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Abstract

Cell voltage inconsistency of battery module is correlated with cell capacity fading inconsistency caused by uneven temperature or improper charge/discharge rate, so it is essential to study on cell voltage inconsistency when establishing a battery module capacity fade model. An accelerated life experiment is conducted on 12-series (12S) LiMn₂O₄ battery. The evaluation index of the voltage inconsistency is given, and the evolution of the voltage of the 12S battery module is obtained. Furthermore, a model of capacity fade of this 12S battery is established based on cell voltage inconsistency, which is extended to any-series battery by means of probability distribution. Based on this extended model, the relationship between the number of cells and the life of the battery is obtained.

Keywords

Battery accelerated life experiment capacity fade model cell voltage inconsistency probability distribution

Introduction

Lithium-ion batteries are promising power sources for electric vehicles considering their high energy, power density, and long cycle life.^1–3 However, the capacity of the battery fades with the increase in the mileage, which has great influence on the performance of electric vehicles. So, it is essential to study the capacity fade when designing the electric vehicle battery system.

When establishing a model for cell capacity fade, there are generally two methods: (1) one method is mainly based on chemical change inside cells, such as the loss of lithium-ion,⁴ the growth of solid–electrolyte interface (SEI) layer,⁵ and the variation law of the impedance of cells.^6,7 However, the aging mechanism is unclear by now and the aging-related chemical kinetic parameters cannot be measured by experiments. And these models cannot directly reflect how external stresses, such as ambient temperature and charge/discharge rate, influence the capacity fade. (2) The other method called empirical life model is mainly based on accelerated life experiments,^8–12 which directly reflects how external stresses influence the capacity fade, and is widely adopted in both academic researches and engineering use.

There have been substantial efforts focused on the development of models to predict capacity fade in lithium-ion batteries. Calendar life and cycle life were adopted to quantify the battery aging. The accelerated calendar and cycle life of lithium-ion cells were studied by Bloom et al.,¹¹ and the experiment data were fitted by the Arrhenius kinetics

Q = A e x p (\frac{- E_{a}}{R T}) t^{z}

(1)

That area-specific impedance (ASI) increase and power fades following (time)^1/2 rule in the calendar life experiments may be due to SEI growth. From the cycle life experiments, the ASI increase data follow (time)^1/2 kinetics also, but there is an apparent change when going from 2% to 6% $Δ SOC$ . Here, the power of time drops to a value less than 0.5,which indicates that the power fade mechanism is more complex than SEI growth. Wang et al.¹² studied capacity fading of LiFePO₄ cells at different temperatures, state of charge (SOC) ranges, depths of discharge (DOD), and discharge rates. Experimental results indicated that at the lower discharge rate, the DOD has little effect on capacity loss. So, they proposed a capacity degradation model following a power law relation with time or charge throughput. And the model corrected with Ah-throughput at high discharge rate can describe capacity loss behavior under different discharge rates better than Bloom’s model.¹¹ Ecker et al.¹³ compared the calendar life and cycle life of NMC/graphite lithium-ion batteries. They found that the capacity degradation was mainly affected by storage time and that the operation of battery helped to reduce the inner resistance. They believe that the calendar life prediction model is enough for battery performance prediction on hybrid electric vehicle. An empirical aging model based on the (time)^1/2 rule is given to predict the capacity loss. Saxena et al.¹⁴ built a power law model for capacity fade of graphite/LiCoO₂ cells under different SOC ranges. But the model is not suitable for dynamic SOC cycling and did not consider discharge current influence. A semi-empirical cycle life model was developed using experimental data by adopting a simple time power law and Arrhenius kinetics described by Bloom et al.,¹¹ but they got a higher power law factor (1.3) compared to the value of 0.5 associated with the general (time)^1/2 rule based on the rate of SEI growth. Intelligent prediction method for battery capacity is also developing, which is also data-driven model but not based on Arrhenius kinetics. A capacity estimation method considering particle filter (PF) and sample entropy was given. Discharge rate and ambient temperatures were also studied, but the method is not fit for dynamic cycle because the ambient temperature and discharge rate are constants in one cycle.¹⁵ Cell inconsistency deteriorates battery module performance. A battery pack capacity estimation method using the measured voltage-to-capacity curve transformation to overlap the standard one provided by the manufactory is proposed.^16,17 Cell life is much longer than battery module life. In the series connection of non-uniform cells, the total capacity of the battery is governed by a cell with the smallest capacity.¹⁸ So, cell inconsistency shall be taken into account when establishing battery module life prediction model.

