Performance evaluation of lithium battery pack based on MATLAB simulation with lumped parameter thermal model

The capacity of lithium batteries varies under different temperature conditions. However, many studies still neglect the influence of temperature on battery capacity. Besides, some studies only considered the combination of the heat and electricity of the battery monomer and failed to study the performance of the battery pack in groups. This paper investigates a new method for dynamic situations that takes the influence of temperature on battery capacity into consideration. After acquiring the parameter through experiment, we can substitute the performance parameters into the thermoelectric coupling model and use MATLAB to realize the iterative calculation of thermoelectric coupling. By combining the grouping mode with the model, we can simulate the thermodynamic and electrical properties of the battery cell in the pack. The novelty is evident because the paper focused on the battery pack instead of a single cell and the lumped parameter thermal model is adopted. Besides, the method proposed in this paper considered the influence of temperature on battery capacity. The program of calculation process can be solidified on the vehicle system. After the simulation, we can get the available capacity of the battery pack at different initial temperatures, with different grouping modes and different inconsistency of parameters. The results predicted by MATLAB model are compared with the results of MATLAB‐FLUENT thermoelectric coupling simulation. The result reflects that the prediction by the MATLAB model about the available capacity, temperature, and residual electricity is of good precision.


| INTRODUCTION
Autonomous underwater vehicles (AUVs) can be divided into two types, that is thermal-powered AUVs and electric-powered AUVs. Electric-powered AUVs have developed fast in recent years due to their significant advantages over thermal-powered AUVs in terms of structure, cost, maintenance, and performance. For instance, it offers low noise and almost no track, and its performance is not influenced by the navigation depth. Therefore, more countries are adopting the electric power mode for AUVs. The power system is the heart of the AUV, and batteries are crucial. Therefore, further development of batteries is significant for the development and application of underwater vehicles. 1 Compared with other secondary batteries, lithium-ion batteries (LIBs) have the advantages of high energy density, high working voltage, low self-discharge rate, convenient use and maintenance, and no memory effect. In recent years, LIBs have been widely used in electronic products, vehicles, and aerospace. [2][3][4][5][6][7] Nowadays, LIBs are used as the power system of most AUVs. To ensure the safety and reliability of the battery system, the monitoring of the basic state of the battery are necessary, including the monitoring of the state of charge (SOC), [8][9][10][11] states of power, 12,13 and state of health. [14][15][16][17] According to the experimental data obtained, the health status and available power of LIBs are significantly affected by temperature. Considering the particular working environment of AUVs power battery packs, such as confined space, heat dissipation becomes a challenge to the battery working under high temperatures. Therefore, the impact of temperatures on the performance of LIBs needs to be further studied.
The field of battery cell modeling and state estimation is vast, and numerous approaches with varying degrees of benefits have been developed in the past [18][19][20][21][22][23][24] Because of their simplicity and ability to capture battery dynamics, equivalent circuit model-based techniques such as Thevenin model, 25,26 the Partnership for a new generation of Vehicles model, 27,28 and the double polarization model have been popular. However, in terms of material qualities and the physical structure of the cell, these models are often difficult to comprehend. Many experimental data sets and estimating methods are required to identify the model parameters. Battery pack modeling is more sophisticated and computationally demanding than single-cell modeling. Detailed physics models, such as electric capacitance tomography and computational fluid dynamics (CFD) models, are highly accurate but on the other hand, computationally expensive. Thus, making them unfeasible for onboard battery pack applications. The majority of the research on battery pack modeling considers very small battery packs with either series or parallel connections [29][30][31][32] However, most studies use electrical modeling to obtain the status and performance parameters of battery cells while ignoring pack-level thermal modeling, which is important in the case of battery packs. 