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Identifying main factors of capacity fading in lithium ion cells using orthogonal design of experiments
Applied Energy 163 (2016) 201–210
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Identifying main factors of capacity fading in lithium ion cells using orthogonal design of experiments Laisuo Su a, Jianbo Zhang a,b, Caijuan Wang c, Yakun Zhang a, Zhe Li a,b,⇑, Yang Song c, Ting Jin c, Zhao Ma d a
Department of Automotive Engineering, State Key Laboratory of Automotive Safety and Energy, Tsinghua University, 100084 Beijing, China Beijing Co-innovation Center for Electric Vehicles, Beijing Institute of Technology, 100081 Beijing, China c General Administration of Quality Supervision, Inspection and Quarantine of China, 215000 Suzhou, China d Dongfeng Motor Corporation Technical Center, 430000 Wuhai, China b
h i g h l i g h t s The effect of seven principal factors on the aging behavior of lithium ion cells is studied. Orthogonal design of experiments is used to reduce the experiment units. Capacity fades linearly during the initial 10% capacity fading period. Statistical methods are used to compare the significance of each principal factor. A multi-factor statistical model is developed to predict the aging rate of cells.
a r t i c l e
i n f o
Article history: Received 4 August 2015 Received in revised form 1 November 2015 Accepted 4 November 2015
Keywords: Lithium-ion cells Aging rate Orthogonal design of experiments Main aging factors
a b s t r a c t The aging rate under cycling conditions for lithium-ion cells is affected by many factors. Seven principal factors are systematically examined using orthogonal design of experiments, and statistical analysis was used to identify the order of principal factors in terms of strength in causing capacity fade. These seven principal factors are: the charge and discharge currents (i1, i2) during the constant current regime, the charge and discharge cut-off voltages (V1, V2) and the corresponding durations (t1, t2) during the constant voltage regime, and the ambient temperature (T). An orthogonal array with 18 test units was selected for the experiments. The test results show that (1) during the initial 10% capacity fading period, the capacity faded linearly with Wh-throughput for all the test conditions; (2) after the initial period, certain cycling conditions exacerbated aging rates, while the others remain the same. The statistical results show that: (1) except for t1, the other six principal factors significantly affect the aging rate; (2) the strength of the principal factors was ranked as: i1 > V1 > T > t2 > V2 > i2 > t1. Finally, a multi-factor statistical aging model is developed to predict the aging rate, and the accuracy of the model is validated. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Lithium-ion batteries have many merits, including high energy density, high coulombic efficiency and comparatively low heat output, which make them highly suitable for electric vehicles (EVs) and electrical energy storage (EES) applications [1]. Although battery manufacturers are making every effort to reduce the cost, including use of cheaper components and materials and innovation in manufacturing processes, the high cost of lithium-ion batteries
⇑ Corresponding author at: Department of Automotive Engineering, State Key Laboratory of Automotive Safety and Energy, Tsinghua University, 100084 Beijing, China. E-mail address: [email protected] (Z. Li). http://dx.doi.org/10.1016/j.apenergy.2015.11.014 0306-2619/Ó 2015 Elsevier Ltd. All rights reserved.
has inhibited widespread use of such batteries in these applications. However, reducing life cycle costs is a way to mitigate the high cost of batteries, so extending the lifespan of the batteries is an effective way to achieve that goal. The lifespan of batteries depends on operating conditions [2,3], such as charging/discharging rates and ambient temperature. Understanding the relationship between the operating conditions and aging behavior of the batteries is an important task to increase the lifespan. However, it is not easy to quantify the effect of each condition on aging behavior, as the aging of batteries involves multiple closely interrelated factors. To solve this problem, three steps are usually practiced [4]. The first step is a screening study to discover factors that affect the battery’s lifespan. The next step is an empirical study to produce an empirical model to describe the
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way the battery lifespan changes with the factors. The last step is a mechanistic study to understand the mechanism and to develop a mechanistic model to describe the causality between lifespan and factors. Following these steps in sequence helps develop a wellestablished mechanistic model efficiently. To date, empirical aging models (the second step) [5–16] and mechanistic aging models (the third step) [17–23] have been studied, while little attention has been paid to the study of the strength of the factors that affect the aging behavior (the first step). The relative strength of these factors needs to be investigated first before focusing on any model development. Otherwise, the validity of the model may be uncertain. To be specific, there are at least seven stress factors that affect the aging behavior of batteries in cycling conditions: the charge and discharge current (i1, i2) during the constant current (CC) regime, the charge and discharge cut-off voltage (V1, V2) and the corresponding durations (t1, t2) in the constant voltage (CV) regime, and the temperature (T) at which battery ages. Wright et al. [7] chose the temperature, initial state of charge (SOC), change in SOC, as the three factors in their battery model to study cycle aging effects. Wang et al. [5] chose the temperature, depth of discharge (DOD), and discharge rate, as the three factors in a similar study. Both studies did not give reasons for their choice of the aging factors. To the best of our knowledge, only two screening studies compared the importance of different factors that affect the aging behavior of batteries. The first one by Prochazka et al. [24] studied the influence of current, temperature, SOC and the change of SOC on the aging of two types of cells using a second-order model and D-optimization method. Their result showed that these factors significantly affect the cells aging behavior. The second one by Chi et al. [25] studied the effect of temperature, DOD, and discharge rate on the aging behavior of cells using Taguchi method. The result showed that the strength of these stress factors on capacity fade follows the order of temperature > discharge rate > DOD. However, both papers arbitrarily constrained the stress factors to three or four without justification. Therefore, more careful screening study (the first step) is needed to identify the main stress factors for cycle aging to lay a foundation for subsequent empirical model studies. This study attempts to quantify the strength of the stress factors and the relationship between aging behavior and operating condition. Since, the real world operating conditions are very complex and vary from cycle to cycle, a simplified protocol using typical CC–CV charge and discharge regimes along with the observations of the effects from the seven factors was used in this study to allow better quantification. This protocol should cover most of the actual cycle aging situations. The rest of this paper is organized as following: in Section 2, the methodology of orthogonal design with orthogonal analysis (OA) L18 is discussed, and the test method is introduced; in Section 3, the test results of cell initial characteristics and cell aging are displayed, and an empirical aging model is developed; in Section 4, we discuss the results in a deeper manner by analyzing the capacity fade of the aging process with statistical methods, including the main effect analysis and analysis of variance (ANOVA), and proposing a multi-factor statistical aging model; in Section 5, we summarize the conclusions that would give guidance for choosing reasonable factors in aging models.
2. Experimental 2.1. Methodology The design of experiments (DOE) is a tool to obtain experimental information using statistical methods. The principles
of DOE include randomization, replication, blocking, and orthogonality. Based on those principles, many methods have been proposed to optimize the DOE for different purposes. For example, factorial experimental design has been proposed to obtain extensive information from only a few sets of experiments compared with the one factor at a time method (OFAT). As a branch of factorial experimental design, the orthogonal design [26] has been extensively used in various fields due to its efficiency. In the orthogonal design, the effect of many factors (and possibly some of their interactions) can be studied simultaneously in a single set of experiments with much fewer experiment units. To apply this method, an orthogonal analysis (OA) needs to be chosen in the first place. In this study, the OA of L18 was chosen because it could be used to study eight factors, among which one factor with two levels and seven with three levels. Moreover, the main effects among these factors can be derived without considering the influence of their interactions since the interactions are uniformly distributed in each column. Table 1 shows nominal specifications of the 18,650 cells made by Samsung. Seven factors were studied for the aging behavior of cells, and three different levels were assigned for each factor. The three levels together should cover most of the actual test conditions based on Table 1. Table 2 shows the three levels of different factors that were chosen for the test. Those levels were distributed in the OA of L18, as shown in Table A.1 in Appendix A. Since only seven factors were considered in this experiment, one column, named as ‘‘blank” in Table A.1, was not allocated to any factor.
2.2. Tests The cells were cycled three times at C/2 to get stable performance before test. Then, the initial performance of 36 cells was tested, including capacity and resistance. Statistics showed that there were no outliers among those cells. The 36 cells were then divided into 18 groups and allocated to the rows in Table A.1 and tested using the levels in Table 2. Cells were taken out to characterize the state of health (SOH) with reference performance tests (RPT) every 50 cycles. The terminating criterion for the cycling of each cell was that the cell capacity degraded to 50% of its initial capacity or the cycle number reached 1000 times. However, parts of the tests that cycled at low rates were forced to stop before they reached the terminating criteria when most of the tests were finished. The overall tests are shown in Fig. 4. All the tests were conducted using a MACCOR Series 4000 and three environmental chambers GDW/JB-0100 (Naya, China). The RPT test was designed to obtain the discharge capacity at C/2, C/50 and the resistance at nine different SOC points. Cells were cycled twice at C/2 followed by a customized test profile, as shown in Fig. 1(a). The capacity at C/2 was chosen as the discharge capacity in the second cycle at C/2. The capacity at C/50 was obtained from the customized profile, instead of the normal CC discharge profile, which would take about 50 h. The customized profile
Table 1 Nominal specifications of the test cell. Item
Specification
Cathode material Anode material Nominal capacity Max. charge current Charge cut-off voltage Max. discharge current Discharge cut-off voltage Operating temperature
LiNixCoyAlxO2 Graphite 2.85 A h (1/3 C) 1C 4.20 ± 0.05 V 2C 2.50 V Charge: 0–45 °C Discharge: 20 to 60 °C
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3.2. Cell aging characteristics
Table 2 Cycling factors and their levels. No.
