# State Estimation of Lithium Batteries for Energy Storage Based on Dual Extended Kalman Filter

Authors: Xinming Xu; Di Wu; Lei Yang; Huai Zhang; Guangjun Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In general, battery packs are monitored by the battery management system (BMS) to ensure the efficiency and reliability of the energy storage system. SOC and SOH represent the battery’s energy and lifetime, respectively. They are the core aspects of the battery BMS. The traditional method assumes that the SOC is determined by the integral of the current input and output from the battery over time, which is an open-loop-based approach and often accompanies by poor estimation accuracy and the accumulation of sensor errors. The contribution of this work is to establish a new equivalent circuit model based on the lithium battery external characteristic, and the battery parameters are identified by considering the influence of capacity fade, voltage rebound, and internal capacitance-resistance performance. The correlation between the ohmic internal resistance and real capacity is obtained by degradation test. Then, the dual extended Kalman filter (DEKF) is used to perform real-time prediction of the lithium battery state. And through the simulation analysis and experiments, the feasibility and precision of the estimation method are well proved.


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## Body

## 1. Introduction

Lithium battery has been widely used in the energy storage field due to its high energy density, long cycle life, high voltage, and outstanding security [1]. In general, in order to ensure the efficiency and reliability of the energy storage system, battery packs are monitored by the battery management system (BMS). And the state estimation of the battery is the essential function of the BMS. The accurate online state estimation can contribute to monitor the battery’s state of charge (SOC) and state of health (SOH) clearly so that the reasonable charge-discharge can be performed to keep its high-efficiency working conditions and long life [2].SOC and SOH represent the battery’s energy and lifetime, respectively. They are the core aspects of the battery BMS. The traditional method assumes that the SOC is determined by the integral of the current input and output from the battery over time. This is an open-loop-based approach, the result of which is often accompanied by poor estimation accuracy and the accumulation of sensor errors [3]. Moreover, in the past, the battery’s capacity variations were not to be taken into account during the lifetime; thus, the estimation method had great deficiencies [4]. In fact, it comes with irreversible chemical reactions and physical changes in the working process of the battery, which make SOC and SOH tightly coupled with each other. Improper battery state estimation method can lead to premature damage and deterioration of the battery [5]. As the impedance and capacity of the battery change with time, the maximum power and energy that can be delivered by the battery will be reduced. This technical problem can be solved by simultaneously estimating the SOC and the SOH of the battery [6].Traditional estimation methods, including the internal resistance method, coulomb counting (CC) method, open-circuit voltage method, and load charge method, have some limitations in practical application due to lack of abilities to correct errors and largely depending on high precision of sensors and low environment influences [7]. Nowadays, with the development of modern control theory, there are some new algorithms which are constantly being used in battery state estimation. Plett [8] proposed the sigma-point Kalman filtering (SPKF) method as applied to BMS algorithms through the analysis of batteries’ nonlinear characteristics. And a better result was obtained by comparison with the traditional Kalman filtering method. Fleischer et al. [9] presented a state-of-available-power prediction method based on the framework of the adaptive neuro-fuzzy inference system. The method achieved the accurate power prediction at low-temperature environment. Lee et al. [10] had researched the linear prediction error method and the neural network method in the battery state estimation and then found the latter had better precision in short-term prediction. Chen et al. [11] employed the genetic algorithm to estimate the battery model parameters based on analysis of the resistance-capacitance circuit model. The precision and robustness of results are much improved, but its real-time performance was limited.The contribution of this paper is to establish a new equivalent circuit model based on the lithium battery external characteristic, and the battery parameters are identified by considering the influence of capacity fade, voltage rebound, and internal capacitance-resistance performance. The correlation between the ohmic internal resistance and real capacity is obtained by degradation test. Then, the dual extended Kalman filter (DEKF) is used to perform real-time prediction of the lithium battery state. And through the simulation analysis and experiments, the feasibility and precision of the estimation method are well proved.The rest of the paper is organized as follows. An equivalent circuit model is established, and the characteristic parameters of the battery are identified in Section2. In Section 3, the method of battery state estimation is proposed, and the computer-aided modeling analysis is carried out. Section 4 provides detailed experimental procedures and results. Section 5 presents some conclusions.


## 2. Battery Model and Parameter Identification

### 2.1. Battery Model

The battery performance is closely related to its internal parameters. And the change rule of characteristic parameters can be analyzed by establishing the reasonable equivalent model [12]. In order to achieve the real-time state analysis, not only the working characteristics, including voltage rebound effect, hysteresis, and electromotive force-SOC (EMF-SOC), need to be taken into consideration in the model but also the complex electrochemistry mechanism should be avoided as much as possible. Therefore, mainly focusing on the analysis of external characteristics, a new circuit containing a controlled source is established as the equivalent circuit model of the lithium battery. The model is based on Thevenin’s theorem and consists of an equivalent voltage source and equivalent impedance, as shown in Figure 1.Figure 1
Equivalent circuit model of the lithium battery.