33 To study the influence of heat generated by discharging on lithium battery pack, it is necessary to establish the system temperature field in the discharge process and analyze how the performance of each battery cell is affected by temperature. Many studies focus on the thermal model to determine the temperature distribution of the battery. LIBs have complicated internal structures and variable application environments, which makes it challenging to determine the interior temperature of the battery. 34,35 Several modeling techniques frequently differ in their dependability and accuracy when simulating the thermal properties of power LIBs. 36,37 Nowadays, methods for measuring the interior temperature of batteries that are often utilized including offline internal temperature prediction techniques and experimental internal temperature measurement techniques. A thermocouple is placed within the battery to measure the interior temperature, according to tests. [38][39][40][41][42] This approach is only appropriate for assessing the pertinent thermal properties of the battery in a test setting. Finite element numerical calculation techniques are typically used for the offline-based internal temperature forecast. The interior temperature of the battery is determined through the development of a battery cell thermal model and offline simulation. [43][44][45] The finite element numerical calculation method has exceptionally high computational complexity and is not appropriate for the actual application and thermal management of batteries. It is primarily utilized for battery cell packaging design and module design. 46 Xifeng Cui proposed an optimized lumped parameter thermal model so that it is able to precisely present heat flow distribution and achieve broad applicability for online battery temperature estimation. Three lumped parameter thermal models for hard-cased lithium-ion batteries are proposed. 47 However, they are only concerned about the thermal field and temperature changes, and seldom conduct real-time coupling calculations of electrical and thermal properties, ignoring the research on the battery's available power. In contrast, the method proposed in this paper comprehensively considers the thermodynamic and electrical characteristics of the battery, and the results are more accurate. To explore the influence of discharging rate on battery capacity, Peukert tested lead-acid batteries at a constant current and determined that a single equation could be used to measure the capacity of the battery. 48 However, lithium-based batteries demonstrate varying available capacities at different temperatures. 49,50 This aspect is completely disregarded in Peukert's equation and can result in significant errors caused by even slight changes in environmental conditions or battery selfheating effects. 51 53 However, they only considered the combination of the heat and electricity of the battery monomer and failed to study the performance of the battery pack in groups. This paper not only coupled the electrical and thermodynamic characteristics of the battery but also took the battery pack in groups as the research object. Combining with the heat conduction model, we carry out a more comprehensive and accurate study of the thermoelectric performance of the battery pack. For the battery pack, we need more studies on a thermoelectric coupling simulation model taking temperature into account because the temperature is the most critical factor affecting battery performance. After the critical parameters are obtained through experiments, the critical parameters can be substituted into the thermoelectric coupling model, and the iterative calculation of thermoelectric coupling can be realized by using relevant calculation software. In this paper, the author selected MATLAB to carry out thermoelectric coupling because MATLAB does not need CFD calculation, thus saving a lot of calculating time. Besides, in the case of onboard vehicles, it can carry out brief calculations through onboard hardware. By combining the grouping mode with the discharging features, the thermoelectric characteristics of the batteries in the battery pack can be simulated. Through simulation, the available capacity of the battery pack at different initial temperatures, with different grouping modes and with different initial resistance inconsistencies can be obtained. Through such prediction in advance, we can have an approximate assessment of the AUVs power system and predict the available range of the AUVs, so as to guide the deployment and use of the equipment.

| Thermal resistance models
In this work, MATLAB was used to construct the lumped parameter thermal model of LIBs, which fully considers the influence of battery temperature on battery performance (Figures 1 and 2). T A B L E 1 Thermal resistance models.

Situation Situation 1 Situation 2
Equations 1. Situation 1 displays that when a battery cell is located adjacent to the bulkhead, the temperature conduction model is as follows: t f 1 is the outside temperature and t w3 is the temperature of the cell adjacent to the cabin, h 2 and h 1 represent the heat transfer coefficient inside and outside of the bulkhead, λ represents the heat conductivity of the bulkhead, and δ represents the thickness of the bulkhead.