Factors
Abbreviation
Level 1
Level 2
Level 3
1 2
Ambient temperature (°C) Charge current during CC process (C) Cut-off voltage of the charge process (V) CV time of the charge process (min) Discharge current during CC process (C) Cut-off voltage of the discharge process (V) CV time of the discharge process (min)
T i1
0 0.2
25 0.5
50 1
V1
4.1
4.2
4.3
t1
0
15
30
i2
0.5
1
2
V2
2.4
2.5
3
t2
0
15
30
3 4 5 6 7
included a CV process at the end of discharge, and the cut-off current was 0.06 A (C/50). The capacity at the low discharge rate of C/50 reflects the thermodynamic characteristic of a cell and was called T-Capacity in this study. The calculation of resistance at different SOC points was illustrated in Fig. 1(b) and could be obtained by formula (1).
Rsoc ¼
V V0 I
ð1Þ
where V is the voltage before discharging at a specific SOC, V0 is the voltage after 60 s of discharging from the specific SOC, and I is the discharge current. Sixty seconds were used here to include the diffusion effect when calculating the resistance. 3. Results 3.1. Cell initial characteristics
Fig. 4 shows the T-Capacity of cells varies with cycle number under different test conditions. In most cases, the T-Capacity decreased linearly with cycle number in aging. L14 and L15 are exceptions; yet display such behavior during the initial 10% capacity fading. Thus, the slope of the linear part of the curves could represent the aging rates at different test conditions. As an energy storage and conversion device, users should concern more with the energy they can extract from the cells through the lifespan rather than the cycle number. Therefore, Whthroughput is a more relevant variable to compare with than cycle number for the aging rate under different conditions. Fig. 5 shows that the Wh-throughputs for the first 50 cycles under the 18 test conditions are different from one another. Fig. 6 displays the retention of T-Capacity as a function of Whthroughput for the 36 cells under 18 test conditions. The TCapacity decreased linearly with the Wh-throughput during the initial 10% capacity fading for all the cells. The slope of this linear part was utilized to represent aging rate at different test conditions. One of the preconditions for the accelerated life test of cells is that the aging mechanism does not change. The relationship between resistance and capacity (R-Cap plot) is a method to differentiate aging mechanism under different test conditions [27]. Since the values of resistance vary little from SOC = 20% to SOC = 90% for the tested cells, as shown in Fig. 7(a), the resistance at SOC = 50% was chosen for further analysis. Fig. 7(b) shows the aging behavior of the 18 groups of tests could be divided into two categories according to the trajectory of resistance vs T-Capacity: aging type I and aging type II. The relationship between T-Capacity and resistance in the initial 10% capacity fading is similar for all the 18 test groups. We infer that the aging mechanism was probably the same. 3.3. Empirical aging model
Fig. 2 displays the distribution of the initial T-Capacity of cells determined from the customized test profile. The T-Capacity is 3.038 ± 0.0051 A h (±0.17%). The curve in the histogram represents the T-Capacity distribution. The probability plot compares the distribution of the T-Capacity of cells with a set of standard normally distributed data. It shows that the T-Capacity is normally distributed with 95% confidence intervals. Fig. 3 shows the distribution of the initial resistance of the cells at SOC = 50%. The distributions of resistance at other SOCs are similar. The resistance at SOC = 50% is 0.058 ± 0.001 X (±1.14%). The probability plot shows the resistance of the tested cells is normally distributed with 95% confidence intervals. The small variances and the normal distribution of both the T-Capacity and the resistance at SOC = 50% signify the high initial quality of the tested cells.
Different aging models have been proposed to describe the aging behavior of lithium ion cells [5–7]. Those models generally use a power law to describe such behavior, as shown below:
y ¼ K xN
ð2Þ
K ¼ kðT; i1 ; i2 ; V 1 ; V 2 ; . . .Þ
ð3Þ
N ¼ nðT; i1 ; i2 ; V 1 ; V 2 ; . . .Þ
ð4Þ
where y is a parameter that can be used to characterize the outcome in the performance of cells, such as capacity, K a coefficient representing the intensity of the aging factors that will influence the outcome y, x an independent variable representing the extension of the
4.10
(a)
V
(b)
60 s
Voltage/V
4.06
4.02
V’ 3.98
3.94
0
30
60
90
120
150
Time/s Fig. 1. (a) Customized test profile for RPT, where the discharge rate is C/2. In each discharge interval, 10% of the nominal capacity was removed. The rest between two discharge intervals is 10 min. The cut-off current in the final CV stage is C/50. (b) The voltage evolution of a cell during a C/2 discharge interval.
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Fig. 2. (a) T-Capacity distribution and (b) normal probability plot of the initial T-Capacity distribution.
3.0
3.0
2.5
2.5
2.5
2.0
2.0
2.0
1.5
L1 L2 L3 L4 L5 L6
1.0 0.5 0.0
0
200
400
600
800 1000
1.5
L7 L8 L9 L10 L11 L12
1.0 0.5 0.0
0
200
Cycle Number
400
600
800 1000
Capacity/Ah
3.0
Capacity/Ah
Capacity/Ah
Fig. 3. (a) Resistance at 50% SOC distribution and (b) normal probability plot of the initial resistance distribution.
1.5
L13 L14 L15 L16 L17 L18
1.0 0.5 0.0 0
200
400
600
800 1000
Cycle Number
Cycle Number
Fig. 4. T-Capacity variations with cycle number for the 18 test conditions.
1200
Cell 1 Cell 2
Wh-throughput/Wh
1000 800
Whi ¼
600
200
L1
L2
L3
L4
L5
L6
L7
L8
L9 L10 L11 L12 L13 L14 L15 L16 L17 L18
Experiment Number Fig. 5. The Wh-throughput for the first 50 cycles for the 18 test conditions.
DF i ¼ aging, such as cycle number, holding time, or Wh-throughput, N is the exponent of x. K and N are functions of aging factors in the cycling process, as shown in formulas (3) and (4). N is constant in most cases and it equals one in this study.
Q loss i ¼ K i Whi
1 Q ¼ Gi Q loss K i loss
i
ð6Þ
As an energy storage and conversion device, users should concern more with the energy they can extract from the cells through the lifespan. Thus, Eq. (5) was transformed into Eq. (6), where Gi is the reciprocal of Ki. The equation expresses the amount of Whthroughput that a cell can transfer before it loses a certain amount of capacity. To visually compare aging rates at different test conditions, a dimensionless index, deceleration factor (DF), was introduced to, as shown in formula (7).
400
0
The capacity of cells fades linearly with Wh-throughput during the initial 10% capacity fading, as shown in Fig. 6. Thus, Eq. (5) describes the capacity fade behavior during aging. where Ki is the slope of the linear part of the curve in Fig. 6 at condition i.
ð5Þ
Gi G0
ð7Þ
where DFi is the decelerated factor of the condition i, Gi is the G value of the test condition i, G0 is the G value of a reference test condition. Fig. 8 shows the DF values of the 18 test conditions. Since this cell aged slowest among the 36 cells, the aging of cell 1 in the cycling condition L1 was chosen as the reference test condition to make the DF values between zero and one.
205
3.0
3.0
2.5
2.5
2.5
2.0 1.5
L1 L2 L3 L4 L5 L6
1.0 0.5 0.0
2.0 1.5
L7 L8 L9 L10 L11 L12
1.0 0.5 0.0
0.3 0.6 0.9 1.2 1.5 1.8
0
Capacity/Ah
3.0
Capacity/Ah
Capacity/Ah
L. Su et al. / Applied Energy 163 (2016) 201–210
0
Wh-throughput/Wh x 10
1.5
L13 L14 L15 L16 L17 L18
1.0 0.5 0.0
0.3 0.6 0.9 1.2 1.5 1.8
Wh-throughput/Wh x 104
2.0
0
0.3 0.6 0.9 1.2 1.5 1.8
Wh-throughput/Wh x 104
4
Fig. 6. The variation of T-Capacity with Wh-throughput for the 18 test conditions.
0.7
0.6
(a)
0.5 0.4 0.3 0.2 0.1 0
(b)
0.5
Resistance/Ohm
Resistance/Ohm
0.6
Aging mode I Aging mode II
0.4 0.3
Inial Aging
0.2 0.1
0
0.2
0.4
0.6
0.8
1
1
1.5
SOC
2
2.5
3
Capacity/Ah
Fig. 7. (a) The variation of resistance with respect to SOC for a fresh cell, and (b) increase of resistance at SOC = 50% with the T-Capacity for the 18 test conditions.
Cell 1 Cell 2
1
DF
0.8
(1) Response: The DF is chosen as the response of tests under different conditions. It can be derived from formula (7). (2) Effects at different levels: The effect of temperature at level 1 (0 °C) is the average of the DF values in those tests whose temperature is 0 °C, as shown in formula (10). The DFik represents the DF value of cell k at test condition i because there were two cells in each test condition. Thus m equals 2. The effects at other levels can be calculated in the same way. The calculated results of temperature at level 1, 2 and 3 are 0.196, 0.413, and 0.265 respectively.
0.6 0.4 0.2 0
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L18
Experiment Number Fig. 8. The DF values for the 18 test conditions.
ETð0 CÞ ¼
4.1. Main effect analysis for capacity fade Main effect analysis compares the strength of different factors to the capacity fade of cells. If the effect of interaction among factors and the effect of errors can by ignored, the effect of the factor A at the level i on the response Y is computed from Eq. (8), where Y AðiÞ is the average of all the response when the factor A is at the level i, regardless the values of other factors.
