b Q
ð6cÞ
ð6aÞ
At the end of the first charge, battery #3 is fully charged, and the SOC for each battery is expressed by:
SOC 31st
ch
¼bþ
I2 t 2 ¼1 Q
ð7aÞ
SOC 21st
ch
¼aþ
I2 t 2 ¼ 1 ðb aÞ Q
ð7bÞ
SOC 11st
ch
¼
I2 t 2 ¼1b Q
ð7cÞ
At the end of the second discharge, battery #1 is fully discharged, and the SOC for each battery can be given by:
I1 t 3 ¼0 Q
SOC 12nd
dch
¼1b
SOC 22nd
dch
¼ 1 ðb aÞ
SOC 32nd
dch
¼1
I1 t 3 ¼a Q
I1 t 3 ¼b Q
SOC 3 ¼ 1
DQ 1 ¼ D3 Q 0 þ d2
ð10cÞ
Three battery will be fully charged at the end of charge since they discharges from fully charged state and their discharged capacity are the same. We can infer that the SOCs for batteries #1, #2, #3 are always 0, D2 and D3 when the battery pack is fully discharged after a number of charge/discharge cycles. The SOC inconsistency of the battery pack will be not magnified by the initial capacity variety if their capacity degradation rates are identical. Similar to Scenario I, the SOC ranges of three batteries are different in operation, and their equivalent charge/discharge rates differ from each other due to initial capacity variations. Both of them result in the capacity fading rate discrepancy even if the battery pack temperature distribution is uniform, which decrease the maximum available capacity/energy of the battery pack. Scenario III: Assume the available discharged capacity Q, resistance R, initial SOC of three batteries are same, but their columbic efficiency g are different. We assume g1 ¼ g0 ; g2 ¼ g0 þ e1 ; g3 ¼ g0 þ e2 , where e1 , e2 are constant, and e2 > e1 > 0, and initial SOC SOC 1 ¼ SOC 2 ¼ SOC 3 ¼ s. The discharge and charge current are I1 and I2 , respectively. At the end of the first discharge, the SOC for each battery is given by:
SOC 11st
ð8bÞ
At the end of the first charge, battery #3 is fully charged first because its charge efficiency is the highest and has the largest effective charge capacity. The SOC for each battery can be described by:
ð8cÞ
Following the recursive method, we assume the SOC of battery #1 as SOC 1n , the SOC for battery #2 and battery #3 after n charge/ discharge cycles can be given by:
SOC 2n ¼ SOC 1n þ a
ð9aÞ
SOC 3n ¼ SOC 1n þ b
ð9bÞ
It can be seen from Eqs. (9a) and (9b) that the SOC of battery #1, #2, #3 at the nth end of discharge are still 0, a and b. This demonstrates that the SOC inconsistency among batteries will not be enlarged by initial SOC variation. In other words, the SOC inconsistency resulted by the initial SOC will be kept constant as the number of battery charge and discharge cycles increases. It is worth noting that three batteries are operated at different SOC ranges, which are [0, 1 b], [a; 1 ðb aÞ], and [b; 1] for battery #1, #2, #3, respectively. Long-term accumulations may make battery degradation rate differ from each other, further extending the battery pack inconsistency. Scenario II: Assume the resistance R, initial SOC and columbic efficiency g of three batteries are the same, but their available capacity are different. Let Q 1 ¼ Q 0 , Q 2 ¼ Q 0 þ d1 ; Q 3 ¼ Q 0 þ d2 , d1 , d2 are constant, and d2 > d1 > 0, g ¼ 1, and initial SOC SOC 1 ¼ SOC 2 ¼ SOC 3 ¼ 1. The discharge and charge current are I1 and I2 , respectively. At the end of the first discharge, three batteries discharged Q 1 ampere hours since battery #1 was fully discharged. The current SOC for each battery can by expressed by:
SOC 1 ¼ 1
I1 t 1 DQ 1 ¼1 ¼0 Q0 Q0
ð10aÞ
SOC 2 ¼ 1
DQ 1 ¼ D2 Q 0 þ d1
ð10bÞ
dch
¼ SOC 21st
dch
¼ SOC 1st 3
dch
I1 t 1 ¼0 Q
ð8aÞ
¼s
ð11Þ
SOC 31st
ch
¼
I2 t 2 ðg0 þ e2 Þ ¼1 Q
ð12aÞ
SOC 21st
ch
¼
I2 t 2 ðg0 þ e1 Þ I2 t 2 ðe2 e1 Þ ¼1 Q Q
ð12bÞ
SOC 11st
ch
¼
I2 t 2 g0 I2 t2 e2 ¼1 Q Q
ð12cÞ
At the end of the second discharge, battery #1 is fully discharged first owing to its SOC lower, and the current SOC for each battery is as follows:
SOC 12nd
dch
¼1
I2 t2 e2 I1 t 3 ¼0 Q Q
ð13aÞ
SOC 22nd
dch
¼1
I2 t2 ðe2 e1 Þ I1 t3 I2 t2 e1 ¼ Q Q Q
ð13bÞ
SOC 32nd
dch
¼1
I1 t3 I2 t2 e2 ¼ Q Q
ð13cÞ
At the end of the second charge, the SOC for each battery is given by:
SOC 32nd
ch
¼
SOC 22nd
ch
¼
SOC 12nd
ch
¼
I2 t 2 e2 I2 t4 ðg0 þ e2 Þ ¼1 þ Q Q
ð14aÞ
I2 t 2 e1 I2 t4 ðg0 þ e1 Þ I2 t2 ðe2 e1 Þ I2 t4 ðe2 e1 Þ ¼1 þ Q Q Q Q ð14bÞ I 2 t 4 g0 I2 t2 e2 I2 t 4 e2 ¼1 Q Q Q
ð14cÞ
Please cite this article in press as: Zhang C et al. Study on battery pack consistency evolutions and equilibrium diagnosis for serial- connected lithium-ion batteries. Appl Energy (2017), http://dx.doi.org/10.1016/j.apenergy.2017.05.176
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Assuming the first charged capacity as I2 t2 ¼ DQ ch 1 , the second charged capacity I2 t4 ¼
DQ ch 2 ,
. . .and the nth charged capacity
I2 t2n ¼ DQ ch n . The SOC for each battery at the end of nth discharge can be described by:
SOC nth 1
SOC nth 2
dch
dch
dch SOC nth 3
¼0
¼
¼
ð15aÞ
ch ch DQ ch 1 þ DQ 2 þ þ DQ n1 e1
ð15bÞ
Q ch ch DQ ch 1 þ DQ 2 þ þ DQ n1 e2
ð15cÞ
Q
The SOC for each battery at the end of nth charge can be described by:
SOC nth 1
ch
¼1
ch ch DQ ch 1 þ DQ 2 þ þ DQ n Þe2 Q
ch SOC nth 2
¼1
SOC nth 3
¼1
ch
ð16aÞ
ch ch DQ ch 1 þ DQ 2 þ þ DQ n ðe2 e1 Þ
ð16bÞ
Q
ð16cÞ
Eqs. (15) and (16) manifest that the SOC inconsistency by battery columbic efficiency variations are amplified with increase of charge and discharge cycles. It is demonstrated that the columbic efficiency difference will make the SOC consistency of the battery group gradually diverge. Scenario IV: Assume the available discharged capacity Q, columbic efficiency g initial SOC of three batteries are the same, but their resistance R are different. Let the initial SOC as SOC 1 ¼ SOC 2 ¼ SOC 3 ¼ s. The discharge and charge current are I1 and I2 , respectively. During the usage of batteries, the terminal voltage is monitored to judge if the discharge and charge process need to be stopped. According to the Thevenin model of battery cell, the voltage drop of the internal resistance can’t be neglected especially when the current is large. The SOC is a function of the open circuit voltage (OCV), the SOC of a battery cell at the cutoff point will be affected by the internal resistance, and the distribution of the SOC at the cutoff point will be further affected by the change of the internal resistance as the number of cycles increase. To illustrate this problem, the SOC value at the end of charge is chosen to analyze the effect on its distribution. Since there exists no straight function of the SOC and the internal resistance, it has difficulty in finding an analytical solutions to study the effect of internal resistance on SOC variation evolutions. A numerical method is therefore employed to investigate the relationship between the internal resistance distribution and the SOC distribution. In the first case, the initial SOC at the end of charge is 100%. In the next cases, the initial SOC value at the end of charge is affected by the internal resistance. The distribution of the internal resistance for each case is shown in Table 1, the mean
and standard deviation of internal resistance increase from Case 1 to Case 10. Both the discharge and charge current are set as I1 ¼ I2 ¼ 30 A. In each case, the battery pack is first discharged to cutoff point and then fully charged. In this calculation, let the SOC value of each cell at the end of charge in the ith case be the SOC value of each cell at the start of discharge in the ði þ lÞth case. The SOC value of each cell at the end of charge is changed gradually due to the change of distribution of the internal resistance and the results are shown in Table 1 based on the SOC track method in Eq. (1). With the increase of the mean and the standard deviation of the internal resistance, the mean SOC at the end of charge is decreasing while the standard deviation and range of the SOC are increasing. However, the variation of the SOC is negligible compared to the mean of SOC. Based on the above discussions, if one assumes that the degradation rate of battery capacity and resistance are same, the degree of SOC inconsistency will be kept constant caused by the initial capacity, resistance and SOC variations. Nevertheless, the initial columbic efficiency varieties will result in the SOC inconsistency of the battery group diverging increasingly, showing prominent accumulative effect. It can be inferred that the columbic efficiency variations may contribute more to the battery SOC inconsistency in use compared to the capacity and the internal resistance variations.