For the case adjacent to the bulkhead, the four equations contained in situation 1 of Table 1 can be rearranged to obtain Among them, can be regarded as a coefficient, k 1 , the Equation (1) can be rewritten as follows: 2. Situation 2 displays that when a battery cell is not adjacent to the bulkhead, the two equations contained in situation 2 of Table 1 can be rearranged to obtain t w5 is the temperature of battery 1 and t w4 is the temperature of the battery 2, among them, can be regarded as a coefficient k 2 ; thus, Equation (3) can be rewritten as follows:

| Heat transfer model
The temperature conduction model is shown in Table 2: The temperature of the battery will be affected by the T A B L E 2 Simplified heat transfer model.

| Electrical model
To the circuit of n series, m parallels shown in Figure 3, each battery cell has the following relationship: where E represents the open circuit voltage of battery cell, U represent the terminal voltage of battery cell, R represents the internal resistance, and I represent the current flowing through the battery. By adding up all the equations in Equation (5), the following relationship in each parallel branch can be obtained: Due to the parallel relationship, follow the following relationship: We can rewrite Equation (6) as: By combining Equation (7) with Equation (8): .
When the open circuit voltages, internal resistances, and external loads of all the battery cells in the battery pack are determined, the current of each branch can be determined accordingly, and the heat generated in each single battery can be calculated. However, when the battery is located in different positions inside the battery compartment, the batteries will be at different temperatures according to the thermal field distribution, which will lead to the corresponding changes in electrical performance. Therefore, it is necessary to study the influence of temperature on the electrical performance of the battery.
As far as we know, the open circuit voltage of the battery will vary with the SOC of the battery. According to the experiments in this paper, the change of the battery's SOC with the current will be affected by the battery's temperature. With the same current and discharging duration, the SOC of the battery with a higher temperature will decrease less than that of the battery with a lower temperature. Therefore, if we want to obtain a better estimation of battery performance, the thermal and electrical properties of the battery need to be coupled in real-time to optimize the approximation of the process of discharging.
When the open circuit voltage and internal resistance of the battery are all known, we need to calculate the current flowing through each battery according to the grouping mode of the battery pack. The SOC of the battery needs to be calculated at time t + 1 according to the SOC of the battery, its temperature, and current at time t.
Through real-time coupling simulation, the estimation of the thermal field and current field can be obtained. In this paper, two methods are used to realize the coupling of multiple physical fields as shown in Figure 4. The first method is to realize the coupling simulation through MATLAB-Fluent joint simulation. The second method is to realize the simplified coupling simulation through MATLAB. The advantage of method one is that it is very highly consistent with the actual situation because its finite element analysis can simulate the thermal performance precisely. But the disadvantage of method one is also obvious, that is it will cost a lot of time to accomplish the calculation.
The advantage of method two is that it can save a lot of calculating time. But because it's developed from a simplified model, it is only an approximate simulation calculation. So it can't simulate the thermal procession as accurately as method one. We need to verify the accuracy of method two by comparing the result with method one.
Step 1: Select the lithium battery cell with the same type as the battery cell in the battery pack to be tested. Then conduct pulse discharge experiments respectively to obtain the R-SOC curve (internal resistance-SOC change curve), R-T curve (internal resistance-temperature change curve), OCV-SOC curve (open circuit voltage-SOC curve), and OCV-T curve (open circuit voltage-temperature curve).
Step 2: Combine the R-SOC curve and the R-T curve to obtain the binary function of the internal resistance variation with temperature and SOC; 2.1. Fit the R-T curve to obtain the function F T ( ) between them; Then, the coefficient of the function f T ( ) is obtained by dividing can be obtained.
Step 4: Obtain the compensation coefficient of ∆SOC with temperature in the discharge duration ∆t under specific discharging current I.
Step 5: Import the model of the battery into the simulation software. The model of battery parameter includes the binary functions U m T n SOC , and the function of SOC changing with temperature in specific discharging current I and discharging duration ∆t.
Step 6: Based on the battery parameters at the time t −1, we can calculate the current I t ( ) and internal heat source  t , average body temperature T ̅ t , and open circuit voltage U t . Internal resistance R t and SOC t at time t. The flow diagram of thermoelectric coupling and it's shown as follows.