ð8Þ
According to the principle knowledge from DOE, the main effect of a factor is defined as the range of the factor’s effects at different levels, as shown in formula (9).
F A ¼ MaxðEAðiÞ Þ MinðEAðiÞ Þ
1 1 m n
X
! DF ik
T¼0 C
3 12 X X 1 1 ¼ DF ik þ DF ik 2 6 i¼1 i¼10
4. Discussion
EAðiÞ ¼ Y AðiÞ
The main effect of temperature is chosen as an example to show the calculation process in more detail.
ð9Þ
! ð10Þ
(3) Main effect: The effect of temperature reaches its maximum at the level 2 and reaches its minimum at the level 1. Therefore, the main effect of the temperature is the difference between the DF values at level 1 and level 2, which is 0.217 as shown in the formula (11).
F T ¼ MaxðETðiÞ Þ MinðETðiÞ Þ ¼ ETð25 CÞ ETð0 CÞ ¼ 0:217 ð11Þ Fig. 9 shows the effect of the seven aging factors at different level. The larger the value, the smaller the aging rate, according to the definition of the DF and the main effect in formula (7) and (10). Those results can be compared with others’ studies to verify the orthogonal design method [24,25,28,29].
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Table 3 shows the main effect of the seven factors and the ‘‘blank” list. The calculated result of each column is composed by three parts: (i) main effect of the studied factor in the column, (ii) part of the interaction effects among the seven factors, and (iii) errors during test. In the blank column, the first part equals zero since no factor is allocated to it. The interaction effects among the seven factors are equally distributed in each column of the L18, and the errors could be considered as the same in each column. Therefore, a factor affects the aging rate of cells only when the main effect of the corresponding column is greater than the main effect of the blank column. According to this principle, except for the factor t1, the other six factors have an effect on the aging rate of cells. In addition, according to the definition of the main effect, the greater is the FA, the larger is the influence of the factor A on the response. Therefore, the effect of the studied factors on the aging rate is ranked: i1 > V1 > t2 > T > V2 > i2 > t1. 4.1.1. Ambient temperature The ambient temperature ranges from 0 °C to 50 °C. Fig. 9 shows that the aging rate is smallest at its middle level (25 °C). The effect of temperature on the aging rate of cells follows Arrhenius’s law [7], which indicates that the aging rate is bigger at a higher temperature. However, lithium plating [30] might appear on the anode at very low temperature during cycling, which would accelerate the aging process. Based on the above analysis, the temperature behavior could be divided into two major domains: A low temperature domain described by a negative activation energy EA,p < 0 and a high temperature domain described by EA,s > 0 [27], as shown in formula (12). Therefore, the aging process of cells would be accelerated when temperature is higher or lower than 25 °C.
r ¼ Ap expfEA;p =ðkB TÞg þ As expfEA;s =ðkB TÞg
ð12Þ
4.1.2. Charge current The charge current ranges from C/5 to 1 C. Fig. 9 shows that the bigger the current, the faster the aging process. Zhang [31] thought that high charging current can cause huge local imbalance between the cell reaction and ionic diffusion on the electrolyte–electrode
Table 3 The main effect of the seven factors. Factors
Blank
T
i1
V1
t1
i2
V2
t2
E1 E2 E3 F Rank
0.361 0.222
0.196 0.413 0.265 0.218 ④
0.484 0.243 0.147 0.337 ①
0.477 0.243 0.154 0.324 ②
0.302 0.290 0.282 0.020 ⑧
0.353 0.324 0.197 0.155 ⑥
0.361 0.172 0.341 0.189 ⑤
0.397 0.177 0.300 0.221 ③
0.138 ⑦
interface and, therefore, cause high polarization. This would result in metallic lithium plating on the anode or overcharge on the cathode, which depends on ratio of the reversible capacity of the cathode to the anode. Therefore, the aging rate relate positively to the current. 4.1.3. Cut-off voltage of the charge stage The cut-off voltage of the charge stage ranges from 4.1 V to 4.3 V. Fig. 9 shows that the higher the voltage, the faster the aging process. The result is in agreement with the conclusions in others’ studies [28,32]. The mechanism to this phenomenon is that high voltage leads to high potential in positive electrode and low potential in negative electrode. The electrolyte is easier to be oxidized by a higher potential in the positive electrode area, and it is easier to be reduced by a lower potential in the negative electrode area. Thus, the aging rate relate positively to the cut-off voltage of the charge stage. 4.1.4. Holding time during the CV stage of the charge stage The holding time during the CV stage ranges from 0 min to 30 min. Fig. 9 shows the holding time has little effect on the aging rate. This result agrees with Fig. 2 of Choi’s study [28] that the holding time during the CV stage at 4.2 V has a relatively small effect on aging rate. However, it contradicts to our intuition that long period at high voltage should speed up the aging process because cells age fast at high voltage. The explanation for this phenomenon could be that the current at the CV stage decreases with time, alleviating the local imbalance between the cell reaction and ionic diffusion, as well as the polarization in both electrodes. Thus,
Fig. 9. The effects of the seven factors at different levels.
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few additional side reactions occur at this period of time, and the aging rate changes little with the holding time at the CV stage. 4.1.5. Discharge current The discharge current ranges from C/2 to 2 C. Fig. 9 shows the aging rate is accelerated by the discharge current. This result agrees with Fig. 4 of Choi’s study [28]. The mechanism to this result is similar with the effect of the charge current to the aging process. However, it should be noted that the discharge current has less effect on aging rate compared with the charge current. 4.1.6. Cut-off voltage of the discharge stage The cut-off voltage of the discharge stage (V2) ranges from 3.0 V to 2.4 V. Fig. 9 shows that the capacity fades fastest when the voltage is at its middle level (2.5 V). This result differs from Fig. 5 of Choi’s study [28] that the DOD has little effect on aging within 500 cycles. The reason for the phenomenon that cells aged faster at V2 = 2.5 V than V2 = 2.4 V is not fully understood. The results need to be reproduced using OFAT method, and will be further explored in our future work.
The total variance is the sum of squares of all the test data, as calculated in formula (13).
SStotal ¼
X
ðDF il DF il Þ
2
ði ¼ 1; 2; . . . ; 18;
l ¼ 1; 2Þ
ð13Þ
ANOVA partitions the total variance into its appropriate components. In this study, the total variance was divided into terms caused by different factors, as shown in formula (14).
SStotal ¼ SST þ SSi1 þ SSV 1 þ SSt1 þ SSi2 þ SSV 2 þ SSt2 þ SSerror
ð14Þ
The variance of a factor, such as temperature, could be calculated in formula (15), which is the sum of squares of the response (DF) among different levels.
SST ¼
3 X 2 ðDF T¼lev el j DF ik Þ
ð15Þ
j¼1
The variance of error is the sum of squares of the response (DF) at the same levels, which can be calculated as follow.
SSerror ¼
3 X
ðDF ik jT¼lev el j DF T¼lev el j Þ
2
ð16Þ
j¼1
4.1.7. Holding time during the CV stage of the discharge stage The holding time during the CV stage (t2) of the discharge stage ranges from 0 min to 30 min. Fig. 9 shows that capacity fades fastest when the time is at its middle level (15 min). This factor has hardly been considered in other researches. To verify the effect of the t2, another OFAT test was conducted. In the test, the t2 were chosen at 0 min, 15 min and 30 min, while The other six factors were kept constant and their values were T = 25 °C, i1 = 1 C, V1 = 4.2 V, t1 = 0 min, i2 = 1 C, V2 = 2.5 V. Fig. 10 shows that the capacity fades fastest when the t2 = 15 min. This result verifies the conclusion from the main effect analysis of t2. The mechanism for this phenomenon will be discussed in the following parts of our series work. 4.2. Analysis of variance for aging rate In the orthogonal design, ANOVA is widely used as a tool to study the effect of factors over a process [33] besides the main effect analysis. By comparing the effect from factors with errors, ANOVA determines whether those factors could be ignored or not. Moreover, its results can be utilized to compare the significance of different factors.
3.0
2.5
Capacity/Ah
2.0
1.5
MSk ¼
SSk DOF k
ð17Þ
The MS can be used to obtain F value of Fisher test (F-test), which is the ratio of the MS of each factor to the MS of the error.
Fk ¼
MSk MSerror
ð18Þ
Finally, p-values can be derived from p-value table based on F value and the DOF of both factor k and error, as shown in formula (19). The p-value determines whether a factor is ‘‘statistically significant” or not. Generally, a factor is significant when the p-value is less than 0.05 [33].
Pk ¼ f ðF k ; DOF k ; DOF error Þ
ð19Þ
Table 4 shows the ANOVA results of the aging behavior based on the values of the DF. There are seven columns in Table 4. The first column shows the factors (factors), the second one is the degrees of freedom (DOF) associated with each factor. The third one represents the sum of squares (sum of sq.) of each factor. The fourth one includes the percentage of the contribution for each factor to the sum of squares. The fifth one is the mean squares (MS). The sixth one represents the Fisher statistics (F-ratio). The last one shows the p-values for the Fisher statistic. The results of ANOVA in Table 4 show that:
Table 4 ANOVA of the DF values.
1.0
t2= 0 min t2= 15 min
0.5
t2= 30 min 0.0
While performing ANOVA, degrees of freedom (DOF) should also be considered together with each sum of squares to remove the effect of the number of samples to the calculated results. For this purpose, mean squares (MS) are introduced, as defined in formula (17).
0
500
1000
1500
2000
Wh-throughput/Wh Fig. 10. The effect of holding time during the CV stage of the discharge stage on the aging process using the one fact at a time (OFAT) method.
