Estimation of the residue capacity of lithium iron phosphate battery based on the internal resistance obtained from charging voltage drop

This study takes the 80 Ah lithium iron phosphate (LFP) prismatic battery that is from the vehicle and is in the middle or end of life as the research target, and the voltage-drop resistance (VDR), which is calculated through the voltage drop at the charging end, is used to for residue capacity estimation. The relationship between the VDR and the residue capacity of the battery is systematically studied, including factors such as charging cut-off voltage, charging current, charging end time, and charging current change mechanism. The results show that the residue capacity is a quadratic function relationship with the charging cut-off voltage and VDR. The VDR is affected by the charging cut-off voltage, charging cut-off current, and charging end time. There is no significant difference in the influence of VDR obtained by the charging modes of step charging and constant current charging. By testing the VDR under different charging conditions and different resting times, and standardizing the VDR, the residue capacity estimation error of 97% of the vehicle’s batteries is within ±5%, which meets the application requirements for residue capacity estimation.


Introduction
Residual capacity is one of the important parameters for the lithium-ion battery.With the aging of lithium-ion batteries, the residue capacity is gradually decreasing, affecting the range and safety of electric vehicles [1][2] .Therefore, the estimation of the residue capacity of the whole vehicle has always been the focus of attention of electric vehicle enterprises.Lithium iron phosphate battery (LFP) is widely used in electric vehicles and energy storage due to its advantages of good safety performance, long cycle life, low price, and low environmental pollution [3][4][5] .However, the voltage platform of the LFP battery is very flat, and its SOC (state of charge) estimation and residual capacity estimation are the focus and difficulty in the industry [6][7] .
The internal resistance of the battery is a parameter to measure the difficulty of carrier movement in the electrode.Numerous experiments show that the internal resistance is closely related to the battery residual capacity [8][9][10] .Generally, the discharge capacity of a battery with a small internal resistance is higher, while the discharge capacity of a battery with a large internal resistance is lower.Especially in the middle or end-of-life stage, the increase of the internal resistance and the decline of the life of the battery are more significant [11] .The internal resistance of the battery is affected by the SOC, current, temperature, and aging degree.By establishing the equivalent model or electrochemical model of the internal resistance of the battery and obtaining the corresponding parameters, the prediction relationship between the internal resistance and the capacity can be obtained [12] .Compared with the capacity estimation method based on the electrochemical model, the performance fitting estimation method based on experimental or vehicle operation data can grasp the internal relationship between battery capacity, battery health status, and key parameters without an in-depth understanding of battery geometric structure, material system and internal mechanism of capacity decay, and use the relationship to estimate the evolution trend of battery residual capacity.
In this study, the 80 Ah LFP prismatic battery from the vehicle that has been run for more than hundreds of thousands of kilometers is taken as the research subject, and the VDR at the charging end is proposed to estimate the residue capacity.The quantitative relationship between the VDR and the residue capacity of the battery is systematically studied.The factors considered include the charging cut-off voltage, charging current, charging end time, and step charging mechanism.The results show that the residual capacity is a quadratic function of the charge cut-off voltage and the VDR.The VDR is affected by the charge cut-off voltage, the charge cut-off current, and the end time of the charge.The variable current charging mechanism in the charging process has little effect on the VDR.By testing the VDR under different charging conditions and different standing times and carrying out standardization treatment on the VDR, the estimation deviation of the residual capacity of 97% of the battery monomers of the whole vehicle is within ±5%, which meets the application requirement of the estimation of the residual capacity.

Battery sample and capacity calibration
To avoid the single working condition of the sample, there are 20 batteries in the experiment, which come from buses in three different cities in China, including Xiamen (7), Beijing (7), and Chongqing (6).After long-term operation of the bus, the battery capacity has different degrees of attenuation.These batteries are produced by a domestic power battery company, all of which are LFP prismatic batteries with a rated capacity of 80 Ah. 1/3C 0 current is used at both the charge and discharge process at the ambient temperature of 25℃ ± 2℃ for capacity calibration, and C 0 is taken as 80 A. The battery charging and discharging instrument is the Sunway XW10005C16, each device has 16 standard channels, the limit current of a single channel is 100 A, the limit voltage is 5 V, the current detection accuracy is 0.001 A, and the voltage detection precision is 1 mV.The sampling interval is 1 s.The constant capacity results are shown in Table 1.The residue capacity interval of 20 batteries is [61.6 Ah, 71.4 Ah], and the corresponding capacity retention rate is [79%, 89%], indicating that the batteries are at the middle or end of life, and also the interval state that needs special attention in practical application.