#### 2.1.1. The Equivalent Voltage Source

The voltage source is related with the SOC, SOH, and working conditions of the battery. Therefore, the equivalent voltage sourceEB consists of electromotive force EMF and hysteresis voltage Vh. As the voltage-controlled voltage source (VCVS), EMF is controlled by voltage VSOC. And Ccap is the capacity of the battery, which is related with the SOH. And the current flowing through Ccap is equal to the working current IB. Therefore, the value of VSOC can be used to represent the battery SOC:(1)VSOC=IBtCcap=Ccap−CremainCcap,where t is the working time and Cremain is the remaining capacity.Hysteresis voltageVhis also a voltage-controlled voltage source (VCVS), which is controlled by voltage VLh. The current flowing through inductance Lh is βIB. Hysteresis voltage is influenced by previous working current, in which the characteristic can be described by a circuit containing the inductance. The value of Vh is determined by charging and discharging conditions:(2)VLh=LhdβIBdt.Therefore, the equivalent voltage source is(3)EB=EMFVSOC+VhVLh.


#### 2.1.2. The Equivalent Impedance

The capacitance and resistance performance of the battery are described by an equivalent impedance submodel, which consists of the ohmic internal resistanceRΩ and RC capacitive-resistance network [13]. RΩ is caused by materials and the specific structure of the battery, which will increase with the decay of SOH. And the RC capacitive-resistance network is used to describe the voltage rebound effect caused by polarization capacitance and polarization resistance [14]. The rebound performance of the model is closer to the real value of the lithium battery with higher the order of the network. Meanwhile, the complexity and computation time of the model will increase with the orders. Considering the accuracy of the model and the computational efficiency, the third-order RC is much reasonable. Then, according to zero-state response of the RC network, the polarization voltage uit can be shown as(4)uit=Ri⋅IBt⋅1−e−t/RiCi,i=1,2,3.Therefore, according to Thevenin's theorem, the complete equivalent circuit model of the lithium battery can be expressed as(5)uBt=EBt−∑i=13uit−IBRΩ.Through the equivalent circuit model, the relations of internal parameters and external working characteristic, including EMF-SOC, voltage hysteresis, rebound effect, and impedance characteristic of the lithium battery, can be described clearly.


### 2.2. Parameter Identification

At the constant temperature of 25°C, a pulse discharge test for the max load 18650 lithium battery is performed to identify the model characteristic parameters. The battery capacity is 2900mAh, and the current is 2.9 A. The test current is shown in Figure2(a). SOC and voltage uB during the test are shown in Figure 2(b).Figure 2
Pulse discharge test. (a) Test current. (b) Voltage and SOC.
(a)
(b)Based on the test data of the balance electromotive forceEB and corresponding SOC, the EB−SOC characteristic curve can be obtained by the polynomial segmentation fitting method, as shown in Figure 3.Figure 3
E
B
−
SOC characteristic curve.The corresponding mathematical expression is(6)f=g11x4+g12x3+g13x2+g14x+g150.904,1.000,g21x4+g22x3+g23x2+g24x+g250.112,0.904,g31x4+g32x3+g33x2+g34x+g350.000,0.112,where G is the coefficient matrix, and(7)G=8916.7−3328546587−28975.16760.70.113−0.3310.2890.3233.702−92042668−287.915.033.368.TheRCnetworks are in the zero-state respond during the voltage rebound period. Therefore, according to equation (5), the output voltage can be shown as(8)uBt=EBt−ΔU1−ΔU2,where ΔU1 and ΔU2 are, respectively, the rebound voltages in the first stage and second stage and can be expressed as(9)ΔU1=IBRΩ,(10)ΔU2=b1e−τ1t+b2e−τ2t+b3e−τ3t,where bi=RiIB, τi=1/CiRi, RΩ=ΔU1/IB, and i=1,2,3.According to test data and equations (8)∼(10), the voltage rebound characteristic curve can be obtained, as shown in Figure 4. And the parameters of the fitting curve are b=33.64823.13613.516 and τ=4.40980.02930.0037. Therefore, the characteristic parameters of the battery can be identified, as shown in Table 1.Figure 4
The rebound characteristic curve.Table 1
The characteristic parameters.
R
Ω
R
1
R
2
R
3
C
1
C
2
C
3
C
cap
101
m
Ω
65
m
Ω
42
m
Ω
24
m
Ω
1258
F
268
F
1581
F
1.61
Ah


## 2.1. Battery Model

The battery performance is closely related to its internal parameters. And the change rule of characteristic parameters can be analyzed by establishing the reasonable equivalent model [12]. In order to achieve the real-time state analysis, not only the working characteristics, including voltage rebound effect, hysteresis, and electromotive force-SOC (EMF-SOC), need to be taken into consideration in the model but also the complex electrochemistry mechanism should be avoided as much as possible. Therefore, mainly focusing on the analysis of external characteristics, a new circuit containing a controlled source is established as the equivalent circuit model of the lithium battery. The model is based on Thevenin’s theorem and consists of an equivalent voltage source and equivalent impedance, as shown in Figure 1.Figure 1
Equivalent circuit model of the lithium battery.


### 2.1.1. The Equivalent Voltage Source

The voltage source is related with the SOC, SOH, and working conditions of the battery. Therefore, the equivalent voltage sourceEB consists of electromotive force EMF and hysteresis voltage Vh. As the voltage-controlled voltage source (VCVS), EMF is controlled by voltage VSOC. And Ccap is the capacity of the battery, which is related with the SOH. And the current flowing through Ccap is equal to the working current IB. Therefore, the value of VSOC can be used to represent the battery SOC:(1)VSOC=IBtCcap=Ccap−CremainCcap,where t is the working time and Cremain is the remaining capacity.Hysteresis voltageVhis also a voltage-controlled voltage source (VCVS), which is controlled by voltage VLh. The current flowing through inductance Lh is βIB. Hysteresis voltage is influenced by previous working current, in which the characteristic can be described by a circuit containing the inductance. The value of Vh is determined by charging and discharging conditions:(2)VLh=LhdβIBdt.Therefore, the equivalent voltage source is(3)EB=EMFVSOC+VhVLh.


