Life cycle testing and reliability analysis of prismatic lithium-iron-phosphate cells

ABSTRACT

A cell’s ability to store energy, and produce power is limited by its capacity fading with age. This paper presents the findings on the performance characteristics of prismatic Lithium-iron phosphate (LiFePO₄) cells under different ambient temperature conditions, discharge rates, and depth of discharge. The accelerated life cycle testing results depicted a linear degradation pattern of up to 300 cycles. Linear extrapolation reveals that at 25°C temperature, an increase in the discharge rate from 0.5 C to 0.8 C reduces the cycle life significantly by 52.9%. On the other hand, at a constant discharge rate, an increase in temperature reduced predicted cycle life in the range of 23.2–41.36%. Lithium-ion cells’ reliability modeling and analysis was carried out using an exponential distribution showing the increasing failure rate with age, with the temperature significantly reducing the expected life of the cells.

KEYWORDS:

• Lithium iron phosphate (LiFePO4) cells
• life cycle testing
• discharge rate
• temperature degradation
• reliability
• exponential distribution

1. Introduction

Lithium-ion batteries (LIBs) are popular due to their higher energy density of 100–265 Wh/kg, long cycle life (typically 800–2500 cycles) relative to lead-acid batteries (Ma et al. 2018). They are used in medium-power traction applications (such as robotics, e-mobility, last-mile delivery, etc.) and heavy-duty traction applications (such as industrial vehicles, marine traction, etc.). Further, the possibility of deep cycling (up to 90% Depth of Discharge (DoD)) makes it possible to use lithium-ion batteries for stationary energy storage applications (Ma et al. 2018). Several operating parameters affect the life of lithium-ion cells, such as cell type and its form factor, environmental temperature, charging and discharging rates, and depth of discharge (Ran et al. 2014; Xiong and O’Connell 2019). All these factors contribute to a gradual loss in cell capacity and higher heat generation rates resulting in a thermal runaway.

The operating temperature has a vital effect on the discharge capacity of the cells (Bandhauer et al. 2011; Sagare et al. 2023). An operating temperature below 25°C diminishes the ionic conductivity of lithium salt-based electrolytes due to increased viscosity (Ma et al. 2018). Temperature lower than the optimal reduces the internal resistance and electrochemical reaction speed, while increasing internal polarization resistance reduces the discharge capacity and energy output (Lv et al. 2021). On the other hand, a temperature greater than 25°C increases the rate of the chemical reaction and helps in faster discharge (Lv et al. 2021). However, continual operation at high temperatures reduces the cycle life by a 1.9% per degree rise in temperature at a 1.8 C rate (Bandhauer et al. 2011). The rise in resistance during charge transfer in LIBs is another key concern contributing to low-temperature performance loss. Zhang et al. (Zhang, Xu, and Jow 2003) demonstrated that the charge-transfer performance of lithium-ion batteries at low temperatures can be improved. If the ambient temperature continuously rises to 60°C, there is a rupture in the lattice of the cathode material, which results in an irreversible drop in the battery capacity. Hence, the use of LIBs at such high temperatures should be avoided. Considering cycle life and discharge efficiency, the most suitable operating temperature of the LIB is 20–50°C (Lv et al. 2021).

Charging and discharging rates govern the amount of internal heat generation inside the lithium-ion cell. This is because the components required for the reaction to occur, do not always have enough time to arrive at their required locations. The recommended charging rate for lithium-ion cells ranges from 0.5 to 1 C (Honsberg and Bowden 2020), with a full charge time of 2–3 h. More energy may be extracted from the battery if it is drained using a low current and a very slow pace, prolonging the battery’s life. Therefore, the battery’s capacity should take into account the rates of charging and discharging (Honsberg and Bowden 2020).

Apart from the operating temperature and charging/discharging rates discussed in the previous section, the battery’s DoD is of paramount importance. For cycle life testing, 80% depth of discharge is recommended. A lithium-ion cell’s cycle life increases as its DoD reduces. Cycling at a lower DoD extends the battery’s cycle life, reduces capacity fading, and slows the changes in the shape of the discharge curves that occur during reference full cycles (Thakur et al. 2020). These alterations are almost definitely produced by stress, which results in local damages during phase transitions at 2.55 V potential. By lowering the DoD, this phase change and the tension that comes with it can be avoided (An, Chen, and Yang 2010). Due to the anode material’s incomplete activation during the early cycles, the battery’s discharge capacity improves a bit. The electrolyte gradually seeps into the inside of the electrode material as the cycle continues. The lithium ions then travel to the inner of the electrode material and then undergo a reversible deintercalation reaction, thus increasing the battery’s capacity. From the factors discussed above, temperature control is crucial in the operation and life of lithium-ion cells. If the temperature is not controlled, it may trigger thermal runaway, a condition where high temperatures promote exothermic reactions in functioning batteries (Liao et al. 2019; Spotnitz and Franklin 2003). These processes produce more heat, accelerating the temperature rise within the batteries. Fire and explosion may occur if such uncontrolled heat generation exceeds the thermal endurance of the batteries (Bandhauer et al. 2011; Spotnitz and Franklin 2003).

