Internal Temperature Estimation of Lithium Batteries Based on a Three-Directional Anisotropic Thermal Circuit Model

Abstract

In order to improve the accuracy of internal temperature estimation in batteries, a 10-parameter time-varying multi-surface heat transfer model including internal heat production, heat transfer and external heat transfer is established based on the structure of a lithium iron phosphate pouch battery and its three directional anisotropic heat conduction characteristics. The entropy heat coefficient, internal equivalent heat capacity and internal equivalent thermal resistance related to the SOC and temperature state of the battery were identified using experimental tests and the least square fitting method, and were then used for online calculation of internal heat production and heat transfer in the battery. According to the time-varying and nonlinear characteristics of the heat transfer between the surface and the environment of the battery, an internal temperature estimation algorithm based on the square root cubature Kalman filter was designed and developed. By iteratively calculating the estimated surface temperature and the measured value, dynamic tracking and online correction of the internal temperature of the battery can be achieved. The verification results using FUDS and US06 dynamic working condition data show that the proposed method can quickly eliminate the influence of initial temperature deviations and accumulated process errors and has the characteristics of a high estimation accuracy and good robustness. Compared with the estimation results of the adaptive Kalman filter, the proposed method improves the estimation accuracy of FUDS and US06 working conditions by 67% and 54%, respectively, with a similar computational efficiency.

1. Introduction

As an important component of new energy vehicles, the temperature state of lithium batteries needs to be accurately and reasonably monitored, which is an important guarantee for the safe use and extended lifespan of lithium batteries [1]. Battery cells under the winding or stacking process generally have a polyhedral structure. In engineering practice, because the internal temperature of the battery cannot be directly measured, the temperature state inside the battery is often replaced by the measurement of the electrode or surface temperature of the battery cell. During the process of battery charging and discharging, heat generation and transfer occur as follows: heat is generated from inside the battery cell and transmitted to the surface of the battery cell along the XYZ coordinate direction, and then convective heat exchange is carried out between each surface and the outer air. Each heat transfer and heat transfer circuit exhibit different heat capacities and resistances, and the temperature hysteresis of heat transfer can lead to an increase in the temperature difference between the inside and outside of the battery during group application. In severe cases, the temperature difference between the inside and outside of the battery can reach over 10 °C [2]. Inaccurate monitoring of the battery cell temperature status not only affects their safety and lifespan, but also affects the estimation accuracy of the remaining state of charge (SOC), actual available capacity, and battery health [3,4,5,6], and may even interfere with the decision-making process of battery balancing and thermal management.

In order to reveal the anisotropic thermal behavior characteristics of lithium batteries, researchers have used three-dimensional computational fluid dynamics simulations (CFD) to calculate the heat generation, transfer, and dissipation processes in batteries, and compared and analyzed the results from the aspects of computational efficiency and solution accuracy. Reference [7] proposed that the thermal conductivity of a battery has anisotropy, and a thermal model was established based on this characteristic. The calculation was carried out using the finite element method, and the results were close to the measured values, indicating a high accuracy. Reference [8] proposed a multidimensional battery thermal model and a new CFD simulation method to calculate the battery temperature, which improves the computational efficiency while ensuring accuracy. In response to the engineering practical solution requirements for dynamic working conditions in vehicles, the equivalent circuit model is applied to real-time solution calculation of the internal temperature of batteries. While considering the anisotropic thermal conductivity characteristics of batteries, the construction and iterative calculation of the temperature difference transfer relationship between the inside and outside of a battery cell under multiple heat exchange surface conditions were carried out to achieve an accurate real-time solution method for tracking the temperature changes inside a battery cell. Some researchers have established equivalent thermoelectric coupling models for batteries with variable parameters, using the extended Kalman filter (EKF) algorithm and the adaptive Kalman filter (AEKF) algorithm, respectively, to estimate the internal temperature of soft pack batteries [9,10,11]. However, this algorithm inevitably introduces errors into the process of linearizing the state equation using first-order Taylor expansion, which leads to a poor solution accuracy. At the same time, the difficulty of establishing a Jacobian matrix and the complexity of the solution process mean that the model used in the algorithm application process can only estimate the temperature measured on a single surface. Otherwise, the thermal property parameters are treated as fixed values or only as temperature-related variables, ignoring the heat exchange between soft pack batteries and the outside world in other directions, resulting in certain errors in the estimation results.