Relationship between charge cut-off voltage, VDR, and residual capacity
The internal resistance of lithium-ion batteries is closely related to SOC, especially at the end of charging, the change of internal resistance is more obvious.Figure 2 fits the relationship between the VDR and residue capacity at different charging cut-off voltages (the charging current is 1/3 C 26.67 A), and it can be seen that the VDR and the residue capacity are highly correlated.To obtain the functional relationship among the VDR, the charging cut-off voltage, and the residual capacity, the curve fitting toolbox in Matlab TM is used for fitting analysis, and it is found that the relationship among the VDR, charge cutoff voltage, and residual capacity meets Equation ( 1): (1) Where is the VDR and is the charge cut-off voltage, indicating that the residue capacity can be estimated if the VDR and charge cut-off voltage are known.

Effect of charging current and standing time after charging on VDR
Considering that the charging current and the standing time after charging vary greatly in the actual application of the vehicle, R (I=26.67 A, t=40 s) is defined as the standard state internal resistance.By studying the relationship between the VDR under other different conditions and the standard state voltage drop internal resistance R (I=26.67 A, t=40 s) , the VDR under different conditions is transformed into the standard state internal resistance R (I=26.67 A, t=40 s) , facilitating the estimation of the residue capacity. Figure 3a is the relationship between the internal resistance R (I=7.5 A, 15 A, 40 A, 50 A, t=40 s ) of the voltage drop at the charging end and the internal resistance R (I=26.67 A, t=40 s) of the voltage drop at the standard state of the same sample under the same cut-off voltage, the same standing time and different charging currents, which is obtained by using the Matlab TM Curve fitting toolbox.It can be seen that there is a linear relationship between the non-standard current internal resistance and the standard current internal resistance, which can be fitted by the linear expression of R (I) = a * R (26.67 A) + B, that is, the non-standard current internal resistance can be converted into the standard current internal resistance.Figure 3b shows that the fitting coefficient a is linearly and positively correlated with the current, while the value of the coefficient B is negatively correlated with the current.Therefore, the conversion Equation ( 2) between the internal resistance of the voltage drop at different currents and the VDR at the current of 26.67A is obtained: (2) The VDR under different charging currents can be converted into the voltage drop internal resistance under 1/3C 0 standard charging by using the relation in Equation ( 2), thereby facilitating the horizontal comparison of the VDR.
Due to the influence of the electrochemical polarization and concentration polarization, the VDR increases exponentially with time.Therefore, the relationship between the VDR at the end of charging and the standing time of each sample under the same charging system is further studied, and the results are shown in Figure 3c-d.It shows that when the standing time is between 20 and 100 s, the VDR of all samples is linearly related to the voltage drop internal resistance when the standing time is 40 s, and the internal resistance conversion Equation (3) under different standing times is obtained by further fitting the correlation coefficient: (3) To prevent the fluctuation of voltage acquisition, Equation (3) can be used to estimate the residue capacity by calculating the average VDR for a period of time, reducing the interference of voltage noise.

Impact of the step charging
To give consideration to both fast charging and life management of electrical vehicles, the step charging mode is generally adopted.At present, the large current is used to charge to 80% -95% SOC, and then the current is reduced to the end of constant current charging [13] .Because the VDR at the charging end is related to the charging state of the battery, whether the current change in the charging process will affect the internal resistance of the voltage drop needs further verification.Figure 4 compares the difference of the VDR under the two different charging modes of step charging and constant current charging.The first group of step charging is to charge the battery to 80% SOC with a current of 40 A, and then to 100% SOC (3.65 V) with a current of 7.5 A. The remaining two groups of step charging are to charge the battery to 90% SOC and 95% SOC respectively with a current of 40 A, and then charge the battery to 100% SOC (3.65 V) with a current of 7.5 A. The battery is charged to 100% SOC (3.65 V) with a standard charging current of 7.5 A. The results show that there is no significant difference between the VDR under step charging mode and the constant current, which means that even if the whole vehicle adopts the step charging mode, the current at the end of charging can be used as the estimation data of the VDR.