Mathematical models of battery capacity fade are useful tools to predict the aging process, thereby enabling accurate determination of cycle life. Electrochemical and machine learning-based models have been used to predict battery degradation and life under various operating conditions. The thermo-electrochemical properties of prismatic lithium-ion batteries were computationally simulated by Kwon and Park (2019) to understand deterioration and thermal runaway across a wide temperature range of 40–80°C and 150°C. Authors report that subzero temperatures greatly reduce charge and discharge efficiency. Nonetheless, at temperatures over 40°C, a significant drop in energy storage capability is expected. When operating at 80°C, the battery may retain 10% of its nominal energy storage capacity after 1000 charge–discharge cycles. Yang et al., (2020) discuss the prediction of lithium-ion battery lifespan using a gradient boosting regression tree (GBRT) model and 72 constructed features from raw battery data. It is shown that the said machine learning model outperforms decision tree, support vector machine (SVM), Random Forest (RF), and Gaussian process regression (GPR) models with an absolute mean average percentage error of roughly 7%. Sureshkumar et al. (2023) report an aging study of a lithium-ion ferrous phosphate prismatic cell for the development of a BMS for the optimal design of battery management systems. The single particle model (SPM) approach was used to analyze battery behaviour during charge–discharge profiles at 0.5, 1, and 2 C ratings. Study identifies the cause of irreversible heating and rise in resistance as temperature driven. Han et al. (2019) outlines the role of loss of lithium-ion inventory, loss of cathode/ anode active material, loss of electrolyte and resistance increment in the degradation which cause capacity fade and power fade in lithium ion cells. The difference between the ageing mechanism in a cell and battery pack is discussed. Though majority of the work reported use simulation models, experimental validation of data predicted using such model is vital to provide confidence in the results.

Many firms are involved in the mass production of lithium-ion cells, and variation in the heat generation characteristics is likely from one manufacturer to the other. It becomes necessary to validate the specifications of the cells and batteries as provided by the manufacturer, as they may differ from actual field data. Testing of cells is therefore mandatory to determine the performance of cells as claimed by manufacturing organizations and provide valuable data otherwise unavailable in open-source literature. From this research, it is shown that an increase in discharge rate from 0.5 to 0.8 C at a temperature of 25°C reduces cycle life significantly by 52.9%, whereas at a temperature of 45°C, the degradation rate with an increased discharge rate was relatively less which is about 38 percent. With increased temperature, the anticipated cycle life decreases by 23–42% at a fixed discharge rate. Such results are valuable for cell selection for a designated application and provide confidence to the battery pack designer/integrator.

2. Methodology

This research is aimed at studying the effect of operational parameters such as ambient temperature, discharge rate, and DoD on the performance of the Li-ion cell by carrying out the cycling test on the cells. However, before conducting a cycling test it is necessary to adjudge the quality of the cells so that cells with internal defects are not tested extensively. To decide which cells from a given sample of 8 cells would be subjected to the cycling test, capacity test, Open Circuit Voltage (OCV), and Direct Current Internal Resistance (DCIR) tests are carried out. The detailed procedure and the sequence in which they are undertaken for testing of cells are elaborated below.

2.1. Cell selection

The lithium iron phosphate battery, also known as the LFP battery, is one of the chemistries of lithium-ion battery that employs a graphitic carbon electrode with a metallic backing as the anode and lithium iron phosphate (LiFePO₄) as the cathode material. Compared to Nickel-Manganese Cobalt oxide (NMC) cells, lithium ferro phosphate (LFP) cells typically have a longer life cycle but relatively lower specific energy. This technology is employed in several applications due to its high specific energy and extended cycle life. Lithium iron phosphate batteries can be used in energy storage applications (such as off-grid systems, stand-alone applications, and self-consumption with batteries) due to their deep cycle capability and long service life. Test results from (Hato et al. 2015) indicate that the capacity loss increases at high temperatures and state of charge (SOC) conditions, and the amount of change in SOC is not related to the loss.

To advance lithium extraction and restoration techniques for enhancing the characteristics of lithium-ion battery cathode, several novel methodologies have emerged in recent research. Du et al. (2023) have developed a novel method for extracting lithium from brine, using wasted LiFePO4 powder without any pretreatment. According to the results, the spent LiFePO4 electrode shows good lithium capacity, excellent separation performance, and low energy consumption. A complex hybrid cathode design for lithium-ion batteries was created by Wang et al. (2022); it combines graphite & LiMn0.7Fe0.3PO4 and permits co-utilisation of cations and anions which improved conductivity and energy density. For spent LiFePO4, Zheng et al. (2023) created an economical, lightning-fast generation method that takes less than 20 s to complete structural restoration and lithium replacement. The resultant material RLFP-800 exhibits outstanding rate performance and long-term cycling stability.