Most of the results of the above focus on the (time)^1/2 rule based on the data fitting. But this rule is not suitable for all external stresses, especially under high discharge rate,¹⁹ and cell inconsistency is seldom considered when establishing the capacity fade model. In this article, the accelerated life test of battery was conducted, and temperature and discharge rate are used to establish 12-series battery capacity fade model. On this basis, the battery module capacity fade model is established considering the cell voltage inconsistency, which is extended to the life prediction model of multi-series battery modules by probability distribution. The relationship between battery module life and cell numbers is clarified.

Capacity fade model under accelerated stress

General directives for battery capacity fade model

Experiments on 12-series LiMn₂O₄ are conducted under conditions of high ambient temperature, normal temperature, high discharge rate, and normal discharge rate. And unknown parameters of capacity fade model can be fitted using the experimental results. Then, the model is established, which directly reflects how ambient temperature or discharge rate or both influence capacity fade.

How battery capacity will fade under external stress can be described by Gauss degradation, that is, given as equation (2)

x (s, t) ~ N (u (s, t), σ^{2} (t))

(2)

where $x (s, t)$ is the cumulative degradation of capacity of t cycles caused by s stress, $u (s, t)$ is the one-cycle average degradation of capacity of tth cycle caused by s stress, and $σ^{2} (t)$ is the variance of cumulative degradation of capacity of the tth cycle.

In the accelerated life experiment, the experimental data can be used to fit the appropriate regression model, which describes the process of accelerated capacity loss under different external stresses. By method of likelihood estimation, the exact amount of capacity loss is calculated from experimental data. And the estimate of characteristic value of this model is $\hat{u} (s, n)$ , that is, also the estimate of average degradation.

A curve-fitting and a regression analysis are performed on $\hat{u} (s, n)$ calculated from experimental data to acquire a function of battery capacity fade given as equation (3)

u (s, t) = f_{i} (s) t^{h_{i} (s)}

(3)

where s is the external stress and it can be temperature T ( $i = 1, h_{1} (T)$ is a constant) or discharge rate ( $i = 2, h_{2} (R_{d})$ is a constant), and $f_{i} (s)$ is called degradation rate function. The establishment of capacity fade model under the external stress of ambient temperature or discharge rate or both is based on equation (3).

Capacity fade model under different ambient temperatures and discharge rates

According to Arrhenius model,²⁰ the fade rate influenced by ambient temperature (stress T) is given as equation (4)

f_{1} (T) = Aexp (\frac{- E_{a}}{RT})

(4)

To make $G = \ln A$ , let $W = - E_{a} / R$ , then equation (5) can be obtained

f_{1} (T) = \exp (G + \frac{W}{T})

(5)

So, under the external stress of ambient temperature (stress T), the battery capacity fade model is given as equation (6)

u (T, t) = f_{1} (T) \cdot t^{h_{1} (T)} = \exp (G + (\frac{W}{T})) \cdot t^{h_{1} (T)}

(6)

The accelerated degradation factor of the stress T is given as equation (7). When temperature is external stress, SEI growth is the main cause of battery aging. It follows the Arrhenius law, however, under the external stress of discharge rate (stress $R_{d}$ ), there are some other mechanisms which play an important role during battery aging,¹¹ which do not follow Arrhenius law. So, a different exponential function is adopted instead of Arrhenius function to fit. The battery capacity fade model under external stress $R_{d}$ is given as equation (8). Where $D_{d}$ , cd, $E_{d}$ , and $h (R_{d})$ are constants. When the temperature and discharge rate are stresses, we assume that their effect on the battery aging is also compound; so, the battery capacity fade model based on both discharge rate and ambient temperature is given as equation (9)

ξ_{T_{2}, T_{1}}^{2} (t) = \frac{u (T_{1}, t)}{u (T_{2}, t)} = \exp (W (\frac{1}{T_{1}} - \frac{1}{T_{2}}))

(7)

u (R_{d}, t) = (D_{d} \cdot R_{d}^{cd} + E_{d}) \cdot t^{h (R_{d})}

(8)

\begin{matrix} u (T, R_{d}, t) = \exp (W (\frac{1}{T_{0}} - \frac{1}{T})) u (R_{d}, t) \\ = (D_{d} \cdot R_{d}^{cd} + E_{d}) \cdot t^{h (R_{d})} \cdot \exp (W (\frac{1}{T_{0}} - \frac{1}{T})) \end{matrix}

(9)

Experiment and model parameters identification

The 12-series 35-Ah LiMn₂O₄ battery is used in the accelerated life experiment, and the parameters of this battery are shown in Table 1.