In order to further improve the estimation accuracy and reduce the difficulty of solving algorithms, reference [12] estimated the internal temperature of a battery using the unscented Kalman filter (UKF) algorithm. Compared with the EKF, the estimation results are more accurate and robust. Considering that in the practical application of batteries, the heat exchange conditions between the battery surface and the external air are variable, there is a high demand for real-time identification of nonlinear time-varying parameters such as the thermal resistance and heat capacity under various heat transfer circuits. Compared with EKF and UKF algorithms, the volume Kalman filter (CKF) algorithm can more efficiently and stably handle high-dimensional state estimation [13], and it has been proven that the CKF can be applied to the estimation of the battery SOC [14,15].

The purpose of this study is to explore and develop a high-dimensional thermal model state estimation algorithm that can deal with nonlinear and time-varying multidimensional parameter identification more efficiently by combining the structural characteristics of batteries and their three-dimensional thermal conductivity characteristics. The initial and process errors of temperature and other state parameters are effectively suppressed. The research object of this study is a 20 Ah lithium iron phosphate soft pack battery. The developed three-dimensional anisotropic thermal model has high-dimensional characteristics of four state variables. Time-varying multi-surface heat transfer model parameter measurement and estimation methods were established, including the internal heat generation, heat transfer, and external heat exchange processes of individual cells. A square root volume Kalman filter (SRCKF) algorithm suitable for high-dimensional thermal model state estimation was designed and developed. The accuracy of the developed algorithm was verified under dynamic operating conditions. The results showed that the method can achieve accurate and robust estimation of internal temperatures. It is of great significance for the accurate monitoring of the battery temperature status.

2. Establishment and Parameterization Analysis of a Battery Thermal Model

2.1. Three-Directional Anisotropic Thermoelectric Coupling Model for Lithium Batteries

The research object of this paper is the lithium iron phosphate soft pack battery produced by A123, and its basic parameters such as capacity and voltage are shown in

Table 1. Battery parameters.

Parameter		Value
Type		LiFePO4
Capacity		20 Ah
Voltage		2.0 V–3.65 V
Size	Length	195 mm
	Width	150 mm
	Height	8 mm

. The battery thermal model consists of the battery heat model and the battery heat transfer model [15]. According to Semenov combustion theory, this paper assumes that the interior of the soft pack battery generates uniform heat and considers the soft pack battery as a symmetrical structure. Based on this, a heat generation model and a heat transfer model are established.

The heat generated in batteries is mainly composed of reversible heat and irreversible heat [16], where the irreversible heat is the Joule heat, which depends on the current and internal resistance. The reversible heat is the entropy heat, which depends on the battery temperature and entropy heat coefficient. The entropy heat coefficient is obtained through experiments. According to the Bernardi heat generation model [17], the heating of the battery is calculated by the following formula:

$$Q=I(V_{OCV}-V_L)+IT\partial V_{OCV}/\partial T$$

(1)

where $I$ represents the battery current; $V_{OCV}$ represents the open circuit voltage of the battery; $V_L$ is the terminal voltage; $T$ is the battery temperature, which represents the internal temperature in this paper; and the entropy heat coefficient $\partial V_{OCV}/\partial T$ is the ratio of open circuit voltage increase to temperature increase.

The heat transfer of the battery is calculated through an equivalent thermal circuit model, which is similar in principle to an equivalent circuit model. The temperature is analogized to voltage, heat generation is analogized to current, and the heat loss during the heat transfer process is calculated through the equivalent thermal resistance and equivalent heat capacity. According to the structure of the pouch battery, all six surfaces exchange heat with the outside world. Considering the symmetry of the pouch battery structure, in order to reduce the number of parameters and computational complexity, a three-directional anisotropic equivalent thermal circuit model is established based on the three surfaces. The schematic diagram of the established three-directional anisotropic equivalent thermal circuit model is shown in Figure 1.

In Figure 1, $T_{in}$, $T_{s1}$, $T_{s2}$ and $T_{s3}$ represent the internal temperature and three surface temperatures of the battery, respectively; $T_{air}$ represents the ambient temperature; $C_c$ represents the equivalent internal heat capacity; $C_{s1}$, $C_{s2}$ and $C_{s3}$ represent the equivalent external heat capacity; $R_{i1}$, $R_{i2}$ and $R_{i3}$ represent the equivalent internal thermal resistance, respectively; and $R_{s1}$, $R_{s2}$ and $R_{s3}$ represent the equivalent external thermal resistance.