### 2.1.2. The Equivalent Impedance

The capacitance and resistance performance of the battery are described by an equivalent impedance submodel, which consists of the ohmic internal resistanceRΩ and RC capacitive-resistance network [13]. RΩ is caused by materials and the specific structure of the battery, which will increase with the decay of SOH. And the RC capacitive-resistance network is used to describe the voltage rebound effect caused by polarization capacitance and polarization resistance [14]. The rebound performance of the model is closer to the real value of the lithium battery with higher the order of the network. Meanwhile, the complexity and computation time of the model will increase with the orders. Considering the accuracy of the model and the computational efficiency, the third-order RC is much reasonable. Then, according to zero-state response of the RC network, the polarization voltage uit can be shown as(4)uit=Ri⋅IBt⋅1−e−t/RiCi,i=1,2,3.Therefore, according to Thevenin's theorem, the complete equivalent circuit model of the lithium battery can be expressed as(5)uBt=EBt−∑i=13uit−IBRΩ.Through the equivalent circuit model, the relations of internal parameters and external working characteristic, including EMF-SOC, voltage hysteresis, rebound effect, and impedance characteristic of the lithium battery, can be described clearly.


## 2.1.1. The Equivalent Voltage Source

The voltage source is related with the SOC, SOH, and working conditions of the battery. Therefore, the equivalent voltage sourceEB consists of electromotive force EMF and hysteresis voltage Vh. As the voltage-controlled voltage source (VCVS), EMF is controlled by voltage VSOC. And Ccap is the capacity of the battery, which is related with the SOH. And the current flowing through Ccap is equal to the working current IB. Therefore, the value of VSOC can be used to represent the battery SOC:(1)VSOC=IBtCcap=Ccap−CremainCcap,where t is the working time and Cremain is the remaining capacity.Hysteresis voltageVhis also a voltage-controlled voltage source (VCVS), which is controlled by voltage VLh. The current flowing through inductance Lh is βIB. Hysteresis voltage is influenced by previous working current, in which the characteristic can be described by a circuit containing the inductance. The value of Vh is determined by charging and discharging conditions:(2)VLh=LhdβIBdt.Therefore, the equivalent voltage source is(3)EB=EMFVSOC+VhVLh.


## 2.1.2. The Equivalent Impedance

The capacitance and resistance performance of the battery are described by an equivalent impedance submodel, which consists of the ohmic internal resistanceRΩ and RC capacitive-resistance network [13]. RΩ is caused by materials and the specific structure of the battery, which will increase with the decay of SOH. And the RC capacitive-resistance network is used to describe the voltage rebound effect caused by polarization capacitance and polarization resistance [14]. The rebound performance of the model is closer to the real value of the lithium battery with higher the order of the network. Meanwhile, the complexity and computation time of the model will increase with the orders. Considering the accuracy of the model and the computational efficiency, the third-order RC is much reasonable. Then, according to zero-state response of the RC network, the polarization voltage uit can be shown as(4)uit=Ri⋅IBt⋅1−e−t/RiCi,i=1,2,3.Therefore, according to Thevenin's theorem, the complete equivalent circuit model of the lithium battery can be expressed as(5)uBt=EBt−∑i=13uit−IBRΩ.Through the equivalent circuit model, the relations of internal parameters and external working characteristic, including EMF-SOC, voltage hysteresis, rebound effect, and impedance characteristic of the lithium battery, can be described clearly.


## 2.2. Parameter Identification

At the constant temperature of 25°C, a pulse discharge test for the max load 18650 lithium battery is performed to identify the model characteristic parameters. The battery capacity is 2900mAh, and the current is 2.9 A. The test current is shown in Figure2(a). SOC and voltage uB during the test are shown in Figure 2(b).Figure 2
Pulse discharge test. (a) Test current. (b) Voltage and SOC.
(a)
(b)Based on the test data of the balance electromotive forceEB and corresponding SOC, the EB−SOC characteristic curve can be obtained by the polynomial segmentation fitting method, as shown in Figure 3.Figure 3
E
B
−
SOC characteristic curve.The corresponding mathematical expression is(6)f=g11x4+g12x3+g13x2+g14x+g150.904,1.000,g21x4+g22x3+g23x2+g24x+g250.112,0.904,g31x4+g32x3+g33x2+g34x+g350.000,0.112,where G is the coefficient matrix, and(7)G=8916.7−3328546587−28975.16760.70.113−0.3310.2890.3233.702−92042668−287.915.033.368.TheRCnetworks are in the zero-state respond during the voltage rebound period. Therefore, according to equation (5), the output voltage can be shown as(8)uBt=EBt−ΔU1−ΔU2,where ΔU1 and ΔU2 are, respectively, the rebound voltages in the first stage and second stage and can be expressed as(9)ΔU1=IBRΩ,(10)ΔU2=b1e−τ1t+b2e−τ2t+b3e−τ3t,where bi=RiIB, τi=1/CiRi, RΩ=ΔU1/IB, and i=1,2,3.According to test data and equations (8)∼(10), the voltage rebound characteristic curve can be obtained, as shown in Figure 4. And the parameters of the fitting curve are b=33.64823.13613.516 and τ=4.40980.02930.0037. Therefore, the characteristic parameters of the battery can be identified, as shown in Table 1.Figure 4
The rebound characteristic curve.Table 1
The characteristic parameters.
R
Ω
R
1
R
2
R
3
C
1
C
2
C
3
C
cap
101
m
Ω
65
m
Ω
42
m
Ω
24
m
Ω
1258
F
268
F
1581
F
1.61
Ah


