CN103472403A

CN103472403A - Composite estimating method of power battery SOC based on PNGV equivalent circuit model

Info

Publication number: CN103472403A
Application number: CN2013104260215A
Authority: CN
Inventors: 李明; 江洋; 郑荐中; 彭筱筱; 朱中文
Original assignee: Zhejiang Province Institute of Metrology
Current assignee: Zhejiang Province Institute of Metrology
Priority date: 2013-09-17
Filing date: 2013-09-17
Publication date: 2013-12-25
Anticipated expiration: 2033-09-17
Also published as: CN103472403B

Abstract

The invention relates to the technical field of electric vehicle power batteries, and discloses a power battery SOC composite estimation method based on the PNGV equivalent circuit model, comprising the following steps: A. Detecting the open circuit voltage of the power battery; B. Using the open circuit voltage method to calculate the initial state of the power battery SOC(t ₀ ); C. During the period of t ₀ -t ₁ , the extended Kalman filter (EKF) algorithm is used to correct the initial SOC (t ₀ ) to obtain SOC (t ₁ ); D. In the period of t ₁ - During the t ₂ time period, use the improved ampere-hour integration method to estimate; E. When continuing to use the power battery, step C and step D are cycled; t ₀ : represents the initial time; t ₁ , t ₂ : time points after t ₀ . The present invention considers the influence of the power battery charging and discharging current, ambient temperature, battery health status and other influencing factors and the correction coefficient, and obtains an improved ampere-hour integration method, which can overcome the defects that the ampere-hour integration method is greatly affected by external factors when estimating the power battery SOC. ; Make full use of the advantage that the Kalman filter method has a strong correction effect on the initial error of the SOC.

Description

A Composite Estimation Method for Power Battery SOC Based on PNGV Equivalent Circuit Model

技术领域technical field

本发明涉及电动汽车动力电池技术领域，尤其涉及了一种基于PNGV等效电路模型的动力电池SOC复合估计方法。The invention relates to the technical field of electric vehicle power batteries, in particular to a composite estimation method for power battery SOC based on a PNGV equivalent circuit model.

背景技术Background technique

电池管理系统是电动汽车的重要组成部分，动力电池荷电状态（State ofCharge，SOC）估计是此系统的关键技术之一。SOC实时估计涉及到动力电池充放电控制和电动汽车的优化管理，直接影响动力电池的使用寿命和动力系统的性能，因此动力电池SOC的准确估计对于电动汽车的运行非常关键。The battery management system is an important part of electric vehicles, and the state of charge (State of Charge, SOC) estimation of power batteries is one of the key technologies of this system. The real-time estimation of SOC involves the charging and discharging control of the power battery and the optimal management of electric vehicles, which directly affects the service life of the power battery and the performance of the power system. Therefore, accurate estimation of the SOC of the power battery is very critical for the operation of electric vehicles.

目前，SOC的估计方法主要有安时积分法、内阻法、开路电压法、神经网络法和卡尔曼滤波法等。但安时积分法存在一些缺陷会导致估计不准确，比如方法本身不能估计SOC初始值，动力电池容量受外界因素影响较大等；开路电压法需要电池长时间静置，不适用于实时预测；神经网络法易受干扰，准确度受训练方法和训练数据的影响很大；卡尔曼滤波法对SOC的初始误差有很强的修正作用，特别适合于电流变化较快的动力电池，但对电池模型准确性要求比较高。At present, SOC estimation methods mainly include ampere-hour integral method, internal resistance method, open circuit voltage method, neural network method and Kalman filter method. However, there are some defects in the ampere-hour integration method that will lead to inaccurate estimates. For example, the method itself cannot estimate the initial value of SOC, and the capacity of the power battery is greatly affected by external factors. The open circuit voltage method requires the battery to stand for a long time and is not suitable for real-time prediction; The neural network method is susceptible to interference, and its accuracy is greatly affected by the training method and training data; the Kalman filter method has a strong correction effect on the initial error of the SOC, and is especially suitable for power batteries with rapid current changes, but it is not effective for battery The model accuracy requirements are relatively high.

