## Abstract

The state of charge (SoC) of the battery is a typical characterization of the operating state of the battery and criterion for the battery management system (BMS) control strategy, which must be evaluated precisely. The establishment of an accurate algorithm of SoC estimation is of great significance for BMS, which can help the driver judge the endurance mileage of electric vehicle (EV) correctly. In this paper, a second-order resistor-capacity (RC) equivalent circuit model is selected to characterize the electrical characteristics based on the electrochemical model of the LiFePO_{4}/graphene (LFP/G) hybrid cathode lithium-ion battery. Moreover, seven open circuit voltage (OCV) models are compared and the best one of them is used to simulate the dynamic characteristics of the battery. It is worth mentioning that an improved test method is proposed, which is combined with least square for parameters identification. In addition, the extended Kalman filter (EKF) algorithm is selected to estimate the SoC during the charging and discharging processes. The simulation results show that the EKF algorithm has the higher accuracy and rapidity than the KF algorithm.

## 1 Introduction

With the development of society and the continuous growth of the economy, the popularity of automobiles and the rapid development of the automobile industry have led to an increasing demand for fossil fuels such as petroleum. However, due to the nonrenewable fossil fuels, the increase in demand and the reduction of surplus fossil fuels have become a society problem that must be solved urgently. In view of the problems caused by automobiles, it’s necessary to find an energy-saving and environmental-friendly energy supply method [1,2]. Electricity is safer and more environmental-friendly than other energy supply methods, so the EVs powered by batteries are immediately generated and concerned [3]. LiFePO_{4} (LFP) battery is one of the most widely used battery because of its good thermal stability and excellent electrochemical property [4]. However, there are also some disadvantages of LFP battery such as the weak electronic conductivity, the restricted Li-ion diffusion coefficient and high heat generation [5–7]. Therefore, it is critical to find a way to enhance the properties of the LFP battery.

The studies [4,8,9] showed that the surface coating is an effective method to modify the electrode material. It is able to enhance the performance of the battery by covering the primary electrode of the battery with a layer of other materials [10], such as graphene. Graphene is a single-atom sheet of graphite with a two-dimensional crystalline layer [11] as shown in Fig. 1(a) [4]. It has the advantages of low resistivity and good electrical conductivity, which can obviously change the obstacles that LFP battery technology has not broken for a long time. One of the most important applications of the graphene is to form the LiFePO_{4}/Graphene (LFP/G) composite cathode. It has a three-dimensional porous structure to reduce the contact resistance [12–14], as shown in Fig. 1(b) [4]. Adding graphene to the electrode can prominently improve battery charging rate, cycle stability, service life, and energy density. Thus, LFP/G hybrid cathode lithium-ion battery has become a breakthrough to solve the problem of slow charging and poor endurance of existing battery because of its large capacity, good conductivity, fast charging rate, and large power-to-weight ratio [9].

A refined BMS is essential to assure safe and effective operation of battery. The SoC is often used to estimate the remaining capacity of the battery, which can directly represent the operating state and usage of the battery pack. The real-time SoC is the basis for the BMS to obtain energy, power, and safety information from the battery and feedback to the vehicle. Therefore, the precise estimation of SoC has a significant impact on the dynamic performance and safety of the vehicle [15].

Several SoC estimation methods have been purposed by researchers, such as density method [16], Coulomb counting method (CCM) or ampere-hour counting method (AHCM) [10], open circuit voltage method (OCVM) [17,18], and model-based method (MBM) [19]. The five methods are compared in Table 1. Among all of these methods, MBM is suitable for EV applications because of its advantages. Some model-based SoC estimation algorithms have been developed in the past few decades, including Kalman filter (KF) [10] which is linear and nonlinear Kalman filter, such as EKF [20,21], unscented Kalman filter (UKF) [22], cubature Kalman filter (CKF) [23], and particle filter [24–27]. EKF algorithm is the most preferred method for the battery parameters and the SoC estimation, which uses partial derivatives and first-order Taylor series expansion [28,29]. Since the computational complexity of the EKF algorithm is moderate, the SoC estimation based on the EKF algorithm has better precision and timeliness. Therefore, the EKF algorithm has been widely used in the case of the high accuracy and appropriate complexity of the battery model.

