# Sensorless Coestimation of Temperature and State-of-Charge for Lithium-Ion Batteries Based on a Coupled Electrothermal Model

Authors: Wenyuan Bai; Xinhui Zhang; Zhen Gao; Shuyu Xie; Ke Peng; Yu Chen
Journal: International Journal of Energy Research
 (2023)

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## Abstract

Accurate estimations of the temperature and the state-of-charge (SOC) are of extreme importance for the safety of lithium-ion battery operation. Traditional battery temperature and SOC estimation methods often omit the relation between battery temperature and SOC, which may lead to significant errors in the estimations. This study presents a coupled electrothermal battery model and a coestimation method for simultaneously estimating the temperature and SOC of lithium-ion batteries. The coestimation method is performed by a coupled model-based dual extended Kalman filter (DEKF). The coupled estimators utilizing electrochemical impedance spectroscopy (EIS) measurements, rather than utilizing direct battery surface measurements, are adopted to estimate the battery temperature and SOC, respectively. The information being exchanged between the temperature estimator and the SOC estimator effectively improves the estimation accuracy. Extensive experiments show that, in contrast with the EKF-based separate estimation method, the DEKF-based coestimation method is more favorable in reducing errors for estimating both the temperature and SOC even if the battery core temperature has increased by 17°C or more during the process of test.


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## Body

## 1. Introduction

In recent years, COVID-19 pandemics have accelerated the structural changes in energy demand and consumption. [1] Lithium-ion batteries, representing a class of clean energy storage devices [2], have attracted widespread attention due to their low self-discharging rate, fast charging rate, and high power density [3]. Cylindrical lithium-ion batteries possess a more secure and stable structure than prismatic and pouch batteries, and they are widely deployed in electric vehicles and energy storage. Currently, LiFePO4 batteries are the most popular and applicable lithium-ion batteries due to their low cost, excellent safety, and high cycle durability. However, cylindrical LiFePO4 batteries contain flammable electrolytes which may lead to accidents if handled incorrectly. Therefore, the precise estimations of the battery states, such as temperature and state-of-charge (SOC), are of critical importance to ensure safe operation. However, the conventional approach to SOC estimation does not consider the core temperature variation during the battery operation, introducing large estimation errors. The incorrect SOC may lead to overdischarge and internal short circuit [4], and the latter could potentially trigger thermal runaway which may result in fires, electrolyte leakage, and explosion [5, 6]. The traditional estimation techniques rely on battery surface measurements to predict the states of the sealed lithium-ion batteries. Therefore, in practical engineering applications, there is an urgent need to improve the estimation techniques to perform multiple estimations using fewer measurements and fewer instruments.Existing lithium-ion battery models have different concentrations on state prediction as they vary in precision and complexity. Under some operating conditions that steady currents are needed, [7] a single electrical model is sufficient. [8] However, when the needed current is large and complicated, more complicated models are needed to provide accurate voltage prediction. Nevertheless, some of the complicated models requiring a higher computational complexity are generally difficult to be parameterized. The conventional approach to balance complexity and fidelity is to utilize the equivalent circuit model (ECM). A couple of resistance-capacitance (RC) pairs are connected in the ECM to describe the electrochemical characteristics and concentrate the polarization characteristics of lithium-ion batteries [9–12]. An additional model is usually introduced to the ECM to estimate the battery SOC under varying temperature conditions. Lin et al. [13] present an electrothermal model in which a thermal model is introduced to perform the SOC estimation considering the temperature effects.The traditional method for estimating temperature is to employ thermal models [14–17]. Such models rely on the feedback from the online sensor, yet the sensor errors will introduce large errors for temperature estimation. Some state estimation methods using online measurements, such as the Kalman filter algorithm and trained neural networks, may perform accurate battery temperature estimations. However, such online state estimations will count on the measurements of surface temperature and cooling fluid temperature [18, 19]. Furthermore, these above temperature estimation methods require temperature sensors on each battery, which means a higher equipment budget. An alternative method for estimating battery temperature is to use electrochemical impedance spectroscopy (EIS) measurements to predict temperature distribution [20, 21]. The method using impedance to obtain the volume-averaged temperature is referred to as “impedance-temperature detection” (ITD). [22] Therefore, the ITD-based thermal model, using EIS measurements as input rather than using the measurements feedback from the online sensors, is capable of predicting the battery core temperature. Some advanced detection technology utilizing power electronics to measure EIS in hybrid electric vehicles or other electric applications have been proposed [23–25]. The EIS measurements could be used in synergistically estimating the battery SOC and temperature because the current measurement is also included in the EIS measurements. The SOC estimation errors caused by temperature variation could be reduced by the coestimation. Research is currently being conducted on SOC estimations considering the effects of temperature. For example, Xing et al. [26] present a temperature-based model and a table for describing the relation among the open circuit voltage, SOC, and the temperature, which can perform SOC estimations at different temperatures. Lee et al. [27] propose a model with temperature compensation to estimate the battery states under different temperature conditions. Li et al. [28] employ a joint SOC estimation method using the adaptive extended Kalman filter (AEKF) and adopt the recursive least square (RLS) method for parameter identification. However, all the above SOC estimation methods rely on the identified battery parameters to involve the effect of temperature variation. This type of estimation method requires both a current sensor and a temperature sensor on every battery, which represents a substantial instrumentation cost. In contrast, the coestimation of SOC and temperature using EIS measurements can reduce the instrumentation cost since temperature sensors are not required.In this paper, to improve the accuracy of the state estimations using fewer measurements and instruments, a coupled electrothermal model is proposed based on the combination of a dual-polarization (DP) electrical submodel and an ITD-based thermal submodel. Furthermore, a coupled model-based dual extended Kalman filter (DEKF) using EIS measurement as inputs is proposed for coestimating the battery core temperature and SOC. Note that both the temperature and the SOC estimation accuracy are improved effectively due to the information being exchanged between the temperature estimator and the SOC estimator. The output of the temperature estimator is used as the input for the SOC estimator, effectively reducing the errors due to temperature variations; and the output of the SOC estimator provides a more accurate open circuit voltage (OCV) for the temperature estimator, which could effectively reduce the temperature estimation error. The estimation results of battery temperature and SOC demonstrate that, compared with the baseline EKF estimator, the DEKF coestimator is effective in improving the accuracy of both the SOC estimation and temperature estimation. Table1 shows the proposed coupled model in comparison with existing coupled models, in which the proposed model based on impedance measurements, rather than battery surface measurements, is highlighted.Table 1
Comparison of coupled models.
Coupled modelsSubmodelsMeasurementThermal submodelElectrical submodelSurface measurementsITDTraditionalNumericalECM✓ElectrochemicalNumericalElectrochemical✓Neural networkSimplifiedECM✓ProposedITD-basedDynamic DP✓This article is structured as follows. The structure of the proposed coupled electrothermal model is introduced in Section2. Section 3 introduces the design process of the coupled estimator based on DEKF. Section 4 exhibits and discusses the validation results by comparing the performance of the proposed DEKF algorithm with that of the baseline EKF algorithm. Finally, conclusions are given in Section 5.