发明内容Contents of the invention

本发明的目的是克服现有的动力电池SOC估计方法存在的不足，提供一种基于PNGV等效电路模型的动力电池SOC复合估计方法，实现了磷酸铁锂动力电池的SOC准确估计。The purpose of the present invention is to overcome the shortcomings of existing power battery SOC estimation methods, provide a power battery SOC composite estimation method based on the PNGV equivalent circuit model, and realize accurate SOC estimation of lithium iron phosphate power batteries.

为了解决上述技术问题，本发明通过下述技术方案得以解决：In order to solve the above technical problems, the present invention is solved through the following technical solutions:

一种基于PNGV等效电路模型的动力电池SOC复合估计方法，包括以下步骤：A kind of power battery SOC compound estimation method based on PNGV equivalent circuit model, comprises the following steps:

A.检测动力电池开路电压；A. Detect the open circuit voltage of the power battery;

B.采用开路电压法计算动力电池初始SOC（t₀）；B. Use the open circuit voltage method to calculate the initial SOC of the power battery (t ₀ );

C.在t₀-t₁时间段内，采用扩展卡尔曼滤波（EKF）算法对初始SOC（t₀）进行修正，得到SOC（t₁）；C. During the time period t ₀ -t ₁ , the extended Kalman filter (EKF) algorithm is used to correct the initial SOC (t ₀ ) to obtain the SOC (t ₁ );

D.在t₁-t₂时间段内，采用改进安时积分法进行估计；D. During the t ₁ -t ₂ time period, use the improved ampere-hour integration method to estimate;

E.继续使用动力电池时，步骤C步骤D进行循环；E. When continuing to use the power battery, step C and step D are cycled;

t₀：表示初始时间；t₁、t₂：在t₀之后的时间点。t ₀ : indicates the initial time; t ₁ , t ₂ : time points after t ₀ .

一种基于PNGV等效电路模型的动力电池SOC复合估计方法，列出开路电压方程如下：A power battery SOC composite estimation method based on the PNGV equivalent circuit model, the open circuit voltage equation is listed as follows:

U(t)＝U_OCV(t)-U_a(t)-U_p(t)-R_oI(t) （1）U(t)＝U _OCV (t)-U _a (t)-U _p (t)-R _o I(t) (1)

${U u}_{a a} ((t t)) = = \frac{11}{{C C}_{a a}} {&Integral; &Integral;}_{{t t}_{00}}^{t t} I I ((t t)) dt dt - - - - - - ((22))$

$I I ((t t)) - - {I I}_{p p} ((t t)) = = I I ((t t)) - - \frac{{U u}_{p p} ((t t))}{{R R}_{p p}} = = {C C}_{p p} \frac{d d {U u}_{p p} ((t t))}{dt dt} - - - - - - ((33))$

U_OCV：理想电压源,表示电池的开路电压；t：时间；U _OCV : ideal voltage source, indicating the open circuit voltage of the battery; t: time;

C_a：电容，电容描述的是因电流的时间累积效应而引起的开路电压的变化；C _a : Capacitance, capacitance describes the change of open circuit voltage caused by the time accumulation effect of current;

U_a：表示电容C_a两端的电压；R_o：电池的欧姆内阻；U _a : indicates the voltage across the capacitor C _a ; R _o : the ohmic internal resistance of the battery;

R_p：电池内部极化电阻；U_p：电池内部极化电阻两端的电压；R _p : the internal polarization resistance of the battery; U _p : the voltage across the internal polarization resistance of the battery;

C_p:电阻R_p的并联电容；C _p : the parallel capacitance of the resistor R _p ;

I:动力电池工作电流；I_p：极化电阻的电流；U:动力电池端电压；I: working current of power battery; I _p : current of polarization resistance; U: terminal voltage of power battery;

列出动力电池开路电压与动力电池SOC的函数关系：List the functional relationship between the open circuit voltage of the power battery and the SOC of the power battery:

U_OCV(t)＝F[SOC(t)] （4） _UOCV (t)＝F[SOC(t)] (4)