## 3. Battery State Estimation

### 3.1. Battery State

Battery state contains the state of charge (SOC) and the state of health (SOH). SOC is the remain charge that can be released from the battery, which is related with the initial charge state, actual capacity, and previous current [15]:(11)SOCt=Qinitial−∫t0tηIBdtCcapnow×100%,where Qinitial is the initial charge state and η is the coefficient of the current ratio.SOH is evaluated based on the battery capacity. According to battery failure standard, the battery is invalid when its actual capacity drops to 80% of rated capacity [16]. Therefore, SOH can be defined as(12)SOH=Ccapnew−CcapnowCcapnew−Ccapold×100%,where Ccapnew,Ccapnow, and Ccapold are, respectively, the rated capacity, actual capacity, and invalid capacity.The lithium ion in the electrolyte is hindered by the solid electrolyte interface (SEI) adhering to the electrode of the lithium battery. With continuous electrochemical reaction, SEI becomes thicker and thicker, which means hindrance to the lithium ion gets more serious. This can be seen from the increased ohmic internal resistance in the macroscopic view [17]. Therefore, the battery capacity gradually decreases, and ohmic internal resistance increases during the battery degradation [18]. In the normal working condition, battery degradation is a long-term and slow process. In this work, the changes of the ohmic internal resistance and capacity are obtained by accelerating the growth of solid electrolyte membranes in the high-temperature working environment. The Ccapnow−RΩ degradation curve is shown in Figure 5.Figure 5
C
cap
now
−
R
Ω degradation curve.The corresponding mathematical expression is(13)Ccapnow=17163RΩ3−1924RΩ2+49RΩ−1.According to the result analysis, SOC and SOH are closely related. The estimation accuracy of SOC can be improved by considering the influence of capacity degradation.


### 3.2. Estimation Principle of DEKF

Extended Kalman filter (EKF) is an improved method based on the traditional Kalman filter that can be used for the estimation of nonlinear state variables [19]. And when two sets of state variables need to be estimated, dual extended Kalman filtering (DEKF) can be used [20]. In this work, the battery SOC is estimated, which is related with the polarization resistances. The state variable is defined as X=SOCu1u2u3T. Besides, SOH can be used to calculate the actual capacity to improve the estimation accuracy of SOC. Although SOH cannot be measured directly, it can be obtained by the estimation of the ohmic internal resistance RΩ due to equation (13). So, another state variable is defined as Y=RΩT. Besides, an observable variable related with state variables is needed in Kalman filter [21]. According to the battery model, the output voltage can be regarded as the observational variable Z=uB for it can be measured by a sensor directly. The measuring equation at time k is shown as(14)uBk=fX1k−X2k−X3k−X4k−RΩkIBk+vk,where fX1k is the EB−SOC characteristic equation and vk is the measurement noise.The battery state estimation based on the DEKF method is shown in Figure6.Figure 6
DEKF procedure for state estimation.According to the result of the last cycle, the battery capacity at timek can be updated. Then, based on the equivalent circuit model, state equations are established, and the preliminary state predictions Xk∗ and Yk∗ are calculated by(15)Xk∗=10000RsCsRsCs+10000RmCmRmCm+10000RlClRlCl+1Xk−1+1CcapRsRsCs+1RmRmCm+1RlRlCl+1IBk−1+ωk−1,Yk∗=Yk−1+rk−1,where IBk−1 is the process variable, ωk−1 is the process noise, and rk−1 is the small perturbation of state Yk. And Xk∗,Yk∗ is the preliminary estimation of state variables.The estimated values obtained by state equations cannot avoid the accumulation of process errors. Besides, the initial errors that may exist is hard to be recognized and corrected [22]. Therefore, an observable variable Zneeds to be established to correct the previous error. This is the key to the algorithm. In the DEKF method, both of the two state variables need to be corrected by the observational variable:(16)Xk=Xk∗+KXkZMk−ZXk,Yk=Yk∗+KYkZMk−ZYk.The error corrections depend on the difference value of observational variableZMk−Zk and the Kalman gain Kk. ZMk=uk is obtained by the voltage sensor, and Zk is calculated by the measuring equation. The Kalman gain Kk is calculated by(17)KXk=PXk∗HXkTHXkPXk∗HXkT+RXk−1,KYk=PYk∗HYkTHYkPYk∗HYkT+RYk−1,where Rk are the covariances of measurement noise that rely on the performance of the sensor. Hk Jacobian matrixes of state variables are shown in equation (18). Pk∗ are the covariance matrixes of the state, which are used to reflect the change of state variables, as shown in equation (19).(18)HXk=∂uBk∂X=∂fX1∂X1−1−1−1,HYk=∂uBk∂Y=−IBk,(19)PXk∗=APk−1AT+QXk−1,PYk∗=Pk−1+QYk−1,where Qk−1 are the covariances of the process noise.


