## 4 Case Study  
### Basic Test Analysis  
The IEEE-9 bus system is firstly taken as the test system in this paper. The single-line diagram of it is given in Figure 2. The classical second-order model of generator considering governor is adopted in simulation. The inertia time constants of G1, G2, and G3 are respectively 47.28, 12.8, 6.02 s; and the damping coefficient is set to 2. The initial values of generator state variables and control variables are set to the values of the system's stable operation before fault occurred. Suppose that in the IEEE 9 bus system, three-phase short-circuit fault occurs at the beginning of branch 4-8 at 0.8 s. After 0.36 s, the switch on both sides of branch jumps off and the fault is removed. The whole simulation time is 6 s.  
The real values of power angle and angular velocity of generator are obtained by Bonneville Power Administration (BPA) simulation software, and the measured values are obtained by adding the real values to the random noise where its mean value obeys normal distribution with zero standard deviation. In this paper, the DSE algorithm is implemented in Matlab. The initial values of the estimated generator state variables are set to stable operation value, and the initial state error covariance matrix is set to unit matrix.  
In order to further compare the difference of filtering results of each algorithm, the root mean square error (_RMSE_) is used  
\[RMSE=\sqrt{\frac{1}{n}\sum_{k=1}^{n}\left(\hat{X}_{\mathrm{ijk}}-X_{k}\right)^ {2}} \tag{66}\]  
where \(n\) is the sampling period.  
The process noise variance matrix and the measurement noise variance matrix of the generator are selected as the corresponding values in [10; 11]. Figure 3 is a comparison of the filtering results of G1 by utilizing the EnKF and EnSRF algorithms under normal noise. Table 1 gives the RMSE results of EnKF and EnSRF algorithms.  
Figure 3: _Cont._  
Figure 2: The single-line diagram of IEEE-9 bus test system.  
As it might be noted from Figure 3 and Table 1, EnKF and AEnSRF both show good filtering effect without deviation of measurement noise. Because the observation value in AEnSRF does not introduce disturbance, the problem of underestimation in EnKF filtering is avoided. The filtering-precision of AEnSRF is improved comparing with EnKF.  
Figure 4 is a sketch of the noise estimator tracking the measured noise in AEnSRF without deviation from the measured noise.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Generator Number** & **Algorithm** & **\(\delta\)/rad** & **\(\alpha\)/pu** \\ \hline \multirow{2}{*}{G1} & EnKF & 0.0345 & 9.7396 \(\times 10^{-4}\) \\  & AEnSRF & 0.0329 & 9.1950 \(\times 10^{-4}\) \\ \hline \multirow{2}{*}{G2} & EnKF & 0.0346 & 10.000 \(\times 10^{-4}\) \\  & AEnSRF & 0.0330 & 9.6680 \(\times 10^{-4}\) \\ \hline \multirow{2}{*}{G3} & EnKF & 0.0354 & 11.000 \(\times 10^{-4}\) \\  & AEnSRF & 0.0344 & 10.000 \(\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The RMSE results of EnKF and EnSRF.  
Figure 4: Measurement noise tracking curve of G1 under AEnSRF: (**a**) Power angle measurement noise tracking curve of G1; (**b**) angular velocity measurement noise tracking curve of G1.  
Figure 3: Comparison of G1 state filtering results under ensemble Kalman filter (EnKF) and AEnSRF: (**a**) Power angle dynamic estimation curve of G1; (**b**) electric angular velocity dynamic estimation curve of G1.  
From the simulation, we can see that in the process of tracking power angle measurement noise by AEnSRF, the standard deviation of 0.14 s is stable at 1.9 degrees, and the angular velocity is too small (almost 0). Combining with Figure 4, it can be remarked that AEnSRF can track the measured noise more accurately.