For the convenience of deriving the equivalent thermal circuit model, the relative temperatures of each part of the battery are defined in Equation (2). The positions of $T_0$, $T_1$, $T_2$ and $T_3$ correspond to $T_{in}$, $T_{s1}$, $T_{s2}$ and $T_{s3}$, respectively.

$$\left\{\begin{array}{l}{T}_0=T_{in}-T_{air}\\ T_1=T_{s1}-T_{air}\\ T_2=T_{s2}-T_{air}\\ T_3=T_{s3}-T_{air}\end{array}\right.$$

(2)

where $T_0$ represents the relative value between the internal temperature of the battery and the ambient temperature and $T_1$, $T_2$ and $T_3$ represent the relative value between the three surface temperatures of the battery and the ambient temperature.

Due to the small proportion of heat radiation inside the battery, and considering that the soft pack battery shell limits the flow of the internal electrolyte, the influence of internal heat convection and heat radiation on the battery heat transfer is ignored. The established equivalent thermal circuit model can be described using Equation (3):

$$\left\{\begin{array}{l}{C}_c\frac{d{T}_0}{dt}=Q-\frac{T_0-T_1}{R_{i1}}-\frac{T_0-T_2}{R_{i2}}-\frac{T_0-T_3}{R_{i3}}\\ C_{s1}\frac{d{T}_1}{dt}=\frac{T_0-T_1}{R_{i1}}-\frac{T_1}{R_{s1}}\\ C_{s2}\frac{d{T}_2}{dt}=\frac{T_0-T_2}{R_{i2}}-\frac{T_2}{R_{s2}}\\ C_{s3}\frac{d{T}_3}{dt}=\frac{T_0-T_3}{R_{i3}}-\frac{T_3}{R_{s3}}\end{array}\right.$$

(3)

where $C_c$ represents the equivalent internal heat capacity and $C_{s1}$, $C_{s2}$ and $C_{s3}$ represent the equivalent external heat capacity.

The equivalent thermal circuit model after discretization is shown in Equation (4). In Equation (4), $\Delta T$ is the sampling time. The input of the model is $u=\left[Q\right]$, the output is the relative surface temperature of the battery $y=[\begin{array}{ccc}{T}_1& T_2& T_3\end{array}{]}^T$, and the state variable is $x=[\begin{array}{cccc}{T}_0& T_1& T_2& T_3\end{array}{]}^T$. The heat transfer model parameters are ${\theta }_T=\left[\begin{array}{cccccccccc}{R}_{i1}& R_{i2}& R_{i3}& R_{s1}& R_{s2}& R_{s3}& C_c& C_{s1}& C_{s2}& C_{s3}\end{array}\right]$.

$$\left\{\begin{array}{l}{T}_0(k)=(1-\frac{\Delta T}{R_{i1}{C}_c}-\frac{\Delta T}{R_{i2}{C}_c}-\frac{\Delta T}{R_{i3}{C}_c})T_0(k-1)+\frac{\Delta T}{R_{i1}{C}_c}{T}_1(k-1)\\ \begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}+\frac{\Delta T}{R_{i2}{C}_c}{T}_2(k-1)+\frac{\Delta T}{R_{i3}{C}_c}{T}_3(k-1)+\frac{\Delta T}{C_c}Q(k-1)\\ T_1(k)=\frac{\Delta T}{R_{i1}{C}_{s1}}{T}_0(k-1)+(1-\frac{\Delta T}{R_{i1}{C}_{s1}}-\frac{\Delta T}{R_{s1}{C}_{s1}})T_1(k-1)\\ T_2(k)=\frac{\Delta T}{R_{i2}{C}_{s2}}{T}_0(k-1)+(1-\frac{\Delta T}{R_{i2}{C}_{s2}}-\frac{\Delta T}{R_{s2}{C}_{s2}})T_2(k-1)\\ T_3(k)=\frac{\Delta T}{R_{i3}{C}_{s3}}{T}_0(k-1)+(1-\frac{\Delta T}{R_{i3}{C}_{s3}}-\frac{\Delta T}{R_{s3}{C}_{s3}})T_3(k-1)\end{array}\right.$$

(4)

Based on the discretized thermal circuit model, a state space equation of battery thermal characteristics is established, as shown in Equations (5) and (6).