## 2. Coupled Electrothermal Model

In this part, we propose a model using EIS measurements as inputs for coestimating battery temperature and SOC simultaneously. The coupled electrothermal model contains two submodels: the ITD-based thermal submodel for temperature estimation and the dynamic DP electrical submodel for SOC estimation. The two submodels can provide accurate state predictions for coestimating the battery temperature and SOC.


### 2.1. Thermal Submodel

The battery electrochemical impedance can be related to the temperature, SOC, and state of health (SOH). The effect of SOC and SOH on electrochemical impedance can be neglected within an appropriate frequency range [29, 30]. The electrochemical impedance is given by
(1)Zω=Z′ω+jZ″ω.The battery temperature distribution can be described by the electrochemical impedance instead of sensor measurements. In this case, the second-order polynomial of battery temperature can be used to infer the real part of the battery admittance (215 Hz), which is depicted as follows:(2)Y′=2ro2∫rirora1+a2Tr+a3T2rdr,where ri and ro are the inside radius and outside radius of the battery, respectively, and a1, a2, and a3 are, respectively, the coefficients of the polynomial for describing the relation between impedance and temperature.The ITD technique uses a thermal model in conjunction with the volume average temperatureT¯EIS based on EIS measurements to estimate the temperature distribution and predict the core temperature, as shown in Figure 1.Figure 1
Schematic of the proposed estimation method based on ITD.A simplified one-dimensional cylindrical battery thermal model is adopted to predict the temperature distribution and infer the core temperature of the battery. The simplified model is extremely effective in capturing the heat generation of the battery, and the boundary value problem of the simplified model is depicted as follows:(3)ρcp∂Tr,t∂t=kt∂2Tr,t∂r2+ktr∂Tr,t∂r+QtVb,where ρ (kg·m-3) represents the density, cp (J·kg-1·K-1) means the specific heat capacity, and kt (W·m-1·K-1) stands for the thermal conductivity. Vb (m3) is the battery volume, and Q (J) represents the heat generation. The energy balance of the battery is defined by the boundary conditions, which are described by
(4)∂T∂r=−hktT−T∞atr=ro,(5)∂T∂r=0atr=ri,where T∞ (K) is the air temperature of the thermal chamber and h (W·m-2) is the convection coefficient. The heat source of a battery can be well captured by
(6)Q=IVOC−VL+IT∂VOC∂T,where VOC (V) is the OCV, VL (V) represents the terminal voltage, I (A) stands for the measured current, and T (K) means the measured temperature. As shown in Equation (6), the two components of the heat generated by the battery are joule heat and entropic heat. For the joule heat, IVOC−VL, VOC is identified by a look-up table of SOC-OCV-temperature, rather than using approximation calculated by a certain SOC. While for the entropic heat, IT∙∂VOC/∂T, it is neglected to simplify the thermal model since the value of ∂VOC/∂T is less than 0.1 mV·K-1 when the battery is operating in the hybrid electric vehicle model. Besides, the battery temperature during a single period of current can be considered a constant; therefore, the average net reversible heat term over a period is almost zero when the battery is operating in the hybrid electric vehicle mode [31]. Thus, the entropic heat is neglected in the evaluation of heat source which is given by Equation (6).The polynomial approximation (PA) method is efficient in estimating the radially distributed temperatures of cylindrical battery, [32] and the PA is also adopted to solve Equation (3) where the temperature distribution can be expressed as
(7)Tr,t=4Tsurf−3T¯−15ro8γ¯+−18Tsurf+18T¯+15ro2γ¯rro2+15Tsurf−15T¯−45ro8γ¯rro4,where Tsurf (K) represents the battery surface temperature, T¯ (K) stands for the volume averaged temperature, and γ¯ (K·m-1) means the temperature gradient. According to Equation (4), the surface temperature is given by
(8)Tsurf=24kt24kt+rohT¯+15ktro48kt+2rohγ¯+roh24kt+rohT∞.Substituting Equation (7) in Equation (2), the real admittance is given by
(9)Y′=a1+a2T¯+3a3T¯2+2a3Tsurf2−4a3T¯Tsurf+15a3ro2γ¯232+15a3roT¯γ¯8−15a3roTsurfγ¯8,where the Y′ is a function of Tsurf, T¯, and γ¯. Note that Tsurf itself is a function of T¯, γ¯, and T∞. Therefore, since the battery parameters have been defined, the battery impedance can be expressed as
(10)Z′=fT¯,γ¯,T∞.According to the above derivation, the state-space equation for temperature estimation is obtained:(11)θ̇=Aθ+Bu,λ=Cθ+Du,where θ=T¯γ¯T is the states of the state-space equation, u=QT∞T is the inputs, and λ=TcoreTsurfT is the outputs. As shown in Equation (6), the heat generation Q is obtained by the VOC and VL. The accurate calculation of VOC and VL is presented in the next subsection.