Factors
DOF
Sum of sq.
%
Mean sq.
F-ratio
P-value
Blank T i1 V1 t1 i2
1 2 2 2 2 2 2 2 20 35
0.17 0.30 0.72 0.67 0.00 0.16 0.26 0.29 0.12 2.70
6.38 10.97 26.76 24.85 0.09 6.07 9.57 10.86 4.45
0.17 0.15 0.36 0.34 0.00 0.08 0.13 0.15 0.01
28.71 24.67 60.20 55.90 0.21 13.64 21.53 24.44
3.03E05 3.99E06 3.44E09 6.47E09 0.81 1.83E04 1.03E05 4.26E06
v2
t2 Error Total
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(1) Except for the factor of t1, the other six factors significantly affect the aging rate for the selected levels, because the p-value is more than 0.05 for the factor of t1 while it is less than 0.05 for the factors of T, i1, i2, t2, V1 and V2. (2) The sequence of the factors that affect aging is i1 > V1 > T > t2 > V2 > i2 > t1. The contribution for each factor to the sum of squares represents the significance of different factors. The higher the contribution, the more significant of the factor to the aging rate. This result is slightly different with the conclusion from the main effect analysis, which shows that i1 > V1 > t2 > T > V2 > i2 > t1. This is due to the different mathematical methods in the two analysis technique. The conclusion of the main effect analysis is calculated from the range among the responses at different levels, while the conclusion of ANOVA is from the variance among them.
Fig. 11. The normal probability plot of the residual of the statistic model.
4.4. Prediction and validation 4.3. Multi-factor statistical aging model The aging rate of cells is affected by many factors. To study the relationship between the aging rate and those factors, a multifactor statistical aging model based on statistical method was developed. The effect of a factor is defined in formula (8). The average of DF for all the 18 test conditions needs to be deduced from the formula to develop the statistical model, as shown in the formula (20).
ai ¼ DF AðiÞ DF
ð20Þ
where DF AðiÞ is the average of the responses when the factor A is at the level i, and DF is the average of the responses collected from all tests. Then, the statistic model can be developed to express the relationship between the response and the factors, as shown in formula (21).
DF ¼ l þ T i þ i1j þ V 1k þ t 1l þ i2m þ V 2n þ t2o þ eijklmno
ð21Þ
where the left part is the DF value when the seven aging factors are at the level of i–o, respectively, l represents the average of the DF values of all the tests, i–o represent the levels of the seven factors, eijklmno represents the residual of the model. Based on the above analysis, the statistic model for this test is shown in formula (22).
DF ¼ 0:2913 þ ½0:09570:1218 0:0261½T
In this study, seven factors were considered and three different levels were assigned for each factor. Thus, there are totally 2187 (37) sets of aging conditions in the combinatorial approach. However, only 18 of those conditions were tested in this study. To predict the aging rates at other conditions, the multi-factor statistical aging model was be used to project the results for these un-tested conditions. Fig. 12 shows the predicted aging rates when considering the effect of T, V1 and i1, while other factors were kept in their middle level. The result shows that the aging rate is the highest at condition (1 C, 4.3 V, 0 °C). The farther a test is away from the condition (1 C, 4.3 V, 0 °C), the less is the aging rate. It should be mentioned that among the 27 conditions, nine of them received negative DF values that were marked by the triangular arrows in Fig. 12. Since the accuracy of the statistical model is ±0.18, the predicted value might be negative when the true DF value is smaller than 0.18. To derive positive DF values for each condition, a technique was used that another statistical model about 1/DF was developed, as shown in the formula (B.1) in Appendix B. The selection of the factors with sufficient orthogonality and the precision and accuracy in the measurements determine the effectiveness of the DOE. We want to claim that the developed statistical model can only be utilized to predict capacity fade under limited conditions because the orthogonal DOE is designed for main factor identification rather than empirical model building. One more test was conducted to validate the accuracy of the multi-factor statistical aging model. The levels of the seven factors were chosen as: T = 25 °C, i1 = 1 C, V1 = 4.2 V, t1 = 15 min, i2 = 2 C,
þ ½0:1925 0:0481 0:1444½i1 50
þ ½0:1861 0:0488 0:1374½V 1
þ ½0:0692 0:11930:0501½V 2 þ ½0:1059 0:11470:0088½t2
ð22Þ
where the matrixes in formula, such as [T], are equal to [1 0 0]t, [0 1 0]t or [0 0 1]t – when factors are in their low, medium or high level, respectively. The distribution of residuals is generally utilized to verify the accuracy of a model. If a model fits to the data quite well, the residuals should distribute randomly. The normal probability plot can be utilized to test the departure of the residuals from normality. Fig. 11 shows the residual of the tested data from the developed multi-factor statistical aging model. It shows that the residuals follows normal distribution with 95% confidence intervals, and the accuracy of the model is ±0.18.
Temperature/°C
þ ½0:0108 0:0014 0:0094½t 1 þ ½0:06120:0327 0:0939½i2
25
0 4.3
Voltage/V
4.2 4.1
0.2
0.5
1
Current/C
Fig. 12. The multi-factor statistical aging model can predict aging rates at different test conditions. The volume of balls is proportional to the aging rates at those conditions.
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3.0
Simulation Test
Capacity/Ah
2.5
2.0
1.5
1.0
0.5
0.0
0
500
1000
1500
2000
2500
3000
3500
Wh-througput/Wh Fig. 13. The result of a verification test is compared with the simulated result from the multi-factor statistical aging model.
V2 = 3.0 V, t2 = 15 min. This test condition was not included in the 18 tests of the orthogonal design. Thus, the result can be utilized to examine the accuracy of the model. Fig. 13 compares the tested data with the simulation results. As we discussed in Section 3.2, the initial 10% capacity fade is chosen for analysis because some groups of cells have different aging modes after the initial 10% capacity fading period. However, the model can also be suitable for the capacity fit over 10% as long as the rate of capacity fade remains the same. The result shows the predict errors are less than 5% within the 20% capacity fading period. The good agreement between the test and predicted results prove the feasibility of the model for predicting the aging rate of cells. 5. Conclusion In this paper, the significance of seven factors to the capacity fade of lithium ion cells was studied using the orthogonal design method. To compare capacity fade under different tests, an empirical aging model was introduced followed with a dimensionless index, deceleration factor (DF). Both the main effect analysis and ANOVA methods were utilized to compare the strength of different factors to the aging of cells. The results show that six out of seven factors significantly affect the aging rate in the chosen range of their levels except for the principal factor of t1. In addition, the importance of these factors to rate of capacity fade is ranked as: i1 > V1 > T > t2 > V2 > i2 > t1. Although the orthogonal DOE is designed for main factor identification rather than empirical aging model, a multi-factor aging model based on statistical method was developed to predict the rate of capacity fade. The accuracy of the model was validated within 5% errors. It should be noted that, the importance of an aging factor depends on its levels chosen in the orthogonal design and a more harsh level could trigger this factor to exhibit more superiority. Therefore, the ranking conclusion in this study is limited in the scope of the levels that we studied. Furthermore, the conclusion from main effect analysis was not affected by the existence of intercalations because the interactions were uniformly separated in each column in the OA L18, which is a characteristic of DOE. The interaction effects to the aging of cells will be considered in our further researches to improve the accuracy of the multifactor statistical aging model. Acknowledgements This research project is funded by Suzhou (Wujing) Automotive Research Institute, Tsinghua University, Project No. 2012WJ-A-01.
Table A.1 Experiment layout using OA L18 (2 37). Exp. No.
Blank
T (°C)
i1 (C)
V1 (V)
t1 (min)
i2 (C)
V2 (V)
t2 (min)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
0 0 0 25 25 25 50 50 50 0 0 0 25 25 25 50 50 50
0.2 0.5 1 0.2 0.5 1 0.2 0.5 1 0.2 0.5 1 0.2 0.5 1 0.2 0.5 1
4.1 4.2 4.3 4.1 4.2 4.3 4.2 4.3 4.1 4.3 4.1 4.2 4.2 4.3 4.1 4.3 4.1 4.2
0 15 30 15 30 0 0 15 30 30 0 15 30 0 15 15 30 0
0.5 1 2 1 2 0.5 2 0.5 1 1 2 0.5 0.5 1 2 2 0.5 1
2.4 2.5 3 3 2.4 2.5 2.5 3 2.4 2.5 3 2.4 3 2.4 2.5 2.4 2.5 3
0 15 30 30 0 15 30 0 15 0 15 30 15 30 0 15 30 0
This research is supported by the National Natural Science Foundation of China under the grant number of 51207080 and General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China, Project No. 2014IK202. Appendix A. Experimental layout Table A.1 shows the experiment layout using L18. The first column is the experiment number list, indicating 18 tests need to be done. The second to the ninth columns are factor lists to express the levels of each factor used in the test. The number ‘‘1”, ‘‘2” and ‘‘3” in the second to the ninth columns represent level 1, level 2 and level 3 of each factor. Appendix B. Statistical model using 1/DF values The multi-factor statistical aging model based on DF is not effective to predict the aging rate when the true DF value is less than 0.18 due to the accuracy of statistical model. To solve this problem, another statistical aging model based on the reciprocal of DF was developed.