The LFP cells have a cycle life from 1800 to 2500 cycles at 80% DoD. Its operational temperature ranges for different conditions are: 10–45°C (−20°C for low-temperature series) for charging and: 20–50°C (extreme – 40°C for low temperature) for discharging. LFP cells have a low internal resistance of about 83 mΩ at −50°C (Yue et al. 2022) for high-power-density batteries. Constant voltage is maintained throughout discharge up to 80% DoD. Therefore, the LFP cells are considered for the testing.

2.2. Cells specifications

Figure 1 illustrates the visual appearance of the 100 Ah 3.2 V (nominal) LFP cells used for testing in this paper. Specifications of the cell are provided in Table 1. Each cell has a rectangular shape of 220 mm in length and 130 mm in width weighing about 2.3 kg. The nominal voltage is 3.2 V, with top of charge voltage of 3.65 V and a discharge cut-off voltage of 2.5 V.

Figure 1. Li-ion phosphate (LFP) cell.

Table 1. Specifications of selected LFP cell.

Parameter	Specification
Form/Shape	Prismatic
Dimensions	220 mm × 130 mm × 40 mm
Rated capacity	100 Ah
Nominal cell voltage	3.2 V
Charge voltage	3.65 V
Discharge cut-off voltage	2.5 V
Impedance	≤0.6 mΩ

2.3. Test plan

Functional quick-testing is required since lithium-ion batteries are widely used and their applications are expanding. The philosophy followed for executing the test plan is illustrated in Figure 2.

Figure 2. Overall test plan for estimating cycle life.

To evaluate the performance and reliability of cells, a range of tests are carried out on them, including capacity, open circuit voltage (OCV), direct current internal resistance (DCIR), and cycling tests. A capacity test determines the number of hours for which a battery can be discharged at a constant current to a defined cutoff voltage. To compare between cells, it is essential to select cells that are similar to each other in terms of internal resistance and capacity. The OCV test measures the voltage in the absence of load and is an indicator of the age and health of a cell. The efficiency of discharge is affected by the internal resistance of the cell and is measured by the value of Direct Current Internal Resistance (DCIR). The variation in DCIR influences cell discharge capacity, and most manufacturers consider indicator of cell performance. The number of charge–discharge cycles a cell can endure before capacity fades to a critical is estimated by cycling tests. Together, these tests guarantee that cells fulfill the requirements and are resilient enough to withstand the intended application. The detailed procedure for each of the tests, along with their significant outcomes, are as elaborated in the following sections.

2.3.1. Capacity test

A capacity test is done to determine the Ah output of the cell on discharge at a fixed current. All eight LFP cells were tested at a constant current (I_d) of 0.5 C (=0.5 × 100 Ah = 50 A) up to a voltage of 2.5 V. Readings of voltage and current are constantly recorded during the test duration with a time interval specified in the datalogging system. The time taken for each cell to get completely discharged (T_d), is recorded and its cumulative product with the constant discharge current is calculated. Mathematically, the Ah capacity of a cell discharged with a constant current (I_d) over the entire discharge time horizon of T_d gives the Ah_out as follows: (1) $\mathrm{A}{\mathrm{h}}_{\mathrm{out}}=\sum _{t=t_i}^{t=T_d}{I}_d\mathrm{\Delta }t$(1) Figure 3 shows the Ah_out values obtained from the capacity test on the 8 LFP cells, where the 6th cell has a maximum capacity of 105.8 Ah, and the 5th cell has the lowest capacity of 102.4 Ah while the average capacity of these cells is 104.6 Ah with a standard deviation of 1.08 Ah.

Figure 3 . Capacity test on 8 LFP cells at 0.5C.

It can thus be concluded that the cells have successfully provided the desired Ah capacity as specified by the manufacturer. After discharge, the cells are once again charged at a 0.5 C rate up to 3.65 VPC.

Energy density provides insight into the amount of energy that can be stored in a cell corresponding to its weight expressed in terms of Wh/kg. A cell with a higher energy density will have a considerably longer discharge time. There may be a risk of a thermal event if the energy density is too great. The energy density of each cell is represented in Figure 4 and is determined by obtaining the ratio of the product of the nominal cell voltage and the cell capacity (obtained from Figure 3) to cell weight. Considering cell number 1, the capacity recorded is 104.18 Ah, and as stated in Section 2.2, the nominal voltage and cell weight specified are 3.2 V and 2.3 kg, respectively. Thus, the energy density of this cell is calculated as (104.18 × 3.2) / 2.3 which is 144.9 Wh/kg. Similarly, the 6th cell has an energy density of 147.297 Wh/kg, and the 5th cell has the lowest energy density of 142.414 Wh/kg, whereas the average energy density of all the cells is 145.6 Wh/kg.