Table 1.

Parameters of 35-Ah LiMn₂O₄ battery module.

Parameter	Value	Parameter	Value
Rated capacity (Ah)	35	Cell quantity	12
Cathode material	LiMn₂O₄	Rated voltage	44.4 V
Anode material	Graphite	Internal resistance	≤12.0 mΩ

As shown in Figure 1, other devices used in the experiment are Digatron EVT500-500 (for discharging or charging the battery), battery management system (BMS), and temperature chamber (to control the ambient temperature).

Figure 1.

The structure of the experiment.

The two factors of ambient temperature and discharge rate are included in the design of the accelerated life experiment. Discharge rate is held constant when testing the effects of ambient temperature on capacity loss and this part of the experimental design is shown in Table 2. Correspondingly, ambient temperature is held constant to test the effects of discharge rate, and this part of the experimental design is shown in Table 3. The SOC range is 0.3–0.8 during the test, and 15 min rest between discharge and charge.

Table 2.

Experimental design on temperature.

Temperature	Charge rate	Discharge rate	Cycle number
20°C	0.5C	1C	500
40°C	0.5C	1C	500

Table 3.

Experimental design on discharge rate.

Temperature	Charge rate	Discharge rate	Cycle number
20°C	0.5C	1C	500
20°C	0.5C	2C	500
20°C	0.5C	3C	500

When the ambient temperature is 20°C and 40°C and the discharge rate is 1C, the experimental results of battery capacity degradation are shown in Table 4. And the battery capacity fade model based on ambient temperature is given as equation (10). Set T = 293 and 313 K, we can get the capacity fade model for the experiment results. And, the factor of capacity loss caused by ambient temperature is given as equation (11)

u (T, t) = \exp (17.26 - (\frac{4218.61}{T})) \cdot t^{0.79}

(10)

ξ_{T_{2}, T_{1}}^{2} (t) = \exp (4218.61 (\frac{1}{T_{1}} - \frac{1}{T_{2}}))

(11)

Table 4.

Capacity loss in 20°C and 40°C.

Temperature (°C)	Cycle for 1 time (Ah)	Cycle for 50 times (Ah)	Cycle for 100 times (Ah)	Cycle for 150 times (Ah)	Cycle for 200 times (Ah)	Cycle for 250 times (Ah)	Cycle for 300 times (Ah)	Cycle for 350 times (Ah)	Cycle for 400 times (Ah)	Cycle for 450 times (Ah)	Cycle for 500 times (Ah)
20	35.47	35.11	34.83	34.58	34.34	34.12	33.91	33.70	33.50	33.31	33.12
40	35.24	34.33	33.62	32.99	32.40	31.84	31.31	30.79	30.29	29.80	29.33

When the ambient temperature is 20°C and the discharge rates are 1C, 2C, and 3C, the experiments are conducted for 500 times for each discharge rate. The relation between discharge rate and the amount of capacity degradation is shown in Figure 2. The battery capacity fade model based on discharge rate is given as equation (12). From equations (5), (9), and (10), the capacity fade model based on temperature and discharge rate is equation (13)

u (R_{d}, t) = (0.53 \cdot R_{d}^{3.375} + 16.96) \cdot t^{0.79}

(12)

u (T, R_{d}, t) = (0.53 \cdot R_{d}^{3.375} + 16.96) \cdot t^{0.79} \cdot \exp (4218.619 (\frac{1}{293} - \frac{1}{T}))

(13)

Figure 2.

Degradation quantity curves of different discharge rates.

There is big difference in capacity when the discharge rate changes and the initial capacity of different discharge rate can be calculated according to Peukert function²¹ as equation (14). The constants 35.01 and −0.038 are fitted results for Peukert function

C (R_{d}, 1) = 35.01 \cdot R_{d}^{- 0.038}

(14)

If the battery life ends when 20% of its capacity is lost, the lost capacity should be $0.2 C (R_{d}, 1) Ah$ according to equation (14), and the one-cycle average degradation of capacity of tth cycle caused by both stresses T and $R_{d}$ is $u (T, R_{d}, t) mAh$ according to equation (13); let $0.2 C (R_{d}, 1) = u (T, R_{d}, t)$ , then the following life-predicting equation (15) is obtained

L = {[\frac{35.01 \cdot R_{d}^{- 0.038} \times 0.2 \times 1000}{(0.53 \cdot R_{d}^{3.375} + 16.96) \cdot \exp (4218.619 (\frac{1}{293} - \frac{1}{T}))}]}^{1.27}

(15)

Furthermore, the number of cycles the battery can go through before its life ends under different conditions of varied ambient temperature and varied discharge rate is shown in Figure 3.