1=DF ¼ 31:21 þ ½51:76 24:52 27:24½T þ ½27:46 20:1247:59½i1 þ ½25:894:9620:93½V 1 þ ½21:275:216:06½t1 þ ½3:13 22:0818:95½i2 þ ½0:32 17:5817:91½V 2 þ ½27:51 15:7043:20½t2
ðB:1Þ
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Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Identifying main factors of capacity fading in lithium ion cells using orthogonal design of experiments Laisuo Su a, Jianbo Zhang a,b, Caijuan Wang c, Yakun Zhang a, Zhe Li a,b,⇑, Yang Song c, Ting Jin c, Zhao Ma d a
Department of Automotive Engineering, State Key Laboratory of Automotive Safety and Energy, Tsinghua University, 100084 Beijing, China Beijing Co-innovation Center for Electric Vehicles, Beijing Institute of Technology, 100081 Beijing, China c General Administration of Quality Supervision, Inspection and Quarantine of China, 215000 Suzhou, China d Dongfeng Motor Corporation Technical Center, 430000 Wuhai, China b
h i g h l i g h t s The effect of seven principal factors on the aging behavior of lithium ion cells is studied. Orthogonal design of experiments is used to reduce the experiment units. Capacity fades linearly during the initial 10% capacity fading period. Statistical methods are used to compare the significance of each principal factor. A multi-factor statistical model is developed to predict the aging rate of cells.
a r t i c l e
i n f o
Article history: Received 4 August 2015 Received in revised form 1 November 2015 Accepted 4 November 2015
Keywords: Lithium-ion cells Aging rate Orthogonal design of experiments Main aging factors
a b s t r a c t The aging rate under cycling conditions for lithium-ion cells is affected by many factors. Seven principal factors are systematically examined using orthogonal design of experiments, and statistical analysis was used to identify the order of principal factors in terms of strength in causing capacity fade. These seven principal factors are: the charge and discharge currents (i1, i2) during the constant current regime, the charge and discharge cut-off voltages (V1, V2) and the corresponding durations (t1, t2) during the constant voltage regime, and the ambient temperature (T). An orthogonal array with 18 test units was selected for the experiments. The test results show that (1) during the initial 10% capacity fading period, the capacity faded linearly with Wh-throughput for all the test conditions; (2) after the initial period, certain cycling conditions exacerbated aging rates, while the others remain the same. The statistical results show that: (1) except for t1, the other six principal factors significantly affect the aging rate; (2) the strength of the principal factors was ranked as: i1 > V1 > T > t2 > V2 > i2 > t1. Finally, a multi-factor statistical aging model is developed to predict the aging rate, and the accuracy of the model is validated. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Lithium-ion batteries have many merits, including high energy density, high coulombic efficiency and comparatively low heat output, which make them highly suitable for electric vehicles (EVs) and electrical energy storage (EES) applications [1]. Although battery manufacturers are making every effort to reduce the cost, including use of cheaper components and materials and innovation in manufacturing processes, the high cost of lithium-ion batteries
⇑ Corresponding author at: Department of Automotive Engineering, State Key Laboratory of Automotive Safety and Energy, Tsinghua University, 100084 Beijing, China. E-mail address: [email protected] (Z. Li). http://dx.doi.org/10.1016/j.apenergy.2015.11.014 0306-2619/Ó 2015 Elsevier Ltd. All rights reserved.
has inhibited widespread use of such batteries in these applications. However, reducing life cycle costs is a way to mitigate the high cost of batteries, so extending the lifespan of the batteries is an effective way to achieve that goal. The lifespan of batteries depends on operating conditions [2,3], such as charging/discharging rates and ambient temperature. Understanding the relationship between the operating conditions and aging behavior of the batteries is an important task to increase the lifespan. However, it is not easy to quantify the effect of each condition on aging behavior, as the aging of batteries involves multiple closely interrelated factors. To solve this problem, three steps are usually practiced [4]. The first step is a screening study to discover factors that affect the battery’s lifespan. The next step is an empirical study to produce an empirical model to describe the
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way the battery lifespan changes with the factors. The last step is a mechanistic study to understand the mechanism and to develop a mechanistic model to describe the causality between lifespan and factors. Following these steps in sequence helps develop a wellestablished mechanistic model efficiently. To date, empirical aging models (the second step) [5–16] and mechanistic aging models (the third step) [17–23] have been studied, while little attention has been paid to the study of the strength of the factors that affect the aging behavior (the first step). The relative strength of these factors needs to be investigated first before focusing on any model development. Otherwise, the validity of the model may be uncertain. To be specific, there are at least seven stress factors that affect the aging behavior of batteries in cycling conditions: the charge and discharge current (i1, i2) during the constant current (CC) regime, the charge and discharge cut-off voltage (V1, V2) and the corresponding durations (t1, t2) in the constant voltage (CV) regime, and the temperature (T) at which battery ages. Wright et al. [7] chose the temperature, initial state of charge (SOC), change in SOC, as the three factors in their battery model to study cycle aging effects. Wang et al. [5] chose the temperature, depth of discharge (DOD), and discharge rate, as the three factors in a similar study. Both studies did not give reasons for their choice of the aging factors. To the best of our knowledge, only two screening studies compared the importance of different factors that affect the aging behavior of batteries. The first one by Prochazka et al. [24] studied the influence of current, temperature, SOC and the change of SOC on the aging of two types of cells using a second-order model and D-optimization method. Their result showed that these factors significantly affect the cells aging behavior. The second one by Chi et al. [25] studied the effect of temperature, DOD, and discharge rate on the aging behavior of cells using Taguchi method. The result showed that the strength of these stress factors on capacity fade follows the order of temperature > discharge rate > DOD. However, both papers arbitrarily constrained the stress factors to three or four without justification. Therefore, more careful screening study (the first step) is needed to identify the main stress factors for cycle aging to lay a foundation for subsequent empirical model studies. This study attempts to quantify the strength of the stress factors and the relationship between aging behavior and operating condition. Since, the real world operating conditions are very complex and vary from cycle to cycle, a simplified protocol using typical CC–CV charge and discharge regimes along with the observations of the effects from the seven factors was used in this study to allow better quantification. This protocol should cover most of the actual cycle aging situations. The rest of this paper is organized as following: in Section 2, the methodology of orthogonal design with orthogonal analysis (OA) L18 is discussed, and the test method is introduced; in Section 3, the test results of cell initial characteristics and cell aging are displayed, and an empirical aging model is developed; in Section 4, we discuss the results in a deeper manner by analyzing the capacity fade of the aging process with statistical methods, including the main effect analysis and analysis of variance (ANOVA), and proposing a multi-factor statistical aging model; in Section 5, we summarize the conclusions that would give guidance for choosing reasonable factors in aging models.
2. Experimental 2.1. Methodology The design of experiments (DOE) is a tool to obtain experimental information using statistical methods. The principles
of DOE include randomization, replication, blocking, and orthogonality. Based on those principles, many methods have been proposed to optimize the DOE for different purposes. For example, factorial experimental design has been proposed to obtain extensive information from only a few sets of experiments compared with the one factor at a time method (OFAT). As a branch of factorial experimental design, the orthogonal design [26] has been extensively used in various fields due to its efficiency. In the orthogonal design, the effect of many factors (and possibly some of their interactions) can be studied simultaneously in a single set of experiments with much fewer experiment units. To apply this method, an orthogonal analysis (OA) needs to be chosen in the first place. In this study, the OA of L18 was chosen because it could be used to study eight factors, among which one factor with two levels and seven with three levels. Moreover, the main effects among these factors can be derived without considering the influence of their interactions since the interactions are uniformly distributed in each column. Table 1 shows nominal specifications of the 18,650 cells made by Samsung. Seven factors were studied for the aging behavior of cells, and three different levels were assigned for each factor. The three levels together should cover most of the actual test conditions based on Table 1. Table 2 shows the three levels of different factors that were chosen for the test. Those levels were distributed in the OA of L18, as shown in Table A.1 in Appendix A. Since only seven factors were considered in this experiment, one column, named as ‘‘blank” in Table A.1, was not allocated to any factor.
2.2. Tests The cells were cycled three times at C/2 to get stable performance before test. Then, the initial performance of 36 cells was tested, including capacity and resistance. Statistics showed that there were no outliers among those cells. The 36 cells were then divided into 18 groups and allocated to the rows in Table A.1 and tested using the levels in Table 2. Cells were taken out to characterize the state of health (SOH) with reference performance tests (RPT) every 50 cycles. The terminating criterion for the cycling of each cell was that the cell capacity degraded to 50% of its initial capacity or the cycle number reached 1000 times. However, parts of the tests that cycled at low rates were forced to stop before they reached the terminating criteria when most of the tests were finished. The overall tests are shown in Fig. 4. All the tests were conducted using a MACCOR Series 4000 and three environmental chambers GDW/JB-0100 (Naya, China). The RPT test was designed to obtain the discharge capacity at C/2, C/50 and the resistance at nine different SOC points. Cells were cycled twice at C/2 followed by a customized test profile, as shown in Fig. 1(a). The capacity at C/2 was chosen as the discharge capacity in the second cycle at C/2. The capacity at C/50 was obtained from the customized profile, instead of the normal CC discharge profile, which would take about 50 h. The customized profile
Table 1 Nominal specifications of the test cell. Item
Specification
Cathode material Anode material Nominal capacity Max. charge current Charge cut-off voltage Max. discharge current Discharge cut-off voltage Operating temperature
LiNixCoyAlxO2 Graphite 2.85 A h (1/3 C) 1C 4.20 ± 0.05 V 2C 2.50 V Charge: 0–45 °C Discharge: 20 to 60 °C
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3.2. Cell aging characteristics
Table 2 Cycling factors and their levels. No.