Figure 4. LFP prismatic cell specific energy / gravimetric energy density.

Following the determination of the energy densities of the cells, the open circuit voltage test is undertaken. This helps to ensure that the appropriate cells are chosen for the cycling test.

2.3.2. Open circuit voltage test

The difference in electrical potential between the negative and positive terminals of a cell when no load is connected is termed the Open Circuit Voltage (OCV) (Malik et al. 2018). Owing to the self-discharge characteristics of batteries, it becomes imperative to test the cells for their OCV values before being subjected to further testing procedures. The motive of the OCV test is to identify any condition of battery degradation over some time due to aging. The data obtained from this test is indicative of the state of charge (SOC) for rechargeable batteries.

The open circuit voltage of the 8 cells as shown in Figure 5, is in the range between 3.58 and 3.6 V with a standard deviation of 0.007 V, illustrating good manufacturing process control.

Figure 5. Variance in Open Circuit Voltage among 8 cells.

After determining the no-load current, the next step is determining the cell’s internal resistance before deciding on the cells that can undergo a cycling test.

2.3.3. Direct current internal resistance test (DCIR)

Direct current internal resistance (DCIR) testing is a non-destructive and non-invasive method of battery inspection that assists in identifying any underlying problem that might impact its performance. It helps to identify cells with high resistances, the reason for which could be internal damage or aging. Using Ohm’s law, the ratio of voltage variations to current variations is evaluated to obtain DCIR. A cell that has less internal resistance can deliver a high current when there is demand for the same, whereas high resistance can cause the cell to heat up and result in a drop in voltage. DCIR of cells tested after OCV measurement was noted similarly. For the DCIR test, cells are charged up to 50% of their capacity, followed by 8 hours resting time, after which a high current charge is applied for 10 s with 2.99 volts.

As noted from Figure 6, the lowest DCIR is 0.66 Ω for cell 7 with a capacity of 104.67 Ah, while the highest value was 6.42 Ω for cell 5 with a 102.36 Ah capacity. Cells with DCIR values that fell in close range, which are cells numbered 2, 3, 4, and 7, were hence selected for the cycling test.

Figure 6. Variation in internal resistance (DCIR) and cell capacity.

2.4. Cycling test

After obtaining results from the DCIR test, the selected cells are subjected to the life cycle testing. For this, a Battery Testing System (BTS) controls the charging/discharging parameters, while a thermal chamber maintains a preset temperature for the tests. Discharging rates, ambient temperature, and Depth of Discharge are the important parameters considered in this study. The equipment details used for testing are described in the succeeding section.

2.4.1. Test equipment

Two main test equipment used are the Battery Testing System and the thermal chamber (oven). The Battery testing system is a computer-controlled electronic device used for testing cells/ batteries at specified conditions, such as the charge/discharge rate(current), time, and voltage. The BTS has 4 channels, with each channel of 18 V, 50 A. For this research, two cells are tested at 50 A and two cells at 80 A. It is possible to connect two channels in a series to get a higher current rating. The system also indicates the type of operation going on in the cells by indicating LEDs with different colours. The green light indicates charging, while the red light indicates discharging. A schematic diagram of the setup is shown in Figure 7, and detailed specifications are listed in Table 2.

Figure 7. Cells testing with thermal chamber and Battery Testing System (BTS).

Table 2. Battery testing system / 18 V, 50A pack tester specification.

Parameter	Specifications
Current range	0.05 A to 50 A
Maximum current range	0.2 A to 200 A
Current measurement accuracy	0.05% FS
Voltage range	0.09 V to 18 V
Maximum voltage range	0.09 V to 18 V
Voltage measurement accuracy	0.05% FS

To obtain the temperature data of the cell, thermocouples are fixed on the surface of the cell at its centre, where the temperature of the cell will be maximum. Each cell is connected to a separate channel of the battery testing system. Precaution has to be taken while connecting cells so that the negative and positive terminals do not come in contact with each other. Each cell is kept away from the other cells by placing them in separate trays to avoid contact between the positive and negative terminals.

A thermal chamber is used for tests at a specified temperature, controlled and maintained for testing cells. In the present study, testing is carried out at temperatures of 25 and 45°C in the thermal chamber. Specifications of the thermal chamber used are presented in Table 3.

Table 3. Thermal chamber / hot air oven specification.

Parameter	Specifications
Manufacturer	Refricon
Dimension	300 mm × 300 mm × 300 mm
Ambient temperature range	5–75°C
Power	16 A, 230 V AC

It is imperative to understand the procedure followed for testing the cells. This is discussed in the subsequent sections.

2.4.2. Life cycle testing procedure

The testing of cells is done on the BTS where cell numbers 2, 3, 4, and 7 having DCIR values close to each other are selected for accelerated life cycle testing. The flow chart shown in Figure 8 outlines the philosophy for carrying out the cycling test.