3. Battery OCV consistency modeling As expressed in Section 2, compared to the capacity and the internal resistance, the SOC inconsistency has the most significant influence on battery pack energy utilization efficiency, which is usually the objective of battery equilibrium and maintenance. The SOC consistency of battery pack can be employed as evaluation index representing the battery consistency level. As is known, the SOC-OCV function is a representative for a particular battery, and is generally a nonlinear monotone function between SOC and OCV for all lithium-ion batteries. The OCV can be used for representing battery SOC, and is easily measured on board. The OCV consistency is hence ultimately chosen as the inconsistency evaluation parameter of the battery pack.
3.1. OCV consistency change under operating conditions The data used in this study are acquired from two dual-source trolley buses (named as Bus #1 and Bus #2) operated for 7 months. The bus is powered by both on-board battery system and grid electricity, which can operate in both charging mode and driving mode at the same time. The operational condition profile of the battery is close to that in a hybrid electric vehicle, and large charge current and discharge current happen and fluctuate severely sometimes. The battery used in the bus is the Nickel Manganese Cobalt Oxide (NMC) & Manganese Oxide cathode and graphite anode based battery. The nominal capacity of the battery pack is 105Ah, three cells with nominal capacity of 35 A h in parallel constitute a battery
Table 1 Distribution of the internal resistance for each case and its impact on SOC distribution. Case number
1
2
3
4
5
6
7
8
9
10
Mean R(X) Std of R Mean SOC (%) Std of SOC Range of SOC (%)
0.0033 0.00015 100.0 0 0
0.0038 0.0002 100.0 0 0
0.0043 0.00025 100.0 0 0
0.0048 0.0003 99.9 0.00158 0.0086
0.0053 0.00035 99.8 0.00645 0.03
0.0058 0.0004 99.45 0.0189 0.1
0.0063 0.00045 99.3 0.0224 0.12
0.0068 0.0005 98.9 0.0328 0.1773
0.0073 0.00055 98.5 0.0483 0.261
0.0078 0.0006 97.7 0.0713 0.386
Please cite this article in press as: Zhang C et al. Study on battery pack consistency evolutions and equilibrium diagnosis for serial- connected lithium-ion batteries. Appl Energy (2017), http://dx.doi.org/10.1016/j.apenergy.2017.05.176
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Bus #1
60.0
21.6
14.4
7.2
0.0 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20
Distribution of OCV(V)
Percentage of Distribution(%)
Percentage of Distribution(%)
28.8
Bus #2
40.0
20.0
0.0 3.85
3.90
3.95
4.00
4.05
4.10
4.15
4.20
Distribution of OCV(V)
Fig. 3. Histogram distributions of average OCV for Bus #1 and Bus #2.
module, and 104 battery modules are then serially connected composed of the battery pack. The bus is off grid when parked at night, and the battery pack are not be charged when standing by. During practical operation, the polarization voltage of battery pack in electric vehicles is eliminated after standing for a whole night. The battery management system (BMS) will record the OCV of each battery module before EV bus start in the following day. The OCV standard deviation of the battery pack, as the inconsistency coefficient on homeostasis parameter characteristics, excluding the impact of the current and the temperature. It is an important factor of the pack inconsistency, not only belong to reversible inconsistent, which can be used as an important reference for improving the battery balancing strategy [33]. Fig. 3 shows the histogram distribution of the average OCV for Bus #1 and Bus #2 before the start of the bus for approximately 160 days. Most OCV values are between 3.95 V and 4.15 V. The SOC-OCV curve of the lithium-ion battery is illustrated in Fig. 4. It can be inferred that before the start of the bus every day, the SOC is mainly distributed ranging from 60% to 95%, which is in a high SOC region. The OCV distribution of battery pack does not always maintain a normal distribution characteristic at each SOC point. Compared to other indexes such as the average value and the range of OCV, the standard deviation (SD) is more stable and reliable for quantifying the amount of variation of the OCV data. This is because the average value and the range of the OCV mainly rely on the SOC of the battery pack, and fluctuate severely as battery SOC varying. The SD is finally selected to represent the battery pack’s consistency, and to characterize the battery pack energy utilization efficiency.
The study is focused on the OCV dispersion and its distribution characteristics are not further discussed here. 3.2. Standard deviation modeling of OCV It has difficulty in getting equivalent full cycling number of the battery at practical application since the battery is seldom fully discharged or charged, and the current fluctuate severely under operating conditions. The absolute cumulative capacity for the battery charging and discharging is employed as an independent variable to model. The total cumulative capacity of two buses for the sampling period are both about 60,000 A h. After removing the bad points of the data, the standard deviation scatter plot of the OCV is shown in Fig. 5. It is found from Fig. 5 that the change of OCV standard deviation for both buses are similar, and both of them are slowing down in the later period. The standard deviation reach around 7–7.5 mV at cumulative capacity of 60,000 A h, which has obvious regularity. Since one month operation data for both two buses are lost, the average of the cumulative capacity of its adjacent two months is regarded as the cumulative capacity for the month of data loss. It can be seen from Fig. 5 that the OCV standard deviation are gradually changing and always keep growing or remain stable in the future, but will not decrease. Both plots manifest a certain behavior that it is easy to use polynomial relationship as the fitting model. The polynomial function is applied for fitting relationship between the OCV standard deviation and the cumulative capacity. The model is given by:
rðQ Þ ¼ k1 þ k2 Q k3
rðQ Þ is the standard deviation of OCV, Q is the cumulative capacity, k1, k2, k3 are the coefficients to be extracted. The model uses the cumulative capacity as the input, which has large numerical value, and the basic functions of the mathematical model is a power function. Compared to the linear function model, the nonlinear fitting requires more input data. In order to make the model more predictable and improve the adaptability of the model, Eq. (17) can be rearranged by:
4.2 4.0
OCV (V)
ð17Þ
3.8 3.6 3.4
rQ ¼ f ðQ lg Þ ¼ k1 þ k2 k3 lg Q
3.2
ð18Þ
After taking denary logarithm of the cumulative capacity as the independent variable, and it is simplified as:
3.0 0
20
40
60
80
SOC (%) Fig. 4. SOC-OCV curve of a lithium-ion battery.
100
rQ ¼ f ðQ lg Þ ¼ k4 þ k5 lg Q
ð19Þ
where Q is the cumulative capacity, k4, k5 are the model coefficients.
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Standard Deviation of OCV (mV)
Standard Deviation of OCV (mV)
8.0 8
7
6
5
Bus #1
4
3
0
12000
40000
60000
80000
7.5 7.0 6.5 6.0 5.5 5.0 4.5
Bus #2 0
17000
45000
60000
Cumulative Capacity (Ah)
Cumulative Capacity (Ah)
Fig. 5. Scatter plot of OCV standard deviation for Bus #1 and Bust #2.