### 2.2. Electrical Submodel

The DP model, as shown in Figure2, is a typical electrical model for terminal voltage prediction. In the DP model, VL (V) is the terminal voltage, VOC (V) is the OCV, and R0 (Ω) is used for simulating the voltage drop caused by the internal resistance of the battery. The paralleled Ri (Ω) and Ci (S) are used for simulating the transient voltage, and the RC pairs are generally used to describe the polarization of the battery.Figure 2
Schematic of the DP model.Since the entropic heat in Equation (6) is neglected, the heat source can be rewritten as
(12)Q=IVOC−VL,where the current I is measured and the terminal voltage VL is predicted by the DP model. The conventional approach to obtain the VOC at a certain SOC is to use the linear interpolation method according to the corresponding look-up table. In general, the VOC is regarded as a function of SOC; however, the OCV will change as the temperature rises at a certain SOC. For example, when the A123 26650 LiFePO4 battery operates at 70% SOC, the open circuit voltage VOC is 3.24 V at −10°C, while the VOC is 3.29 V at 30°C. Therefore, temperature-dependent OCV is essential to estimate the battery states accurately.To build the OCV-SOC-temperature model, we process the measurement data of the Incremental OCV tests for temperatures ranging from −10°C to 50°C at each 10°C interval [33]. The procedure for measuring OCV is designed as follows: (i) fully charge the battery to 100% SOC and discharge with relaxation duration at every 10% SOC and record the corresponding OCV, (ii) charge the battery to 100% SOC with the same duration of 10% SOC and record the corresponding OCV, and (iii) perform (1) and (2) repeatedly from −10°C to 50°C at each 10°C interval to build the table of OCV-SOC-temperature by applying averaging method and linear interpolation. Notably, to obtain the OCV as accurately as possible, the measurement data should be collected until the OCV has reached a steady open-circuit state. To obtain accurate OCV for online SOC estimation, we build the look-up tables of OCV-SOC-temperature according to the measurement data. The look-up table is built as a three-dimensional (3-D) form to contain the OCV values at different temperatures and different SOC. Therefore, when the battery temperature changes at different SOC, the accurate VOC can be obtained by 3-D linear interpolation according to the look-up tables of OCV-SOC-temperature. As shown in Figure 3, the 3-D look-up tables are visualized with a 3-D response surface, and the extracted OCV map describes the OCV change with the temperature rising at different SOC.Figure 3
The 3-D OCV response surface.The DP models are required to predict the terminal voltage,VL:
(13)V̇1=−V1R1C1+IC1,V̇2=−V2R2C2+IC2,VL=VOC−V1−V2−IR0,where the RC parameters are obtained by parameter identification for lithium-ion battery. The RLS method, a procedure to find the optimal match for the data by minimizing the sum of the square of errors, is efficient in identifying the match parameters of the battery [34–36]. The RC parameters of the DP model are generally preset for a specific temperature (e.g., 25°C). However, the temperature and SOC will affect the battery internal reactions during battery (dis)charging. To perform accurate voltage prediction, the RC parameters characterizing the battery polarization should be updated when the battery’s temperature changes at different SOC. The relation among the RC parameters, SOC, and temperature can be expressed as
(14)ParameterR,C=fTcore,SOC.To describe the relation above, a dynamic parameter model is built based on the experimental data under different temperatures and SOC. The experimental data is obtained by the pulse-relaxation tests conducted on 2.3 Ah A123 26650 LiFePO4 batteries to identify the RC parameters accurately. The experiment temperatures range from 5°C to 45°C with an interval of 10°C, and SOC ranges from 10% to 90% with an interval of 10% SOC. The RLS method adopted for the RC parameter identification is summarized in Algorithm 1. The relation among RC parameters, SOC, and temperature is formulated in (14), and the identification results of varying temperatures and SOC are shown in Figure 4. Series look-up tables of parameter-SOC-temperature are constructed based on the identification results. The accurate RC parameter values could be obtained by using the linear interpolation method according to the look-up tables when the battery temperature changes at different SOC.Algorithm 1:The procedure of parameter identification using RLS.
Step 1: applying the Laplace transform to the terminal voltage calculation Equation (13), the following equation can be obtained:VOCs−VLs=IsR0+R1/R1C1s+1+R2/R2C2s+1.Step 2: deriving the transfer function:Gs=VOCs−VLs/Is=R0s21/τ1τ2R0τ1+R0τ2+R1τ2+R2τ1s+R0+R1+R2/τ1τ2/s2+τ1+τ2s/τ1τ2+1/τ1τ2Step 3: applying bilinear transformation for discretization:Lets=2/Δt⋅1−z−1/1+z−1, Gz−1=a3+a4z−1+a5z−2/1−a1z−1−a2z−2, where aii=1,2,⋯,5 are constants.Step 4: transforming the discrete system transfer function into a time-domain difference equation:yt=a1yt−1+a2yt−2+a3It+a4It−1+a5It−2,whereyt is the voltage drop and yt=VOCt−VLt, It is the input current.Step 5: definingφt=yt−1yt−2ItIt−1It−2Tθ=a1a2a3a4a5.Step 6: definingyt=φTtθ+et, where et is the sampling error.Step 7: computing the recursive termination conditionsJθ:Jθ=∑i=1tY−Φθ2=∑i=1tei+22, where Y=y3y4y5⋯yN+2Te=e3e4e5⋯eN+2T.Step 8: deriving the recursion formula:θ^t+1=θ^t+Kt+1yt+1−ΦTt+1θ^t,Kt+1=Pt+1Φt+11+ΦTt+1PtΦt+1−1,Pt+1=I−Kt+1ΦTt+1Pt.Step 9: substitutingθ^=ΦTΦ−1ΦTY into Step 8, and obtaining the value of parameter ai.Step 10: applying the inverse bilinear transformation toG(z-1) in Step 3, where z−1=1−ΔT/2s/1+ΔT/2s:Gs=a3−a4+a5/1+a1−a2s2+4a3−a5/ΔT1+a1−a2s+4a3−a4+a5/ΔT21+a1−a2/s2+41+a2/ΔT1+a1−a2s+41−a1−a2/ΔT21+a1−a2Step 11: deriving the value ofR0,R1,R2,C1, and C2 by comparing the corresponding coefficients in Step 2 and Step 10.Figure 4
The 3-D response surfaces of the DP model’s parameters. (a)R0, (b) R1, (c) R2, (d) C1, and (e) C2.
(a)(b)(c)(d)(e)A dynamic DP model is proposed according to the parameter identification, as shown in Figure5, which utilizes dynamic RC parameters instead of using the values identified at a specific temperature. The values of R0 (DYN), R1 (DYN), R2 (DYN), C1 (DYN), and C2 (DYN) are obtained by linear interpolation according to the real-time temperature and SOC. The values of RC parameters could be calibrated during the estimation, and the errors in terminal voltage prediction and heat generation could be further reduced.Figure 5
Schematic of the dynamic DP model.This dynamic DP model is also used to derive the state equations for the SOC estimation. Substituting Kirchhoff’s current law to Equation (13):
(15)dVC1dt=1C1I−1R1C1VC1,dVC2dt=1C2I−1R2C2VC2.Substituting the expression of SOC into Equation (15), the continuous-time state-space model can be expressed as
(16)dSOCdtdVC1dtdVC2dt=0000−1R1C1000−1R2C2SOCVC1VC2+−1Cbat1C11C2I.Applying the closed-form discretization formula to Equation (16), the discrete state-space model is defined as
(17)SOCn+1VC1,n+1VC2,n+1=1000e−T/R1C1000e−T/R2C2SOCnVC1,nVC2,n+−TCbatR11−e−T/C1R1R21−e−T/C2R2I.According to Equation (17), the state-space equations of the dynamic DP model are depicted as follows:
(18)ẋ=Ax+Bu,y=Cx+Du,where x represents the state of the state-space equations which is given by xn=SOCnVC1,nVC2,nT, where u stands for the exogenous input and y means the output.