Figure 8. Cell life cycle testing procedure.

Out of the above four cells, two cells are tested at 0.5 C rate, at 25 and 45°C corresponding to 100% DoD, while the remaining two cells are tested at 0.8 C rate, at 45 and 25°C corresponding to 80% DoD. The choice of testing temperature of 25°C is justified as this is typically the standard rating temperature specified by the manufacturer. The higher temperature is chosen from the point of view of the typical highest ambient temperatures in summer months in tropical climates typical to south and south-east Asia. Moreover, 45°C is also an industry standard temperature for testing cells. Also, the discharge cut-off voltage is 2.5 V, corresponding to 80% Depth of Discharge for 3.2 V cells. Testing at 45°C is done in the thermal chamber for 300 cycles. The results obtained after carrying out the cycling test are illustrated in the subsequent sections.

3. Accelerated life cycle test

The cycle life test is conducted at different charge and discharge rates, temperatures, and depth of discharge, as specified in Figure 8. For all the cells, about 300 cycles (equal to one year of operation) were completed using a fully automated battery testing setup. A detailed description of each test is provided in the next sections.

3.1. Testing at 0.5 C and 100% DoD

Two cells are simultaneously cycled at 0.5 C rate in a controlled atmosphere of 25 and 45°C using an environmental chamber. In a cycling test, cells are discharged up to a specific voltage for every cycle corresponding to 100% depth of discharge or rather, a 0% S.O.C (State of Charge). The Ah discharged is recorded on the system datalogger for every discharge. Ah discharged is calculated using Equation (1) (Section 2.3.1). During the recharge, the energy supplied to the cell in Ah is similarly recorded over a time horizon of T_c, by discretely recording the charging current (I_c) for every time step Δt (e.g. 1–10 minutes). Thus, Ah_i is determined as: (2) $\mathrm{A}{\mathrm{h}}_i=\sum _{t=t_i}^{t=T_c}{I}_c\mathrm{\Delta t}$(2) The Ah or coulombic efficiency (η_co) is the ratio of Ah discharged to the Ah charged and given by (3) (3) ${\eta }_{\mathrm{co}}={\displaystyle \frac{\mathrm{A}{\mathrm{h}}_o}{\mathrm{A}{\mathrm{h}}_i}}$(3) The output charge that can be derived from a fully charged cell is determined by the ampere-hour (Ah) efficiency or coulombic efficiency. Change in capacity in every cycle is calculated from the Ah obtained during the discharge time period of T_d.

The variation of the Ah charged and discharged for the first 20 cycles is plotted as shown in Figure 9. It is noted that Ah efficiency is almost about 99% for all the cycles indicating little loss of charge during the charge/discharge process. Similar Ah efficiency is noted for 0.8 C.

Figure 9. Record of Ah charged and discharged during the cycling process at (a) 0.5 C and 25°C and (b) 0.5 C and 45°C for prismatic LFP cell.

Figure 10 illustrates the trend of capacity fade obtained by plotting the discharge capacity (Ah_o) in every cycle for the test at 25 and 45°C. In the initial 100 cycles, the capacity obtained by discharging the cell at 45°C was higher than that at 25°C. However, the rate of capacity degradation is higher when the cell operates at an ambient temperature of 45°C relative to that at 25°C.

Figure 10. Relative capacity fade at 0.5 C and 100% DoD at 25 and 45°C ambient temperature for prismatic lithium-ion cells.

The capacity degradation in about 310 cycles was observed to be about 5.2% at 45°C and 3.1% at 25°C. It was also observed that when ambient conditions were maintained at 25°C, the cell temperature rose to 38°C during the charge–discharge process. However, when cycling at 45°C, the maximum cell temperature rose to 51°C.

The trend of capacity fade of lithium-ion cells is mainly governed by cell chemistry, charge/discharge rate, DoD, and temperature. Though full cycle life tests are desirable, they are prohibitive from the time and expenses to be dedicated for the same. Battery system integrators are interested in understanding the useful cycle life of cells through a quick estimate with reasonable accuracy. Relevant literature (Diao, Saxena, and Pecht 2019; Preger et al. 2020) shows that in the range of temperature and discharge rates tested, the capacity fade trend can be fitted to a linear approximation. Therefore, a linear approximation was used for the predictions. The linear trendline prediction has a coefficient of determination of about 97–99%. The extrapolated trends for cell life at 0.5 C, 100% DoD, and at temperatures of 25 and 45°C respectively are plotted as shown in Figure 11, which illustrates the development of linear regression models for all cells depicting residual capacity with the number of cycles. The linear regression model is useful for extrapolation and predicting the cell capacity for several cycles for the specified residual capacity of the cell since the value of Karl Pearson’s correlation coefficient (R²) is close to one for all the cells (Mekonnen, Aburbu, and Sarwat 2018)., (Roy, Patil, and Sen 2022), (Kaneko 2021).