### 3.3. Simulation Model and Analysis

The real cycle test is time consuming. And it is more efficient for the algorithm analysis to simulate the battery working conditions with the help of computer simulation technique [23]. Therefore, according to the principle of the battery equivalent circuit, a simulation model is established by using mathematical logic modules and circuit element modules in Matlab Simulink, as shown in Figure 7. In order to ensure the reliability of simulation analysis and reflect the real working characteristics of the battery, the parameters of the simulation model are set up corresponding with Table 1.Figure 7
The battery simulation model.The SOC estimation is performed by using the DEKF method in the 1C intermittent discharge working condition as shown in Figure8. The measurement noise and process noise are simulated by Gaussian white noises. According to the sensor performance, the noise covariances are set as Q=0.039 and R=2.5×10−5. To consider the ability of error correction, four situations are simulated, where the initial errors are, respectively, set as 0%, 5%, 10%, and 15%, and the step size k is 0.5s. The simulation results are shown in Figure 9.Figure 8
The simulated discharge condition.Figure 9
The simulation results. (a) SOC. (b) Estimation errors.
(a)
(b)In Figure9, the initial errors of DEKF1∼ DEKF4 are, respectively, 0%, 5%, 10%, and 15%. It is seen that, with the discharge process, in the DEKF1 situation, which has no initial error, the estimation is still near the real value. And in terms of DEKF2∼ DEKF4 that contain different initial errors, their estimations are gradually close to real value. Therefore, the results show that DEKF has good convergence and stability and can correct the initial errors in the state estimation of the battery.


## 3.1. Battery State

Battery state contains the state of charge (SOC) and the state of health (SOH). SOC is the remain charge that can be released from the battery, which is related with the initial charge state, actual capacity, and previous current [15]:(11)SOCt=Qinitial−∫t0tηIBdtCcapnow×100%,where Qinitial is the initial charge state and η is the coefficient of the current ratio.SOH is evaluated based on the battery capacity. According to battery failure standard, the battery is invalid when its actual capacity drops to 80% of rated capacity [16]. Therefore, SOH can be defined as(12)SOH=Ccapnew−CcapnowCcapnew−Ccapold×100%,where Ccapnew,Ccapnow, and Ccapold are, respectively, the rated capacity, actual capacity, and invalid capacity.The lithium ion in the electrolyte is hindered by the solid electrolyte interface (SEI) adhering to the electrode of the lithium battery. With continuous electrochemical reaction, SEI becomes thicker and thicker, which means hindrance to the lithium ion gets more serious. This can be seen from the increased ohmic internal resistance in the macroscopic view [17]. Therefore, the battery capacity gradually decreases, and ohmic internal resistance increases during the battery degradation [18]. In the normal working condition, battery degradation is a long-term and slow process. In this work, the changes of the ohmic internal resistance and capacity are obtained by accelerating the growth of solid electrolyte membranes in the high-temperature working environment. The Ccapnow−RΩ degradation curve is shown in Figure 5.Figure 5
C
cap
now
−
R
Ω degradation curve.The corresponding mathematical expression is(13)Ccapnow=17163RΩ3−1924RΩ2+49RΩ−1.According to the result analysis, SOC and SOH are closely related. The estimation accuracy of SOC can be improved by considering the influence of capacity degradation.


## 3.2. Estimation Principle of DEKF

Extended Kalman filter (EKF) is an improved method based on the traditional Kalman filter that can be used for the estimation of nonlinear state variables [19]. And when two sets of state variables need to be estimated, dual extended Kalman filtering (DEKF) can be used [20]. In this work, the battery SOC is estimated, which is related with the polarization resistances. The state variable is defined as X=SOCu1u2u3T. Besides, SOH can be used to calculate the actual capacity to improve the estimation accuracy of SOC. Although SOH cannot be measured directly, it can be obtained by the estimation of the ohmic internal resistance RΩ due to equation (13). So, another state variable is defined as Y=RΩT. Besides, an observable variable related with state variables is needed in Kalman filter [21]. According to the battery model, the output voltage can be regarded as the observational variable Z=uB for it can be measured by a sensor directly. The measuring equation at time k is shown as(14)uBk=fX1k−X2k−X3k−X4k−RΩkIBk+vk,where fX1k is the EB−SOC characteristic equation and vk is the measurement noise.The battery state estimation based on the DEKF method is shown in Figure6.Figure 6
DEKF procedure for state estimation.According to the result of the last cycle, the battery capacity at timek can be updated. Then, based on the equivalent circuit model, state equations are established, and the preliminary state predictions Xk∗ and Yk∗ are calculated by(15)Xk∗=10000RsCsRsCs+10000RmCmRmCm+10000RlClRlCl+1Xk−1+1CcapRsRsCs+1RmRmCm+1RlRlCl+1IBk−1+ωk−1,Yk∗=Yk−1+rk−1,where IBk−1 is the process variable, ωk−1 is the process noise, and rk−1 is the small perturbation of state Yk. And Xk∗,Yk∗ is the preliminary estimation of state variables.The estimated values obtained by state equations cannot avoid the accumulation of process errors. Besides, the initial errors that may exist is hard to be recognized and corrected [22]. Therefore, an observable variable Zneeds to be established to correct the previous error. This is the key to the algorithm. In the DEKF method, both of the two state variables need to be corrected by the observational variable:(16)Xk=Xk∗+KXkZMk−ZXk,Yk=Yk∗+KYkZMk−ZYk.The error corrections depend on the difference value of observational variableZMk−Zk and the Kalman gain Kk. ZMk=uk is obtained by the voltage sensor, and Zk is calculated by the measuring equation. The Kalman gain Kk is calculated by(17)KXk=PXk∗HXkTHXkPXk∗HXkT+RXk−1,KYk=PYk∗HYkTHYkPYk∗HYkT+RYk−1,where Rk are the covariances of measurement noise that rely on the performance of the sensor. Hk Jacobian matrixes of state variables are shown in equation (18). Pk∗ are the covariance matrixes of the state, which are used to reflect the change of state variables, as shown in equation (19).(18)HXk=∂uBk∂X=∂fX1∂X1−1−1−1,HYk=∂uBk∂Y=−IBk,(19)PXk∗=APk−1AT+QXk−1,PYk∗=Pk−1+QYk−1,where Qk−1 are the covariances of the process noise.