$$\left\{\begin{array}{l}x(k+1)=Ax(k)+Bu(k)\\ y(k+1)=Cx(k+1)\end{array}\right.$$

(5)

$$\left\{\begin{array}{l}A=\left(\begin{array}{cccc}1-\frac{\Delta T}{R_{i1}{C}_c}-\frac{\Delta T}{R_{i2}{C}_c}-\frac{\Delta T}{R_{i3}{C}_c}& \frac{\Delta T}{R_{i1}{C}_c}& \frac{\Delta T}{R_{i2}{C}_c}& \frac{\Delta T}{R_{i3}{C}_c}\\ \frac{\Delta T}{R_{i1}{C}_{s1}}& 1-\frac{\Delta T}{R_{i1}{C}_{s1}}-\frac{\Delta T}{R_{s1}{C}_{s1}}& 0& 0\\ \frac{\Delta T}{R_{i2}{C}_{s2}}& 0& 1-\frac{\Delta T}{R_{i2}{C}_{s2}}-\frac{\Delta T}{R_{s2}{C}_{s2}}& 0\\ \frac{\Delta T}{R_{i3}{C}_{s3}}& 0& 0& 1-\frac{\Delta T}{R_{i3}{C}_{s3}}-\frac{\Delta T}{R_{s3}{C}_{s3}}\end{array}\right)\\ B={\left(\begin{array}{cccc}\frac{\Delta T}{C_c}& 0& 0& 0\end{array}\right)}^T\\ C=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\\ \end{array}\right.$$

(6)

where A, B, C are the coefficients of the difference equation the coefficient matrix of the state space equation.

2.2. Parameterization of the Equivalent Thermal Circuit Model

There are a total of 10 parameters in the equivalent thermal circuit model, requiring a large amount of identification calculations which can reduce the identification accuracy of each parameter. Therefore, different methods are used to obtain different parameters. The equivalent external heat capacity is the heat capacity of the soft pack battery housing. Due to the compact structure, clear size and material properties of the aluminum–plastic film, it can be obtained through specific heat capacity and mass calculations [18]: $C_{s1}$ = 21.960 J/K, $C_{s2}$ = 1.025 J/K, $C_{s3}$ = 0.769 J/K.

By conducting parameter identification experiments using the equivalent thermal circuit model combined with the least squares method, the remaining thermal parameters are identified. The difference equation of the equivalent thermal circuit model is established for parameter identification, as shown in Equations (7) and (8).

$$y_T(k)=K_1{y}_T(k-1)+K_2{y}_T(k-2)+K_{3}Q(k-2)$$

(7)

$$\left\{\begin{array}{l}{y}_T={\left(\begin{array}{ccc}{T}_1& T_2& T_3\end{array}\right)}^T\\ K_1=\left(\begin{array}{ccc}{a}_1+b_2& 0& 0\\ 0& a_1+c_2& 0\\ 0& 0& a_1+d_2\end{array}\right)\\ K_2=\left(\begin{array}{ccc}{a}_2{b}_1-a_1{b}_2& a_3{b}_1& a_4{b}_1\\ a_2{c}_1& a_2{c}_1-a_1{c}_2& a_4{c}_1\\ a_2{d}_1& a_3{d}_1& a_2{d}_1-a_1{d}_2\end{array}\right)\\ K_3={\left(\begin{array}{ccc}{a}_5{b}_1& a_5{c}_1& a_5{d}_1\end{array}\right)}^T\end{array}\right.$$

(8)

where K₁, K₂, and K₃ are the coefficients of the difference equation,

The symbols in the coefficient matrix can be calculated from Equation (9). Meanwhile, the coefficients in Equation (8) can be fitted using the least squares method. Furthermore, the thermal model parameters of the battery can be calculated by the relationship between the coefficients in Equation (9) and the thermal model parameters. The equivalent internal heat capacity $C_c$ and equivalent internal thermal resistance $[\begin{array}{ccc}{R}_{i1}& R_{i2}& R_{i3}\end{array}{]}^T$ are three-dimensional relationships between the SOC and temperature. The equivalent external thermal resistance represents the thermal resistance $[\begin{array}{ccc}{R}_{s1}& R_{s2}& R_{s3}\end{array}{]}^T$ between the battery surface and the environment, which has a minor relationship with the battery SOC and is related to the ambient temperature. It is identified as a temperature-dependent function. During the calculation process of the thermal model, the above parameters are obtained by looking up the table based on the actual SOC and the predicted internal temperature of the battery.