### 2.3. Coupled Electrothermal Model

In general, the models for temperature estimation and SOC estimation use disparate data sources. However, the ITD-based temperature estimation method uses EIS measurements as inputs, and the EIS measurements also could be used in the SOC estimation. In other words, the EIS measurements could be used to synergistically estimate temperature and SOC. Therefore, a coupled electrothermal model using EIS measurements is proposed by integrating the above two submodels shown in Sections2.1 and 2.2.The parameters of the proposed model consist of structural and physical parameters, and Table2 shows the parameters of the two submodels, as well as the parameters shared by the two submodels. The two submodels share the updated physical parameters to improve the estimation accuracy during the process each estimation.Table 2
The structural and physical parameters of the model.
ModelsParametersStructural parametersPhysical parametersThermal submodelRadius of the batteryrDensity ρHeat capacity cpThermal conductivity ktBattery volume VbAir temperature of the thermal chamberT∞Convection coefficient hBattery temperature Tsurf,TcoreVolume averaged temperature T¯Temperature gradient γ¯Heat generation QElectrical submodelOhmic internal resistanceR0Polarization resistance R1 and R2Battery polarization capacitance C1 and C2Terminal voltageVLOpen circuit voltage VOCCurrent IState of charge SOCCoupled electrothermal model/Heat generationQOpen circuit voltage VOCBattery core temperature TcoreCurrent IState of charge SOCAccording to Figure6, the thermal submodel and the electrical submodel interact through a two-way coupling. Once the measurement data is received by the coupled model, the electrical model will perform the predictions of SOC and voltage using the measured current I and the updated RC parameters. Then, the VOC with a smaller error is obtained by linear interpolation according to the estimated SOC and the table of OCV-SOC-temperature. The heat generation Q is calculated using the difference between VL and VOC, along with I. Then, the thermal model performs battery temperature estimation, and the estimated temperature and SOC can be further used to update the electrical RC parameters; therefore, the electrical submodel using dynamic RC parameters will reduce the errors in the SOC estimation. The VOC with a smaller error is used to calculate the VL and Q, which will reduce the errors of the temperature estimation. Thus, it can be seen that the coupling of the two submodels improves the accuracy of both temperature estimation and SOC estimation.Figure 6
Coupling between the thermal model and the electrical model.


## 2.1. Thermal Submodel

The battery electrochemical impedance can be related to the temperature, SOC, and state of health (SOH). The effect of SOC and SOH on electrochemical impedance can be neglected within an appropriate frequency range [29, 30]. The electrochemical impedance is given by
(1)Zω=Z′ω+jZ″ω.The battery temperature distribution can be described by the electrochemical impedance instead of sensor measurements. In this case, the second-order polynomial of battery temperature can be used to infer the real part of the battery admittance (215 Hz), which is depicted as follows:(2)Y′=2ro2∫rirora1+a2Tr+a3T2rdr,where ri and ro are the inside radius and outside radius of the battery, respectively, and a1, a2, and a3 are, respectively, the coefficients of the polynomial for describing the relation between impedance and temperature.The ITD technique uses a thermal model in conjunction with the volume average temperatureT¯EIS based on EIS measurements to estimate the temperature distribution and predict the core temperature, as shown in Figure 1.Figure 1
Schematic of the proposed estimation method based on ITD.A simplified one-dimensional cylindrical battery thermal model is adopted to predict the temperature distribution and infer the core temperature of the battery. The simplified model is extremely effective in capturing the heat generation of the battery, and the boundary value problem of the simplified model is depicted as follows:(3)ρcp∂Tr,t∂t=kt∂2Tr,t∂r2+ktr∂Tr,t∂r+QtVb,where ρ (kg·m-3) represents the density, cp (J·kg-1·K-1) means the specific heat capacity, and kt (W·m-1·K-1) stands for the thermal conductivity. Vb (m3) is the battery volume, and Q (J) represents the heat generation. The energy balance of the battery is defined by the boundary conditions, which are described by
(4)∂T∂r=−hktT−T∞atr=ro,(5)∂T∂r=0atr=ri,where T∞ (K) is the air temperature of the thermal chamber and h (W·m-2) is the convection coefficient. The heat source of a battery can be well captured by
(6)Q=IVOC−VL+IT∂VOC∂T,where VOC (V) is the OCV, VL (V) represents the terminal voltage, I (A) stands for the measured current, and T (K) means the measured temperature. As shown in Equation (6), the two components of the heat generated by the battery are joule heat and entropic heat. For the joule heat, IVOC−VL, VOC is identified by a look-up table of SOC-OCV-temperature, rather than using approximation calculated by a certain SOC. While for the entropic heat, IT∙∂VOC/∂T, it is neglected to simplify the thermal model since the value of ∂VOC/∂T is less than 0.1 mV·K-1 when the battery is operating in the hybrid electric vehicle model. Besides, the battery temperature during a single period of current can be considered a constant; therefore, the average net reversible heat term over a period is almost zero when the battery is operating in the hybrid electric vehicle mode [31]. Thus, the entropic heat is neglected in the evaluation of heat source which is given by Equation (6).The polynomial approximation (PA) method is efficient in estimating the radially distributed temperatures of cylindrical battery, [32] and the PA is also adopted to solve Equation (3) where the temperature distribution can be expressed as
(7)Tr,t=4Tsurf−3T¯−15ro8γ¯+−18Tsurf+18T¯+15ro2γ¯rro2+15Tsurf−15T¯−45ro8γ¯rro4,where Tsurf (K) represents the battery surface temperature, T¯ (K) stands for the volume averaged temperature, and γ¯ (K·m-1) means the temperature gradient. According to Equation (4), the surface temperature is given by
(8)Tsurf=24kt24kt+rohT¯+15ktro48kt+2rohγ¯+roh24kt+rohT∞.Substituting Equation (7) in Equation (2), the real admittance is given by
(9)Y′=a1+a2T¯+3a3T¯2+2a3Tsurf2−4a3T¯Tsurf+15a3ro2γ¯232+15a3roT¯γ¯8−15a3roTsurfγ¯8,where the Y′ is a function of Tsurf, T¯, and γ¯. Note that Tsurf itself is a function of T¯, γ¯, and T∞. Therefore, since the battery parameters have been defined, the battery impedance can be expressed as
(10)Z′=fT¯,γ¯,T∞.According to the above derivation, the state-space equation for temperature estimation is obtained:(11)θ̇=Aθ+Bu,λ=Cθ+Du,where θ=T¯γ¯T is the states of the state-space equation, u=QT∞T is the inputs, and λ=TcoreTsurfT is the outputs. As shown in Equation (6), the heat generation Q is obtained by the VOC and VL. The accurate calculation of VOC and VL is presented in the next subsection.