3.3. Parameters extraction and model validation Levenberg-Marquardt (L-M) algorithm is one of the most effective methods for nonlinear model parameter fitting. L-M algorithm
Initialization k1,k2,k3,J,H,e,e_lm,λ
Build the initial simplex
Lamda= /10; e=e_lm
Calculate J,H
No Calculate k1,k2,k3,e_lm
e_lm
Yes Cycle index
No Output Fig. 6. Procedure of the L-M algorithm.
has the advantage of fast convergence similar to the Gauss-Newton steepest descent method, and has good adaptability like neural network at the same time. The procedure of the L-M algorithm for extracting parameters is shown in Fig. 6. Where J is the Jacobian matrix of model functions, H = J0 ⁄ J, k1 , k2 , k3 are the fitting parameters of the model, e is the initial calculation error, elm is the calculation error after the parameter is updated, k is the damping factor, n is the number of the iterations. According to this method, coefficients can be identified effectively. Eqs. (17) and (19) are identified using LM algorithm. The input variables are the first 80% of the standard deviation of OCV and the cumulative capacity. After the fitted parameters are obtained, the latter 20% of the data is predicted. The goodness of fit is calculated by R-squared coefficient of determination, a statistical measure of how close the data are to the fitted regression line. The prediction accuracy is represented by the maximum relative error. Table 2 shows the fitted parameters of the model. The values of the parameters of two buses are not exactly same, indicating that even if the battery type are same, the variation of parameters still exist due to the different operating conditions. The fitted results using two models are illustrated in Figs. 7 and 8, in which the abscissae of Figs. 7 and 8 are cumulative capacity and cumulative capacity taking denary logarithm, respectively. The results of the goodness of fit are shown in Table 3, and the maximum predictive relative error is reported in Table 4. Both models can fit the training data and predict the validation data accurately. The estimation error fluctuation of Bus #1 is larger than that of Bus #2 due to its greater volatility. From Fig. 7 and Table 4, it can be seen that the fitted and predicted results using L-M algorithm manifest better accuracy compared to the least square (LS) method. This is because the iterative algorithm in L-M algorithm cause the fitting model more adaptable for predicting. This is a great advantage under the practical operating condition discussed in the study. After taking denary logarithm of the accumulated power, the standard deviation of OCV keeps a good linear change. The result further confirms that the two variables have a correlation of polynomial functions. The goodness of fit and the prediction precision
Table 2 Fitted parameters of the two models.
rQ ¼ f ðQ lg Þ ¼ k4 þ k5 lgQ
rðQ Þ ¼ k1 þ k2 Q k3 Coefficient
k1
k2
k3
k4
k5
Bus #1 Bus #2
0.0020 0.0034
0.00016 0.00016
0.32 0.29
0.0046 0.0014
0.0024 0.0018
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8.0 7.5
8.0
Standard Deviation of OCV (mV)
Standard Deviation of OCV (mV)
8.5
Bus #1
7.0 6.5 6.0 5.5 5.0
Validating data Training data LM fitted LS fitted
4.5 4.0 3.5 3.0 2.5
0 12000
40000
60000
80000
Bus #2
7.5 7.0 6.5 6.0
Validating data Training data LM fitted LS fitted
5.5 5.0 4.5
0
17000
45000
60000
75000
Cumulative Capacity (Ah)
Cumulative Capacity (Ah)
Fig. 7. Fitted results using the polynomial function.
0%~80% SOC 80%~100% SOC Fitted Values
0.008
0.006
Standard Deviation of OCV (V)
Standard Deviation of OCV (V)
0.008
0.004
0.002
0.000 3.0
3.2
3.4
3.6
3.8
4.0
4.50
4.75
5.00
Cumulative Capacity
0.006
0.004
0%~80% SOC 80%~100% SOC Fitted Values
0.002
0.000
3.15
3.50
3.9
4.2
4.5
4.8
Cumulative Capacity (Ah)
Fig. 8. Fitted results after taking denary logarithm of the cumulative capacity.
Table 3 Goodness of fit of the model for the first 80% of data. Model
rðQ Þ ¼ k1 þ k2 Q k3
Method
LM
LS
Bus #1 Bus #2
0.894 0.969
0.899 0.938
rQ ¼ f ðQ lg Þ ¼ k4 þ k5 Q lg 0.874 0.955
Table 4 Maximum relative error of the predictive result for the latter 20% of data. Model
rðQ Þ ¼ k1 þ k2 Q k3
Method
LM
LS
Bus #1 Bus #2
9.58% 1.32%
11.41% 5.14%
rQ ¼ f ðQ lg Þ ¼ k4 þ k5 Q lg 7.02% 3.14%
SOC consistency for estimating the energy utilization efficiency of the battery pack. The detailed calculation method of energy utilization efficiency is introduced in Ref. [32]. The inputs of the calculation in Ref. [32] are the capacity, internal resistance and SOC distributions of batteries in the pack. In this study, the data of the capacity and internal resistance distribution are from the data mentioned in Section 2.1 and the SOC distribution is calculated through the OCV distribution. To calculate the SOC distribution based on the OCV distribution, a series of voltage data is firstly randomly generated which has the same OCV distribution of the battery pack when the battery pack is fully charged. Then a SOC-OCV model in Ref. [34] is used to calculate the SOC distribution. To determine the parameters in the model, we firstly calculate the SOC-OCV curves of all batteries and then average them as the SOC-OCV curve for fitting the curve in the model. The proposed SOC-OCV model is shown the following Equation: 0:478
V ocv ¼ 2:549 0:4851 ð ln sÞ of linear model do not decrease. The linear model can effectively improve the adaptability and has a wider application scenario compared to the non-linear model.
4. Application of battery equilibrium diagnosis The energy utilization efficiency is used to describe the performance of battery pack and is also a judgment base for equilibrium in this paper. The predicted results using the OCV consistency model described in Section 3 need to be transformed into the
1:221 s þ 2:206 e0:4ðs1Þ ð20Þ
where V ocv and s represent the OCV and SOC of the battery, respectively, and 0 < s 6 1. Eq. (20) is a transcendental equation and its inverse function cannot be accessible. Therefore, it is impossible to calculate the exact SOC value of a given OCV by using the inverse function of Eq. (20). To solve this problem, the dichotomy method is employed. The basic idea of the dichotomy method is to produce a series of convergent data {xk } to approach the true value x of a specific equation. Since the proposed SOC-OCV model is
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30 Mean 86.12 StDev 9.13 N 95
Relative frequency%
25 20 15 10 5 0
60
65
70
75
80
85
90
95
100
SOC% Fig. 9. Simulated results of the SOC distribution.
Table 5 Simulated results of the battery pack energy utilization efficiency. Current
10 A
20 A
30 A
Simulated True Estimate error
65.81% 65.03% 0.78%
65.28% 64.38% 0.9%
64.75% 63.73% 1.02%
monotonous increasing and there is only one SOC value s if given a specific OCV value V ocv , the process of producing the series of convergent data {xk } can be concluded as follow: Step 1: Denote the function as Eq. (21) 0:478
f ðsÞ ¼ 2:549 0:4851 ð ln sÞ
1:221 s þ 2:206 e0:4ðs1Þ V ocv
9
distribution properties of the capacity, internal resistance and the SOC of a battery pack. It concludes that the SOC variation contributions the most to battery consistency from the perspective of energy utilization efficiency, and hence is employed as evaluation index characterizing battery consistency level. The evolutions of the SOC variation are deduced as the battery usage to investigate the impact of various parameters divergence on battery pack inconsistency. It is indicated the initial columbic efficiency variety will result in the SOC inconsistency diverging increasingly, showing evident accumulative effect. Two OCV standard deviation models for describing battery consistency are proposed based on the electric buses operating data, and L-M algorithm is utilized to extract the model parameters. The L-M algorithm possesses predominant estimating accuracy for both the fitted and predicted results compared to the least square method. It is noteworthy that the maximum relative errors for Bus #1 and Bus #2 exhibit obvious different due to greater data fluctuation of Bus #2. The stability and reliability of the data have significant influence on the model, and is worthy to be further discussed. To estimate the energy utilization efficiency of the battery pack, the mapping relation between the OCV distribution and the SOC distribution of the battery pack by using the dichotomy method is developed. The developed method that estimate the energy utilization efficiency is demonstrated with a high accuracy, and provide a foundation for the battery equilibrium diagnosis. In the future work, the impact of temperature and columbic efficiency on the SOC consistency and degradation consistency will be further investigated.