Figure 11. Cycle life prediction at various temperatures and 0.5 C charge/discharge rate and 100% DoD.

The cycle life is calculated based on 80% of the original capacity of the cell (Kaneko 2021). The cell cycled at 45°C has an initial capacity of 104.22 Ah. Hence its life cycle would be the number of charge/discharge cycles endured till the capacity reaches 80% of 104.22 which is 83.4 Ah For instance, using the linear relationship predicted from the trend line y = −0.0151 x + 103.57 and substituting for y as 83.4 Ah, yields the cycle life as 1354. Similarly, the cell subjected to 25°C ambient conditions has a life cycle corresponding to 82.8 Ah for an initial capacity of 103.53 Ah.

From Figure 11, it is noted that while the predicted life of an LFP prismatic cell operating at 0.5 C charge and discharge at an ambient temperature of 45°C is about 1354 cycles (∼4.5 years), the predicted life under similar conditions but at an ambient temperature of 25°C is almost double and equal to 2309 cycles (∼7.7 years). Note, in both cases, the DoD is constant and equal to 100%. The initial 100 cycles are not included in the prediction for better accuracy.

3.2. Testing at 0.8 C and 80% DoD

In certain applications such as stationary storage, it is not desirable to entirely discharge the cell up to 100% DoD. Therefore, tests were performed at a lower DoD of 80%. Two cells (cell no 3 and 4) were cycled at different ambient temperatures viz., 25°C and 45°C, 0.8 C rate and 80% DoD for about 300 cycles in a controlled environment thermal chamber.

As the cell operates at 20% lower capacity, the discharge capacity is obtained in the range of 82–85 Ah in the first few discharges. The chronological variation of discharge capacity is depicted in Figure 12. In line with earlier observations, it is noted that in this case, the capacity fade at higher temperatures was higher. While at 25°C, cell degraded in capacity at a rate of 5.8%, the cell that were operated at 45°C degraded at a rate of 8.4%. Though the effect of higher operating temperature is higher discharge capacity in the interim, the capacity fade rate is also higher leading to lower cycle life. Figure 13 depicts the predicted cell cycle life at 0.8 C and 80% DoD based on linear extrapolation of the data that is presented in Figure 12, where 80% capacity corresponds to 65.1 and 67.3 Ah for 25 and 45°C respectively.

Figure 12. Comparison of capacity degradation at 0.8 C and 80% DoD at temperatures of 25 and 45^oC.

Figure 13. Effect of operating temperature on predicted cycle at 0.8 C and 80% DoD.

The predicted life at these conditions is found to be 1086 cycles at 25°C and 834 cycles at 45°C, indicating a 23.2% reduction in cell life, which excluded the initial 100 cycles to achieve better accuracy. Table 4 presents a summary of the results recorded through the above experimental investigations.

Table 4. Summary of results from life cycle testing.

Cell number	Charging rate	Discharging rate	DOD %	Ambient temperature ^oC	Maximum temperature ^oC	Predicted life, Cycles	% reduction
1	0.5 C	0.5 C	100	25	38	2309	–
2				45	51	1354	41.36
3	0.8 C	0.8 C	80	25	43	1086	–
4				45	59	834	23.20

The effect of a higher charge/discharge rate is to increase the maximum cell temperature. At 0.5 C, the maximum cell temperature increases from 38 to 43°C which is a 13% rise. While at 0.8 C, an increase in operating temperature, causes a rise in the maximum temperature from 51 to 59°C (15.6%). At a constant ambient temperature, the reduction in cycle life is predicted to be 52.96% at 25°C and 38.4% at 45°C. For a constant charge/discharge rate, the operating temperature has a major effect on the predicted cycle life. About 42% decrease in predicted life is indicated when operating at a temperature of 45°C and 0.5 C. On the other hand, the reduction in predicted life at a higher charging /discharging rate is less pronounced. This is because a higher discharge rate inherently leads to an operation with a relatively higher cell temperature. Thus, the difference between the ambient temperatures produces a 23.2% change in the predicted cycle life.

4. Reliability modelling and analysis

The life of lithium-ion batteries is commonly expressed in Cycles-To-Failure (CTF), which is utilised in this research for analysis (Álvarez et al. 2003; Liu, Wang, and Chen 2019). The CTF is the number of charge–discharge cycles completed by the cell before it fails to produce 80% of its designed capacity; which is referred to as residual capacity in this context (S. Batteries Committee of the IEEE Power and E. Society 2011). Even though the CTF is the proper approach to estimating the life of a Li-ion cell, obtaining CTF data is a time-consuming procedure. According to this viewpoint, a small sample of four Li-ion (LFP) cells (B1, B2, B3, and B4) was tested on the experimental setup for roughly 300 cycles, and residual capacity data were collected at regular intervals. Exponential probability distribution is widely used for reliability analysis (Gaonkar et al. 2021), (Patil, Kothavale, and Waghmode 2019), (Patil et al. 2017), especially in the case of limited data (Ganjeizadeh, Tapananon, and Lei 2014), (Vignarooban et al. 2016).