Method | Advantage | Disadvantage | Application | Ref. |
---|---|---|---|---|

Density method | Reliable | Poor timeliness | Laboratory | [16] |

AHCM | High precision Simple | Error accumulation | EV Laboratory | [10] |

CCM | Simple computation | Error accumulation | Small electronic devices | [10] |

OCVM | High precision | Multiple influencing factors | Inappropriate for online SoC estimation | [17,18] |

MBM | High precision Low complexity | Time-consuming process | All kinds of batteries EV | [19] |

Method | Advantage | Disadvantage | Application | Ref. |
---|---|---|---|---|

Density method | Reliable | Poor timeliness | Laboratory | [16] |

AHCM | High precision Simple | Error accumulation | EV Laboratory | [10] |

CCM | Simple computation | Error accumulation | Small electronic devices | [10] |

OCVM | High precision | Multiple influencing factors | Inappropriate for online SoC estimation | [17,18] |

MBM | High precision Low complexity | Time-consuming process | All kinds of batteries EV | [19] |

In this paper, seven OCV models are compared and setup in Sec. 2 to simulate the characteristics of the LiFePO_{4}/graphene hybrid cathode lithium-ion battery. An improved test method is proposed to identify the parameters and verify the OCV model. The EKF algorithm is adopted in Sec. 3 to estimate the SoC, and its astringency and reliability are verified. The results and discussion are shown in Sec. 4 as well as the conclusion is drawn in Sec. 5.

## 2 Battery Model and Parameters Identification

### 2.1 Open Circuit Voltage Model.

The equivalent circuit model uses electrical components to represent the electrical characteristics of the battery. The existing studies [16,30] show that SoC is related to the battery terminal voltage, charge–discharge current, and temperature. For instance, if the temperature is too high or too low, the charge or discharge efficiency of the battery will be reduced. The change of efficiency will have a great impact on the cumulative deviation of the SoC in practical applications [31], thereby affecting the accuracy of the SoC estimation. Therefore, the experimental temperature is set at 25 °C, in which case the equivalent circuit model based on electrical theory is suitable for the high-precision SoC estimation algorithm studied. In this paper, second-order RC equivalent circuit model shown in Fig. 2 is selected to simulate the dynamic characteristics of the LiFePO_{4}/graphene hybrid cathode lithium-ion battery.

*U*

_{OCV}is the OCV,

*U*

_{t}is the terminal voltage,

*C*

_{p1}and

*C*

_{p2}are polarization capacitance in two RC networks,

*R*

_{p1}and

*R*

_{p2}are polarization resistance in two RC networks,

*U*

_{p1},

*U*

_{p2}are the polarization voltage across

*C*

_{p1},

*C*

_{p2}, respectively,

*R*

_{0}is the ohmic resistance, and

*I*

_{L}is the load current.

A large number of experiments [31–34] show that there is a fixed functional relationship between the OCV and the SoC. Although OCV is related to SoC closely, it shows a strong nonlinear characteristic. The OCV of the battery can be obtained by measuring the battery terminal voltage [35].

Based on the studies presented, seven OCV models are selected and compared in Table 2. Where *s* is the SoC, *K*_{i}, *α*_{i}, *β*_{i} (*i* = 0 − 9) are linear parameters and nonlinear parameters, respectively, and *m*, *n* are parameters of OCV models, all of which are determined by ORIGIN curve fitting toolbox [36].