Figure 3.

Battery life map influenced by temperature and discharge rate.

Battery capacity fade model considering cell voltage inconsistency

Analysis of influencing factors of voltage inconsistency

The battery system in electric vehicles is composed of cells in series–parallel connection. Because of cell voltage inconsistency, the battery capacity loss cannot be calculated by just adding all the cell capacity loss. Through the analysis of the evolution of cell voltage inconsistency combined with the analysis of external stress in the accelerated life experiment, a comprehensive model of battery capacity fade can be established.

Two indexes are introduced to describe the degree of cell voltage inconsistency: (1) relative standard deviation of cell voltage and (2) relative range of cell voltage. The (1) index can be calculated in equation (16)

ζ_{σ} = {[\frac{\sum_{i = 1}^{n} {(V_{i} - \bar{V})}^{2}}{(n - 1) {\bar{V}}^{2}}]}^{1 / 2} \times 100 %

(16)

where n is the number of cells inside battery module, and $V_{i}$ is the voltage of ith cell, and $\bar{V}$ is the average cell voltage. The (2) index can be calculated in equation (17)

ζ_{Δ} = \frac{V_{\max} - V_{\min}}{\bar{V}} \times 100 %

(17)

One of the factors causing cell voltage inconsistency is initial cell voltage inconsistency, which is due to the manufacturing process. Other factors causing cell voltage inconsistency include discharge rate, temperature, and SOC, which are shown in Figures 4 –6, respectively.

Figure 4.

The change law of battery inconsistency in different discharge rates.

Figure 5.

The change law of battery inconsistency in different temperatures.

Figure 6.

The change law of battery inconsistency in different SOC ranges.

From Figures 4 –6, the consistency of the battery module is very poor at the later stage during discharge, and with the increase in the discharge rate, the consistency becomes worse. With the decrease in the temperature, the consistency of the battery pack becomes worse. Meanwhile, the consistency is better and more uniform when SOC is between 0.4 and 0.8.

In this article, when establishing the battery module capacity fade model based on the evolution of cell inconsistency, the effects of temperature and discharge rate on the battery capacity loss are analyzed with 50% DOD.

Battery capacity fade model based on cell voltage inconsistency

Based on the model established in the accelerated life experiment, battery capacity fade model based on cell voltage inconsistency is given as equation (18). In this model, the influence of inconsistency is considered as an aging factor, which is put in denominator

u' (T, R_{d}, t) = \frac{u (T, R_{d}, t)}{(1 - ζ_{1}) {(1 - ζ')}^{t - 1}}

(18)

where t is the number of cycles and $u (T, R_{d}, t)$ is the capacity fade model based on both factors of temperature and discharge rate. $ζ_{1}$ is the factor of initial cell voltage inconsistency and $ζ'$ is the factor of the evolution of cell voltage inconsistency.

According to the formula calculating the factor of capacity degradation caused by temperature in the accelerated life experiment, the formula calculating the factor of evolution of cell voltage inconsistency caused by temperature is deduced as equation (19)

λ_{T, T_{1}} = \frac{ζ (T)}{ζ (T_{1})} = \frac{ζ (T)}{ζ (293)}

(19)

The factor of cell voltage inconsistency caused by both temperature and discharge rate is calculated using the formula as equation (20)

ζ (T, R_{d}) = λ_{T, T_{1}} ζ (R_{d}) = \frac{ζ (T)}{ζ (293)} ζ (R_{d})

(20)

Taking the cumulative effect of capacity loss into consideration, equation (21) is obtained

(1 - ζ_{1}) (1 - ζ')^{n - 1} = ζ_{n} - ζ_{1}

(21)

where $ζ_{n}$ is the nth life cycle $ζ_{n} (T, R_{d})$ and $ζ_{1}$ is the first life cycle $ζ_{1} (T, R_{d})$ . Other factors in those formulas can also be calculated using the two indexes stated above describing the degree of cell voltage inconsistency.