When a cell’s capacity drops below 80%, it deteriorates more quickly and is more likely to fail unexpectedly because of a temperature rise that causes a higher discharge rate. As a result, the end of life for Li-ion cells is considered to be at 80% of their intended capacity, and the CTF is the equivalent number of cycles (Mekonnen, Aburbu, and Sarwat 2018). In this analysis, four cells, namely B1, B2, B3, and B4, are tested at different C-rates and temperatures, and their expected life, i.e. Mean-Cycle-To-Failure (MCTF), is given in Table 5.

Table 5. Mean-Cycle-To-Failure (MCTF) of cells tested at different C-rates and temperatures.

Cell ID	C-rate	Temperature (^oC)	MCTF (cycles)
B1	0.5 C	25	2309
B2	0.5 C	45	1354
B3	0.8 C	25	1086
B4	0.8 C	45	834

4.1. Selection of probability distribution for reliability modelling

The exponential distribution is used for Li-ion cell reliability analysis due to its mathematical properties and its applicability to modeling its failure behaviour. The exponential distribution is a memoryless distribution, which means that the time to failure is independent of the time already spent in operation. In the case of Li-ion cells, this property implies that the probability of failure at any given time is constant, regardless of the age or usage history of the cell. This assumption is reasonable for many Li-ion batteries, as they typically experience wear-out failures rather than gradual deterioration. The exponential distribution is relatively simple to work with mathematically, making it a convenient choice for modeling reliability. Its probability density function (PDF) and cumulative distribution function (CDF) have simple and well-defined mathematical forms, allowing for straightforward calculations and analysis. Li-ion cells do not typically have a well-defined wear-out period, where failures occur due to aging mechanisms or material degradation. Instead, failure events in Li-ion cells are often sudden and unpredictable, resulting from a variety of factors such as manufacturing defects, overheating, or electrical stress. The exponential distribution, with its constant failure rate over time, can be a reasonable approximation for such failure processes. The exponential distribution has been widely used in reliability engineering and is often employed as an initial approximation when analyzing the failure behaviour of Li-ion cells, especially in the presence of a limited data set. Consequently, there is a significant amount of historical failure data available that supports the use of the exponential distribution for battery reliability analysis. The mathematical equations for various measures of exponential distribution are given in Table 6.

Table 6. The mathematical equations for exponential distribution.

Reliability measure	Equation	Reliability measure	Equation
Reliability	$R\left(t\right)=e^{-\mathrm{\lambda t}},$	PDF	$f\left(t\right)=\lambda e^{-\mathrm{\lambda t}}$
Design life:	$t_R={\displaystyle \frac{\ln R}{-\lambda }}$	Mean cycle to failure	$\mathrm{MCTF}={\displaystyle \frac{1}{\lambda }}$
CDF	$F\left(t\right)=1-e^{-\mathrm{\lambda t}}$	Median cycle to failure	$t_{\mathrm{median}}={\displaystyle \frac{0.69315}{\lambda }}$

The MTCF, failure rate, and median time to failure information of cells B1, B2, B3, and B4 are shown in Table 7. It depicts that as the operating temperature of the Li-ion cell increases, the failure rate increases, and the median time to failure decreases. It shows for cells B1 and B2 that both the MTCF and median time to failure decreases nearly by 40% for 0.5 C rate when operating temperature increases from 25 to 45°C. Furthermore, the failure rate also increases by nearly 70%. Similarly, for cells B3 and B4, the MTCF and median time to failure decreased by nearly 23%, and the failure rate increased by nearly 30%. Therefore, it can be concluded that the temperature has a significant impact on the expected life and failure rate of the Li-ion cells. The comparative reliability vs. time and Cumulative Distribution Function (CDF) or unreliability vs. time plots are shown in Figure 14 and Figure 15, respectively. It depicts that cell B1 has the highest reliability and cell B4 has the least reliability.

Figure 14. Comparative reliability vs. time plot.

Figure 15. Comparative Cumulative Distribution Function (CDF) vs. time plot.

Table 7. Reliability metric for Li-ion cells B1, B2, B3, and B4.