$$\left\{\begin{array}{l}{a}_1=1-\frac{\Delta T}{R_{i1}{C}_c}-\frac{\Delta T}{R_{i2}{C}_c}-\frac{\Delta T}{R_{i3}{C}_c}\\ \begin{array}{cc}{a}_2=\frac{\Delta T}{R_{i1}{C}_c}& a_3=\frac{\Delta T}{R_{i2}{C}_c}\end{array}\\ \begin{array}{cc}{a}_4=\frac{\Delta T}{R_{i3}{C}_c}& a_5=\frac{\Delta T}{C_c}\end{array}\\ \begin{array}{cc}{b}_1=\frac{\Delta T}{R_{i1}{C}_{s1}}& b_2=1-\frac{\Delta T}{R_{i1}{C}_{s1}}-\frac{\Delta T}{R_{s1}{C}_{s1}}\end{array}\\ \begin{array}{cc}{c}_1=\frac{\Delta T}{R_{i2}{C}_{s2}}& c_2=1-\frac{\Delta T}{R_{i2}{C}_{s2}}-\frac{\Delta T}{R_{s2}{C}_{s2}}\end{array}\\ \begin{array}{cc}{d}_1=\frac{\Delta T}{R_{i3}{C}_{s3}}& d_2=1-\frac{\Delta T}{R_{i3}{C}_{s3}}-\frac{\Delta T}{R_{s3}{C}_{s3}}\end{array}\end{array}\right.$$

(9)

3. Experiment of Thermal Model Parameter Identification

The parameters of the thermal model are identified through experimental data, and Figure 2 shows the test platform. Four thermocouples are arranged at the inner center, maximum outer surface center, side center, and bottom center of the battery to measure four temperature points $[\begin{array}{cccc}{T}_0& T_1& T_2& T_3\end{array}{]}^T$. During the experiment, the battery is placed in a thermostatic box, which is used to adjust the external environmental temperature and conduct experiments at different ambient temperatures. The charging and discharging equipment is connected to the battery, and the charging and discharging experiment is conducted through the charging and discharging regulations input by the main computer. The current, voltage, and temperature data during the process are collected through the charging and discharging equipment and temperature data acquisition and input into the main computer for recording. Since the sampling frequency of temperature is 1 s in engineering practice [19], the sampling frequency of all tests in this paper is 1 s.

3.1. Identification Experiment of Open Circuit Voltage

The reversible heat in the battery is calculated from the difference between the open circuit voltage and the terminal voltage, as well as the current. The purpose of the open circuit voltage identification experiment is to provide a data basis for the calculation of reversible heat in the battery heat generation model. The open circuit voltage of a battery is a function of temperature and SOC, and it is identified through a mixed power pulse test. The specific experimental steps are as follows:

(1)

At an ambient temperature of 25 °C, the battery is fully charged through a constant current, constant voltage (CCCV) test, where the current rate is 1 C and the cut-off voltage is 3.65 V;

(2)

Let the battery stand for 1 h at four temperature points: 15 °C, 25 °C, 35 °C, and 45 °C;

(3)

At intervals of 10% SOC, the battery starts discharging from 100% SOC. For each discharge of 10% SOC, let the battery stand for 1 h and record the terminal voltage value until the SOC reaches 0%.

The terminal voltage of the battery recorded during the experiment is considered as the open circuit voltage at the current temperature and SOC. All open circuit voltage values were recorded and displayed in the form of a three-dimensional graph, as shown in Figure 3. In the thermal model calculation process, the open circuit voltage is obtained by looking up the SOC and internal temperature table, and the reversible heat is calculated according to Equation (1).

3.2. Determination of the Entropy Heat Coefficient

The entropy heat coefficient of the battery refers to the derivative of the open circuit voltage with respect to temperature. During the low-rate charging and discharging process, irreversible heat is the main source of battery heat generation. The purpose of this section’s entropy heat coefficient identification experiment is to obtain the entropy heat coefficient in the irreversible heat calculation formula in Equation (1), whose value is a function uniquely related to the SOC [20]. The battery was set to discharge at a rate of 1 C at an ambient temperature of 25 °C, the voltage variation with temperature was recorded at different SOCs, and the entropy heat coefficient was obtained, as shown in Figure 4.