According to the Arrhenius performance formula of the battery, the SOC at time t + 1 in Figure 5 is determined by the SOC and temperature at time t. According to the model of Joule heat, the temperature at time t + 1 is calculated by the temperature distribution at time t and the internal heat source. Then the electrical model is used to calculate the current distribution of each circuit of the battery pack according to the battery OCV and internal resistance at each time. Keep the calculation process iterating until the battery pack reaches the cut-off conditions.

| EXPERIMENTAL AND DISCUSSION
To observe the influence of temperature and discharging rate on the capacity and energy of the battery, we chose the LF105 battery as the experiment object, which is shown in Figure 6. The parameters of battery LF105 are listed in Table 3. The discharging experiments were conducted at nine different temperatures (−20°C, −10°C, 0°C, 10°C, 15°C, 25°C, 35°C, 45°C, and 55°C) and at four different rates (1/3C, 1/2C, 1C, and 2C). The voltage and current of the battery were measured every second. Thermocouples were used to record the temperatures of the positive electrode, the negative electrode, and the shell of the battery.
Generally speaking, at the typical discharge rate for electric vehicles, the capacity is closely related to the selfgenerated heat. 54 Even if a very high convective cooling rate is used to maintain temperature, there would still be a temperature difference between the surface and the center. 55 Therefore, the ambient temperature is difficult to characterize the temperature of the battery precisely. Based on the abovementioned conditions, this paper measured the temperature at the positive electrode, negative electrode, and shell of the battery at each time during discharging and took the average of them as the approximate temperature of the whole battery. The curve of the battery capacity changing with the battery body temperature is shown in Figure 7A. The curves of the 2C rate in Figure 7A cross each other when the ambient temperature is very low (−10°C, −20°C). When the discharging rate of the battery is high, the heating rate of the battery will increase significantly. As a result, there is F I G U R E 5 The flow diagram of the thermoelectric coupling. a relatively large deviation between the actual temperature of the battery and the measured temperature on the battery surface. In an environment of low temperature, the capacity will be significantly affected by temperature deviation. Thus, leading to the overlapping of curves. The results of the experiments are listed in Figure 7. Figure 7 shows that the impact of the discharge rate on the battery capacity is very minimal and can almost be ignored when the ambient temperature is higher than 25°C. When the surrounding temperature is below 25°C, temperature variations have a significant impact on battery capacity, and capacity displays a trend of first decreasing and then increasing with the rise of discharging rate. The average temperature of the battery has a relatively significant impact than discharging rate on its capacity. In the high-temperature stage, the capacity is less sensitive to the temperature. However, in the lowtemperature stage, the capacity tends to be significantly affected by the change of the average temperature of the battery. To quantify the effect of temperature, we introduce the coefficient k, which we will describe later. (Table 4).
The obtained data indicate that the available power of the battery changes with its temperature. An improved model is proposed to define the available capacity of the battery by the author. The electricity discharged by the battery at a specific rate from the full state is defined as the battery's available capacity in this paper. The end of the discharging is considered to occur when the battery's output voltage reaches the cut-off voltage, and the amount of electricity discharged during this process is defined as the battery's usable capacity. This presumption is based on the idea that the battery's absolute maximum residual capacity can never be less than zero when the depth of discharging reaches 100%. Considering the influence of various discharging conditions, the way of the discharging procedure affects the residual capacity can be listed as follows: r r (11) where C t r is the effective capacity of the battery at time t, C t r +1 indicates the effective capacity of the battery at time t + 1. As mentioned earlier, the power loss per unit battery at each time interval will be affected by the battery discharge current (I) and battery temperature (T): From the experimental data of the battery listed before, we can obtain the capacity-temperature relationship. At high temperatures, the temperature variation has little impact on the battery capacity, but at low temperatures, the battery capacity will be more sensitive to the temperature variation. In this paper, an Arrhenius equation about temperature T is introduced to characterize the equivalent capacity consumed at a certain time.