Model | OCV model expression | Ref. |
---|---|---|

1 | $UOCV=K0+K1lns+K2ln(1\u2212s)$ | [43] |

2 | U_{OCV} = K_{0} + K_{1}s + K_{2}s^{2} + K_{3}s^{3} + K_{4} ln s + K_{5} ln(1 − s) | [44] |

3 | U_{OCV} = K_{0} + K_{1}s + K_{2}s^{2} + K_{3}s^{3} + K_{4} exp(K_{5}s) | [45] |

4 | U_{OCV} = K_{0} + K_{1}s + K_{2}(1 − ln s)^{m} + K_{3} exp(n(s − 1)) | [46,47] |

5 | $UOCV=K0+K1s+K2s2+K3s3+K4s4+K5s5+K6s6+K7s7+K8s8$ | [48] |

6 | $UOCV=K0+K1s+K2s2+K3s3+K4s4+K5s5+K6s6+K7s7+K8s8+K9s9$ | [8] |

7 | $UOCV=K0+K1(1+exp(\alpha 1(s\u2212\beta 1)))\u22121+K2(1+exp(\alpha 2s))\u22121+K3(1+exp(\alpha 3(s\u2212\beta 3)))\u22121+K4(1+exp(\alpha 4(s\u22121)))\u22121+K5$ | [9] |

Model | OCV model expression | Ref. |
---|---|---|

1 | $UOCV=K0+K1lns+K2ln(1\u2212s)$ | [43] |

2 | U_{OCV} = K_{0} + K_{1}s + K_{2}s^{2} + K_{3}s^{3} + K_{4} ln s + K_{5} ln(1 − s) | [44] |

3 | U_{OCV} = K_{0} + K_{1}s + K_{2}s^{2} + K_{3}s^{3} + K_{4} exp(K_{5}s) | [45] |

4 | U_{OCV} = K_{0} + K_{1}s + K_{2}(1 − ln s)^{m} + K_{3} exp(n(s − 1)) | [46,47] |

5 | $UOCV=K0+K1s+K2s2+K3s3+K4s4+K5s5+K6s6+K7s7+K8s8$ | [48] |

6 | $UOCV=K0+K1s+K2s2+K3s3+K4s4+K5s5+K6s6+K7s7+K8s8+K9s9$ | [8] |

7 | $UOCV=K0+K1(1+exp(\alpha 1(s\u2212\beta 1)))\u22121+K2(1+exp(\alpha 2s))\u22121+K3(1+exp(\alpha 3(s\u2212\beta 3)))\u22121+K4(1+exp(\alpha 4(s\u22121)))\u22121+K5$ | [9] |

### 2.2 Open Circuit Voltage–State of Charge Curve Fitting.

Open circuit voltage model is a key parameter in the SoC estimation process. As shown in Table 2, there are overt differences in the structures of these OCV models. Thus, we give a comparison on these models shown as Fig. 3. In Fig. 3, model 7 fits better than other models, which accurately describes the variation of OCV with SoC during the initial and termination phases of the voltage. Therefore, model 7 is selected to be the OCV model in this paper. The charging process OCV–SoC curve and the discharging OCV–SoC curve are compared in Fig. 4. The model errors of charging and discharging processes are shown in Fig. 5. The results show that the maximum errors of the charging and discharging processes are 0.64% and 0.41%, which mean that model 7 can fit accurately. Since the OCV is related to ambient temperatures [9], the experiments are performed under 25 °C to reduce the influence of temperature, and the number of cycles of charge and discharge are 60 and 23, respectively in this study.

In order to clearly explain the effect of each variable and random error, the difference between the data point and its corresponding position on the regression line is statistically referred to as the residual. The sum of each residual square is called the sum squared residual (SSE), which represents the effect of random errors [37]. The SSE and the maximum errors are shown in Table 3, which can measure the fitting performance. The smaller the SSE is, the better the curve fits.

### 2.3 Improved Test Method and Parameters Identification.

Since the electrical parameters of the model components cannot be directly measured by the multimeter, and each parameter displays nonlinear characteristics as the external factors such as temperature, charging and discharging degree, lifetime, and remaining capacity change, it is necessary to identify the conditions under different charging states. The parameters identification of the electrical components of the equivalent circuit can be based on the standard identification method [38] and the specific process.

#### 2.3.1 Improved Test Method.

#### 2.3.2 Parameters Identification.

The component parameters are relatively stable during the plateau period where the SoC is between 0.1 and 0.9 as well as vary sharply at the beginning and end of the charge and discharge phases, which exhibit similar characteristic to the OCV-SoC curve. Based on the actual working conditions, this paper mainly studies the parameters identification of the interval of SoC = 0.1–0.9 by using off-line identification method.