3.3. Identification Experiment of Equivalent Thermal Circuit Model Parameters

At different temperatures, the peak temperature increase and the speed of temperature increase during battery charging and discharging are different, indicating that the equivalent heat capacity and equivalent thermal resistance vary with temperature. At the same time, some studies have also shown that the SOC is also an important factor affecting the equivalent heat capacity and equivalent thermal resistance [21]. Therefore, when identifying the parameters of the equivalent thermal circuit model, the influence of the SOC and temperature should be considered simultaneously. The purposes of the parameter identification experiment for the equivalent thermal circuit model in this section are to obtain the thermal property parameters of the heat transfer model in Equation (4) and to determine the relationship between equivalent heat capacity, equivalent thermal resistance, SOC, and temperature. Equivalent thermal circuit model identification experiments at different temperatures and SOCs were designed, with the specific steps as follows:

(1)

Arrange the insulated thermocouple in the center of the three surfaces of the soft pack battery and seal the cut with epoxy resin;

(2)

At an ambient temperature of 25 °C, fully charge the battery during the constant current, constant voltage (CCCV) test with a current rate of 1 C and a cut-off voltage of 3.65 V;

(3)

Let the battery stand for 1 h at four temperature points: 15 °C, 25 °C, 35 °C, and 45 °C;

(4)

Discharge at a current rate of 1 C; let the battery stand for 1 h every 10% SOC. Then conduct a pulse charging and discharging test with 2 C charging for 10 s and 2 C discharging for 10 s in a cycle. After that, let the battery stand for 1 h.

The thermocouple used in the test is a T-type thermocouple; the model is TT-T-36, the applicable temperature range is −200 °C–260 °C, and the diameter is 0.5 mm.

Experimental data for identifying the equivalent thermal circuit model parameters of the battery at four temperature points were obtained through the above experiments. Combined with Equations (7)–(9), the identified equivalent thermal circuit model parameters of the battery are shown in Figure 5.

4. Internal Temperature Estimation of the Battery

According to the battery thermal model, the internal temperature of the battery can be preliminarily calculated, but the calculation results are greatly affected by the initial value and the accuracy of the model. When there are initial value errors and cumulative errors, the accuracy of the internal temperature calculated directly through the thermal model is poor. Therefore, an algorithm needs to be designed to correct the initial value errors and cumulative errors.

4.1. Internal Temperature Estimation Algorithm

The cubature Kalman filter (CKF) based on the radial spherical volume rule approximates the nonlinear function using the probability density. Compared with the extended Kalman filter (EKF), the CKF does not ignore higher-order Taylor expansion terms and has a higher accuracy. In addition, the computational efficiency of the EKF for high-dimensional state matrices is low, which is not suitable for the thermal model proposed in this paper. However, the traditional CKF needs to square the covariance of the system state, which increases computing costs and reduces stability. The square root cubature Kalman filter (SRCKF) introduces QR decomposition during filtering, and the SRCKF directly calculates the square root of the covariance matrix and iteratively updates it, ensuring the positive definiteness and symmetry of the error covariance matrix in the CKF recursive process. Therefore, the SRCKF can improve the filtering stability and numerical accuracy, and its problem statement and algorithm steps are as follows:

(1)

State equation:

$$x_k=A{x}_{k-1}+B{u}_{k-1}+{\omega }_{k-1}$$

(10)

Output equation:

$$y_k=C{x}_k+D{u}_k+{\nu }_k$$

(11)

where $\omega \left(k\right)$ and $\nu \left(k\right)$ are system noise.

Initialize:

$${\widehat{x}}_0=E(x_0)$$

(12)

$$p_0=E[(x_0-{\widehat{x}}_0){(x_0-{\widehat{x}}_0)}^T]$$

(13)

(2)

Calculate the error covariance matrix:

$$P_{k-1}=S_{k-1}{(S_{k-1})}^T$$

(14)

(3)

Propagate the cubature points:

$$x_{i,k-1}=S_{k-1}{\xi }_i+{\widehat{x}}_{k-1},i=1,2,3,\cdots m$$

(15)

$$x_{i,k|k-1}^{*}=A_{k-1}{x}_{i,k-1}+B_{k-1}{u}_{k-1}$$

(16)

(4)

A priori estimate o the state:

$${\widehat{x}}_{k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{x}_{i,k|k-1}^{*}}$$

(17)

(5)

Take the square root of the prior error covariance matrix:

$$X_{k|k-1}^{*}=\frac{1}{\sqrt{2n}}\times \left[x_{1,k|k-1}^{*}-{\widehat{x}}_{k|k-1},x_{2,k|k-1}^{*}-{\widehat{x}}_{k|k-1},\cdots ,x_{2n,k|k-1}^{*}-{\widehat{x}}_{k|k-1}\right]$$

(18)

$$S_{k|k-1}^x=qr(X_{k|k-1}^{*},sqrt(Q_{k-1}))$$

(19)

(6)