Sr. No.	Reliability measure	B1	B2	Percentage change $\left({\displaystyle \frac{B1-B2}{B1}}\right)\times 100$	B3	B4	Percentage change $\left({\displaystyle \frac{B3-B4}{B3}}\right)\times 100$
1	MTCF	2309	1354	41.36	1086	834	23.20
2	Failure rate (per cycle)	0.0004331	0.0007386	−70.54	0.0009208	0.001199	−30.21
3	Median time to failure	1596	938	41.23	753	578	23.24

The comparative Probability Density Function (PDF) vs. time plot for the four Li-ion cells is shown in Figure 16. The PDF shows the data is right-skewed distribution, indicates ‘infant mortality’, and implies that the failure events occur more frequently at earlier stages or shorter durations and become less likely as time progresses. This can indicate issues such as manufacturing defects, material weakness, design flaws, or inadequate quality control, which result in a higher failure rate during the early life of the system. It emphasises the importance of thorough testing and quality assurance measures to mitigate early failures. It highlights the need for burn-in or stress testing to identify and eliminate potential early failures. In some cases, the right-skewed distribution may reflect a wear-out phase, especially if the system has been in operation for an extended period. As the system ages, gradual degradation, material fatigue, or other wear-out mechanisms can lead to an increasing failure rate over time. This can result in a right-skewed distribution, with a higher probability of failures occurring at longer durations. The right-skewed distribution might suggest the presence of multiple failure mechanisms or a mix of different failure modes within the system. These mechanisms can have varying rates of occurrence, resulting in a distribution that is skewed to the right. Understanding and identifying the underlying failure mechanisms are crucial for effective reliability improvement strategies.

Figure 16. Comparative Probability Density Function (PDF) vs. time plot.

The comparative failure rate vs. time plots for the Li-ion cells is shown in Figure 17. The increasing hazard rate indicates the probability of failure per unit of time. This implies that the Li-ion cells become more prone to failure as it ages. An increasing hazard rate often suggests that the Li-ion cell is approaching or entering a wear-out phase. Over time, the degradation or aging mechanisms can accumulate, leading to an increased likelihood of failure. The increasing hazard rate in the Li-ion cell may indicate the accumulation of stress or dendrite formation within the cell. This can occur when stressors, such as mechanical or electrical, or thermal stress, accumulate with each operation, leading to an increased probability of failure. This interpretation emphasises the importance of proactive maintenance and monitoring strategies to mitigate the risk of failure.

Figure 17. Comparative failure rate vs. time plot.

5. Conclusion

Operation at higher ambient temperatures is one of the major causes of premature failure of lithium-ion cells. Experimental evidence is crucial in justifying the predicted cycle life of the cells at higher operational temperatures. This research reports the results of testing lithium iron phosphate prismatic cells at laboratory conditions by varying the discharge rate, depth of discharge and operational temperature. The cells are cycled in a computerised programmable battery test set up for 300 cycles at temperatures of 25°C and 45°C at discharge rates of 0.5 and 0.8 C with 80% and 100% depth of discharge. Experimental data obtained from the above 300 cycles are further extrapolated to obtain a prediction of the useful life of the cells under varying operational conditions. Results from experimental investigations reveal the following for prismatic lithium-ion cells:

• The tested cells depicted a linear capacity fade trend, with the highest capacity degradation rate of 8.4% being observed for 45°C, 0.8 C and 80% DoD in 300 cycles.

• At a constant discharge rate, an increase in temperature led to a reduction in predicted cycle life in the range of 23.2–41.36%.

• At a constant temperature increase in discharge rate from 0.5 C to 0.8 C reduced estimated cycle life by about 52.9–38.4%.

The reliability modeling and analysis of the Li-ion cells were carried out using exponential distribution due to their mathematical properties and their applicability to modeling its failure behaviour. The analysis shows that as the temperature of the Li-ion cell increases, the failure rate increases by nearly 70% at 0.5 C rate and by 30% at 0.8 C rate. Furthermore, the MTCF and median cycle to failure decreased by 41% for the 0.5 C rate and 23% for the 0.8 C rate. The PDF of the Li-ion cell shows the right-skewed distribution, indicates ‘infant mortality’, and may occur due to reasons such as manufacturing defects, material weakness, design flaws, or inadequate quality control. It emphasises the importance of thorough testing and quality assurance measures to mitigate early failures. The hazard rate of the Li-ion cells increases per unit of time, and this implies that the Li-ion cells become more prone to failure as they age.

Acknowledgements

The authors are thankful to the R&D support from Customized Energy Solutions Limited for support in the experimental work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials cited within the text.

Nomenclature
Ahi	=	Ampere hour charged
Aho	=	Ampere hour discharged
Tc	=	Charging time duration
Td	=	Discharging time duration
Δt	=	time step for discharge/charge
ηco	=	Coulombic (Ampere-hour) efficiency
DCIR	=	Direct Circuit Internal Resistance
LIB	=	Lithium Ion batteries
LFP	=	Lithium Iron Phosphate
OCV	=	Open Circuit Voltage
Ah	=	Ampere hour
DoD	=	Depth of Discharge
SOC	=	State of Charge
VPC	=	Voltage per cell
BTS	=	Battery Testing System
CTF	=	Cycles-To-Failure
MCTF	=	Mean-Cycle-To-Failure
CDF	=	Cumulative Distribution Function
PDF	=	Probability Density Function
Ic	=	Charging Current
Id	=	Discharging Current