Factors
Abbreviation
Level 1
Level 2
Level 3
1 2
Ambient temperature (°C) Charge current during CC process (C) Cut-off voltage of the charge process (V) CV time of the charge process (min) Discharge current during CC process (C) Cut-off voltage of the discharge process (V) CV time of the discharge process (min)
T i1
0 0.2
25 0.5
50 1
V1
4.1
4.2
4.3
t1
0
15
30
i2
0.5
1
2
V2
2.4
2.5
3
t2
0
15
30
3 4 5 6 7
included a CV process at the end of discharge, and the cut-off current was 0.06 A (C/50). The capacity at the low discharge rate of C/50 reflects the thermodynamic characteristic of a cell and was called T-Capacity in this study. The calculation of resistance at different SOC points was illustrated in Fig. 1(b) and could be obtained by formula (1).
Rsoc ¼
V V0 I
ð1Þ
where V is the voltage before discharging at a specific SOC, V0 is the voltage after 60 s of discharging from the specific SOC, and I is the discharge current. Sixty seconds were used here to include the diffusion effect when calculating the resistance. 3. Results 3.1. Cell initial characteristics
Fig. 4 shows the T-Capacity of cells varies with cycle number under different test conditions. In most cases, the T-Capacity decreased linearly with cycle number in aging. L14 and L15 are exceptions; yet display such behavior during the initial 10% capacity fading. Thus, the slope of the linear part of the curves could represent the aging rates at different test conditions. As an energy storage and conversion device, users should concern more with the energy they can extract from the cells through the lifespan rather than the cycle number. Therefore, Whthroughput is a more relevant variable to compare with than cycle number for the aging rate under different conditions. Fig. 5 shows that the Wh-throughputs for the first 50 cycles under the 18 test conditions are different from one another. Fig. 6 displays the retention of T-Capacity as a function of Whthroughput for the 36 cells under 18 test conditions. The TCapacity decreased linearly with the Wh-throughput during the initial 10% capacity fading for all the cells. The slope of this linear part was utilized to represent aging rate at different test conditions. One of the preconditions for the accelerated life test of cells is that the aging mechanism does not change. The relationship between resistance and capacity (R-Cap plot) is a method to differentiate aging mechanism under different test conditions [27]. Since the values of resistance vary little from SOC = 20% to SOC = 90% for the tested cells, as shown in Fig. 7(a), the resistance at SOC = 50% was chosen for further analysis. Fig. 7(b) shows the aging behavior of the 18 groups of tests could be divided into two categories according to the trajectory of resistance vs T-Capacity: aging type I and aging type II. The relationship between T-Capacity and resistance in the initial 10% capacity fading is similar for all the 18 test groups. We infer that the aging mechanism was probably the same. 3.3. Empirical aging model
Fig. 2 displays the distribution of the initial T-Capacity of cells determined from the customized test profile. The T-Capacity is 3.038 ± 0.0051 A h (±0.17%). The curve in the histogram represents the T-Capacity distribution. The probability plot compares the distribution of the T-Capacity of cells with a set of standard normally distributed data. It shows that the T-Capacity is normally distributed with 95% confidence intervals. Fig. 3 shows the distribution of the initial resistance of the cells at SOC = 50%. The distributions of resistance at other SOCs are similar. The resistance at SOC = 50% is 0.058 ± 0.001 X (±1.14%). The probability plot shows the resistance of the tested cells is normally distributed with 95% confidence intervals. The small variances and the normal distribution of both the T-Capacity and the resistance at SOC = 50% signify the high initial quality of the tested cells.
Different aging models have been proposed to describe the aging behavior of lithium ion cells [5–7]. Those models generally use a power law to describe such behavior, as shown below:
y ¼ K xN
ð2Þ
K ¼ kðT; i1 ; i2 ; V 1 ; V 2 ; . . .Þ
ð3Þ
N ¼ nðT; i1 ; i2 ; V 1 ; V 2 ; . . .Þ
ð4Þ
where y is a parameter that can be used to characterize the outcome in the performance of cells, such as capacity, K a coefficient representing the intensity of the aging factors that will influence the outcome y, x an independent variable representing the extension of the
4.10
(a)
V
(b)
60 s
Voltage/V
4.06
4.02
V’ 3.98
3.94
0
30
60
90
120
150
Time/s Fig. 1. (a) Customized test profile for RPT, where the discharge rate is C/2. In each discharge interval, 10% of the nominal capacity was removed. The rest between two discharge intervals is 10 min. The cut-off current in the final CV stage is C/50. (b) The voltage evolution of a cell during a C/2 discharge interval.
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Fig. 2. (a) T-Capacity distribution and (b) normal probability plot of the initial T-Capacity distribution.
3.0
3.0
2.5
2.5
2.5
2.0
2.0
2.0
1.5
L1 L2 L3 L4 L5 L6
1.0 0.5 0.0
0
200
400
600
800 1000
1.5
L7 L8 L9 L10 L11 L12
1.0 0.5 0.0
0
200
Cycle Number
400
600
800 1000
Capacity/Ah
3.0
Capacity/Ah
Capacity/Ah
Fig. 3. (a) Resistance at 50% SOC distribution and (b) normal probability plot of the initial resistance distribution.
1.5
L13 L14 L15 L16 L17 L18
1.0 0.5 0.0 0
200
400
600
800 1000
Cycle Number
Cycle Number
Fig. 4. T-Capacity variations with cycle number for the 18 test conditions.
1200
Cell 1 Cell 2
Wh-throughput/Wh
1000 800
Whi ¼
600
200
L1
L2
L3
L4
L5
L6
L7
L8
L9 L10 L11 L12 L13 L14 L15 L16 L17 L18
Experiment Number Fig. 5. The Wh-throughput for the first 50 cycles for the 18 test conditions.
DF i ¼ aging, such as cycle number, holding time, or Wh-throughput, N is the exponent of x. K and N are functions of aging factors in the cycling process, as shown in formulas (3) and (4). N is constant in most cases and it equals one in this study.
Q loss i ¼ K i Whi
1 Q ¼ Gi Q loss K i loss
i
ð6Þ
As an energy storage and conversion device, users should concern more with the energy they can extract from the cells through the lifespan. Thus, Eq. (5) was transformed into Eq. (6), where Gi is the reciprocal of Ki. The equation expresses the amount of Whthroughput that a cell can transfer before it loses a certain amount of capacity. To visually compare aging rates at different test conditions, a dimensionless index, deceleration factor (DF), was introduced to, as shown in formula (7).
400
0
The capacity of cells fades linearly with Wh-throughput during the initial 10% capacity fading, as shown in Fig. 6. Thus, Eq. (5) describes the capacity fade behavior during aging. where Ki is the slope of the linear part of the curve in Fig. 6 at condition i.
ð5Þ
Gi G0
ð7Þ
where DFi is the decelerated factor of the condition i, Gi is the G value of the test condition i, G0 is the G value of a reference test condition. Fig. 8 shows the DF values of the 18 test conditions. Since this cell aged slowest among the 36 cells, the aging of cell 1 in the cycling condition L1 was chosen as the reference test condition to make the DF values between zero and one.
205
3.0
3.0
2.5
2.5
2.5
2.0 1.5
L1 L2 L3 L4 L5 L6
1.0 0.5 0.0
2.0 1.5
L7 L8 L9 L10 L11 L12
1.0 0.5 0.0
0.3 0.6 0.9 1.2 1.5 1.8
0
Capacity/Ah
3.0
Capacity/Ah
Capacity/Ah
L. Su et al. / Applied Energy 163 (2016) 201–210
0
Wh-throughput/Wh x 10
1.5
L13 L14 L15 L16 L17 L18
1.0 0.5 0.0
0.3 0.6 0.9 1.2 1.5 1.8
Wh-throughput/Wh x 104
2.0
0
0.3 0.6 0.9 1.2 1.5 1.8
Wh-throughput/Wh x 104
4
Fig. 6. The variation of T-Capacity with Wh-throughput for the 18 test conditions.
0.7
0.6
(a)
0.5 0.4 0.3 0.2 0.1 0
(b)
0.5
Resistance/Ohm
Resistance/Ohm
0.6
Aging mode I Aging mode II
0.4 0.3
Inial Aging
0.2 0.1
0
0.2
0.4
0.6
0.8
1
1
1.5
SOC
2
2.5
3
Capacity/Ah
Fig. 7. (a) The variation of resistance with respect to SOC for a fresh cell, and (b) increase of resistance at SOC = 50% with the T-Capacity for the 18 test conditions.
Cell 1 Cell 2
1
DF
0.8
(1) Response: The DF is chosen as the response of tests under different conditions. It can be derived from formula (7). (2) Effects at different levels: The effect of temperature at level 1 (0 °C) is the average of the DF values in those tests whose temperature is 0 °C, as shown in formula (10). The DFik represents the DF value of cell k at test condition i because there were two cells in each test condition. Thus m equals 2. The effects at other levels can be calculated in the same way. The calculated results of temperature at level 1, 2 and 3 are 0.196, 0.413, and 0.265 respectively.
0.6 0.4 0.2 0
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L18
Experiment Number Fig. 8. The DF values for the 18 test conditions.
ETð0 CÞ ¼
4.1. Main effect analysis for capacity fade Main effect analysis compares the strength of different factors to the capacity fade of cells. If the effect of interaction among factors and the effect of errors can by ignored, the effect of the factor A at the level i on the response Y is computed from Eq. (8), where Y AðiÞ is the average of all the response when the factor A is at the level i, regardless the values of other factors.
ð8Þ
According to the principle knowledge from DOE, the main effect of a factor is defined as the range of the factor’s effects at different levels, as shown in formula (9).