## 3.3. Simulation Model and Analysis

The real cycle test is time consuming. And it is more efficient for the algorithm analysis to simulate the battery working conditions with the help of computer simulation technique [23]. Therefore, according to the principle of the battery equivalent circuit, a simulation model is established by using mathematical logic modules and circuit element modules in Matlab Simulink, as shown in Figure 7. In order to ensure the reliability of simulation analysis and reflect the real working characteristics of the battery, the parameters of the simulation model are set up corresponding with Table 1.Figure 7
The battery simulation model.The SOC estimation is performed by using the DEKF method in the 1C intermittent discharge working condition as shown in Figure8. The measurement noise and process noise are simulated by Gaussian white noises. According to the sensor performance, the noise covariances are set as Q=0.039 and R=2.5×10−5. To consider the ability of error correction, four situations are simulated, where the initial errors are, respectively, set as 0%, 5%, 10%, and 15%, and the step size k is 0.5s. The simulation results are shown in Figure 9.Figure 8
The simulated discharge condition.Figure 9
The simulation results. (a) SOC. (b) Estimation errors.
(a)
(b)In Figure9, the initial errors of DEKF1∼ DEKF4 are, respectively, 0%, 5%, 10%, and 15%. It is seen that, with the discharge process, in the DEKF1 situation, which has no initial error, the estimation is still near the real value. And in terms of DEKF2∼ DEKF4 that contain different initial errors, their estimations are gradually close to real value. Therefore, the results show that DEKF has good convergence and stability and can correct the initial errors in the state estimation of the battery.


## 4. Experiment

### 4.1. Experimental Setup

To verify the feasibility and accuracy of the DEKF method in the real SOC estimation, the experiments are performed by the BMS circuit. The circuit structure consists of a lithium battery pack, the battery monitoring circuit (BMC), the battery control unit (BCU), and the upper computer, as shown in Figure10(a). In the experiment, the battery pack consists of 12 MAX LOAD 18650 cells connected in series. The LTC6804-1 chip is used to monitor the battery data. BCU is a Linduino microcontroller. The upper computer and BCU are communicated by the serial peripheral interface (SPI), as shown in Figure 10(b). The load is a resistor, and the current is controlled by a sliding rheostat.Figure 10
Experimental circuit and its structure. (a) The circuit structure. (b) Experimental setup.
(a)
(b)The SOC estimation experiments are performed by, respectively, using the traditional CC method and DEKF method in the working conditions of the open-circuit state, constant-current discharge state, and intermittent discharge state.


### 4.2. Results and Discussion

#### 4.2.1. Open-Circuit State

When the initial charge is 92%, the SOC estimation is performed in the open-circuit state. The measuring voltage nearly remains constant due to the zero working current, as shown in Figure11. When initial error is 0, the SOC estimated by both methods is close to 92% with a little fluctuate, as shown in Figure 12(a). Because the current is zero, CC-SOC and REAL-SOC have the same value at this time, and the two curves are basically coincident. And when 2% initial error exists, the SOC estimated by the DEKF method can converge to real value, but the estimation still contains initial error by using the CC method as shown in Figure 12(b).Figure 11
Measuring voltage in the open-circuit state.Figure 12
SOC estimation in the open-circuit state. (a) 0 initial error. (b) 2% initial error.
(a)
(b)


#### 4.2.2. Constant-Current Discharge State

The measuring voltage constantly decreases in 1.5 C constant-current discharge state, as shown in Figure13. It is seen that the gradient is great in the high and low SOC stages, while the gradient is much less in the middle stages, which reflects the EB−SOC characteristic of the lithium battery. When initial error is 0, the SOC estimated by the DEKF method is near the real value, while the value by the CC method gradually deviated from the real value, as shown in Figure 14(a). And when 10% initial error is introduced, the estimation of the CC method still contains nearly whole error, while the error is reduced obviously by the DEKF method, as shown in Figure 14(b).Figure 13
Measuring voltage in the constant-current discharge state.Figure 14
SOC estimation in the constant-current discharge state. (a) 0 initial error. (b) 10% initial error.
(a)
(b)