式中，F[SOC(t)]是一个非线性函数；In the formula, F[SOC(t)] is a nonlinear function;

根据原始安时积分法估算SOC公式列出在时间t₀-t₁的安时积分法估算SOC公式：The formula for estimating SOC based on the original ampere-hour integration method is listed in the ampere-hour integration method at time t ₀ -t ₁ to estimate the SOC formula:

$SOC SOC (({t t}_{11})) = = SOC SOC (({t t}_{00})) - - \frac{11}{{Q Q}_{R R}} {&Integral; &Integral;}_{{t t}_{00}}^{{t t}_{11}} I I ((t t)) dt dt - - - - - - ((55))$

式中，Q_R为动力电池额定容量；In the formula, Q _R is the rated capacity of the power battery;

考虑动力电池充放电电流λ_c、环境温度λ_T以及电池健康状态λ_SOH等影响因素修正系数的影响，可得到改进的安时积分法：Considering the influence of the correction coefficient of the charging and discharging current _λc of the power battery, the ambient temperature _λT , and the state of health of the battery _λSOH , an improved ampere-hour integration method can be obtained:

$SOC SOC (({t t}_{11})) = = SOC SOC (({t t}_{00})) - - \frac{11}{λ λ {Q Q}_{R R}} {&Integral; &Integral;}_{{t t}_{00}}^{{t t}_{11}} I I ((t t)) dt dt - - - - - - ((66))$

式中：λ＝λ_c×λ_T×λ_SOH，Where: λ=λ _c ×λ _T ×λ _SOH ,

λ为修正系数；λ_c为动力电池充放电电流系数；λ_T为环境温度系数；λ_SOH为电池健康状态系数；λ is the correction coefficient; λ _c is the charge and discharge current coefficient of the power battery; λ _T is the ambient temperature coefficient; λ _SOH is the battery state of health coefficient;

将公式（1）（3）（6）离散化，Discretize formulas (1) (3) (6),

U(k)＝F[SOC(k)]-U_a(k)-U_p(k)-R_oI(k) （7）U(k)＝F[SOC(k)]-U _a (k)-U _p (k)-R _o I(k) (7)

$I I ((k k)) - - \frac{{U u}_{p p} ((k k))}{{R R}_{p p}} = = {C C}_{p p} \frac{{U u}_{p p} ((k k + + 11)) - - {U u}_{p p} ((k k))}{Δt Δt} - - - - - - ((88))$

$((k k + + 11)) = = SOC SOC ((k k)) - - \frac{Δt Δt}{λ λ {Q Q}_{R R}} ((k k)) - - - - - - ((99))$

最终得到离散时间状态空间模型如下：Finally, the discrete-time state-space model is obtained as follows:

$[\begin{matrix} SOC SOC ((k k + + 11)) \\ {U u}_{p p} ((k k + + 11)) \end{matrix}] = = [\begin{matrix} 11 & 00 \\ 00 & 11 - - \frac{Δt Δt}{{R R}_{p p} {C C}_{p p}} \end{matrix}] [\begin{matrix} SOC SOC ((k k)) \\ {U u}_{p p} ((k k)) \end{matrix}] + + [\begin{matrix} - - \frac{Δt Δt}{λ λ {Q Q}_{R R}} \\ \frac{Δt Δt}{{C C}_{p p}} \end{matrix}] I I ((k k)) + + [\begin{matrix} {w w}_{11} ((k k)) \\ {w w}_{22} ((k k)) \end{matrix}] - - - - - - ((1010))$

U(k)＝F[SOC(k)]-U_a(k)-U_p(k)-R_oI(k)+v(k) （11）U(k)＝F[SOC(k)]-U _a (k)-U _p (k)-R _o I(k)+v(k) (11)

该状态空间模型中，I(k)为输入量，表示工作电流；U(k)为输出量，表示端电压； $[\begin{matrix} w_{1} (k) \\ w_{2} (k) \end{matrix}]$ 为系统噪声；v(k)为观测噪声；Δt表示时间差；In the state-space model, I(k) is the input quantity, representing the working current; U(k) is the output quantity, representing the terminal voltage; $[\begin{matrix} w_{1} (k) \\ w_{2} (k) \end{matrix}]$ is the system noise; v(k) is the observation noise; Δt is the time difference;