F A ¼ MaxðEAðiÞ Þ MinðEAðiÞ Þ
1 1 m n
X
! DF ik
T¼0 C
3 12 X X 1 1 ¼ DF ik þ DF ik 2 6 i¼1 i¼10
4. Discussion
EAðiÞ ¼ Y AðiÞ
The main effect of temperature is chosen as an example to show the calculation process in more detail.
ð9Þ
! ð10Þ
(3) Main effect: The effect of temperature reaches its maximum at the level 2 and reaches its minimum at the level 1. Therefore, the main effect of the temperature is the difference between the DF values at level 1 and level 2, which is 0.217 as shown in the formula (11).
F T ¼ MaxðETðiÞ Þ MinðETðiÞ Þ ¼ ETð25 CÞ ETð0 CÞ ¼ 0:217 ð11Þ Fig. 9 shows the effect of the seven aging factors at different level. The larger the value, the smaller the aging rate, according to the definition of the DF and the main effect in formula (7) and (10). Those results can be compared with others’ studies to verify the orthogonal design method [24,25,28,29].
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Table 3 shows the main effect of the seven factors and the ‘‘blank” list. The calculated result of each column is composed by three parts: (i) main effect of the studied factor in the column, (ii) part of the interaction effects among the seven factors, and (iii) errors during test. In the blank column, the first part equals zero since no factor is allocated to it. The interaction effects among the seven factors are equally distributed in each column of the L18, and the errors could be considered as the same in each column. Therefore, a factor affects the aging rate of cells only when the main effect of the corresponding column is greater than the main effect of the blank column. According to this principle, except for the factor t1, the other six factors have an effect on the aging rate of cells. In addition, according to the definition of the main effect, the greater is the FA, the larger is the influence of the factor A on the response. Therefore, the effect of the studied factors on the aging rate is ranked: i1 > V1 > t2 > T > V2 > i2 > t1. 4.1.1. Ambient temperature The ambient temperature ranges from 0 °C to 50 °C. Fig. 9 shows that the aging rate is smallest at its middle level (25 °C). The effect of temperature on the aging rate of cells follows Arrhenius’s law [7], which indicates that the aging rate is bigger at a higher temperature. However, lithium plating [30] might appear on the anode at very low temperature during cycling, which would accelerate the aging process. Based on the above analysis, the temperature behavior could be divided into two major domains: A low temperature domain described by a negative activation energy EA,p < 0 and a high temperature domain described by EA,s > 0 [27], as shown in formula (12). Therefore, the aging process of cells would be accelerated when temperature is higher or lower than 25 °C.
r ¼ Ap expfEA;p =ðkB TÞg þ As expfEA;s =ðkB TÞg
ð12Þ
4.1.2. Charge current The charge current ranges from C/5 to 1 C. Fig. 9 shows that the bigger the current, the faster the aging process. Zhang [31] thought that high charging current can cause huge local imbalance between the cell reaction and ionic diffusion on the electrolyte–electrode
Table 3 The main effect of the seven factors. Factors
Blank
T
i1
V1
t1
i2
V2
t2
E1 E2 E3 F Rank
0.361 0.222
0.196 0.413 0.265 0.218 ④
0.484 0.243 0.147 0.337 ①
0.477 0.243 0.154 0.324 ②
0.302 0.290 0.282 0.020 ⑧
0.353 0.324 0.197 0.155 ⑥
0.361 0.172 0.341 0.189 ⑤
0.397 0.177 0.300 0.221 ③
0.138 ⑦
interface and, therefore, cause high polarization. This would result in metallic lithium plating on the anode or overcharge on the cathode, which depends on ratio of the reversible capacity of the cathode to the anode. Therefore, the aging rate relate positively to the current. 4.1.3. Cut-off voltage of the charge stage The cut-off voltage of the charge stage ranges from 4.1 V to 4.3 V. Fig. 9 shows that the higher the voltage, the faster the aging process. The result is in agreement with the conclusions in others’ studies [28,32]. The mechanism to this phenomenon is that high voltage leads to high potential in positive electrode and low potential in negative electrode. The electrolyte is easier to be oxidized by a higher potential in the positive electrode area, and it is easier to be reduced by a lower potential in the negative electrode area. Thus, the aging rate relate positively to the cut-off voltage of the charge stage. 4.1.4. Holding time during the CV stage of the charge stage The holding time during the CV stage ranges from 0 min to 30 min. Fig. 9 shows the holding time has little effect on the aging rate. This result agrees with Fig. 2 of Choi’s study [28] that the holding time during the CV stage at 4.2 V has a relatively small effect on aging rate. However, it contradicts to our intuition that long period at high voltage should speed up the aging process because cells age fast at high voltage. The explanation for this phenomenon could be that the current at the CV stage decreases with time, alleviating the local imbalance between the cell reaction and ionic diffusion, as well as the polarization in both electrodes. Thus,
Fig. 9. The effects of the seven factors at different levels.
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L. Su et al. / Applied Energy 163 (2016) 201–210
few additional side reactions occur at this period of time, and the aging rate changes little with the holding time at the CV stage. 4.1.5. Discharge current The discharge current ranges from C/2 to 2 C. Fig. 9 shows the aging rate is accelerated by the discharge current. This result agrees with Fig. 4 of Choi’s study [28]. The mechanism to this result is similar with the effect of the charge current to the aging process. However, it should be noted that the discharge current has less effect on aging rate compared with the charge current. 4.1.6. Cut-off voltage of the discharge stage The cut-off voltage of the discharge stage (V2) ranges from 3.0 V to 2.4 V. Fig. 9 shows that the capacity fades fastest when the voltage is at its middle level (2.5 V). This result differs from Fig. 5 of Choi’s study [28] that the DOD has little effect on aging within 500 cycles. The reason for the phenomenon that cells aged faster at V2 = 2.5 V than V2 = 2.4 V is not fully understood. The results need to be reproduced using OFAT method, and will be further explored in our future work.
The total variance is the sum of squares of all the test data, as calculated in formula (13).
SStotal ¼
X
ðDF il DF il Þ
2
ði ¼ 1; 2; . . . ; 18;
l ¼ 1; 2Þ
ð13Þ
ANOVA partitions the total variance into its appropriate components. In this study, the total variance was divided into terms caused by different factors, as shown in formula (14).
SStotal ¼ SST þ SSi1 þ SSV 1 þ SSt1 þ SSi2 þ SSV 2 þ SSt2 þ SSerror
ð14Þ
The variance of a factor, such as temperature, could be calculated in formula (15), which is the sum of squares of the response (DF) among different levels.
SST ¼
3 X 2 ðDF T¼lev el j DF ik Þ
ð15Þ
j¼1
The variance of error is the sum of squares of the response (DF) at the same levels, which can be calculated as follow.
SSerror ¼
3 X
ðDF ik jT¼lev el j DF T¼lev el j Þ
2
ð16Þ
j¼1
4.1.7. Holding time during the CV stage of the discharge stage The holding time during the CV stage (t2) of the discharge stage ranges from 0 min to 30 min. Fig. 9 shows that capacity fades fastest when the time is at its middle level (15 min). This factor has hardly been considered in other researches. To verify the effect of the t2, another OFAT test was conducted. In the test, the t2 were chosen at 0 min, 15 min and 30 min, while The other six factors were kept constant and their values were T = 25 °C, i1 = 1 C, V1 = 4.2 V, t1 = 0 min, i2 = 1 C, V2 = 2.5 V. Fig. 10 shows that the capacity fades fastest when the t2 = 15 min. This result verifies the conclusion from the main effect analysis of t2. The mechanism for this phenomenon will be discussed in the following parts of our series work. 4.2. Analysis of variance for aging rate In the orthogonal design, ANOVA is widely used as a tool to study the effect of factors over a process [33] besides the main effect analysis. By comparing the effect from factors with errors, ANOVA determines whether those factors could be ignored or not. Moreover, its results can be utilized to compare the significance of different factors.
3.0
2.5
Capacity/Ah
2.0
1.5
MSk ¼
SSk DOF k
ð17Þ
The MS can be used to obtain F value of Fisher test (F-test), which is the ratio of the MS of each factor to the MS of the error.
Fk ¼
MSk MSerror
ð18Þ
Finally, p-values can be derived from p-value table based on F value and the DOF of both factor k and error, as shown in formula (19). The p-value determines whether a factor is ‘‘statistically significant” or not. Generally, a factor is significant when the p-value is less than 0.05 [33].
Pk ¼ f ðF k ; DOF k ; DOF error Þ
ð19Þ
Table 4 shows the ANOVA results of the aging behavior based on the values of the DF. There are seven columns in Table 4. The first column shows the factors (factors), the second one is the degrees of freedom (DOF) associated with each factor. The third one represents the sum of squares (sum of sq.) of each factor. The fourth one includes the percentage of the contribution for each factor to the sum of squares. The fifth one is the mean squares (MS). The sixth one represents the Fisher statistics (F-ratio). The last one shows the p-values for the Fisher statistic. The results of ANOVA in Table 4 show that:
Table 4 ANOVA of the DF values.
1.0
t2= 0 min t2= 15 min
0.5
t2= 30 min 0.0
While performing ANOVA, degrees of freedom (DOF) should also be considered together with each sum of squares to remove the effect of the number of samples to the calculated results. For this purpose, mean squares (MS) are introduced, as defined in formula (17).
0
500
1000
1500
2000
Wh-throughput/Wh Fig. 10. The effect of holding time during the CV stage of the discharge stage on the aging process using the one fact at a time (OFAT) method.