Factor of initial cell voltage inconsistency $ζ_{1} (T, R_{d})$

When the discharge rate is 1C and the temperature values are 0°C, 20°C, and 40°C, the experiment based on 12-series battery is conducted to analyze how the temperature influences the initial relative standard deviation and initial relative range, and the results can be described in equation (22)

{\begin{matrix} ζ_{1, σ} (T) = 0.61239 - 0.00398 T + 6.5 \times 10^{- 6} T^{2} \\ ζ_{1, Δ} (T) = 2.1355 - 0.014 T + 2.28 \times 10^{- 5} T^{2} \end{matrix}

(22)

When the temperature is 20°C and the discharge rates are 1C, 2C, and 3C, the experiment based on 12-series battery is conducted to analyze how the discharge rate influence the initial relative standard deviation and initial relative range, and the results are described in equation (23)

{\begin{matrix} ζ_{1, σ} (R_{d}) = 0.0035 - 7.5 \times 10^{- 4} R_{d} + 3.5 \times 10^{- 4} R_{d}^{2} \\ ζ_{1, Δ} (R_{d}) = 0.0139 - 0.0067 R_{d} + 0.00235 R_{d}^{2} \end{matrix}

(23)

So, the factor of initial cell voltage inconsistency caused by both temperature and discharge rate can be calculated using the relative standard deviation and relative range

{\begin{matrix} ζ_{1, σ} (T, R_{d}) = \frac{ζ_{1, σ} (T)}{ζ_{1, σ} (293)} ζ_{1, σ} (R_{d}) \\ ζ_{1, Δ} (T, R_{d}) = \frac{ζ_{1, Δ} (T)}{ζ_{1, Δ} (293)} ζ_{1, Δ} (R_{d}) \end{matrix}

(24)

Based on experiments of 12-series, 48-series, and 120-series battery pack when discharge rate is 1C, initial relative standard deviation is 0.0030, 0.0031, and 0.0030, respectively, and the initial relative range is 0.0096, 0.0095, and 0.0096, respectively, so the influence of series number on initial cell voltage inconsistency is very small.

Evolutionary factor of cell voltage inconsistency $ζ' (T, R_{d})$

When discharge rate is 1C and the temperature values are 0°C, 20°C, and 40°C, the battery goes through 500 cycles to acquire the relation between temperature and relative range, which is described in equation (25)

{\begin{matrix} ζ_{500, σ} (T) = 1.031 \times 10^{- 5} - 9.801 \times 10^{- 8} T \\ + 2.311 \times 10^{- 10} T^{2} \\ ζ_{500, Δ} (T) = 1.059 \times 10^{- 5} - 1.29 \times 10^{- 7} T \\ + 3.752 \times 10^{- 10} T^{2} \end{matrix}

(25)

When temperature is 20°C and the discharge rates are 1C, 2C, and 3C, the battery also goes through 500 cycles to acquire the relation between discharge rate and relative standard deviation, which is described in equation (26)

{\begin{matrix} ζ_{500, σ} (R_{d}) = 1.354 \times 10^{- 6} - 6.25 \times 10^{- 8} R_{d} \\ + 1.185 \times 10^{- 7} {R_{d}}^{2} \\ ζ_{500, Δ} (R_{d}) = 2.8668 \times 10^{- 6} - 2.0835 \times 10^{- 6} R_{d} \\ + 0.501 {R_{d}}^{2} \end{matrix}

(26)

According to the calculation method of factor of initial cell voltage inconsistency, the factor of cell voltage inconsistency based on standard deviation and relative range can be calculated using equation (24) with the experimental data of the 500 cycles which is described in equation (27)

{\begin{matrix} ζ_{500, σ} (T, R_{d}) = \frac{ζ_{500, σ} (T)}{ζ_{500, σ} (293)} ζ_{500, σ} (R_{d}) \\ ζ_{500, Δ} (T, R_{d}) = \frac{ζ_{500, Δ} (T)}{ζ_{500, Δ} (293)} ζ_{500, Δ} (R_{d}) \end{matrix}

(27)

Then, the evolution factor $ζ'$ of inconsistency based on temperature and discharge rate can be calculated according to equation (20).

Capacity fade model based on probability distribution

The capacity fade model based on the 12-series battery can be extended to any-series battery by means of probability theory. In this article, the probability statistics method is applied to the voltage of battery cell, and the normal distribution method is used to establish the capacity fade model of any-series battery on the basis of reliability.