Except for the capacity-temperature relationship, the inconsistency of internal resistance is also closely related to other battery parameters. Due to the inconsistency of internal resistance, the current flowing through different batteries will be different. Furthermore, the difference in current will lead to inconsistency in Joule heat. For battery cells in parallel lines, larger current flows through the branch with smaller internal resistance. According to the experiment reported by Gogoana et al., 56 in the parallel circuits with inconsistent internal resistance, the heat generation rate of the battery with small internal resistance is higher than that of other branches, and the temperature rise of the battery is higher than that of other cells. Since the internal resistance is known to decrease with the rise of temperature, the branch with a small internal resistance will lead to lower resistance due to drastic temperature changes. 57 This phenomenon results in a more severe current imbalance and heating imbalance between parallel branches. This phenomenon promotes the deterioration of inconsistency between batteries. 58 Between different batteries placed in series, because the current flowing through different batteries is equal, more heat is generated in the battery with high resistance, resulting in a greater temperature rise and reduced internal resistance. Thus, inhibiting the expansion of inconsistency between batteries. Therefore, arranging the cells in the grouping mode of the first series and then parallel helps reduce the current inconsistency between the different branches. At the end of discharging, the internal resistance of the battery with lower SOC increases fast. If the battery with lower SOC is in a circuit of parallel, the current flowing through the battery will be reduced due to the parallel relationship. At the same time, other branches with higher SOC will bear a larger current. Thus, the SOC inconsistency between the different branches can be reduced. To quantify the influence of temperature on the performance of the battery pack and to study the SOC inconsistency and temperature inconsistency, we need to use MATLAB to simulate the effect of heat exchange between batteries. We supposed a battery pack composed of 18 battery cells and take the grouping mode of 3P6S/ 6S3P into consideration.
We set the temperature of the environment is 25°C and the discharging rate is 1C. We assumed that the capacities of 18 batteries conform to the normal distribution of N(105,0.5) Ah, and the internal resistance conforms to a normal distribution of N(0.0005, 0.00005) Ω. As for the discharge cut-off conditions of the battery pack, we divide the standard into two situations for analysis. For parallel circuits, we define that when all the branches are discharged, the discharge of parallel circuits is over. For series circuits, we define that when any of the series batteries in the circuit is fully discharged, the series circuit is regarded as reaching the termination of discharging. Except that, the main parameters in the simulation can be listed as follows: ρ = 2120kg/m 3 , c = 1040 J/kg K, T 0 = 25°C.
To quantify the inconsistency of the residual electricity, we take the mean square deviation of the residual power of all batteries as the measurement index, which is shown in Figure 9. In this formula, the Q i represents the residual electricity of battery i.
The result of the Cloud diagram at the end of the simulation can be shown in Figure 8. The curve of electricity inconsistency during the discharging is shown in Figure 9.
It can be seen from Figures 8 and 9 that the results calculated by the simplified model of MATLAB are highly consistent with the results calculated by the joint simulation of MATLAB-FLUENT. That means the single MATLAB method can act as an ideal substitute for MATLAB-FLUENT joint simulation. The cloud map distribution of the remaining electricity at the end of the discharging and the temperature cloud map distribution, as well as the temperature/residual electricity inconsistency curve in the whole discharge process, showed a high degree of similarity. From Figure 10, we can see that both the temperature deviation of 6S*3P and 3P*6S is smaller than 0.25°C. This shows that the approximation effect of the simplified model is ideal, and it can be used as an alternative method of joint simulation to analyze the internal inconsistency of the battery pack and evaluate the battery efficiency.