*U*is the instantaneous voltage drop,

*R*

_{0}is the ohmic resistance.

*U*

_{c}is the sum of the voltages on capacitance

*C*

_{p1}and

*C*

_{p2}, and

*τ*is the time constant.

*U*

_{p1},

*U*

_{p2},

*τ*

_{1},

*τ*

_{2}are fitted and identified by least squares method. Eq. (5) is used as the fitting equation for curve fitting, and

*U*

_{p1},

*U*

_{p2},

*τ*

_{1},

*τ*

_{2}are substituted as the undetermined parameters for parameters identification to get

*τ*

_{1},

*τ*

_{2}. The mathematical expression for the zero- response is

*U*

_{k}is the terminal voltage of the battery after the current loaded. Eq. (8) is used as the fitting equation for curve fitting, and

*R*

_{p1},

*R*

_{p2}are substituted as the undetermined parameters to get

*R*

_{p1},

*R*

_{p2}by using ORIGIN curve fitting toolbox [36]. According to Eqs. (6) and (7) as well as

*τ*

_{1},

*τ*

_{2}gotten before, we can get

*C*

_{p1},

*C*

_{p2}and parameters identification is completed. The fitting results of parameters identification which used the sixth-order polynomial are shown in Fig. 8.

## 3 State of Charge Estimation

The continuous development of intelligent algorithms makes the modeling approach easier to implement. The fundamental principle of linear KF and nonlinear KF algorithm is to recursively estimate the current state using previously estimated states and current measurement signals. The self-correcting properties of these algorithms make it suitable for model-based online SoC estimation for EV applications. The contrast of the model-based SoC estimation algorithms we mentioned above is shown in Table 4.

Algorithm | Advantage | Disadvantage | Ref. |
---|---|---|---|

KF | Bound the minimum mean square error | Not suitable for the highly nonlinear system | [10] |

EKF | Moderate computational complexity | Accuracy depends on battery model parameter | [20,21] |

UKF | High accuracy, robustness and convergence rate | Not appropriate for high measurement noise | [22] |

CKF | High accuracy | More computation time | [23] |

Algorithm | Advantage | Disadvantage | Ref. |
---|---|---|---|

KF | Bound the minimum mean square error | Not suitable for the highly nonlinear system | [10] |

EKF | Moderate computational complexity | Accuracy depends on battery model parameter | [20,21] |

UKF | High accuracy, robustness and convergence rate | Not appropriate for high measurement noise | [22] |

CKF | High accuracy | More computation time | [23] |

*x*

_{k}is the system state vector,

*y*

_{k}is the observation quantity,

*u*

_{k}is the system input,

*w*

_{k}is the system noise, and

*v*

_{k}is the observation noise, and

*f*(

*x*

_{k},

*u*

_{k}) and

*g*(

*x*

_{k},

*u*

_{k}) are nonlinear functions. The block diagram of discrete nonlinear system is shown in Fig. 9.

*x*

_{k},

*u*

_{k}), the linearized system state-output equation is:

*A*

_{k}and

*C*

_{k}are the derivation matrices of

*f*(

*x*

_{k},

*u*

_{k}) and

*g*(

*x*

_{k},

*u*

_{k}) with respect to system state vector

*x*

_{k}, respectively.

The essence of the EKF algorithm is the recursive operation based on covariance correction. The general principle of the EKF algorithm is summarized in Table 5. Due to the adaptability of the EKF, accurate values can still be approximated even in the case of initial error. Once the initial state variable $x0\u2227$ and the initial covariance matrix *P*_{(0)} are given, the state variable $xk\u2227$ can be calculated by recursion on the basis of the observation quantity *y*_{k}. Where *Q* is the system noise covariance matrix, *R* is the measurement noise covariance matrix, *I* is the unit matrix, *P* is the covariance matrix, *K* is the Kalman gain vector, and *k* is the sample point.