Assume that the battery module is connected in series with n cells and the initial capacity of ith cell is $C_{S} (i, 1)$ whose distribution function is $F_{S, 1} (c)$ , where S means series. The capacity degradation increases with the increase in the cycle number. Let the capacity of ith cell is $C_{S} (i, t)$ when the battery is going through tth cycle, where its capacity degradation is $x_{S} (i, t)$ . For the battery module, let the initial capacity is $C_{S} (1)$ , and after t cycles, the capacity degradation is $x_{S} (t)$ and the capacity is $C_{S} (t)$ .

The theoretical capacity of the battery module is the minimum of all cell capacities. As the initial capacity of the cell capacity is very close, the rate of degradation of battery cell is only influenced by cycle parameters during test, which cannot be influenced by initial capacity, and that is to say, $C_{S} (i, 1)$ and $x_{S} (i, t)$ are independent of each other. Assuming that $C_{S} (i, 1)$ and $x_{S} (i, t)$ follow normal distribution

C_{S} (i, 1) ~ N (u_{c 1}, σ_{c 1}^{2})

(28)

x_{S} (i, t) ~ N (u_{x}, σ_{x}^{2})

(29)

According to the distribution of independent random variables, equation (30) is obtained

C_{S} (i, t) = C_{S} (i, 1) - x_{S} (i, t) ~ N (u_{c 1} - u_{x}, σ_{c 1}^{2} + σ_{x}^{2})

(30)

For the battery with cells in series connection, assuming that the battery life ends when 20% of its capacity is lost, the failure value is $l_{i}$ and the failure condition is $x_{S} (t) \geq l_{i}$ , namely, $l_{i} = 0.2 C_{S} (1)$ . In practical application, the value of failure is generally determined in advance, so it is determined that $l_{i} = 0.2 E (C_{S} (1)) = 0.2 u_{c 1}$ , and $C_{S} (t)$ can be selected as the residual capacity of the battery. And failure value of $C_{S} (t)$ is assumed to be $l'$ and the failure condition is $C_{S} (t) \leq l'$ , where $l' = u_{c 1} - l_{i}$ , and the distribution of battery module life is described in equation (31)

F_{LS} (t) = P (L \leq t) = 1 - {[1 - ϕ (\frac{u_{x} - l_{i}}{\sqrt{σ_{c 1}^{2} + σ_{x}^{2}}})]}^{n}

(31)

where n is the number of cells and L is the cycle life. The reliability of this battery in series connection at moment t is

R_{S} (t) = 1 - F_{LS} (t) = {[1 - ϕ (\frac{u_{x} - l_{i}}{\sqrt{σ_{c 1}^{2} + σ_{x}^{2}}})]}^{n}

(32)

This model establishes the relation between life distribution and number of cells. When t is an integer, the probability of battery failure at k can be described in equation (33)

P (L = k) = F_{LS} (k + 0.5) - F_{LS} (k - 0.5)

(33)

So, the average life of this battery in series connection is

\begin{matrix} E (L) = \sum_{k = 1}^{+ \infty} P (L = k) \cdot k \\ = \sum_{k = 1}^{+ \infty} [F_{LS} (k + 0.5) - F_{LS} (k - 0.5)] \cdot k \end{matrix}

(34)

The model of 12-series battery life is built based on the experimental data of accelerated life experiment using the probability theory. The calculation process is as follows

u_{c 1} = \frac{1}{12} \sum_{i = 1}^{12} C_{S} (i, 1) = 35, 455.9

(35)

σ_{c 1} = \frac{1}{12} \sum_{i = 1}^{12} {(C_{S} (i, 1) - u_{c 1})}^{2} = 20, 035.67

(36)

C_{s} (i, 1) ~ N (35, 455.9, 20, 035.67)

(37)

When the temperature is 20°C and the discharge rate is 0.5C and 1C, the distribution factor of battery capacity fade is

u_{x} = 17.49 t^{0.79}

(38)

σ_{x}^{2} = 200.9 t + 1509

(39)

x_{s} (i, t) ~ N (17.49 t^{0.79}, 200.9 t + 1509)

(40)

Assuming that battery life ends when it lost 20% of its capacity, the initial capacity is

35.01 \cdot R_{d}^{- 0.038} = 35, 010 mAh

(41)

And, the reliability of battery after t cycles is

R_{S} (t) = 1 - F_{LS} (t) = {[1 - ϕ (\frac{17.49 t^{0.79} - 35010 \times 0.2}{\sqrt{200.9 t + 21544.67}})]}^{n}

(42)

When the battery is composed of 12, 48, 120, 204, and 240 cells in series connection, the relation between reliability and battery life is shown in Figure 7.