From the result of two different grouping methods, we can see that the available capacity of the 3P6S battery pack is improved compared with that of the 6S3P battery pack. It can be seen from the curve that no matter which grouping mode is used, the inconsistency of the residual electricity first increases and then decreases. Focused on the inconsistency of the residual electricity at the cut-off time, the inconsistency of the 3P*6S grouping method is relatively low, which means that when the cell with the lowest SOC in the battery pack reached the cut-off condition, there will be more electricity remained in other batteries, which leads to the increase of the residual electricity in the battery pack. Thus, the energy efficiency of the battery pack becomes lower too. In theory, when the initial capacities of all the cells in the battery pack are constant value, and the current flowing through several cells in the same series branch is equal, according to the experimental results in this paper, it can be inferred that if the temperature of a monomer is higher, the effective capacity of the battery cell in the discharge process decreases less, and the final remaining power of the battery will be higher. Under this premise, the residual power distribution at the end of the discharging of the battery pack will show an obvious linear relationship with the temperature distribution of the battery. But in this paper, the initial setting of the simulation process is that the initial capacities of the batteries conform to normal distribution rather than completely equal. Therefore, after the nonrandom interference of the initial capacity, there is no obvious correlation between the residual electricity quantity distribution at the end of the discharge and the temperature distribution of the battery, which is shown in Figure 8. Thus, the final remaining power of the battery doesn't show an obvious relationship with the temperature distribution of the battery.
For the parallel circuit, when the SOC of batteries is high, the voltages of batteries are not sensitive to their SOC variation. This cause the open circuit voltage of the parallel battery to be nearly the same. On this premise, if one of the branches has an internal resistance smaller than other branches, the current flowing through the branch will be larger than the others. Because the heating of the battery has a quadratic relationship with the current flowing through, this will cause the temperature of the branch to be higher than that of other branches. Accordingly, the high temperature will further reduce the internal resistance of the battery, thus causing a greater current imbalance. At the same time, due to the increase in current inconsistency, the power inconsistency between battery cells will also increase correspondingly, which is the reason for the increase of the remaining power inconsistency of the battery in the early stage. As a result, we can see from Figure 9 that the residual electricity inconsistency of the batteries increases at first. With the power consumption of the battery and the SOC approaches 0.1, the open-circuit voltage of the batteries will drop significantly as the SOC continues to decrease, and the current flowing by the battery with the lowest SOC in the parallel branch will be significantly reduced due to the voltage drop, thus reducing the SOC difference with other branches, which will lead to the reduction of the inconsistency of the residual power at the end. Therefore, the inconsistency of the remaining electricity of the battery will increase first and then decrease. From the discharge duration obtained from the simulation test, we can also see that the dischargeable power in the 3P*6S group mode is greater than that in the 6S*3P group mode. This shows that the grouping mode of first parallel and then serial has a positive effect on making full use of the electric energy of the battery pack.

| CONCLUSION
We developed a new real-time simulation with lumped parameter thermal model for dynamic situations. We use MATLAB to realize the thermoelectric coupling. According to the relationship between the voltage, the internal resistance, and SOC, we can build an electrical model for the battery monomer. Combining the grouping mode of the battery pack and the thermoelectric coupling model, we can realize the iterative calculation of thermoelectric coupling. Through real-time simulation, the temperature distribution and the distribution of remaining electricity in the discharging process are obtained. Thus, we can improve the accuracy of the simulation. We developed two methods of thermoelectric coupling. The first method is MATLAB-FLUENT joint simulation which has high precision but costs a lot of time and calculation in simulation. We use Fluent for thermal calculation and then substitute the thermal result into MATLAB to calculate the temperature-dependent electrical property. After the calculation of electrical property, substitute the result into fluent for thermal calculation. By repeating the cycle, we can get the performance evaluation of the lithium battery pack. The second method is simplified coupling simulation through MATLAB. MATLAB was used to construct the lumped parameter thermal model of LIBs which can realize the iterative calculation of thermoelectric coupling. The advantage of method two is that it can save a lot of calculating time. Through verification, we can come to the conclusion that the simulation result of the simplified model in the discharging process is highly consistent with the result of the MATLAB-FLUENT joint simulation. So the simplified model can be chosen as an ideal method to predict the performance of the battery pack which has good precision and saves computational cost.