Acknowledgement
ð21Þ
and denote the SOC value of a given OCV as s . Step 2: Set the range which includes s as I0 ¼ ½a; b; set the midpoint of I0 as s0 ¼ 0:5ða þ bÞ; Calculate f ðs0 Þ. Since 0 < s 6 1, in this situation, a ¼ 0 and b ¼ 1; Step 3: Estimate the value of f ðs0 Þ, If f ðs0 Þ = 0, thus x ¼ x0 and stop calculation; If f ðs0 Þf ðaÞ < 0, thus s 2 ½a; s0 and let I1 ¼ ½a; s0 ; If f ðs0 Þf ðaÞ > 0, thus, s 2 ½s0 ; b and let I1 ¼ ½s0 ; b; Step 4: Define I1 ¼ ½a1 ; b1 and set the midpoint of I1 as s0 ¼ 0:5ða1 þ b1 Þ;If js1 s0 j < e, s s1 and stop calculation, where e represents the required SOC estimation accuracy. In this situation, e ¼ 0:001. Otherwise, replace I0 with I1 and go back to Step 2. Fig. 9 displays the simulated results of the SOC distribution based on the OCV distribution. The mean and standard deviation of the SOC data from simulation are 91.76% and 8.16% respectively. Compared with the real SOC distributions in Section 2.1, the error of the mean value is within 6% and the error of standard deviation is within 1%. Based on the battery capacity, internal resistance and OCV distributions, the battery pack energy utilization efficiency is calculated and the result is shown in Table 5. For the given three current rates, the estimation error is within 1.5%, demonstrating a satisfactory agreement with the true values. If the threshold of the energy utilization efficiency for balancing is set in advanced, such estimation result can be applied for equilibrium diagnosis. 5. Conclusion The article systematically analyzes the influence of parameters variation on battery pack consistency based on the statistical
The authors are grateful for financial support via the National Key Research and Development Program of China (2016YFB0101800), Science and Technology Program of SGCC (Operation Safety and Interconnection Technology for Electric Infrastructure).
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Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Study on battery pack consistency evolutions and equilibrium diagnosis for serial- connected lithium-ion batteries Caiping Zhang ⇑, Yan Jiang, Jiuchun Jiang ⇑, Gong Cheng, Weiping Diao, Weige Zhang National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing 100044, China Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Jiaotong University, Beijing 100044, China
h i g h l i g h t s Impact sensitivity of parameters variety on energy utilization ratio is analyzed. The law of SOC consistency evolutions is deduced for parameter variations. The OCV consistency model of the battery pack is proposed. The mapping between OCV and SOC distribution of the battery pack is established.
a r t i c l e
i n f o
Article history: Received 16 February 2017 Received in revised form 23 May 2017 Accepted 26 May 2017 Available online xxxx Keywords: Lithium-ion battery Battery pack consistency Sensitivity analysis Modeling Equilibrium diagnosis
a b s t r a c t The consistency among lithium-ion battery pack is an important factor affecting their performance. The paper analyzes the impact sensitivity of parameters consistency including capacity, internal resistance and state of charge (SOC) on energy utilization efficiency of the battery pack. It turns out that SOC variations is the most significant influence on battery consistency, and hence is employed as evaluation index to characterize battery consistency level. Then the SOC evolution is explored under four scenarios, and the result reflects that the columbic efficiency is associated with prominent accumulative effect on SOC divergence of the battery group in use. The OCV consistency model is established based on longterm battery data of two trolley buses. It is observed that the fitted results match with the operational data very well. Finally, the mapping relation between the OCV distribution and the SOC distribution of the battery pack using dichotomy method is proposed. The calculation error of battery pack energy utilization efficiency by using such method is within 1.5%, which can be used for battery equilibrium diagnosis and prognosis at a certain aging state. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction To deal with the energy crisis and environmental pollution, vehicle electrification is one of the effective approaches. Since lithium-ion batteries possess high energy density, high terminal voltage, long life and none memory effect [1,2], they are widely used as power sources in electric vehicles (EVs). With the prevailing application of lithium-ion batteries, the durability and safety for battery packs receive more and more attentions from researchers and engineers since the improper usage of batteries will shorten the battery life [3–6] and even sometimes cause severe fire hazards [7,8]. ⇑ Corresponding authors at: National Active Distribution Network Technology Research Center (NANTEC), Beijing Jiaotong University, Beijing 100044, China. E-mail addresses: [email protected] (C. Zhang), [email protected] (J. Jiang).
In the practical applications, batteries are usually series connected to form a high voltage battery group in order to meet the power and energy requirements. The inconsistency among battery cells is a key factor influencing the performance of battery packs. Battery inconsistency, which is also called cell to cell variations, origins from two main factors. One happens in the procedure of battery production [9–11], such as coating, ingredients and unevenness of impure contents of batteries which give rise to the difference in the battery initial performance like original capacity, resistance, coulombic efficiency and self-discharge rate. Although such differences are minute, huge variations will occur during the operation of batteries. For example, researchers in Ref. [12] reported that 48 cells from a mass production were tested under equal conditions but a 10% capacity range showed after 1000 cycles due to the production inconsistency. Another occurs in usage. Inconsistency of battery original performance will cause difference in equivalent current rate, SOC usage range among cells. On
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the other side, battery pack thermal distribution is always nonuniform, causing temperature variations among cells. The aforementioned factors will generate variations in SOC and aging (e.g. capacity fade, power fade) of the batteries [13]. The former will influence the utilization efficiency of the battery pack, the latter will impact the maximum available energy of the battery pack. In addition, both of them have an impact on the durability of battery packs. Experimental investigations on battery consistency have been implemented from various perspectives. In Ref. [14], Gogoana et al. reported that there is an additional 40% lifetime reduction when there is a 20% mismatch in the ohmic resistance for parallel connected batteries. During practical operation, the battery inconsistency becomes more and more severe because the temperature ingredient leads to different aging behaviors of batteries [5,6,15–17]. Su et al. [5] reported that batteries were more susceptible to temperature other than discharge current and discharge cutoff voltage. Susana et al. [15] compared the stability of the cathode material of lithium-ion batteries with blended cathode materials under 25 °C and 40 °C, and the experimental results showed that the elevated temperature is detrimental to the long term cycling stability of the cathode. Ganesan et al. developed an electrochemical-thermal coupled model for a battery pack to analyze the battery pack performance under various rates and temperatures. An additional 5% capacity loss of the battery pack was obtained when there is a temperature difference of 15 °C among the cells [18]. Before assembling the battery cells into a practical battery pack, batteries are usually connected in parallel in the first place. According to Ref. [19–21], either the internal resistance or the capacity mismatch would cause an unbalancing current distribution among parallel connected cells. These accelerate the capacity fade of a specific cell in the battery pack. Battery inconsistency directly influences the energy and capacity utilization efficiency of a battery pack. Different approaches have been developed to reduce inconsistency before battery pack assembling [22,23]. In Ref. [22], experimental charge/discharge voltage signals were analyzed by using discrete wavelet transform-based feature extraction method to select batteries with similar electrochemical characteristics. Although the initial performance of the battery pack was guaranteed, the above method failed to predict the consistency divergence along with usage. For a battery pack without regularly maintenance and balancing, the capacity and energy utilization ratio will be affected by internal resistance [14,19], capacity [24], coulombic efficiency [25–27] and SOC [28] variations of battery cells during the operation. Generally speaking, there are two ways to investigate the impact of the battery pack way on inconsistency. One way is based on all individual cells [9,14,22,23,27,29,30] and another way is to just choose some representative cells in a pack [31]. Jiang et al. [30] developed a novel energy utilization efficiency evaluation method based on the distribution of the capacity. This method took into account the influence of capacity variations but neglected the internal resistance and SOC variations. Moreover, the capacities of all batteries are different in a battery pack. Schuster et al. [9,29] studied the correlation among capacity, the impedance, the researched evolution of capacity and the impedance distribution of battery cells during operation. The aforementioned approach could predict the inconsistency divergence based on experimental results obtained from lab, however, the impedance data was not available on board. Ouyang et al. [31] presented a battery pack capacity estimation method by determining the remaining charging and discharging electric quantities of batteries which are the firstly reached to charge and discharge cutoff voltages respectively. A merit of this method consists in one only needs to pay attention to two specific cells, but the estimation error could be large if the battery group only operates in a small SOC range. In applications, the commonly used battery pack SOC estimation method is to
use current integration together with frequently calibration associated with OCV methods [24]. This would lead to a sudden drop or increase of the battery pack SOC when one specific battery voltage reaches beforehand to the cutoff conditions. The previous studies have achieved notable progresses, but there are also some limitations. One weakness of the existed literature is without considering the online application evaluation and the prediction of battery inconsistency. Most researches analyze the battery inconsistency and its influence on battery packs based on the battery capacity and the impedance. However, it is difficult to measure the impedance on board. Moreover, the online capacity estimation errors of battery cells can be large if battery packs operate in a small and almost fixed SOC range - such as the working conditions of electric buses. Additionally, there is little literature which clearly states and predicts the battery pack performance along with the inconsistency evolution of the battery capacity, the internal resistance, SOC and the coulombic efficiency. Since only the battery voltage, the current and the temperature data are measured on board, effective methods to evaluate the battery pack consistency and predict how the consistency will evolve are urgently needed. 1.1. Contributions of the paper The paper makes original contributions in the following aspects: (i) Sensitivity analysis of the impact of parameters consistency on energy utilization efficiency of the battery pack is performed. The SOC variation is the most significant influence on battery consistency, and hence is employed as evaluation index to characterize battery consistency level. (ii) The law of SOC consistency evolution is deduced for parameter variations of the capacity, the internal resistance, the columbic efficiency, and the initial SOC. Battery columbic efficiency shows prominent accumulative effect on SOC divergence of the battery group in use. (iii) Two OCV standard deviation models for describing the battery consistency are proposed based on the electric buses operating data, and L-M algorithms are used to identify the parameters. (iv) A mapping relationship between the OCV distribution and the SOC distribution of a battery pack is established by using dichotomy methods, and is applied for battery equilibrium diagnosis. 1.2. Organization of the paper The rest of the paper is organized as follows. Section 2 introduced the theoretical analysis method of battery consistency. Detailed sensitivity analysis of the impact of the parameters consistency on energy utilization efficiency of the battery pack is described, and one conclude that SOC consistency is the most significant factor on energy utilization efficiency. Battery SOC consistency evolutions under four scenarios are discussed, which shows that columbic efficiency will lead to prominently accumulative effect on SOC divergence. The OCV consistency model of the battery pack and identification algorithm are introduced in Section 3 and the corresponding accuracy is demonstrated. The mapping relationship between the OCV distribution and the SOC distribution of the battery pack using dichotomy method is presented in Section 4. The proposed method is validated by experimental data. Finally, Section 5 summarizes the main findings of this paper and related open issues. 2. Theoretical analysis of battery inconsistency 2.1. Sensitivity analysis of parameters consistency to energy utilization efficiency At a specified battery aging state, the distributions of parameters such as capacity, internal resistance and SOC will directly
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Fig. 1. Parameter distribution of the battery pack with 95 cells.