## 2.2. Electrical Submodel

The DP model, as shown in Figure2, is a typical electrical model for terminal voltage prediction. In the DP model, VL (V) is the terminal voltage, VOC (V) is the OCV, and R0 (Ω) is used for simulating the voltage drop caused by the internal resistance of the battery. The paralleled Ri (Ω) and Ci (S) are used for simulating the transient voltage, and the RC pairs are generally used to describe the polarization of the battery.Figure 2
Schematic of the DP model.Since the entropic heat in Equation (6) is neglected, the heat source can be rewritten as
(12)Q=IVOC−VL,where the current I is measured and the terminal voltage VL is predicted by the DP model. The conventional approach to obtain the VOC at a certain SOC is to use the linear interpolation method according to the corresponding look-up table. In general, the VOC is regarded as a function of SOC; however, the OCV will change as the temperature rises at a certain SOC. For example, when the A123 26650 LiFePO4 battery operates at 70% SOC, the open circuit voltage VOC is 3.24 V at −10°C, while the VOC is 3.29 V at 30°C. Therefore, temperature-dependent OCV is essential to estimate the battery states accurately.To build the OCV-SOC-temperature model, we process the measurement data of the Incremental OCV tests for temperatures ranging from −10°C to 50°C at each 10°C interval [33]. The procedure for measuring OCV is designed as follows: (i) fully charge the battery to 100% SOC and discharge with relaxation duration at every 10% SOC and record the corresponding OCV, (ii) charge the battery to 100% SOC with the same duration of 10% SOC and record the corresponding OCV, and (iii) perform (1) and (2) repeatedly from −10°C to 50°C at each 10°C interval to build the table of OCV-SOC-temperature by applying averaging method and linear interpolation. Notably, to obtain the OCV as accurately as possible, the measurement data should be collected until the OCV has reached a steady open-circuit state. To obtain accurate OCV for online SOC estimation, we build the look-up tables of OCV-SOC-temperature according to the measurement data. The look-up table is built as a three-dimensional (3-D) form to contain the OCV values at different temperatures and different SOC. Therefore, when the battery temperature changes at different SOC, the accurate VOC can be obtained by 3-D linear interpolation according to the look-up tables of OCV-SOC-temperature. As shown in Figure 3, the 3-D look-up tables are visualized with a 3-D response surface, and the extracted OCV map describes the OCV change with the temperature rising at different SOC.Figure 3
The 3-D OCV response surface.The DP models are required to predict the terminal voltage,VL:
(13)V̇1=−V1R1C1+IC1,V̇2=−V2R2C2+IC2,VL=VOC−V1−V2−IR0,where the RC parameters are obtained by parameter identification for lithium-ion battery. The RLS method, a procedure to find the optimal match for the data by minimizing the sum of the square of errors, is efficient in identifying the match parameters of the battery [34–36]. The RC parameters of the DP model are generally preset for a specific temperature (e.g., 25°C). However, the temperature and SOC will affect the battery internal reactions during battery (dis)charging. To perform accurate voltage prediction, the RC parameters characterizing the battery polarization should be updated when the battery’s temperature changes at different SOC. The relation among the RC parameters, SOC, and temperature can be expressed as
(14)ParameterR,C=fTcore,SOC.To describe the relation above, a dynamic parameter model is built based on the experimental data under different temperatures and SOC. The experimental data is obtained by the pulse-relaxation tests conducted on 2.3 Ah A123 26650 LiFePO4 batteries to identify the RC parameters accurately. The experiment temperatures range from 5°C to 45°C with an interval of 10°C, and SOC ranges from 10% to 90% with an interval of 10% SOC. The RLS method adopted for the RC parameter identification is summarized in Algorithm 1. The relation among RC parameters, SOC, and temperature is formulated in (14), and the identification results of varying temperatures and SOC are shown in Figure 4. Series look-up tables of parameter-SOC-temperature are constructed based on the identification results. The accurate RC parameter values could be obtained by using the linear interpolation method according to the look-up tables when the battery temperature changes at different SOC.Algorithm 1:The procedure of parameter identification using RLS.
Step 1: applying the Laplace transform to the terminal voltage calculation Equation (13), the following equation can be obtained:VOCs−VLs=IsR0+R1/R1C1s+1+R2/R2C2s+1.Step 2: deriving the transfer function:Gs=VOCs−VLs/Is=R0s21/τ1τ2R0τ1+R0τ2+R1τ2+R2τ1s+R0+R1+R2/τ1τ2/s2+τ1+τ2s/τ1τ2+1/τ1τ2Step 3: applying bilinear transformation for discretization:Lets=2/Δt⋅1−z−1/1+z−1, Gz−1=a3+a4z−1+a5z−2/1−a1z−1−a2z−2, where aii=1,2,⋯,5 are constants.Step 4: transforming the discrete system transfer function into a time-domain difference equation:yt=a1yt−1+a2yt−2+a3It+a4It−1+a5It−2,whereyt is the voltage drop and yt=VOCt−VLt, It is the input current.Step 5: definingφt=yt−1yt−2ItIt−1It−2Tθ=a1a2a3a4a5.Step 6: definingyt=φTtθ+et, where et is the sampling error.Step 7: computing the recursive termination conditionsJθ:Jθ=∑i=1tY−Φθ2=∑i=1tei+22, where Y=y3y4y5⋯yN+2Te=e3e4e5⋯eN+2T.Step 8: deriving the recursion formula:θ^t+1=θ^t+Kt+1yt+1−ΦTt+1θ^t,Kt+1=Pt+1Φt+11+ΦTt+1PtΦt+1−1,Pt+1=I−Kt+1ΦTt+1Pt.Step 9: substitutingθ^=ΦTΦ−1ΦTY into Step 8, and obtaining the value of parameter ai.Step 10: applying the inverse bilinear transformation toG(z-1) in Step 3, where z−1=1−ΔT/2s/1+ΔT/2s:Gs=a3−a4+a5/1+a1−a2s2+4a3−a5/ΔT1+a1−a2s+4a3−a4+a5/ΔT21+a1−a2/s2+41+a2/ΔT1+a1−a2s+41−a1−a2/ΔT21+a1−a2Step 11: deriving the value ofR0,R1,R2,C1, and C2 by comparing the corresponding coefficients in Step 2 and Step 10.Figure 4
The 3-D response surfaces of the DP model’s parameters. (a)R0, (b) R1, (c) R2, (d) C1, and (e) C2.
(a)(b)(c)(d)(e)A dynamic DP model is proposed according to the parameter identification, as shown in Figure5, which utilizes dynamic RC parameters instead of using the values identified at a specific temperature. The values of R0 (DYN), R1 (DYN), R2 (DYN), C1 (DYN), and C2 (DYN) are obtained by linear interpolation according to the real-time temperature and SOC. The values of RC parameters could be calibrated during the estimation, and the errors in terminal voltage prediction and heat generation could be further reduced.Figure 5
Schematic of the dynamic DP model.This dynamic DP model is also used to derive the state equations for the SOC estimation. Substituting Kirchhoff’s current law to Equation (13):
(15)dVC1dt=1C1I−1R1C1VC1,dVC2dt=1C2I−1R2C2VC2.Substituting the expression of SOC into Equation (15), the continuous-time state-space model can be expressed as
(16)dSOCdtdVC1dtdVC2dt=0000−1R1C1000−1R2C2SOCVC1VC2+−1Cbat1C11C2I.Applying the closed-form discretization formula to Equation (16), the discrete state-space model is defined as
(17)SOCn+1VC1,n+1VC2,n+1=1000e−T/R1C1000e−T/R2C2SOCnVC1,nVC2,n+−TCbatR11−e−T/C1R1R21−e−T/C2R2I.According to Equation (17), the state-space equations of the dynamic DP model are depicted as follows:
(18)ẋ=Ax+Bu,y=Cx+Du,where x represents the state of the state-space equations which is given by xn=SOCnVC1,nVC2,nT, where u stands for the exogenous input and y means the output.