Figure 7.

The relationship between series battery pack reliability and battery number, under the condition of 20°C and 1C discharge.

According to battery life model based on probability theory and capacity decline model based on cell voltage inconsistency, the model of capacity fade of any-series battery is established as follows

u' (T, R_{d}, t) = η \frac{u (T, R_{d}, t)}{(1 - ζ_{1}) {(1 - ζ')}^{t - 1}}

(43)

where $η$ is the ratio of life of any-series battery which can be calculated from equations (31) and (33) and the life of 12-series battery when reliability is 50%.

Result analysis from different models

According to equation (13) of capacity fade model based on accelerated life experiments and equation (18) of capacity fade model based on cell voltage inconsistency (including two conditions of standard deviation and relative range), the characteristics of capacity fade are analyzed based on 12-series battery—see operating modes 1, 2, and 3:

Operating mode 1: 20°C, 1C discharge, 0.5C charge, 300 times, 500 times, 1000 times, 1500 times, and the capacity loss of battery in the end of life are shown in Table 5. From Table 5, we can see that the capacity of the battery in different life cycles can be obtained by the model, and the maximum battery life is acquired using the model based on the accelerated life experiment, and the minimum life acquired using the model based on relative range. Take 1500 cycles as an example, the capacity loss calculated by capacity fade model based on relative range is 0.6% higher than the model based on accelerated life experiment and 0.4% higher than the model based on standard deviation. Therefore, capacity loss is higher because of battery inconsistency and inconsistency described by relative range is much harsh than by standard deviation.

Operating mode 2: 20°C, 2C discharge, 0.5C charge, 300 times, 500 times, 1000 times, the relation between reliability and battery life is shown in Figure 8, and the capacity loss of battery in the end of life are shown in Table 6. It can be seen when compared with operating mode 1 that with the increase in the discharge rate, the battery life shortens. For example, in the 1000th cycle, capacity fade rate calculated by model based on standard deviation rises from 11.8% in operating mode 1 to 15.5% in operating mode 2. From Table 6 and Figure 8, it can be seen that the life cycles is about 1400 times calculated by model based on probability distribution when reliability is 80%,which is very close to the result from the model based on inconsistency.

Operating mode 3: 40°C, 1C discharge, 0.5C charge, 300 times, 500 times, the relation between reliability and battery life is shown in Figure 9, and the capacity loss of battery in the end of life is shown in Table 7. It can be seen that the capacity severely declines when the temperature is 40°C. For example, in the 300th cycle, the capacity fade rate calculated by model based on standard deviation rises from 4.54% when temperature is 20°C to 11.5% when temperature is 40°C. From Table 5 and Figure 9, it is obvious that the life cycle is almost half of operating mode 1. Therefore, high temperature plays a very important role in battery capacity fade.

Table 5.

Comparisons of capacity fade models under work condition 1.

Cycle times	Capacity fade model based on accelerated life experiment		Capacity fade model based on standard deviation of cell voltage		Capacity fade model based on relative range of cell voltage
	Deterioration capacity (Ah)	Deterioration ratio (%)	Deterioration capacity (Ah)	Deterioration ratio (%)	Deterioration capacity (Ah)	Deterioration ratio (%)
300	1.58	4.52	1.59	4.54	1.63	4.65
500	2.37	6.77	2.38	6.80	2.44	6.96
1000	4.10	11.71	4.13	11.80	4.23	12.1
1500	5.65	16.1	5.70	16.3	5.84	16.7
Life	2019 times		1946 times		1884 times

Figure 8.

The relationship between series battery pack reliability and battery number, under the condition of 20 °C and 2C discharge.

Table 6.

Comparisons of capacity fade models under work condition 2.

Cycle times	Capacity fade model based on accelerated life experiment		Capacity fade model based on standard deviation of cell voltage		Capacity fade model based on relative range of cell voltage
Cycle times	Capacity loss (Ah)	Capacity loss ratio (%)	Capacity loss (Ah)	Capacity loss ratio (%)	Capacity loss (Ah)	Capacity loss ratio (%)
300	2.03	5.96	2.04	5.99	2.09	6.12
500	3.04	8.9	3.06	9	3.13	6.96
1000	5.26	15.4	5.3	15.5	5.42	15.89
Life	1421 times		1374 times		1310 times

Figure 9.