Q j ¼ DSOCðjÞ C j ¼ C j ½SOC I ðjÞ f
1
ðU cutoff ðdisÞ þ I Rj Þ
ð1Þ
where {C j ; j ¼ 1; . . . ; N} and {Rj ; j ¼ 1; . . . ; N} are the current nominal capacities and internal resistances of the N battery cells, respectively, U cutoff ðdisÞ is the discharge cut-off voltage, SOC I ðjÞ is the initial 1
SOC of the jth battery cell, f ðU OCV Þ is addressed as a function of SOC. Thus the available capacity of the battery pack Q A discharged at the discharge current I is given by:
Q A ¼ min Q j
ð2Þ
16j6N
The SOC of the jth battery cell when the discharge process of the pack is cut off, is given by:
SOC 0 ðjÞ ¼ SOC I ðjÞ
QA CðjÞ
ð3Þ
Thus, in this case, the available SOC range of the jth battery cell is SOC 0 ðjÞ SOC I ðjÞ. The available energy of the jth battery cell being discharged in a battery pack is described by:
Z Ea ðjÞ ¼
Z U OCVj ðtÞIdt
Rj I2 dt jSOCðjÞ¼½SOC 0 ðjÞ;SOC I ðjÞ
ð4Þ
The energy utilization efficiency of a battery pack is defined as in Eq. (5).
PN EUE ¼
j¼1 Ea ðjÞ
EM
100%
ð5Þ
where EM is the total energy that can be stored in a battery pack. To investigate the most significant factor for the energy utilization efficiency, four cases are studied based on the battery data illustrated in Fig. 1. Case 1 represents the original distribution of
80
Energy Utilization Efficiency(%)
affect the energy utilization efficiency of a battery pack. Analyzing the sensitivity of the distributions of the three parameters on energy utilization efficiency of the battery pack is necessary to determine a battery consistency evaluation index. A group of battery data is collected from a company’s EV operated for three years is used in the study. The inconsistency of the battery capacity, the internal resistance and the initial SOC of discharge is illustrated in Fig. 1. In this battery pack, the ratios of the range to mean value for the capacity, the internal resistance and the initial SOC of discharge are 16%, 21%, 34%, respectively. Each of the parameter variations has an impact on the energy utilization efficiency of the battery pack. To investigate the corresponding effect, a calculation method has been introduced [32]. The available capacity of each battery cell discharged individually at the discharge current I is Qj {j = 1, . . . , N}, which is given by Eq. (1).
70 60 50 40 30 20 10 0
Case 1
Case 2
Case 3
Case 4
Fig. 2. Comparisons of energy utilization efficiency at various parameters distributions.
batteries including the inconsistency of the internal resistance, capacity and SOC. We assume that the inconsistency of the internal resistance, capacity and SOC is eliminated in Case 2, Case 3 and Case 4, respectively, and the other two parameter distributions are the same as the original distribution shown in Fig. 1. As shown in Fig. 2, the energy utilization efficiency of Case 1 is almost the same as that of Case 2. In case 3, the energy utilization efficiency is increased by 3.72% compared with Case 1. Furthermore, when the inconsistency of the initial SOCs of discharge (Case 4) is eliminated, the energy utilization efficiency is increased by 16.01% compared with Case1. Comparing the contributions of each parameter’s inconsistency to the energy utilization efficiency, it is obvious that the SOC variation has the most impact while the internal resistance has the least influence on the energy utilization efficiency of the battery pack. 2.2. Deduction of SOC consistency evolutions for a battery group As discussed in Section 2.1, the SOC variations has the most significant effect on the energy utilization efficiency of a battery pack. The evolution of SOC variations of the battery group are therefore deduced as battery usage to investigate the impact of various parameters divergence on battery inconsistency. Three batteries connected in series are used in the following scenarios, and their capacity, resistance, columbic efficiency are symbolized as Q , R, and g, respectively. Scenario I: Suppose the available discharged capacity Q , resistance R and columbic efficiency of three batteries are the same, but their initial SOC are different. Let SOC 1 ¼ s, SOC 2 ¼ s þ a, SOC 3 ¼ s þ b, where a and b are constant, and b > a > 0, g ¼ 1, and let the discharge and charge current be I1 and I2 , respectively.