将输出方程中的F[SOC(t)]进行线性化处理后，状态空间模型（10）、（11）中的A(k)，B(k)与C(k)分别为：After linearizing F[SOC(t)] in the output equation, A(k), B(k) and C(k) in state space models (10) and (11) are respectively:

$A A ((k k)) = = [\begin{matrix} 11 & 00 \\ 00 & 11 - - \frac{Δt Δt}{{R R}_{p p} {C C}_{p p}} \end{matrix}],, B B ((k k)) = = [\begin{matrix} - - \frac{Δt Δt}{λ λ {Q Q}_{R R}} \\ \frac{Δt Δt}{{C C}_{p p}} \end{matrix}],, C C ((k k)) = = {[[\frac{&PartialD; &PartialD; F f ((SOC SOC ((k k))]]}{&PartialD; &PartialD; SOC SOC ((k k))}]]}_{X x ((k k)) = = \overset{^^}{X x} ((k k))}$

EKF算法公式如下：The EKF algorithm formula is as follows:

$\overset{^^}{X x} ((k k | | k k - - 11)) = = A A ((k k - - 11)) \overset{^^}{X x} ((k k - - 11)) + + B B ((k k - - 11)) I I ((k k - - 11)) - - - - - - ((1212))$

$\overset{^^}{X x} ((k k)) = = \overset{^^}{X x} ((k k | | k k - - 11)) + + K K ((k k)) [[U u ((k k)) - - \overset{^^}{U u} ((k k))]] - - - - - - ((1313))$

P(k|k-1)＝A(k-1)P(k-1)A^T(k-1)+Q （14）P(k|k-1)＝A(k-1)P(k-1) ^AT (k-1)+Q (14)

K(k)＝P(k|k-1)C^T(k)[C(k)P(k|k-1)C^T(k)+R]^-1 （15）K(k)＝P(k|k-1)C ^T (k)[C(k)P(k|k-1)C ^T (k)+R] ^-1 (15)

P(k)＝[1-K(k)C(k)]P(k|k-1) （16）P(k)＝[1-K(k)C(k)]P(k|k-1) （16）

EKF算法由滤波器计算和滤波器增益计算两部分组成，滤波器计算由式（11）-（13）完成，滤波器先根据式（12）由前一时刻的结果

得到状态变量的预测值

再根据输出方程（11）得到系统观测量的预测值

跟实际观测值U(k)比较后得到预测误差，然后根据误差由式（13）对状态变量的预测值修正，得到新的滤波结果

滤波器增益计算由式（14）-（16）完成，式中P(k|k-1)和P(k)分别是状态变量预测误差和滤波误差的方差阵，K(k)为滤波器增益，Q和R分别是噪声w(k)和v(k)的方差阵。The EKF algorithm consists of two parts: filter calculation and filter gain calculation. The filter calculation is completed by formula (11)-(13).

Get the predicted value of the state variable

Then according to the output equation (11), the predicted value of the system observation is obtained

After comparing with the actual observed value U(k), the prediction error is obtained, and then according to the error, the predicted value of the state variable is corrected by formula (13) to obtain a new filtering result

The calculation of filter gain is completed by equations (14)-(16), where P(k|k-1) and P(k) are the variance matrix of state variable prediction error and filter error respectively, and K(k) is the filter Gains, Q and R are variance matrices of noise w(k) and v(k), respectively.