The relationship between series battery pack reliability and battery number, under the condition of 40°C and 1C discharge.

Table 7.

Comparisons of capacity fade models under work condition 3.

Cycle times	Capacity fade model based on accelerated life experiment		Capacity fade model based on standard deviation of cell voltage		Capacity fade model based on relative range of cell voltage
Cycle times	Capacity loss (Ah)	Capacity loss ratio (%)	Capacity loss (Ah)	Capacity loss ratio (%)	Capacity loss (Ah)	Capacity loss ratio (%)
300	3.97	11.35	3.99	11.5	4.08	11.64
500	5.95	16.9	5.98	17.0	6.11	17.45
Life	628 times		611 times		594 times

In the above models, the capacity degradation and life of 12-series battery can be acquired quantitatively, but these models do not apply to any-series battery. However, using equation (31), based on probability statistics, the life of any-series battery can be calculated, but it cannot analyze capacity degradation quantitatively. So, combine them to create equation (43) to acquire the characteristics of capacity degradation of any-number series battery.

Figure 10 shows the life of any-series battery under the above three operating modes calculated by model based on probability distribution. It can be seen that 240S battery goes through 320 less cycles than 48S battery in operating mode 1, indicating a 17.8% decrease, and 110 less cycles in operating mode 3, indicating a 17% decrease.

Figure 10.

Life of series battery modules calculated by probability statistical model.

Table 8 shows the characteristics of capacity loss based on a combined model consisting of model based on standard deviation and model based on probability distribution. It can be seen that cycle number of 240S battery is 1457 times, indicating a 25.1% decrease when compared with 12S battery. So, the number of cells in series connection is also an important factor in determining the battery life. This may due to consistency become worse when the battery cell number increase. So, the discharge rate, temperature, and number of cells in series should be in a proper range when designing a battery pack.

Table 8.

Comparisons of battery modules capacity fade.

Cycle times	12S battery module		204S battery module		240S battery module
Cycle times	Capacity loss (Ah)	Capacity loss ratio (%)	Capacity loss (Ah)	Capacity loss ratio (%)	Capacity loss (Ah)	Capacity loss ratio (%)
300	1.59	4.54	1.92	5.63	1.99	5.75
500	2.38	6.80	2.90	8.36	2.96	8.53
1000	4.13	11.80	5.0	14.46	5.16	14.75
1500	5.70	16.3	6.99	20.1	7.13	20.8
Life	1946 times		1488 times		1457 times

Conclusion

Based on the experiment of 12S LiMn₂O₄ battery, life-predicting models of battery are established taking temperature or discharging rate or both of them into consideration. Relative standard deviation of cell voltage and relative range of cell voltage are introduced to describe cell voltage inconsistency. Another two models, one based on relative range of cell voltage and the other based on standard deviation of cell voltage, are established to analyze the capacity fade of battery. The results of models are compared with each other through experiments, which show that maximum battery life is acquired using model based on accelerated life experiment and the minimum life acquired using the model based on relative range. The relative difference is less than 0.6%.With the increase in the discharge rate or temperature, the battery life shortens. Furthermore, a life-predicting model based on probability distribution is established to explain the relation between number of cells in series connection and the life of battery, which shows that the more the number of cell in series, the shorter the battery life. 240S battery life decreases 25.1% compared with 12S battery. It is clear that battery life is influenced by both number of cells in series connection and cell voltage inconsistency, so a model incorporating the two is good at predicting the life of battery.

Footnotes

Appendix 1

Academic Editor: Jose Antonio Tenreiro Machado

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the National Key R&D Program of China under the contract of No. 2016YFB0100305.

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Battery module capacity fade model based on cell voltage inconsistency and probability distribution

Abstract

Keywords

Introduction

Capacity fade model under accelerated stress

General directives for battery capacity fade model

Capacity fade model under different ambient temperatures and discharge rates

Experiment and model parameters identification

Battery capacity fade model considering cell voltage inconsistency

Analysis of influencing factors of voltage inconsistency

Battery capacity fade model based on cell voltage inconsistency

Factor of initial cell voltage inconsistency ζ 1 ( T , R d )

Evolutionary factor of cell voltage inconsistency ζ ′ ( T , R d )

Capacity fade model based on probability distribution

Result analysis from different models

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References

Factor of initial cell voltage inconsistency $ζ_{1} (T, R_{d})$

Evolutionary factor of cell voltage inconsistency $ζ' (T, R_{d})$