## 2.3. Coupled Electrothermal Model

In general, the models for temperature estimation and SOC estimation use disparate data sources. However, the ITD-based temperature estimation method uses EIS measurements as inputs, and the EIS measurements also could be used in the SOC estimation. In other words, the EIS measurements could be used to synergistically estimate temperature and SOC. Therefore, a coupled electrothermal model using EIS measurements is proposed by integrating the above two submodels shown in Sections2.1 and 2.2.The parameters of the proposed model consist of structural and physical parameters, and Table2 shows the parameters of the two submodels, as well as the parameters shared by the two submodels. The two submodels share the updated physical parameters to improve the estimation accuracy during the process each estimation.Table 2
The structural and physical parameters of the model.
ModelsParametersStructural parametersPhysical parametersThermal submodelRadius of the batteryrDensity ρHeat capacity cpThermal conductivity ktBattery volume VbAir temperature of the thermal chamberT∞Convection coefficient hBattery temperature Tsurf,TcoreVolume averaged temperature T¯Temperature gradient γ¯Heat generation QElectrical submodelOhmic internal resistanceR0Polarization resistance R1 and R2Battery polarization capacitance C1 and C2Terminal voltageVLOpen circuit voltage VOCCurrent IState of charge SOCCoupled electrothermal model/Heat generationQOpen circuit voltage VOCBattery core temperature TcoreCurrent IState of charge SOCAccording to Figure6, the thermal submodel and the electrical submodel interact through a two-way coupling. Once the measurement data is received by the coupled model, the electrical model will perform the predictions of SOC and voltage using the measured current I and the updated RC parameters. Then, the VOC with a smaller error is obtained by linear interpolation according to the estimated SOC and the table of OCV-SOC-temperature. The heat generation Q is calculated using the difference between VL and VOC, along with I. Then, the thermal model performs battery temperature estimation, and the estimated temperature and SOC can be further used to update the electrical RC parameters; therefore, the electrical submodel using dynamic RC parameters will reduce the errors in the SOC estimation. The VOC with a smaller error is used to calculate the VL and Q, which will reduce the errors of the temperature estimation. Thus, it can be seen that the coupling of the two submodels improves the accuracy of both temperature estimation and SOC estimation.Figure 6
Coupling between the thermal model and the electrical model.