Initialization | |

For k = 0, set | |

$x^0=E[x0]$ | |

$P(0)=E[(x0\u2212x^0)(x0\u2212x^0)T]$ | |

Computation | |

For k = 1, 2, 3,…, compute | |

$State estimate time update:x^k/k\u22121=f(x^k\u22121/k\u22121,uk\u22121)$ | |

$Error covariance time updatePk/k\u22121=Ak\u22121Pk\u22121/k\u22121Ak\u22121T+Qk\u22121$ | |

$Kalman gain matrix:Kk=Pk/k\u22121CkT(CkPk/k\u22121CkT+Rk)\u22121$ | |

State estimate measurement update: $x^k/k=x^k/k\u22121+Kk(yk\u2212g(x^k/k\u22121,uk))$ | |

$Error covariance measurement update:Pk/k=(I\u2212KkCk)Pk/k\u22121)$ |

Initialization | |

For k = 0, set | |

$x^0=E[x0]$ | |

$P(0)=E[(x0\u2212x^0)(x0\u2212x^0)T]$ | |

Computation | |

For k = 1, 2, 3,…, compute | |

$State estimate time update:x^k/k\u22121=f(x^k\u22121/k\u22121,uk\u22121)$ | |

$Error covariance time updatePk/k\u22121=Ak\u22121Pk\u22121/k\u22121Ak\u22121T+Qk\u22121$ | |

$Kalman gain matrix:Kk=Pk/k\u22121CkT(CkPk/k\u22121CkT+Rk)\u22121$ | |

State estimate measurement update: $x^k/k=x^k/k\u22121+Kk(yk\u2212g(x^k/k\u22121,uk))$ | |

$Error covariance measurement update:Pk/k=(I\u2212KkCk)Pk/k\u22121)$ |

Taking the practicability of the algorithm as well as the accuracy and complexity of the second-order RC equivalent circuit model into consideration, the algorithm is applied in the SoC = 0.1 ∼ 0.9 interval. The EKF algorithm combining with battery model is summarized in Table 6 [15]. Where *s* is the SoC, *η* is the Coulombic efficiency, *t* is the sampling interval, *Q*_{n} is the nominal capacity, *i*_{k} is the current, and *y*_{k} is the predicted voltage. The parameters *R*_{0}, *R*_{p1}, *R*_{p2}, *C*_{p1}, *C*_{p2}, and *U*_{OCV} we obtained before can be substituted into Eqs. (20)–(28) directly.

Combined model 7 |

$sk+1=sk\u2212(\eta tQn)ik$ |

Defined the state vector |

$xk=[sk,Up1,k,Up2,k]T$ |

$f(xk,uk)=[1000exp(\u2212t/\tau 1)000exp(\u2212t/\tau 2)][skUp1,kUp2,k]+[\u2212\eta t/QnRp1(1\u2212exp(\u2212t/\tau 1))Rp2(1\u2212exp(\u2212t/\tau 2))]ik$ |

$g(xk,uk)=UOCV(sk)\u2212ikR0\u2212Up1,k\u2212Up2,k$ |

Matrix computation |

$Ak=\u2202f\u2202xk|x=x^k+=[1000exp(\u2212t/\tau 1)000exp(\u2212t/\tau 2)]$ |

$Ck=\u2202g\u2202x|x=x^k\u2212=[dUOCV(s)ds\u22121\u22121]$ |

Discretized battery terminal voltage |

$[sk+1Up1,k+1Up2,k+1]=[1000exp(\u2212t/\tau 1)000exp(\u2212t/\tau 2)][skUp1,kUp2,k]+[\u2212\eta t/QnRp1(1\u2212exp(\u2212t/\tau 1))Rp2(1\u2212exp(\u2212t/\tau 2))]ik+wk$ |