本发明由于采用了以上技术方案，具有显著的技术效果：The present invention has remarkable technical effect owing to adopted above technical scheme:

本发明考虑动力电池充放电电流、环境温度以及电池健康状态等影响因素修正系数的影响，得到改进的安时积分法，可以克服安时积分法估计动力电池SOC时受外界因素影响较大等缺陷；对于车载的动力电池系统而言,由于汽车负载的不断变化,所适用的电池模型应当是动态模型，而研究人员通过恒流放电试验和脉冲放电试验对上述模型静态和动态特性进行对比分析，确认PNGV模型就是这样一种能比较准确描述电池动态充放电特性的等效电路模型。综合采用开路电压法、改进安时积分法以及EKF算法进行估计，充分利用卡尔曼滤波法对SOC的初始误差有很强的修正效果的优势。可以有效提高估计方法的准确性。The present invention considers the influence of the power battery charging and discharging current, ambient temperature, battery health status and other influencing factors and the correction coefficient, and obtains an improved ampere-hour integration method, which can overcome the defects that the ampere-hour integration method is greatly affected by external factors when estimating the power battery SOC. ; For the vehicle-mounted power battery system, due to the continuous change of vehicle load, the applicable battery model should be a dynamic model, and the researchers compared the static and dynamic characteristics of the above models through constant current discharge tests and pulse discharge tests. It is confirmed that the PNGV model is such an equivalent circuit model that can accurately describe the dynamic charging and discharging characteristics of the battery. The open-circuit voltage method, the improved ampere-hour integral method and the EKF algorithm are used for estimation, and the Kalman filter method is fully utilized to have a strong correction effect on the initial error of the SOC. It can effectively improve the accuracy of the estimation method.

附图说明Description of drawings

图1是本发明所采用的PNGV等效电路模型；Fig. 1 is the PNGV equivalent circuit model that the present invention adopts;

图2是本发明中计算SOC的流程图。Fig. 2 is a flow chart of calculating SOC in the present invention.

具体实施方式Detailed ways

下面结合附图与实施例对本发明作进一步详细描述。The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments.

实施例1Example 1

下面结合附图1、2对本发明进一步详细描述：Below in conjunction with accompanying drawing 1,2 the present invention is described in further detail:

一种基于PNGV等效电路模型的动力电池SOC复合估计方法，如图2所示，包括以下步骤：A power battery SOC composite estimation method based on the PNGV equivalent circuit model, as shown in Figure 2, includes the following steps:

一种基于PNGV等效模型的动力电池SOC复合估计方法，PNGV推荐的电池模型简洁清晰，可以直接引用，如图1所示，为动力电池PNGV等效电路模型的数学模型，根据电路可以得出开路电压方程如下：A power battery SOC composite estimation method based on the PNGV equivalent model. The battery model recommended by PNGV is concise and clear, and can be directly quoted. As shown in Figure 1, it is the mathematical model of the power battery PNGV equivalent circuit model. According to the circuit, it can be obtained The open circuit voltage equation is as follows:

U_OCV(t)＝F[SOC(t)] （4） _UOCV (t)＝F[SOC(t)] (4)

式中：λ＝λ_c×λ_T×λ_SOH，Where: λ=λ _c ×λ _T ×λ _SOH ,

将公式（1）（3）（6）离散化，Discretize formulas (1) (3) (6),

EKF算法公式如下：The EKF algorithm formula is as follows:

P(k)＝[1-K(k)C(k)]P(k|k-1) （16）P(k)＝[1-K(k)C(k)]P(k|k-1) （16）

得到状态变量的预测值

再根据输出方程（11）得到系统观测量的预测值

Get the predicted value of the state variable

本发明提出的SOC复合估计算法（IAh-EKF）综合采用开路电压法、改进安时积分法结合EKF算法进行估计，流程如下：The SOC composite estimation algorithm (IAh-EKF) proposed by the present invention comprehensively uses the open circuit voltage method, the improved ampere-hour integral method combined with the EKF algorithm for estimation, and the process is as follows:

当电动汽车刚启动时，首先检测动力电池开路电压，利用开路电压法估计刚启动时的SOC初始值SOC（t₀），然后在t₀-t₁时间段内由EKF算法对电池初始值SOC（t₀）进行修正得到SOC（t₁），最后根据修正后的SOC（t₁）值，采用改进安时积分法对t₁后的SOC进行估计，采用改进安时积分法估计时间段t₁-t₂后，如果继续使用动力电池，则再采用EKF算法对电池SOC进行修正。When the electric vehicle is just started, first detect the open circuit voltage of the power battery, use the open circuit voltage method to estimate the initial value of SOC (t ₀ ) at the beginning of the start, and then use the EKF algorithm to calculate the initial value of the battery SOC during the time period of t ₀ -t ₁ (t ₀ ) to obtain SOC (t ₁ ), and finally according to the revised SOC (t ₁ ) value, use the improved ampere-hour integration method to estimate the SOC after t ₁ , and use the improved ampere-hour integration method to estimate the time period t After ₁ -t ₂ , if you continue to use the power battery, then use the EKF algorithm to correct the battery SOC.

总之，以上所述仅为本发明的较佳实施例，凡依本发明申请专利范围所作的均等变化与修饰，皆应属本发明专利的涵盖范围。In a word, the above descriptions are only preferred embodiments of the present invention, and all equivalent changes and modifications made according to the scope of the patent application of the present invention shall fall within the scope of the patent of the present invention.

Claims

1. a kind of power battery SOC composite estimation method based on PNGV equivalent circuit model, it is characterized in that comprising the following steps:

A. Detect the open circuit voltage of the power battery;

B. Use the open circuit voltage method to calculate the initial SOC of the power battery (t ₀ );

C. During the time period t ₀ -t ₁ , the extended Kalman filter (EKF) algorithm is used to correct the initial SOC (t ₀ ) to obtain the SOC (t ₁ );

D. During the t ₁ -t ₂ time period, use the improved ampere-hour integration method to estimate;

E. When continuing to use the power battery, step C and step D are cycled;

t ₀ : indicates the initial time; t ₁ , t ₂ : time points after t ₀ .

2. A kind of power battery SOC composite estimation method based on PNGV equivalent circuit model according to claim 1, is characterized in that: list open circuit voltage equation as follows:

U(t)＝U _OCV (t)-U _a (t)-U _p (t)-R _o I(t) (1)

{U u}_{a a} ((t t)) = = \frac{11}{{C C}_{a a}} {&Integral; &Integral;}_{{t t}_{00}}^{t t} I I ((t t)) dt dt - - - - - - ((22))

I I ((t t)) - - {I I}_{p p} ((t t)) = = I I ((t t)) - - \frac{{U u}_{p p} ((t t))}{{R R}_{p p}} = = {C C}_{p p} \frac{d d {U u}_{p p} ((t t))}{dt dt} - - - - - - ((33))

U _OCV : ideal voltage source, indicating the open circuit voltage of the battery; t: time;

C _a : Capacitance, capacitance describes the change of open circuit voltage caused by the time accumulation effect of current;

U _a : indicates the voltage across the capacitor C _a ; R _o : the ohmic internal resistance of the battery;

R _p : the internal polarization resistance of the battery; U _p : the voltage across the internal polarization resistance of the battery;

C _p : the parallel capacitance of the resistor R _p ;

I: working current of power battery; I _p : current of polarization resistance; U: terminal voltage of power battery;

List the functional relationship between the open circuit voltage of the power battery and the SOC of the power battery:

_UOCV (t)＝F[SOC(t)] (4)

In the formula, F[SOC(t)] is a nonlinear function;

The formula for estimating SOC based on the original ampere-hour integration method is listed in the ampere-hour integration method at time t ₀ -t ₁ to estimate the SOC formula:

SOC SOC (({t t}_{11})) = = SOC SOC (({t t}_{00})) - - \frac{11}{{Q Q}_{R R}} {&Integral; &Integral;}_{{t t}_{00}}^{{t t}_{11}} I I ((t t)) dt dt - - - - - - ((55))

In the formula, Q _R is the rated capacity of the power battery;

Considering the influence of the correction coefficient of the charging and discharging current _λc of the power battery, the ambient temperature _λT , and the state of health of the battery _λSOH , the improved ampere-hour integration method is obtained:

SOC SOC (({t t}_{11})) = = SOC SOC (({t t}_{00})) - - \frac{11}{λ λ {Q Q}_{R R}} {&Integral; &Integral;}_{{t t}_{00}}^{{t t}_{11}} I I ((t t)) dt dt - - - - - - ((66))