## 3. Design of Estimators

The Kalman filter (KF) is an effective algorithm for estimating the state of the battery, and the EKF or unscented KF is generally employed for a nonlinear estimation. The coupled electrothermal model contains two states to be estimated: temperature and SOC. In this study, a DEKF is designed to estimate the temperature and the SOC synergistically. The DEKF consists of two extended Kalman filters, one is based on the ITD-based thermal model for temperature estimation, and the other is based on the DP model for SOC estimation.By combining Equations (11) and (18), the state-space models of the DEKF are obtained:
(19)xn+1=gxn,un+enwyn=hxn,un+envθn+1=gθn,un+enr,λn=hθn,un+ens,where the model denoted by x and y is used to estimate the SOC, while the model denoted by θ and λ is used to estimate the temperature, and en represents the noises of the covariance matrices ∑ for SOC estimation and temperature estimation. The DEKF algorithm can be summarized as Algorithm 2.Algorithm 2: The procedure of DEKF algorithm.
Definitions of the system matrices:An−1=∂gxn−1,un−1/∂xn−1xn−1=x^n−1+Cnx=∂hxn,un/∂xnxn=x^n−Bn−1=∂gθn−1,un−1/∂θn−1θn−1=θ^n−1+Cnθ=∂hθn,un/∂θnθn=θ^n−Computing forn=1,2,⋯Time updateState prediction is as follows:x^n−=gx^n−1+,un−1,θ^n−=gθ^n−1+,un−1.Error covariance is as follows:Σx~,n−=An−1Σx~,n−1+An−1T+Σw,Σθ~,n−=Bn−1Σθ~,n−1+Bn−1T+Σr.Measurement updateKalman gain update is as follows:Lnx=Σx~,n−CnxTCnxΣx~,n−CnxT+Σv−1Lnθ=Σθ~,n−CnθTCnθΣθ~,n−CnθT+Σs−1.State update is as follows:x^n+=x^n−+Lnxyn−hx^n−,un,θ^n+=θ^n−+Lnθλn−hθ^n−,un.Error covariance is as follows:Σx~,n+=I−LnxCnxΣx~,n−,Σθ~,n+=I−LnθCnθΣθ~,n−.Parameter update is as follows:R0orR1,R2,C1,C2=fx^n+,θ^n+.The detailed signal flow of the proposed DEKF is shown in Figure7 by a block diagram. It is clearly shown that two integrated EKFs are adopted to simultaneously estimate the temperature and SOC, with information being exchanged between the filters. Both filters perform a parameter update during each cycle to ensure sufficient accuracy when the battery operates at varying temperatures.Figure 7
Diagram of the DEKF algorithm.The DEKF is designed according to the two-way coupling, as shown in Figure6, to verify the effectiveness of the coupled electrothermal model. The DEKF consists of two EKFs used for battery state estimations: the EKFx used for estimating SOC and the EKFθ used for estimating temperature. The information being exchanged between the two EKFs includes the SOC, the temperature, and the predicted terminal voltage, all of which are able to improve the accuracy in estimating both temperature and SOC. The detailed comparisons of the algorithms are discussed in the following section.