#### 4.2.3. Intermittent Discharge State

The change of measuring voltage in 1C intermittent discharge state is shown in Figure15. It is seen that the voltage is in a decreasing tendency, but some rebounds and step changes occur in the intermittent phase due to the rebound characteristic of the battery.Figure 15
Measuring voltage in the intermittent discharge state.When initial errors are 0 and 10%, the estimation results of SOC are shown in Figure16. It is seen that the SOC can keep track with real value both in cutoff and discharge stage, while the CC method is serious influenced by errors.Figure 16
SOC estimation in the intermittent discharge state. (a) 0 initial error. (b) 10% initial error.
(a)
(b)


#### 4.2.4. Error Analysis

In the open-circuit state, the accuracy of the CC method completely depends on initial error because of zero working current, while DEKF can correct error within a short time. In working conditions, even if the initial error is 0, the maximum errors of CC, respectively, reach 8% and 12%, while the DEKF method is less than 2%. In terms of situation that 10% initial error is introduced, and the error of the CC method is still very big in all of the experiment processes. But, the DEKF method has good convergence, and the initial error can gradually be decreased by correction of the state equation and observational equation.The average errors of two methods in different conditions are shown in Table2. It is seen that whatever the initial errors exist or not, average errors of DEKF are less than 2.7%, which is also much less than the errors of the CC method. Besides, the DEKF has good robustness and can well deal with the influence of initial errors, while the CC method much relies on the stable initial value and the precision of sensors. Therefore, it is proved that the DEKF performs well in the SOC estimation of the lithium battery.Table 2
The average estimation error.
Open-circuit state
Constant-current discharge state
Intermittent discharge state
0 initial error (%)
2% initial error (%)
0 initial error (%)
10% initial error (%)
0 initial error (%)
10% initial error (%)
DEKF
0.2
0.2
0.8
1.3
1.8
2.7
CC
0
1.8
4.2
11.4
6.5
13.3


## 4.1. Experimental Setup

To verify the feasibility and accuracy of the DEKF method in the real SOC estimation, the experiments are performed by the BMS circuit. The circuit structure consists of a lithium battery pack, the battery monitoring circuit (BMC), the battery control unit (BCU), and the upper computer, as shown in Figure10(a). In the experiment, the battery pack consists of 12 MAX LOAD 18650 cells connected in series. The LTC6804-1 chip is used to monitor the battery data. BCU is a Linduino microcontroller. The upper computer and BCU are communicated by the serial peripheral interface (SPI), as shown in Figure 10(b). The load is a resistor, and the current is controlled by a sliding rheostat.Figure 10
Experimental circuit and its structure. (a) The circuit structure. (b) Experimental setup.
(a)
(b)The SOC estimation experiments are performed by, respectively, using the traditional CC method and DEKF method in the working conditions of the open-circuit state, constant-current discharge state, and intermittent discharge state.


## 4.2. Results and Discussion

### 4.2.1. Open-Circuit State

When the initial charge is 92%, the SOC estimation is performed in the open-circuit state. The measuring voltage nearly remains constant due to the zero working current, as shown in Figure11. When initial error is 0, the SOC estimated by both methods is close to 92% with a little fluctuate, as shown in Figure 12(a). Because the current is zero, CC-SOC and REAL-SOC have the same value at this time, and the two curves are basically coincident. And when 2% initial error exists, the SOC estimated by the DEKF method can converge to real value, but the estimation still contains initial error by using the CC method as shown in Figure 12(b).Figure 11
Measuring voltage in the open-circuit state.Figure 12
SOC estimation in the open-circuit state. (a) 0 initial error. (b) 2% initial error.
(a)
(b)


### 4.2.2. Constant-Current Discharge State

The measuring voltage constantly decreases in 1.5 C constant-current discharge state, as shown in Figure13. It is seen that the gradient is great in the high and low SOC stages, while the gradient is much less in the middle stages, which reflects the EB−SOC characteristic of the lithium battery. When initial error is 0, the SOC estimated by the DEKF method is near the real value, while the value by the CC method gradually deviated from the real value, as shown in Figure 14(a). And when 10% initial error is introduced, the estimation of the CC method still contains nearly whole error, while the error is reduced obviously by the DEKF method, as shown in Figure 14(b).Figure 13
Measuring voltage in the constant-current discharge state.Figure 14
SOC estimation in the constant-current discharge state. (a) 0 initial error. (b) 10% initial error.
(a)
(b)


### 4.2.3. Intermittent Discharge State

The change of measuring voltage in 1C intermittent discharge state is shown in Figure15. It is seen that the voltage is in a decreasing tendency, but some rebounds and step changes occur in the intermittent phase due to the rebound characteristic of the battery.Figure 15
Measuring voltage in the intermittent discharge state.When initial errors are 0 and 10%, the estimation results of SOC are shown in Figure16. It is seen that the SOC can keep track with real value both in cutoff and discharge stage, while the CC method is serious influenced by errors.Figure 16
SOC estimation in the intermittent discharge state. (a) 0 initial error. (b) 10% initial error.
(a)
(b)