Where: λ=λ _c ×λ _T ×λ _SOH ,

λ is the correction coefficient; λ _c is the charge and discharge current coefficient of the power battery; λ _T is the ambient temperature coefficient; λ _SOH is the battery state of health coefficient;

Discretize formulas (1) (3) (6),

U(k)＝F[SOC(k)]-U _a (k)-U _p (k)-R _o I(k) (7)

I I ((k k)) - - \frac{{U u}_{p p} ((k k))}{{R R}_{p p}} = = {C C}_{p p} \frac{{U u}_{p p} ((k k + + 11)) - - {U u}_{p p} ((k k))}{Δt Δt} - - - - - - ((88))

((k k + + 11)) = = SOC SOC ((k k)) - - \frac{Δt Δt}{λ λ {Q Q}_{R R}} ((k k)) - - - - - - ((99))

Finally, the discrete-time state-space model is obtained as follows:

[\begin{matrix} SOC SOC ((k k + + 11)) \\ {U u}_{p p} ((k k + + 11)) \end{matrix}] = = [\begin{matrix} 11 & 00 \\ 00 & 11 - - \frac{Δt Δt}{{R R}_{p p} {C C}_{p p}} \end{matrix}] [\begin{matrix} SOC SOC ((k k)) \\ {U u}_{p p} ((k k)) \end{matrix}] + + [\begin{matrix} - - \frac{Δt Δt}{λ λ {Q Q}_{R R}} \\ \frac{Δt Δt}{{C C}_{p p}} \end{matrix}] I I ((k k)) + + [\begin{matrix} {w w}_{11} ((k k)) \\ {w w}_{22} ((k k)) \end{matrix}] - - - - - - ((1010))

U(k)＝F[SOC(k)]-U _a (k)-U _p (k)-R _o I(k)+v(k) (11)

In the state-space model, I(k) is the input quantity, representing the working current; U(k) is the output quantity, representing the terminal voltage;

[\begin{matrix} w_{1} (k) \\ w_{2} (k) \end{matrix}]

is the system noise; v(k) is the observation noise; Δt is the time difference;

After linearizing F[SOC(t)] in the output equation, A(k), B(k) and C(k) in state space models (10) and (11) are respectively:

A A ((k k)) = = [\begin{matrix} 11 & 00 \\ 00 & 11 - - \frac{Δt Δt}{{R R}_{p p} {C C}_{p p}} \end{matrix}],, B B ((k k)) = = [\begin{matrix} - - \frac{Δt Δt}{λ λ {Q Q}_{R R}} \\ \frac{Δt Δt}{{C C}_{p p}} \end{matrix}],, C C ((k k)) = = {[[\frac{&PartialD; &PartialD; F f ((SOC SOC ((k k))]]}{&PartialD; &PartialD; SOC SOC ((k k))}]]}_{X x ((k k)) = = \overset{^^}{X x} ((k k))}

The EKF algorithm formula is as follows:

\overset{^^}{X x} ((k k | | k k - - 11)) = = A A ((k k - - 11)) \overset{^^}{X x} ((k k - - 11)) + + B B ((k k - - 11)) I I ((k k - - 11)) - - - - - - ((1212))

\overset{^^}{X x} ((k k)) = = \overset{^^}{X x} ((k k | | k k - - 11)) + + K K ((k k)) [[U u ((k k)) - - \overset{^^}{U u} ((k k))]] - - - - - - ((1313))

P(k|k-1)＝A(k-1)P(k-1) ^AT (k-1)+Q (14)

K(k)＝P(k|k-1)C ^T (k)[C(k)P(k|k-1)C ^T (k)+R] ^-1 (15)

P(k)＝[1-K(k)C(k)]P(k|k-1) （16）

The EKF algorithm consists of two parts: filter calculation and filter gain calculation. The filter calculation is completed by formula (11)-(13). Get the predicted value of the state variable Then according to the output equation (11), the predicted value of the system observation is obtained