## 4. Validation

The validation data was measured from experiments conducted on 2.3 AhA123 26650 LiFePO4 batteries. The tests were carried out using the Artemis HEV drive cycle and federal urban driving schedule (FUDS), and the profiles are shown in Figure 8. The single-frequency impedance measurements were conducted at 215 Hz, and a thermocouple was inserted into the core of the battery to measure the core temperatures. The temperature of the experiments was controlled with a Boyi B-TH-1000D thermal chamber. The detailed experimental setup is depicted in Figure 9. The first 3500 s current profile was generated according to the Artemis HEV drive cycle to simulate the driving situation of the urban road, rural road, and motorway. The initial value of temperature estimation was set to 8°C (the same as the real battery temperature). The second current profile was generated according to the FUDS, and the 6400 s current profile was generated to validate the coupled model across wider intervals of SOC (10%SOC–90%SOC). The effect of inaccurate initial temperature was also considered, and an incorrect initial value of temperature estimation was provided as 28°C (the initial error is 20°C).Figure 8
Driving cycle with current and voltage. (a) Artemis HEV drive cycle. (b) FUDS.
(a)(b)Figure 9
Schematic diagram of experimental setup.The preset parameters are shown in Table3, and these structure parameters have been identified and validated in the previous work [18, 37]. To verify the effectiveness of the DEKF-based coestimation method, we made several estimation comparisons of the performance of the DEKF and baseline EKF algorithms whereby the same parameters are provided in Table 3. The two baseline EKFs are designed according to the ITD-based thermal model and the standard DP model, respectively, the RC parameters are provided with the identified results at 8°C, and the RC parameters are assumed fixed (the initial battery temperature is 8°C). The proposed DEKF is based on the coupled model, while the RC parameters are provided with the same identified results at 8°C but the RC parameters are not fixed.Table 3
Parameters of the battery.
ParameterUnitsValuesktW·m-1·K-10.404ρkg·m-32107hW·m-239.3cpJ·kg-1·K-11171.6We first investigate the proposed DEKF’s performance for temperature estimation. Figure10 depicts the core temperature estimation results of the two algorithms, and Figure 11 depicts the errors of the core temperature estimations. Figure 10(b) shows that the DEKF takes less than 100 s to make the estimated values converge to the actual values, while the baseline EKF takes more than 300 s to converge to the actual values. The evaluation criteria, including the maximum error, the mean absolute error (MAE), and root mean squared error (RMSE), are used to quantify the estimation errors. The comparison of core temperature estimation errors is shown in Table 4. For the Artemis HEV drive cycle, the maximum error of core temperature estimation drops from 1.60% to 0.60%, the MAEs drop from 0.69% to 0.14%, and the RMSEs drop from 0.81% to 0.20%. For the FUDS drive cycle, the maximum error drops from 1.81% to 0.40%, the MAEs drop from 0.92% to 0.16%, and the RMSEs drop from 1.06% to 0.23%.Figure 10
Core temperature estimation results. (a) Artemis HEV drive cycle. (b) FUDS.
(a)(b)Figure 11
Errors of core temperature estimation. (a) Artemis HEV drive cycle. (b) FUDS.
(a)(b)Table 4
Comparison of core temperature estimation errors.
EKF (Artemis)DEKF (Artemis)EKF (FUDS)DEKF (FUDS)Maximum (%)1.600.601.810.40MAE (%)0.690.140.920.16RMSE (%)0.810.201.060.23Next, we investigate the proposed DEKF’s performance for estimating SOC. Figure12 shows the SOC estimation results of the Artemis HEV drive cycle, and Figure 13 shows the results of the FUDS drive cycle. The comparison of SOC estimation errors is shown in Table 5. For the Artemis HEV drive cycle, the maximum error of SOC estimation drops from 1.26% to 0.38%, the MAEs drop from 0.58% to 0.21%, and the RMSEs drop from 0.68% to 0.23%. For the FUDS drive cycle, maximum error drops from 1.90% to 0.44%, the MAEs drop from 1.06% to 0.15%, and the RMSEs drop from 1.14% to 0.20%.Figure 12
SOC results for EKF and DEKF (Artemis HEV drive cycle). (a) EKF. (b) DEKF. (c) Errors of estimation.
(a)(b)(c)Figure 13
SOC results for EKF and DEKF (FUDS). (a) SOC estimation. (b) Errors of estimation.
(a)(b)Table 5
Comparison of SOC estimation errors.
EKF (Artemis)DEKF (Artemis)EKF (FUDS)DEKF (FUDS)Maximum (%)1.260.381.900.44MAE (%)0.580.211.060.15RMSE (%)0.680.231.140.20The temperature and SOC estimation errors of the two separate EKFs are higher than those of the DEKF, as shown in Tables4 and 5. This is expected for that the RC parameters of the SOC EKF are fixed on the identified results at 8°C, whereas the real core temperature will increase due to the heat generation accumulated inside the battery. The deviations of the RC parameters may introduce errors in SOC estimation, and the deviations of SOC estimation would further introduce the errors in temperature estimation. In contrast, the coupled model-based DEKF can update the RC parameters according to the temperature and SOC and, thus, improve the accuracy of both temperature and SOC estimations.To further investigate the effect of coupling in the coestimation, we calculated the estimation deviations corrected by the DEKF. The corrected deviations can be expressed as(20)DT=TEKF−TDEKFTtrue,DSOC=SOCEKF‐SOCDEKFSOCtrue,whereDT and DSOC are the corrected deviations of temperature and SOC, respectively; TEKF and TDEKF are the core temperature estimated by EKF and DEKF, respectively; Ttrue is the temperature measured by the thermocouple; SOCEKF and SOCDEKF are the SOC estimated by EKF and DEKF, respectively; and SOCtrue is the SOC calculated by the measured data.The same trend can be observed in the corrected deviations of temperature and SOC, as shown in Figure14. The reason for this trend is that the SOC estimation deviations are almost caused by temperature change. The deviations would be corrected once the matched parameter is identified by the coupled electrothermal model. The temperature estimation deviations would also be corrected once the accurate SOC value is identified, and the theory for reducing the estimation errors is introduced in detail in Section 2.3.Figure 14
Deviations of temperature and SOC estimation. (a) Artemis HEV drive cycle. (b) FUDS.
(a)(b)To further validate the performance of the proposed method, we compare the RMSEs of temperature estimation and SOC estimation based on the different coestimation methods. All the experiments involved in the comparison were conducted using the same type of battery over a similar temperature range under driving schedule (DS) cycles. The performance of the proposed coupled model is compared with the performance of other models in Refs. [13, 14, 38]. In Ref. [13], a second-order RC model and a two-state thermal model are parameterized to capture the core temperature and the terminal voltage of the battery. In Ref. [14], a hybrid electrochemical-thermal-neural-network (ETNN) model is designed for coestimating core temperature and SOC, and the model is validated under the FUDS cycle and urban dynamometer driving schedule (UDDS) cycle. In Ref. [38], a thermoelectric model with hysteresis states for coestimating core temperature and SOC is validated under the FUDS cycle. Therefore, to maintain similar validation conditions, we use the results of the proposed model under the FUDS cycle for comparison, and the numerical results of the comparison are shown in Table 6.Table 6
Comparison of different coestimation methods in terms of RMSEs.
Coupled modelsRMSE SOC (%)RMSETcore (°C)RMSEVL (mV)2nd-order RC and thermal [13]—0.5645.4Hysteresis and thermal [38]2.310.44—ETNN [14]0.650.4731.2Proposed0.200.2337.1As shown in Table6, the RMSEs of both the battery SOC and core temperature of the proposed model are smaller than those of other models. As for the terminal voltage estimation, the data-driven electrochemical model possesses smaller errors than the model-driven electrical model. Although the precision of the electrochemical model is better than the electrical model, the structure of the electrochemical model is more complicated than that of the electrical model, which means the electrochemical model requires much more computing sources. Furthermore, the data-driven models lack supports of specific physical concepts, whereas the model-driven models with specific physical structures are more convincing. Overall, the proposed coupled model is capable of accurately estimating the SOC and temperature with acceptable voltage estimation errors.As mentioned above, the proposed coestimation method is effective in reducing the errors for SOC and temperature estimations simultaneously. And the estimation performance of DEKF is better than that of two separate EKFs using the same EIS measurements as the inputs. Moreover, in comparison with other coestimation methods, the proposed coestimation method can perform more accurate estimations for both temperature and SOC.


## 5. Conclusion

Knowledge of accurate SOC and temperatures of lithium-ion batteries is of extreme significance for safe operation. This study proposes a novel method for coestimating both the core temperature and the SOC based on EIS measurements rather than using direct battery measurements. The coupled electrothermal model is proposed by integrating the DP electrical submodel with the ITD-based thermal submodel. The ITD-based thermal submodel enables accurate temperature estimation using the EIS measurements instead of temperature sensors’ measurements. And the proposed temperature-dependent DP electrical submodel enables accurate SOC estimation at varying temperatures.The DEKF, designed in accordance with the coupled model, is capable of estimating the SOC and core temperatures accurately. The DEKF can accurately update the parameters of the coupled model when the battery temperature is changing during the test. The performance of the proposed DEKF using EIS measurements is better than that of the two separate EKFs using the same input. Further investigation of the error convergence proves that the proposed coestimation method is more favorable for error control. Moreover, the coestimation method requires fewer measurements and fewer instruments. In addition, since the coupling method is capable of improving the performance of the submodels, incorporating more relevant states for joint estimations can further reduce the estimation errors. Such multiple states joint estimation is demanded by high-performance applications such as the battery management system.Future work will be conducted to investigate the coestimation method for the individual batteries in a battery pack, as well as investigate the application of the coestimation method for both the SOH and SOC to enable a more comprehensive state estimation of the lithium-ion battery. In addition, to investigate the applicability of the proposed estimation method, we will conduct research on other types of batteries such as magnesium and iron-based batteries.



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*Source: 4021256-2023-02-06.xml*