### 4.2.4. Error Analysis

In the open-circuit state, the accuracy of the CC method completely depends on initial error because of zero working current, while DEKF can correct error within a short time. In working conditions, even if the initial error is 0, the maximum errors of CC, respectively, reach 8% and 12%, while the DEKF method is less than 2%. In terms of situation that 10% initial error is introduced, and the error of the CC method is still very big in all of the experiment processes. But, the DEKF method has good convergence, and the initial error can gradually be decreased by correction of the state equation and observational equation.The average errors of two methods in different conditions are shown in Table2. It is seen that whatever the initial errors exist or not, average errors of DEKF are less than 2.7%, which is also much less than the errors of the CC method. Besides, the DEKF has good robustness and can well deal with the influence of initial errors, while the CC method much relies on the stable initial value and the precision of sensors. Therefore, it is proved that the DEKF performs well in the SOC estimation of the lithium battery.Table 2
The average estimation error.
Open-circuit state
Constant-current discharge state
Intermittent discharge state
0 initial error (%)
2% initial error (%)
0 initial error (%)
10% initial error (%)
0 initial error (%)
10% initial error (%)
DEKF
0.2
0.2
0.8
1.3
1.8
2.7
CC
0
1.8
4.2
11.4
6.5
13.3


## 4.2.1. Open-Circuit State

When the initial charge is 92%, the SOC estimation is performed in the open-circuit state. The measuring voltage nearly remains constant due to the zero working current, as shown in Figure11. When initial error is 0, the SOC estimated by both methods is close to 92% with a little fluctuate, as shown in Figure 12(a). Because the current is zero, CC-SOC and REAL-SOC have the same value at this time, and the two curves are basically coincident. And when 2% initial error exists, the SOC estimated by the DEKF method can converge to real value, but the estimation still contains initial error by using the CC method as shown in Figure 12(b).Figure 11
Measuring voltage in the open-circuit state.Figure 12
SOC estimation in the open-circuit state. (a) 0 initial error. (b) 2% initial error.
(a)
(b)


## 4.2.2. Constant-Current Discharge State

The measuring voltage constantly decreases in 1.5 C constant-current discharge state, as shown in Figure13. It is seen that the gradient is great in the high and low SOC stages, while the gradient is much less in the middle stages, which reflects the EB−SOC characteristic of the lithium battery. When initial error is 0, the SOC estimated by the DEKF method is near the real value, while the value by the CC method gradually deviated from the real value, as shown in Figure 14(a). And when 10% initial error is introduced, the estimation of the CC method still contains nearly whole error, while the error is reduced obviously by the DEKF method, as shown in Figure 14(b).Figure 13
Measuring voltage in the constant-current discharge state.Figure 14
SOC estimation in the constant-current discharge state. (a) 0 initial error. (b) 10% initial error.
(a)
(b)


## 4.2.3. Intermittent Discharge State

The change of measuring voltage in 1C intermittent discharge state is shown in Figure15. It is seen that the voltage is in a decreasing tendency, but some rebounds and step changes occur in the intermittent phase due to the rebound characteristic of the battery.Figure 15
Measuring voltage in the intermittent discharge state.When initial errors are 0 and 10%, the estimation results of SOC are shown in Figure16. It is seen that the SOC can keep track with real value both in cutoff and discharge stage, while the CC method is serious influenced by errors.Figure 16
SOC estimation in the intermittent discharge state. (a) 0 initial error. (b) 10% initial error.
(a)
(b)


## 4.2.4. Error Analysis

In the open-circuit state, the accuracy of the CC method completely depends on initial error because of zero working current, while DEKF can correct error within a short time. In working conditions, even if the initial error is 0, the maximum errors of CC, respectively, reach 8% and 12%, while the DEKF method is less than 2%. In terms of situation that 10% initial error is introduced, and the error of the CC method is still very big in all of the experiment processes. But, the DEKF method has good convergence, and the initial error can gradually be decreased by correction of the state equation and observational equation.The average errors of two methods in different conditions are shown in Table2. It is seen that whatever the initial errors exist or not, average errors of DEKF are less than 2.7%, which is also much less than the errors of the CC method. Besides, the DEKF has good robustness and can well deal with the influence of initial errors, while the CC method much relies on the stable initial value and the precision of sensors. Therefore, it is proved that the DEKF performs well in the SOC estimation of the lithium battery.Table 2
The average estimation error.
Open-circuit state
Constant-current discharge state
Intermittent discharge state
0 initial error (%)
2% initial error (%)
0 initial error (%)
10% initial error (%)
0 initial error (%)
10% initial error (%)
DEKF
0.2
0.2
0.8
1.3
1.8
2.7
CC
0
1.8
4.2
11.4
6.5
13.3


## 5. Conclusions

(1)
A new equivalent circuit model that can describe the working characteristics of the lithium battery is established based on Thevenin’s theorem. The internal parameters, including capacity, the ohmic internal resistance, polarization resistance, and capacitance, are identified by fitting characteristic curve. The relation of equivalent internal resistance and capacity is obtained by battery life experiments.
(2)
The DEKF principle using in battery state estimation is analyzed based on the relation of SOC and SOH. A simulation model is established to analyze the estimation ability with the influence of different initial errors.
(3)
The SOC estimation experiments are performed by, respectively, using the traditional CC method and DEKF method in three different working conditions. By comparison with the estimation results of the CC method, the results of DEKF are still near the real values, even if the errors of sensors exist. When 10% initial errors are introduced, the DEKF can correct the errors, and the estimation can fast converge to real values. The average errors are less than 3% in all kinds of working conditions, which verifies the feasibility of the algorithm.



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*Source: 6096834-2020-04-11.xml*