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At the end of the first discharge, battery #1 is fully discharged, and the SOC for each battery can be given by:
I1 t 1 ¼0 Q
SOC 11st
dch
¼s
SOC 21st
dch
¼sþa
I1 t 1 ¼a Q
ð6bÞ
SOC 31st
dch
¼sþb
I1 t 1 ¼b Q
ð6cÞ
ð6aÞ
At the end of the first charge, battery #3 is fully charged, and the SOC for each battery is expressed by:
SOC 31st
ch
¼bþ
I2 t 2 ¼1 Q
ð7aÞ
SOC 21st
ch
¼aþ
I2 t 2 ¼ 1 ðb aÞ Q
ð7bÞ
SOC 11st
ch
¼
I2 t 2 ¼1b Q
ð7cÞ
At the end of the second discharge, battery #1 is fully discharged, and the SOC for each battery can be given by:
I1 t 3 ¼0 Q
SOC 12nd
dch
¼1b
SOC 22nd
dch
¼ 1 ðb aÞ
SOC 32nd
dch
¼1
I1 t 3 ¼a Q
I1 t 3 ¼b Q
SOC 3 ¼ 1
DQ 1 ¼ D3 Q 0 þ d2
ð10cÞ
Three battery will be fully charged at the end of charge since they discharges from fully charged state and their discharged capacity are the same. We can infer that the SOCs for batteries #1, #2, #3 are always 0, D2 and D3 when the battery pack is fully discharged after a number of charge/discharge cycles. The SOC inconsistency of the battery pack will be not magnified by the initial capacity variety if their capacity degradation rates are identical. Similar to Scenario I, the SOC ranges of three batteries are different in operation, and their equivalent charge/discharge rates differ from each other due to initial capacity variations. Both of them result in the capacity fading rate discrepancy even if the battery pack temperature distribution is uniform, which decrease the maximum available capacity/energy of the battery pack. Scenario III: Assume the available discharged capacity Q, resistance R, initial SOC of three batteries are same, but their columbic efficiency g are different. We assume g1 ¼ g0 ; g2 ¼ g0 þ e1 ; g3 ¼ g0 þ e2 , where e1 , e2 are constant, and e2 > e1 > 0, and initial SOC SOC 1 ¼ SOC 2 ¼ SOC 3 ¼ s. The discharge and charge current are I1 and I2 , respectively. At the end of the first discharge, the SOC for each battery is given by:
SOC 11st
ð8bÞ
At the end of the first charge, battery #3 is fully charged first because its charge efficiency is the highest and has the largest effective charge capacity. The SOC for each battery can be described by:
ð8cÞ
Following the recursive method, we assume the SOC of battery #1 as SOC 1n , the SOC for battery #2 and battery #3 after n charge/ discharge cycles can be given by:
SOC 2n ¼ SOC 1n þ a
ð9aÞ
SOC 3n ¼ SOC 1n þ b
ð9bÞ
It can be seen from Eqs. (9a) and (9b) that the SOC of battery #1, #2, #3 at the nth end of discharge are still 0, a and b. This demonstrates that the SOC inconsistency among batteries will not be enlarged by initial SOC variation. In other words, the SOC inconsistency resulted by the initial SOC will be kept constant as the number of battery charge and discharge cycles increases. It is worth noting that three batteries are operated at different SOC ranges, which are [0, 1 b], [a; 1 ðb aÞ], and [b; 1] for battery #1, #2, #3, respectively. Long-term accumulations may make battery degradation rate differ from each other, further extending the battery pack inconsistency. Scenario II: Assume the resistance R, initial SOC and columbic efficiency g of three batteries are the same, but their available capacity are different. Let Q 1 ¼ Q 0 , Q 2 ¼ Q 0 þ d1 ; Q 3 ¼ Q 0 þ d2 , d1 , d2 are constant, and d2 > d1 > 0, g ¼ 1, and initial SOC SOC 1 ¼ SOC 2 ¼ SOC 3 ¼ 1. The discharge and charge current are I1 and I2 , respectively. At the end of the first discharge, three batteries discharged Q 1 ampere hours since battery #1 was fully discharged. The current SOC for each battery can by expressed by:
SOC 1 ¼ 1
I1 t 1 DQ 1 ¼1 ¼0 Q0 Q0
ð10aÞ
SOC 2 ¼ 1
DQ 1 ¼ D2 Q 0 þ d1
ð10bÞ
dch
¼ SOC 21st
dch
¼ SOC 1st 3
dch
I1 t 1 ¼0 Q
ð8aÞ
¼s
ð11Þ
SOC 31st
ch
¼
I2 t 2 ðg0 þ e2 Þ ¼1 Q
ð12aÞ
SOC 21st
ch
¼
I2 t 2 ðg0 þ e1 Þ I2 t 2 ðe2 e1 Þ ¼1 Q Q
ð12bÞ
SOC 11st
ch
¼
I2 t 2 g0 I2 t2 e2 ¼1 Q Q
ð12cÞ
At the end of the second discharge, battery #1 is fully discharged first owing to its SOC lower, and the current SOC for each battery is as follows:
SOC 12nd
dch
¼1
I2 t2 e2 I1 t 3 ¼0 Q Q
ð13aÞ
SOC 22nd
dch
¼1
I2 t2 ðe2 e1 Þ I1 t3 I2 t2 e1 ¼ Q Q Q
ð13bÞ
SOC 32nd
dch
¼1
I1 t3 I2 t2 e2 ¼ Q Q
ð13cÞ
At the end of the second charge, the SOC for each battery is given by:
SOC 32nd
ch
¼
SOC 22nd
ch
¼
SOC 12nd
ch
¼
I2 t 2 e2 I2 t4 ðg0 þ e2 Þ ¼1 þ Q Q
ð14aÞ
I2 t 2 e1 I2 t4 ðg0 þ e1 Þ I2 t2 ðe2 e1 Þ I2 t4 ðe2 e1 Þ ¼1 þ Q Q Q Q ð14bÞ I 2 t 4 g0 I2 t2 e2 I2 t 4 e2 ¼1 Q Q Q
ð14cÞ
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Assuming the first charged capacity as I2 t2 ¼ DQ ch 1 , the second charged capacity I2 t4 ¼
DQ ch 2 ,
. . .and the nth charged capacity
I2 t2n ¼ DQ ch n . The SOC for each battery at the end of nth discharge can be described by:
SOC nth 1
SOC nth 2
dch
dch
dch SOC nth 3
¼0
¼
¼
ð15aÞ
ch ch DQ ch 1 þ DQ 2 þ þ DQ n1 e1
ð15bÞ
Q ch ch DQ ch 1 þ DQ 2 þ þ DQ n1 e2
ð15cÞ
Q
The SOC for each battery at the end of nth charge can be described by:
SOC nth 1
ch
¼1
ch ch DQ ch 1 þ DQ 2 þ þ DQ n Þe2 Q
ch SOC nth 2
¼1
SOC nth 3
¼1
ch
ð16aÞ
ch ch DQ ch 1 þ DQ 2 þ þ DQ n ðe2 e1 Þ
ð16bÞ
Q
ð16cÞ
Eqs. (15) and (16) manifest that the SOC inconsistency by battery columbic efficiency variations are amplified with increase of charge and discharge cycles. It is demonstrated that the columbic efficiency difference will make the SOC consistency of the battery group gradually diverge. Scenario IV: Assume the available discharged capacity Q, columbic efficiency g initial SOC of three batteries are the same, but their resistance R are different. Let the initial SOC as SOC 1 ¼ SOC 2 ¼ SOC 3 ¼ s. The discharge and charge current are I1 and I2 , respectively. During the usage of batteries, the terminal voltage is monitored to judge if the discharge and charge process need to be stopped. According to the Thevenin model of battery cell, the voltage drop of the internal resistance can’t be neglected especially when the current is large. The SOC is a function of the open circuit voltage (OCV), the SOC of a battery cell at the cutoff point will be affected by the internal resistance, and the distribution of the SOC at the cutoff point will be further affected by the change of the internal resistance as the number of cycles increase. To illustrate this problem, the SOC value at the end of charge is chosen to analyze the effect on its distribution. Since there exists no straight function of the SOC and the internal resistance, it has difficulty in finding an analytical solutions to study the effect of internal resistance on SOC variation evolutions. A numerical method is therefore employed to investigate the relationship between the internal resistance distribution and the SOC distribution. In the first case, the initial SOC at the end of charge is 100%. In the next cases, the initial SOC value at the end of charge is affected by the internal resistance. The distribution of the internal resistance for each case is shown in Table 1, the mean
and standard deviation of internal resistance increase from Case 1 to Case 10. Both the discharge and charge current are set as I1 ¼ I2 ¼ 30 A. In each case, the battery pack is first discharged to cutoff point and then fully charged. In this calculation, let the SOC value of each cell at the end of charge in the ith case be the SOC value of each cell at the start of discharge in the ði þ lÞth case. The SOC value of each cell at the end of charge is changed gradually due to the change of distribution of the internal resistance and the results are shown in Table 1 based on the SOC track method in Eq. (1). With the increase of the mean and the standard deviation of the internal resistance, the mean SOC at the end of charge is decreasing while the standard deviation and range of the SOC are increasing. However, the variation of the SOC is negligible compared to the mean of SOC. Based on the above discussions, if one assumes that the degradation rate of battery capacity and resistance are same, the degree of SOC inconsistency will be kept constant caused by the initial capacity, resistance and SOC variations. Nevertheless, the initial columbic efficiency varieties will result in the SOC inconsistency of the battery group diverging increasingly, showing prominent accumulative effect. It can be inferred that the columbic efficiency variations may contribute more to the battery SOC inconsistency in use compared to the capacity and the internal resistance variations.
3. Battery OCV consistency modeling As expressed in Section 2, compared to the capacity and the internal resistance, the SOC inconsistency has the most significant influence on battery pack energy utilization efficiency, which is usually the objective of battery equilibrium and maintenance. The SOC consistency of battery pack can be employed as evaluation index representing the battery consistency level. As is known, the SOC-OCV function is a representative for a particular battery, and is generally a nonlinear monotone function between SOC and OCV for all lithium-ion batteries. The OCV can be used for representing battery SOC, and is easily measured on board. The OCV consistency is hence ultimately chosen as the inconsistency evaluation parameter of the battery pack.