Factors
DOF
Sum of sq.
%
Mean sq.
F-ratio
P-value
Blank T i1 V1 t1 i2
1 2 2 2 2 2 2 2 20 35
0.17 0.30 0.72 0.67 0.00 0.16 0.26 0.29 0.12 2.70
6.38 10.97 26.76 24.85 0.09 6.07 9.57 10.86 4.45
0.17 0.15 0.36 0.34 0.00 0.08 0.13 0.15 0.01
28.71 24.67 60.20 55.90 0.21 13.64 21.53 24.44
3.03E05 3.99E06 3.44E09 6.47E09 0.81 1.83E04 1.03E05 4.26E06
v2
t2 Error Total
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(1) Except for the factor of t1, the other six factors significantly affect the aging rate for the selected levels, because the p-value is more than 0.05 for the factor of t1 while it is less than 0.05 for the factors of T, i1, i2, t2, V1 and V2. (2) The sequence of the factors that affect aging is i1 > V1 > T > t2 > V2 > i2 > t1. The contribution for each factor to the sum of squares represents the significance of different factors. The higher the contribution, the more significant of the factor to the aging rate. This result is slightly different with the conclusion from the main effect analysis, which shows that i1 > V1 > t2 > T > V2 > i2 > t1. This is due to the different mathematical methods in the two analysis technique. The conclusion of the main effect analysis is calculated from the range among the responses at different levels, while the conclusion of ANOVA is from the variance among them.
Fig. 11. The normal probability plot of the residual of the statistic model.
4.4. Prediction and validation 4.3. Multi-factor statistical aging model The aging rate of cells is affected by many factors. To study the relationship between the aging rate and those factors, a multifactor statistical aging model based on statistical method was developed. The effect of a factor is defined in formula (8). The average of DF for all the 18 test conditions needs to be deduced from the formula to develop the statistical model, as shown in the formula (20).
ai ¼ DF AðiÞ DF
ð20Þ
where DF AðiÞ is the average of the responses when the factor A is at the level i, and DF is the average of the responses collected from all tests. Then, the statistic model can be developed to express the relationship between the response and the factors, as shown in formula (21).
DF ¼ l þ T i þ i1j þ V 1k þ t 1l þ i2m þ V 2n þ t2o þ eijklmno
ð21Þ
where the left part is the DF value when the seven aging factors are at the level of i–o, respectively, l represents the average of the DF values of all the tests, i–o represent the levels of the seven factors, eijklmno represents the residual of the model. Based on the above analysis, the statistic model for this test is shown in formula (22).
DF ¼ 0:2913 þ ½0:09570:1218 0:0261½T
In this study, seven factors were considered and three different levels were assigned for each factor. Thus, there are totally 2187 (37) sets of aging conditions in the combinatorial approach. However, only 18 of those conditions were tested in this study. To predict the aging rates at other conditions, the multi-factor statistical aging model was be used to project the results for these un-tested conditions. Fig. 12 shows the predicted aging rates when considering the effect of T, V1 and i1, while other factors were kept in their middle level. The result shows that the aging rate is the highest at condition (1 C, 4.3 V, 0 °C). The farther a test is away from the condition (1 C, 4.3 V, 0 °C), the less is the aging rate. It should be mentioned that among the 27 conditions, nine of them received negative DF values that were marked by the triangular arrows in Fig. 12. Since the accuracy of the statistical model is ±0.18, the predicted value might be negative when the true DF value is smaller than 0.18. To derive positive DF values for each condition, a technique was used that another statistical model about 1/DF was developed, as shown in the formula (B.1) in Appendix B. The selection of the factors with sufficient orthogonality and the precision and accuracy in the measurements determine the effectiveness of the DOE. We want to claim that the developed statistical model can only be utilized to predict capacity fade under limited conditions because the orthogonal DOE is designed for main factor identification rather than empirical model building. One more test was conducted to validate the accuracy of the multi-factor statistical aging model. The levels of the seven factors were chosen as: T = 25 °C, i1 = 1 C, V1 = 4.2 V, t1 = 15 min, i2 = 2 C,
þ ½0:1925 0:0481 0:1444½i1 50
þ ½0:1861 0:0488 0:1374½V 1
þ ½0:0692 0:11930:0501½V 2 þ ½0:1059 0:11470:0088½t2
ð22Þ
where the matrixes in formula, such as [T], are equal to [1 0 0]t, [0 1 0]t or [0 0 1]t – when factors are in their low, medium or high level, respectively. The distribution of residuals is generally utilized to verify the accuracy of a model. If a model fits to the data quite well, the residuals should distribute randomly. The normal probability plot can be utilized to test the departure of the residuals from normality. Fig. 11 shows the residual of the tested data from the developed multi-factor statistical aging model. It shows that the residuals follows normal distribution with 95% confidence intervals, and the accuracy of the model is ±0.18.
Temperature/°C
þ ½0:0108 0:0014 0:0094½t 1 þ ½0:06120:0327 0:0939½i2
25
0 4.3
Voltage/V
4.2 4.1
0.2
0.5
1
Current/C
Fig. 12. The multi-factor statistical aging model can predict aging rates at different test conditions. The volume of balls is proportional to the aging rates at those conditions.
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3.0
Simulation Test
Capacity/Ah
2.5
2.0
1.5
1.0
0.5
0.0
0
500
1000
1500
2000
2500
3000
3500
Wh-througput/Wh Fig. 13. The result of a verification test is compared with the simulated result from the multi-factor statistical aging model.
V2 = 3.0 V, t2 = 15 min. This test condition was not included in the 18 tests of the orthogonal design. Thus, the result can be utilized to examine the accuracy of the model. Fig. 13 compares the tested data with the simulation results. As we discussed in Section 3.2, the initial 10% capacity fade is chosen for analysis because some groups of cells have different aging modes after the initial 10% capacity fading period. However, the model can also be suitable for the capacity fit over 10% as long as the rate of capacity fade remains the same. The result shows the predict errors are less than 5% within the 20% capacity fading period. The good agreement between the test and predicted results prove the feasibility of the model for predicting the aging rate of cells. 5. Conclusion In this paper, the significance of seven factors to the capacity fade of lithium ion cells was studied using the orthogonal design method. To compare capacity fade under different tests, an empirical aging model was introduced followed with a dimensionless index, deceleration factor (DF). Both the main effect analysis and ANOVA methods were utilized to compare the strength of different factors to the aging of cells. The results show that six out of seven factors significantly affect the aging rate in the chosen range of their levels except for the principal factor of t1. In addition, the importance of these factors to rate of capacity fade is ranked as: i1 > V1 > T > t2 > V2 > i2 > t1. Although the orthogonal DOE is designed for main factor identification rather than empirical aging model, a multi-factor aging model based on statistical method was developed to predict the rate of capacity fade. The accuracy of the model was validated within 5% errors. It should be noted that, the importance of an aging factor depends on its levels chosen in the orthogonal design and a more harsh level could trigger this factor to exhibit more superiority. Therefore, the ranking conclusion in this study is limited in the scope of the levels that we studied. Furthermore, the conclusion from main effect analysis was not affected by the existence of intercalations because the interactions were uniformly separated in each column in the OA L18, which is a characteristic of DOE. The interaction effects to the aging of cells will be considered in our further researches to improve the accuracy of the multifactor statistical aging model. Acknowledgements This research project is funded by Suzhou (Wujing) Automotive Research Institute, Tsinghua University, Project No. 2012WJ-A-01.
Table A.1 Experiment layout using OA L18 (2 37). Exp. No.
Blank
T (°C)
i1 (C)
V1 (V)
t1 (min)
i2 (C)
V2 (V)
t2 (min)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
0 0 0 25 25 25 50 50 50 0 0 0 25 25 25 50 50 50
0.2 0.5 1 0.2 0.5 1 0.2 0.5 1 0.2 0.5 1 0.2 0.5 1 0.2 0.5 1
4.1 4.2 4.3 4.1 4.2 4.3 4.2 4.3 4.1 4.3 4.1 4.2 4.2 4.3 4.1 4.3 4.1 4.2
0 15 30 15 30 0 0 15 30 30 0 15 30 0 15 15 30 0
0.5 1 2 1 2 0.5 2 0.5 1 1 2 0.5 0.5 1 2 2 0.5 1
2.4 2.5 3 3 2.4 2.5 2.5 3 2.4 2.5 3 2.4 3 2.4 2.5 2.4 2.5 3
0 15 30 30 0 15 30 0 15 0 15 30 15 30 0 15 30 0
This research is supported by the National Natural Science Foundation of China under the grant number of 51207080 and General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China, Project No. 2014IK202. Appendix A. Experimental layout Table A.1 shows the experiment layout using L18. The first column is the experiment number list, indicating 18 tests need to be done. The second to the ninth columns are factor lists to express the levels of each factor used in the test. The number ‘‘1”, ‘‘2” and ‘‘3” in the second to the ninth columns represent level 1, level 2 and level 3 of each factor. Appendix B. Statistical model using 1/DF values The multi-factor statistical aging model based on DF is not effective to predict the aging rate when the true DF value is less than 0.18 due to the accuracy of statistical model. To solve this problem, another statistical aging model based on the reciprocal of DF was developed.
1=DF ¼ 31:21 þ ½51:76 24:52 27:24½T þ ½27:46 20:1247:59½i1 þ ½25:894:9620:93½V 1 þ ½21:275:216:06½t1 þ ½3:13 22:0818:95½i2 þ ½0:32 17:5817:91½V 2 þ ½27:51 15:7043:20½t2
ðB:1Þ
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