# Online SoC Estimation of Lithium-Ion Batteries Using a New Sigma Points Kalman Filter
## 2 Modeling and Parameter Identification
### Parameter Identification  
According to the dynamic stress condition (DST) test rules mentioned in the \"USABC Electric Vehicle Battery Test Manual\", the battery sample was charged and discharged. Based on the charge and discharge current and voltage data under the DST condition, the unknown parameters (\(R_{0}\), \(R_{1}\), \(R_{2}\), \(C_{1}\), \(C_{2}\)) in the state equation were identified using the recursive least square method. The parameter estimation matrix (Equation (9)), observation matrix (Equation (10)) and transfer function (Equation (12)) are as follows:  
\[\theta(k)=\left[a_{1},a_{2},a_{3},a_{4},a_{5}\right]^{T} \tag{9}\]  
\[\varphi(k)=\left[\Delta V(k-1),\Delta V(k-2),I(k),I(k-1),I(k-2)\right]^{T} \tag{10}\]  
where the differential voltage is given by  
\[\Delta V(k)=U_{OCV}(k)-U_{L}(k) \tag{11}\]  
\[G(s)=\frac{U_{OCV}(s)-U_{L}(s)}{I_{L}(s)}=\frac{R_{0}s^{2}+\frac{1}{\tau_{1} \tau_{2}}(R_{0}\tau_{1}+R_{0}\tau_{2}+R_{2}\tau_{1}+R_{1}\tau_{2})s+\frac{R_{ 0}+R_{1}+R_{2}}{\tau_{1}\tau_{2}}}{s^{2}+\frac{(\tau_{1}+\tau_{2})}{\tau_{1} \tau_{2}}} \tag{12}\]  
The parameter identification steps of the recursive least squares method with the forgetting factor are as follows:  
\[K(k)=\frac{P(k-1)\varphi(k)}{\varphi(k)^{T}P(k-1)\varphi(k)+\lambda} \tag{13}\]  
\[\varepsilon(k)=y(k)-\theta(k)^{T}\cdot\varphi(k) \tag{14}\]  
\[P(k)=\lambda^{-1}(P(k-1)-K(k)\varphi(k)^{T}P(k-1)) \tag{15}\]  
\[\theta(k)=\theta(k-1)+K(k)\varepsilon(k) \tag{16}\]  
where, \(K(k)\) is the RLS gain matrix, \(\varepsilon(k)\) is the output estimation error matrix, \(P(k)\) is the covariance matrix, \(\varphi(k)\) is the observation matrix, \(\theta(k)\) is the parameter estimation matrix and \(\lambda\) is the forgetting factor.  
The functional relationship between the five parameters (\(R_{0}\), \(R_{1}\), \(R_{2}\), \(C_{1}\), \(C_{2}\)) and \(\theta(k)=[a_{1},a_{2},a_{3},a_{4},a_{5}]^{T}\) is derived as follows.  
\[R_{0}=\frac{a_{3}-a_{4}+a_{5}}{1+a_{1}-a_{2}} \tag{17}\]  
\[\tau_{1}\times\tau_{2}=\frac{T^{2}\times(1+a_{1}-a_{2})}{4\times(1-a_{1}-a_{2 })} \tag{18}\]  
\[\tau_{1}+\tau_{2}=\frac{T\times(1+a_{2})}{1-a_{1}-a_{2}} \tag{19}\]  
\[R_{0}+R_{1}+R_{2}=\frac{a_{3}+a_{4}+a_{5}}{1-a_{1}-a_{2}} \tag{20}\]  
\[R_{0}\tau_{1}+R_{0}\tau_{2}+R_{2}\tau_{1}+R_{1}\tau_{2}=\frac{T\times(a_{3}-a _{5})}{1-a_{1}-a_{2}} \tag{21}\]  
\(T\) is the sampling time.  
HPPC is a pulse constant current discharge condition. The HPPC condition is used to identify the parameters of the equivalent circuit model in the literature. The real operating conditions of EVs are much more complicated than HPPC, and the parameters identified by the HPPC operating condition may not be able to estimate the battery status under complex operating conditions well. In this paper, the DST operating condition is selected as the model parameter identification of dynamic battery behavior. A complete DST operating condition cycle is 360 s, as shown in Figure 4.  
First, the discrete-time state-space representation (Equation (7)) and measurement equation (Equation (8)) were established. Then, based on the SoC-OCV mapping relationship curve, the equation parameter identification was carried out. The parameter identification results are shown in Table 4.