3.1. OCV consistency change under operating conditions The data used in this study are acquired from two dual-source trolley buses (named as Bus #1 and Bus #2) operated for 7 months. The bus is powered by both on-board battery system and grid electricity, which can operate in both charging mode and driving mode at the same time. The operational condition profile of the battery is close to that in a hybrid electric vehicle, and large charge current and discharge current happen and fluctuate severely sometimes. The battery used in the bus is the Nickel Manganese Cobalt Oxide (NMC) & Manganese Oxide cathode and graphite anode based battery. The nominal capacity of the battery pack is 105Ah, three cells with nominal capacity of 35 A h in parallel constitute a battery
Table 1 Distribution of the internal resistance for each case and its impact on SOC distribution. Case number
1
2
3
4
5
6
7
8
9
10
Mean R(X) Std of R Mean SOC (%) Std of SOC Range of SOC (%)
0.0033 0.00015 100.0 0 0
0.0038 0.0002 100.0 0 0
0.0043 0.00025 100.0 0 0
0.0048 0.0003 99.9 0.00158 0.0086
0.0053 0.00035 99.8 0.00645 0.03
0.0058 0.0004 99.45 0.0189 0.1
0.0063 0.00045 99.3 0.0224 0.12
0.0068 0.0005 98.9 0.0328 0.1773
0.0073 0.00055 98.5 0.0483 0.261
0.0078 0.0006 97.7 0.0713 0.386
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Bus #1
60.0
21.6
14.4
7.2
0.0 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20
Distribution of OCV(V)
Percentage of Distribution(%)
Percentage of Distribution(%)
28.8
Bus #2
40.0
20.0
0.0 3.85
3.90
3.95
4.00
4.05
4.10
4.15
4.20
Distribution of OCV(V)
Fig. 3. Histogram distributions of average OCV for Bus #1 and Bus #2.
module, and 104 battery modules are then serially connected composed of the battery pack. The bus is off grid when parked at night, and the battery pack are not be charged when standing by. During practical operation, the polarization voltage of battery pack in electric vehicles is eliminated after standing for a whole night. The battery management system (BMS) will record the OCV of each battery module before EV bus start in the following day. The OCV standard deviation of the battery pack, as the inconsistency coefficient on homeostasis parameter characteristics, excluding the impact of the current and the temperature. It is an important factor of the pack inconsistency, not only belong to reversible inconsistent, which can be used as an important reference for improving the battery balancing strategy [33]. Fig. 3 shows the histogram distribution of the average OCV for Bus #1 and Bus #2 before the start of the bus for approximately 160 days. Most OCV values are between 3.95 V and 4.15 V. The SOC-OCV curve of the lithium-ion battery is illustrated in Fig. 4. It can be inferred that before the start of the bus every day, the SOC is mainly distributed ranging from 60% to 95%, which is in a high SOC region. The OCV distribution of battery pack does not always maintain a normal distribution characteristic at each SOC point. Compared to other indexes such as the average value and the range of OCV, the standard deviation (SD) is more stable and reliable for quantifying the amount of variation of the OCV data. This is because the average value and the range of the OCV mainly rely on the SOC of the battery pack, and fluctuate severely as battery SOC varying. The SD is finally selected to represent the battery pack’s consistency, and to characterize the battery pack energy utilization efficiency.
The study is focused on the OCV dispersion and its distribution characteristics are not further discussed here. 3.2. Standard deviation modeling of OCV It has difficulty in getting equivalent full cycling number of the battery at practical application since the battery is seldom fully discharged or charged, and the current fluctuate severely under operating conditions. The absolute cumulative capacity for the battery charging and discharging is employed as an independent variable to model. The total cumulative capacity of two buses for the sampling period are both about 60,000 A h. After removing the bad points of the data, the standard deviation scatter plot of the OCV is shown in Fig. 5. It is found from Fig. 5 that the change of OCV standard deviation for both buses are similar, and both of them are slowing down in the later period. The standard deviation reach around 7–7.5 mV at cumulative capacity of 60,000 A h, which has obvious regularity. Since one month operation data for both two buses are lost, the average of the cumulative capacity of its adjacent two months is regarded as the cumulative capacity for the month of data loss. It can be seen from Fig. 5 that the OCV standard deviation are gradually changing and always keep growing or remain stable in the future, but will not decrease. Both plots manifest a certain behavior that it is easy to use polynomial relationship as the fitting model. The polynomial function is applied for fitting relationship between the OCV standard deviation and the cumulative capacity. The model is given by:
rðQ Þ ¼ k1 þ k2 Q k3
rðQ Þ is the standard deviation of OCV, Q is the cumulative capacity, k1, k2, k3 are the coefficients to be extracted. The model uses the cumulative capacity as the input, which has large numerical value, and the basic functions of the mathematical model is a power function. Compared to the linear function model, the nonlinear fitting requires more input data. In order to make the model more predictable and improve the adaptability of the model, Eq. (17) can be rearranged by:
4.2 4.0
OCV (V)
ð17Þ
3.8 3.6 3.4
rQ ¼ f ðQ lg Þ ¼ k1 þ k2 k3 lg Q
3.2
ð18Þ
After taking denary logarithm of the cumulative capacity as the independent variable, and it is simplified as:
3.0 0
20
40
60
80
SOC (%) Fig. 4. SOC-OCV curve of a lithium-ion battery.
100
rQ ¼ f ðQ lg Þ ¼ k4 þ k5 lg Q
ð19Þ
where Q is the cumulative capacity, k4, k5 are the model coefficients.
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Standard Deviation of OCV (mV)
Standard Deviation of OCV (mV)
8.0 8
7
6
5
Bus #1
4
3
0
12000
40000
60000
80000
7.5 7.0 6.5 6.0 5.5 5.0 4.5
Bus #2 0
17000
45000
60000
Cumulative Capacity (Ah)
Cumulative Capacity (Ah)
Fig. 5. Scatter plot of OCV standard deviation for Bus #1 and Bust #2.
3.3. Parameters extraction and model validation Levenberg-Marquardt (L-M) algorithm is one of the most effective methods for nonlinear model parameter fitting. L-M algorithm
Initialization k1,k2,k3,J,H,e,e_lm,λ
Build the initial simplex
Lamda= /10; e=e_lm
Calculate J,H
No Calculate k1,k2,k3,e_lm
e_lm
Yes Cycle index
No Output Fig. 6. Procedure of the L-M algorithm.
has the advantage of fast convergence similar to the Gauss-Newton steepest descent method, and has good adaptability like neural network at the same time. The procedure of the L-M algorithm for extracting parameters is shown in Fig. 6. Where J is the Jacobian matrix of model functions, H = J0 ⁄ J, k1 , k2 , k3 are the fitting parameters of the model, e is the initial calculation error, elm is the calculation error after the parameter is updated, k is the damping factor, n is the number of the iterations. According to this method, coefficients can be identified effectively. Eqs. (17) and (19) are identified using LM algorithm. The input variables are the first 80% of the standard deviation of OCV and the cumulative capacity. After the fitted parameters are obtained, the latter 20% of the data is predicted. The goodness of fit is calculated by R-squared coefficient of determination, a statistical measure of how close the data are to the fitted regression line. The prediction accuracy is represented by the maximum relative error. Table 2 shows the fitted parameters of the model. The values of the parameters of two buses are not exactly same, indicating that even if the battery type are same, the variation of parameters still exist due to the different operating conditions. The fitted results using two models are illustrated in Figs. 7 and 8, in which the abscissae of Figs. 7 and 8 are cumulative capacity and cumulative capacity taking denary logarithm, respectively. The results of the goodness of fit are shown in Table 3, and the maximum predictive relative error is reported in Table 4. Both models can fit the training data and predict the validation data accurately. The estimation error fluctuation of Bus #1 is larger than that of Bus #2 due to its greater volatility. From Fig. 7 and Table 4, it can be seen that the fitted and predicted results using L-M algorithm manifest better accuracy compared to the least square (LS) method. This is because the iterative algorithm in L-M algorithm cause the fitting model more adaptable for predicting. This is a great advantage under the practical operating condition discussed in the study. After taking denary logarithm of the accumulated power, the standard deviation of OCV keeps a good linear change. The result further confirms that the two variables have a correlation of polynomial functions. The goodness of fit and the prediction precision
Table 2 Fitted parameters of the two models.
rQ ¼ f ðQ lg Þ ¼ k4 þ k5 lgQ
rðQ Þ ¼ k1 þ k2 Q k3 Coefficient
k1
k2
k3
k4
k5
Bus #1 Bus #2
0.0020 0.0034
0.00016 0.00016
0.32 0.29
0.0046 0.0014
0.0024 0.0018
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8.0 7.5
8.0
Standard Deviation of OCV (mV)
Standard Deviation of OCV (mV)
8.5
Bus #1
7.0 6.5 6.0 5.5 5.0
Validating data Training data LM fitted LS fitted
4.5 4.0 3.5 3.0 2.5
0 12000
40000
60000
80000
Bus #2
7.5 7.0 6.5 6.0
Validating data Training data LM fitted LS fitted
5.5 5.0 4.5
0
17000
45000
60000
75000
Cumulative Capacity (Ah)
Cumulative Capacity (Ah)
Fig. 7. Fitted results using the polynomial function.
