# A New Constrained State Estimation Method Based on Unscented Hoo Filtering
## 3 Simulation Results
### State of Charge Estimation for LITHIUM-Ion Batteries  
The proposed methods are applied to the SOC estimation for a lithium battery. The models are listed as follows [25]:  
State model:  
\[x_{k+1}=f(x_{k},i_{k})=x_{k}-(\frac{\eta i_{k}\Delta t}{Q_{n}}) \tag{48}\]  
Measurement model:  
\[y_{k}=h(x_{k},i_{k})=K_{0}-Ri_{k}-\frac{K_{1}}{x_{k}}-K_{2}x_{k}+K_{3}\ln x_{k}+ K_{4}\ln(1-x_{k}) \tag{49}\]  
where \(x_{k}\) is the SOC at iteration \(k\). When \(x_{k}=1\) represents the battery being fully charged, and \(x_{k}=0\) represents the battery being fully discharged, the state variable \(x_{k}\) should be limited between 0 and 1. \(i_{k}\) is the battery current. \(\Delta t\) is the sampling interval and usually equals one. \(\eta\) is the proportion coefficient, which describes the significant influence of the discharging rate to the SOC. \(Q_{n}\) is the nominal capacity of the battery. \(y_{k}\) is the measured value of the battery terminal voltage. \(R\) is the battery internal resistance, \(P=[K_{0},R,K_{1},K_{2},K_{3},K_{4}]^{T}\) represents the model parameters which can be determined by the least square method [7; 26]. Once the model parameters are obtained, they are fixed and used for all batteries of this type. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Mean Square Error (P\({}_{\mathbf{A}}\))** & **Mean Square Error (P\({}_{\mathbf{B}}\))** \\ \hline unconstrained UHF & 0.5348 & 0.5326 \\ \hline constrained UHF & 0.1922 & 0.1976 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Estimated mean square errors of the phases in the reversible reaction example for the unconstrained UHF and constrained UHF. Figure 3: State estimation results for the reversible reaction example under the unconstrained UHF and the constrained UHF: (**a**) the unconstrained UHF; (**b**) the constrained UHF. The proposed algorithm is tested on a Lithium battery with a nominal capacity of 15 Ah and a nominal voltage of 3.2 V, manufactured by Wanxiang Group. The model parameters in Equation (49) were obtained as in Table 2.  
In order to simulate the real working states of the battery, two driving cycles, Highway Fuel Economy Driving Schedule (HWFET) and United States Advanced Battery Consortium (USABC), are applied to the Lithium battery. The test temperature is \(25\,^{\circ}\mathrm{C}\) and the sampling time is \(1\,\mathrm{s}\). The battery current measured in the HWFET is shown in Figure 4a, and the battery current measured in the USABC is shown in Figure 4b. The proposed method is applied to the SOC estimation in the HWFET and USABC. The initial state covariance is set to \(0.01\). The process noise covariance is set to \(0.01\) and the measurement noise covariance is set to \(1.0\times 10^{-6}\). The initial value of the battery SOC is set to \(0.8\). The initial value of the real SOC is \(1\). In the SOC estimation example, the generated sigma points which are greater than \(1\) will be projected onto the constraint boundary. The SOC estimation results and errors of different methods under the same conditions are shown in Figures 5 and 6, respectively. Tables 3 and 4 show the data of the mean square errors (MSE), the maximum errors, and the convergence time of different methods. The following conclusions can be obtained from the results: (1) the constrained UHF has better SOC estimation accuracy than the unconstrained UHF and the UKF; (2) the SOC estimation error of the constrained UHF distributes in a small region near zero, which shows the excellent stability of the proposed algorithm; (3) the constrained UHF has a faster convergence rate. By comparing the state estimation errors in Tables 3 and 4, we may also conclude that the factor most influencing the performance of the constrained and unconstrained state estimation methods is the system dynamics.