$yk=UOCV(sk)\u2212ikR0\u2212Up1,k\u2212Up2,k+vk$ |

Combined model 7 |

$sk+1=sk\u2212(\eta tQn)ik$ |

Defined the state vector |

$xk=[sk,Up1,k,Up2,k]T$ |

$g(xk,uk)=UOCV(sk)\u2212ikR0\u2212Up1,k\u2212Up2,k$ |

Matrix computation |

$Ak=\u2202f\u2202xk|x=x^k+=[1000exp(\u2212t/\tau 1)000exp(\u2212t/\tau 2)]$ |

$Ck=\u2202g\u2202x|x=x^k\u2212=[dUOCV(s)ds\u22121\u22121]$ |

Discretized battery terminal voltage |

$yk=UOCV(sk)\u2212ikR0\u2212Up1,k\u2212Up2,k+vk$ |

## 4 Results and Discussion

The residual is the difference between the actual observed value and the estimated value or fitted value, which contains the important information about the basic assumptions of the model [40]. Using the information provided by the residual to examine the rationality of the model hypothesis and the reliability of the data is called the residual analysis [41]. It is the analysis of the reliability, periodicity, or other interference of the data. The functional relationships between the residual, the SoC, and the fitted OCV are shown in Figs. 10 and 11, respectively. The results show that the maximum residuals during the charging and discharging processes are 0.01613 and 0.0299, which mean that the actual observed value and the estimated value are within an allowable error range. The reliability of the model is proven.

Based on the SoC estimation algorithm, this paper designs a new test experiment method, which combines the high-current method with the low-current method, to simulate the charging and discharging conditions. The experiment data used by this paper are acquired through the test bench. It is composed of the battery test system, a thermostat for temperature control, a computer for operation, and LFP/G hybrid cathode lithium-ion batteries. The terminal voltage between the measured data and the simulated data by using matlab software [42] is compared in Fig. 12. The low-current process lasted longer during the charging process than the discharging, and the hysteresis phenomenon is significantly better than the high-current process.

The SoC estimation between the measured data and the simulated data are compared in Figs. 13(a) and 13(c). The results show that the SoC based on the EKF algorithm has the accurate values, although there are fluctuations, it always occurs near the measured values. The EKF algorithm has good convergence, which can continuously correct the battery parameters to reduce the estimated errors. It can be seen in Figs. 13(b) and 13(d) that the maximum SoC estimation errors under the EKF algorithm of the two processes are 0.027% and 2%, respectively, it can meet the requirements of the BMS.

In summary, the second-order RC equivalent circuit model selected in this paper can accurately describe the battery characteristics and is convenient for parameters measurement. The OCV model we chose has higher precision than other models mentioned in this paper, which provides a solid foundation for the establishment of the SoC estimation algorithm. SoC is considered as one of the critical and significant factors in BMS. Accurate estimation of the SoC not only helps to reflect information about the current and remaining capacity of battery but also provides a guarantee for the reliable and safe operation of EV.

## 5 Conclusion

In this paper, a second-order RC equivalent circuit model is established by combining an electrochemical-thermal model of the LiFePO_{4}/graphene hybrid cathode lithium-ion battery. Seven OCV models are compared and the best of them is selected with consideration of model accuracy and complexity. The functional relationships between the residual, the SoC, and the fitted OCV in Figs. 10 and 11 indicate that the OCV model we opted has high accuracy and reliability. Moreover, an improved test method is purposed, which combines the high-current process with the low-current process, to verify the OCV model based on the battery. The test method we developed has less loss to the battery. The parameters of the battery are fulfilled offline identification, which are fitted by using least squares method.

Based on the second-order RC equivalent circuit model and the parameters obtained by identification, the estimation model is established by matlab software. The principle of the EKF algorithm is explained and the error bounds are also generated. The simulation experiments are designed to verify the feasibility and effectiveness of the EKF algorithm. As it can be seen, the EKF algorithm has high accuracy with the high precision of the battery model and can correct the initial value even if it is wrong. The predicted SoC is nearly consistent with the measured data. In addition, a long-term prediction is also achieved with increased accuracy.

In conclusion, the EKF algorithm is a preferred approach to meet the algorithmic requirements of the BMS of EV.

## Acknowledgment

This study was funded by the Natural Science Foundation of Inner Mongolia Autonomous Region (2017ZD02).

## References

_{4}Battery Pack Capacity Estimation for Electric Vehicles Based on Charging Cell Voltage Curve Transformation

_{4}/Graphene Hybrid Cathode Lithium-Ion Battery

_{4}Battery via Dual Extended Kalman Filter and Charging Voltage Curve