Propagate the cubature points:

$$x_{i,k|k-1}=S_{k|k-1}^x{\xi }_i+{\widehat{x}}_{k|k-1},i=1,2,3,\cdots m$$

(20)

$$y_{i,k|k-1}^{*}=C_{k|k-1}{x}_{i,k|k-1}+D_{k|k-1}{u}_{k-1}$$

(21)

(7)

A priori estimate the measurement:

$${\widehat{y}}_{k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{y}_{i,k|k-1}^{*}}$$

(22)

(8)

Calculate the auto covariance matrix and cross-covariance matrix:

$$Y_{k|k-1}^{*}=\frac{1}{\sqrt{2n}}\times \left[y_{1,k|k-1}^{*}-{\widehat{y}}_{k|k-1},y_{2,k|k-1}^{*}-{\widehat{y}}_{k|k-1},\cdots ,y_{2n,k|k-1}^{*}-{\widehat{y}}_{k|k-1}\right]$$

(23)

$$S_{k|k-1}^{yy}=qr(Y_{k|k-1}^{*},sqrt(R_{k-1}))$$

(24)

$$P_{yy,k|k-1}=S_{k|k-1}^{yy}{(S_{k|k-1}^{yy})}^T$$

(25)

$$P_{xy,k|k-1}=X_{k|k-1}^{*}{(Y_{k|k-1}^{*})}^T$$

(26)

(9)

Calculate the innovation and Kalman gain:

$$e_k=y_k-{\widehat{y}}_{k|k-1}$$

(27)

$$K_k=P_{xy,k|k-1}{(P_{yy,k|k-1})}^{-1}$$

(28)

(10)

Update the states and square root of the error covariance matrix:

$${\widehat{x}}_k={\widehat{x}}_{k|k-1}+K_k{e}_k$$

(29)

$$S_k=qr(X_{k|k-1}^{*}-K_k{Y}_{k|k-1}^{*},K_{k}sqrt(R_{k-1}))$$

(30)

(11)

Steps (2)–(5) are the time update process, and Steps (6)–(10) are the state update process. The time update process and the state update process are repeated to achieve the SRCKF algorithm.

The SRCKF outputs the predicted internal temperature value and inputs it into the thermal model to update the thermal model parameters. The algorithm process is shown in Figure 6. The experiment can provide a parameter basis for the thermal model. The inputs of the thermal model are the predicted values of current, the actual SOC, and the predicted internal temperature. The outputs of the thermal model are the calculated values of the heat generation rate and three relative surface temperatures. The output heat generation rate is input into the SRCKF for prior estimation calculation of state variables. The relative surface temperature calculated by the thermal model is input into the measurement update for prediction of the measurement values, that is, the predicted values of the three relative surface temperatures. Further information is obtained by the difference between the measured and predicted relative surface temperature values. Finally, by updating the state variables and error covariance matrix through the Kalman gain, the predicted internal temperature values are updated, and the thermal model parameters are updated by inputting them into the thermal model.

4.2. Accuracy Verification and Robustness Analysis of Estimation Results

The feasibility of the internal temperature estimation method was verified through two dynamic operating conditions. Simulations and experimental verification of the internal temperature were carried out under FUDS and US06 operating conditions, respectively. The simulation environment temperature was set to 25°C, and the simulation and experimental results are shown in Figure 7 and Figure 8. The SRCKF algorithm used in this paper was compared with the adaptive extended Kalman filter (AEKF) algorithm. The advantage of the AEKF algorithm is that it has a higher computational efficiency when the dimension of the state matrix is low. The state matrix of the thermal model in this article is four-dimensional. As the EKF algorithm needs to calculate the Jacobian matrix, this will significantly reduce the computational efficiency of the algorithm. Based on

Table 2. Comparison of internal temperature estimation errors.

Algorithm	Condition	Maximum Error (°C)	Average Error (°C)	Calculation Time (s)
SRCKF	FUDS	0.153	0.023	0.54
	US06	0.096	0.030	0.40
AEKF	FUDS	0.213	0.070	0.50
	US06	0.144	0.065	0.33