0%~80% SOC 80%~100% SOC Fitted Values
0.008
0.006
Standard Deviation of OCV (V)
Standard Deviation of OCV (V)
0.008
0.004
0.002
0.000 3.0
3.2
3.4
3.6
3.8
4.0
4.50
4.75
5.00
Cumulative Capacity
0.006
0.004
0%~80% SOC 80%~100% SOC Fitted Values
0.002
0.000
3.15
3.50
3.9
4.2
4.5
4.8
Cumulative Capacity (Ah)
Fig. 8. Fitted results after taking denary logarithm of the cumulative capacity.
Table 3 Goodness of fit of the model for the first 80% of data. Model
rðQ Þ ¼ k1 þ k2 Q k3
Method
LM
LS
Bus #1 Bus #2
0.894 0.969
0.899 0.938
rQ ¼ f ðQ lg Þ ¼ k4 þ k5 Q lg 0.874 0.955
Table 4 Maximum relative error of the predictive result for the latter 20% of data. Model
rðQ Þ ¼ k1 þ k2 Q k3
Method
LM
LS
Bus #1 Bus #2
9.58% 1.32%
11.41% 5.14%
rQ ¼ f ðQ lg Þ ¼ k4 þ k5 Q lg 7.02% 3.14%
SOC consistency for estimating the energy utilization efficiency of the battery pack. The detailed calculation method of energy utilization efficiency is introduced in Ref. [32]. The inputs of the calculation in Ref. [32] are the capacity, internal resistance and SOC distributions of batteries in the pack. In this study, the data of the capacity and internal resistance distribution are from the data mentioned in Section 2.1 and the SOC distribution is calculated through the OCV distribution. To calculate the SOC distribution based on the OCV distribution, a series of voltage data is firstly randomly generated which has the same OCV distribution of the battery pack when the battery pack is fully charged. Then a SOC-OCV model in Ref. [34] is used to calculate the SOC distribution. To determine the parameters in the model, we firstly calculate the SOC-OCV curves of all batteries and then average them as the SOC-OCV curve for fitting the curve in the model. The proposed SOC-OCV model is shown the following Equation: 0:478
V ocv ¼ 2:549 0:4851 ð ln sÞ of linear model do not decrease. The linear model can effectively improve the adaptability and has a wider application scenario compared to the non-linear model.
4. Application of battery equilibrium diagnosis The energy utilization efficiency is used to describe the performance of battery pack and is also a judgment base for equilibrium in this paper. The predicted results using the OCV consistency model described in Section 3 need to be transformed into the
1:221 s þ 2:206 e0:4ðs1Þ ð20Þ
where V ocv and s represent the OCV and SOC of the battery, respectively, and 0 < s 6 1. Eq. (20) is a transcendental equation and its inverse function cannot be accessible. Therefore, it is impossible to calculate the exact SOC value of a given OCV by using the inverse function of Eq. (20). To solve this problem, the dichotomy method is employed. The basic idea of the dichotomy method is to produce a series of convergent data {xk } to approach the true value x of a specific equation. Since the proposed SOC-OCV model is
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30 Mean 86.12 StDev 9.13 N 95
Relative frequency%
25 20 15 10 5 0
60
65
70
75
80
85
90
95
100
SOC% Fig. 9. Simulated results of the SOC distribution.
Table 5 Simulated results of the battery pack energy utilization efficiency. Current
10 A
20 A
30 A
Simulated True Estimate error
65.81% 65.03% 0.78%
65.28% 64.38% 0.9%
64.75% 63.73% 1.02%
monotonous increasing and there is only one SOC value s if given a specific OCV value V ocv , the process of producing the series of convergent data {xk } can be concluded as follow: Step 1: Denote the function as Eq. (21) 0:478
f ðsÞ ¼ 2:549 0:4851 ð ln sÞ
1:221 s þ 2:206 e0:4ðs1Þ V ocv
9
distribution properties of the capacity, internal resistance and the SOC of a battery pack. It concludes that the SOC variation contributions the most to battery consistency from the perspective of energy utilization efficiency, and hence is employed as evaluation index characterizing battery consistency level. The evolutions of the SOC variation are deduced as the battery usage to investigate the impact of various parameters divergence on battery pack inconsistency. It is indicated the initial columbic efficiency variety will result in the SOC inconsistency diverging increasingly, showing evident accumulative effect. Two OCV standard deviation models for describing battery consistency are proposed based on the electric buses operating data, and L-M algorithm is utilized to extract the model parameters. The L-M algorithm possesses predominant estimating accuracy for both the fitted and predicted results compared to the least square method. It is noteworthy that the maximum relative errors for Bus #1 and Bus #2 exhibit obvious different due to greater data fluctuation of Bus #2. The stability and reliability of the data have significant influence on the model, and is worthy to be further discussed. To estimate the energy utilization efficiency of the battery pack, the mapping relation between the OCV distribution and the SOC distribution of the battery pack by using the dichotomy method is developed. The developed method that estimate the energy utilization efficiency is demonstrated with a high accuracy, and provide a foundation for the battery equilibrium diagnosis. In the future work, the impact of temperature and columbic efficiency on the SOC consistency and degradation consistency will be further investigated.
Acknowledgement
ð21Þ
and denote the SOC value of a given OCV as s . Step 2: Set the range which includes s as I0 ¼ ½a; b; set the midpoint of I0 as s0 ¼ 0:5ða þ bÞ; Calculate f ðs0 Þ. Since 0 < s 6 1, in this situation, a ¼ 0 and b ¼ 1; Step 3: Estimate the value of f ðs0 Þ, If f ðs0 Þ = 0, thus x ¼ x0 and stop calculation; If f ðs0 Þf ðaÞ < 0, thus s 2 ½a; s0 and let I1 ¼ ½a; s0 ; If f ðs0 Þf ðaÞ > 0, thus, s 2 ½s0 ; b and let I1 ¼ ½s0 ; b; Step 4: Define I1 ¼ ½a1 ; b1 and set the midpoint of I1 as s0 ¼ 0:5ða1 þ b1 Þ;If js1 s0 j < e, s s1 and stop calculation, where e represents the required SOC estimation accuracy. In this situation, e ¼ 0:001. Otherwise, replace I0 with I1 and go back to Step 2. Fig. 9 displays the simulated results of the SOC distribution based on the OCV distribution. The mean and standard deviation of the SOC data from simulation are 91.76% and 8.16% respectively. Compared with the real SOC distributions in Section 2.1, the error of the mean value is within 6% and the error of standard deviation is within 1%. Based on the battery capacity, internal resistance and OCV distributions, the battery pack energy utilization efficiency is calculated and the result is shown in Table 5. For the given three current rates, the estimation error is within 1.5%, demonstrating a satisfactory agreement with the true values. If the threshold of the energy utilization efficiency for balancing is set in advanced, such estimation result can be applied for equilibrium diagnosis. 5. Conclusion The article systematically analyzes the influence of parameters variation on battery pack consistency based on the statistical
The authors are grateful for financial support via the National Key Research and Development Program of China (2016YFB0101800), Science and Technology Program of SGCC (Operation Safety and Interconnection Technology for Electric Infrastructure).
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Please cite this article in press as: Zhang C et al. Study on battery pack consistency evolutions and equilibrium diagnosis for serial- connected lithium-ion batteries. Appl Energy (2017), http://dx.doi.org/10.1016/j.apenergy.2017.05.176