Verification based on vehicle operation data
To verify the reliability of this method, 100 LFP batteries are disassembled from the battery pack of the whole vehicle, the VDR is tested and the residual capacity is estimated according to the above method, as shown in Figure 5.The column height in Figure 5b represents the proportion of battery cells in different error intervals, and the dotted line data represents the cumulative growth of cells from -6% error to 5% error interval.The results show that the deviation between the real residual capacity and the estimated residual capacity of 97% of the batteries is within ± 5%, and the error of the residual capacity of most of the batteries is between -2% and + 5%, which meet the application requirements of residual capacity estimation.The method is simple to operate and can be used for estimating and correcting the residual capacity of the BMS battery at the vehicle end.

Conclusion
In this study, the 80 Ah LFP prismatic battery in the middle or end of life is taken as the research subject, the residual capacity is estimated by the VDR, and the quantitative relationship between the internal resistance of the battery and the residual capacity is systematically studied.At the same time, the influence of the charge cut-off voltage, the charge current, the charge end time, and the charge current change mechanism on the relationship between the VDR and the residue capacity is investigated.At room temperature, the charging cut-off voltage, charging cut-off current, and charging end time affect the VDR, and there is no significant difference between the VDR obtained by the step charging mode and the constant current charging mode.The method realizes the standardized treatment of the VDR by testing the VDR under different charging conditions and standing times and finally completes the capacity estimation of the whole vehicle under different use conditions, and the method can realize that the error of more than 97% of samples is within ±5%.

Figure 1
shows the charging curves and the change of the voltage at the end of charging of the same sample at different charging currents and different charging cut-off voltages.It can be seen that the VDR at the end of the charge of the same sample is consistent with the previous internal resistance model, and the VDR is related to the current, the charge cut-off voltage, and standing time, so the control variable method is used to study the corresponding relationship between the VDR and the residue capacity under different charge systems, and ultimately achieves the purpose of estimating the residue capacity through the VDR.

Figure 1 .
Figure 1.Static characteristics of charging terminal voltage with different charging cut-off voltages and charging currents.

Figure 2 .
Figure 2. VDR-residue capacity fitting relationship under different charging cut-off voltages.

Figure 3 .
Figure 3. (a) The conversion relationship of VDR with different charging currents; (b) The relationship between the parameters in R(I)-R(26.67A) fitting equation and current I; (c) The VDR transformation relationship at different times; (d) The relationship between the parameters in R(t)-R(40 s) fitting equation and time t.

Figure 4 .
Figure 4. Comparison of VDR under variable current and constant current test conditions.

Figure 5 .
Figure 5. (a) Comparison of actual residue capacity and estimated residue capacity of batteries from electrical vehicle; (b) Capacity estimation error statistics.

Table 1 .
The Capacity Distribution of 20 Batteries Used in the Experiment.Galvanostatic charge test.To simulate the different charging states of the whole vehicle during use, the samples after capacity calibration shall be charged according to the matrix listed in Table2.In order to prevent test fluctuation, each data point is tested twice.To detect whether the capacity is attenuated, the sample capacity is re-calibrated after each set of current tests.According to Table2, 20 cells are tested for constant current charging at different currents, and the data of the standing process are obtained.

Table 2 .
Battery Constant Current Charging Test Matrix.Step charging test.In order to compare the difference of the VDR under the two different charging modes of step charging and constant current charging, one battery from each of the six different residue capacity sections in Table1(6 batteries in total) is taken for the comparison test.In the step charging mode, the battery is charged to 80%, 90%, and 95% SOC respectively with a current of 40 A. Then the battery is charged to 100% SOC (3.65 V) with a current of 7.5 A. The data of the charging, discharging, and standing processes are recorded, and the internal resistance of voltage drop under 100% SOC is calculated.In the constant current charging mode, the battery is charged to 100% SOC (3.65 V) at a constant current of 7.5 A. The charge-discharge and standing data are also recorded, and the VDR at 100% SOC is calculated.