, in terms of calculation time, the SRCKF and AEKF have similar calculation times, while the SRCKF has a slightly longer calculation time. However, the accuracy of the SRCKF is much higher than that of the AEKF. In the simulation under two dynamic working conditions, the average error and maximum error of the SRCKF are both smaller than those of the AEKF. For the FUDS operating condition, the maximum errors of SRCKF and AEKF are 0.153 °C and 0.213 °C, respectively; the average errors are 0.023 °C and 0.070 °C, respectively; and the estimation accuracy of the SRCKF is improved by more than 60%. For the US06 operating condition, the maximum errors of SRCKF and AEKF are 0.096 °C and 0.144 °C, respectively. The average errors are 0.030 °C and 0.065 °C, respectively, and the estimation accuracy of SRCKF is improved by more than 50%. In the early stage of calculating the two sets of operating conditions, the AEKF algorithm has significant errors, which is due to the inherent error of the built thermal model itself, and the inherent flaw of ignoring high-order Taylor expansions in the calculation of nonlinear systems in the EKF algorithm, resulting in significant errors in the early stage of the AEKF. In summary, the SRCKF algorithm achieved an accuracy improvement of over 50% with a computational efficiency similar to the AEKF, making it more suitable for internal temperature estimation.

In practical applications, temperature sensors cannot be installed inside a battery, so the initial value of the internal temperature is difficult to obtain. If the initial value error is ignored, there will be an offset in the internal temperature calculation results, which will affect the accuracy of temperature estimation. One important role of the SRCKF algorithm is that it solves the poor accuracy of internal temperature estimation caused by initial value errors [14]. This paper verifies the robustness of the estimation method by changing the initial value of internal temperature. The experimental data used are two dynamic operating conditions of batteries, FUDS and US06, at an ambient temperature of 25 °C. Different initial values of the internal temperature were set at 28 °C and 22 °C to verify the robustness of the estimation method in the presence of initial errors. The results are shown in Figure 9. Under different operating conditions, when the internal temperature has an initial error, the estimation results can be quickly adjusted back and can be adjusted back to the true value within 1200 s under the two operating conditions. The error of the thermal parameters in the temperature range of 22 °C to 25 °C is slightly higher than that in the temperature range of 25 °C to 28 °C, resulting in a slower internal temperature correction speed when the initial temperature is 22 °C under the two operating conditions. As the experiments continue, the two internal temperature curves with different initial values gradually approach, until the middle and later stages of the experiment, they almost completely overlap. This indicates that the SRCKF algorithm can solve the problem of initial value errors and can ensure the robustness of the result.

The estimation method proposed in this paper has a high accuracy of internal temperature estimation, and the estimation results have a sufficient accuracy and robustness. In the subsequent application to a multi-cell pack, it can estimate the internal temperature of the cell, where it is not easy to install a temperature sensor, and calculate the temperature difference between the center and the edge of the pack. In the actual application process, in order to improve the accuracy of monitoring the internal temperature of the battery module, it is still necessary to establish a thermal model of the battery module combined with the battery installation mode and the thermal management mode, which is a future research direction.

5. Conclusions

An accurate estimation of the internal temperature can ensure the reliability of battery operation. This article considers the structure of soft pack batteries to establish a variable parameter thermal model, and the square root volume Kalman filter (SRCKF) is proposed for online estimation of internal temperatures. The accuracy and robustness of the estimation method are verified through dynamic operating condition experiments. The main conclusions are as follows:

(1)

A battery thermal model is established from the perspectives of heat generation and transfer. The battery heat generation model considers the reversible and irreversible heat of the battery. In addition, based on the structure of the soft pack battery, an equivalent thermal circuit model considering three surface heat exchanges is established. The thermal model parameters are variables related to the internal temperature and SOC.

(2)

An experiment for determining the open circuit voltage, entropy heat coefficient, and equivalent thermal circuit model parameters is designed to parameterize the thermal model. The difference equation of the equivalent thermal circuit model for parameter identification is derived. Considering the large number of parameters, the material characteristics are used to empirically determine some parameters, reducing the difficulty of parameter identification.

(3)

The SRCKF is proposed for internal temperature estimation, and its accuracy is verified through two dynamic operating conditions: FUDS and US06. Compared with the AEKF, the SRCKF has a similar computational efficiency, with an accuracy improvement of over 50%. In addition, the robustness of the estimation method is verified by changing the initial internal temperature value.

In the past, data on the temperature difference between the inside and outside of battery cells needed to be obtained through three-dimensional CFD simulations. The proposed thermal model mentioned in this paper, which combines three-dimensional anisotropic heat generation, heat transfer, and heat dissipation, can be extended to the module level to estimate the relative surface temperature difference between the center and edges of the battery. Furthermore, it can provide model and algorithm support for charging current control and battery thermal management based on model predictive control, and can even be combined with balanced control to provide a basis for energy flow path planning. In addition, the proposed internal temperature estimation method can be validated on other high-capacity batteries and optimized by combining the estimation results of battery SOC, state of health and other states.