## 3 Identification of SSO Parameters of the Fitting-Based ERA
### The Basic Idea of the Fitting-Based ERA
#### 3.3.2 Solution of the SSO Parameters  
Based on Section 3.3.1, the angular frequencies \(\omega_{0}\) and \(\omega_{\mathrm{s}}\) of the fundamental component and oscillation component can be obtained. Subsequently, the comprehensive methodology for deriving the frequencies, magnitudes and phases of individual components is outlined as follows.  
First, the four constants \(R_{1}\), \(R_{2}\), \(R_{3}\) and \(R_{4}\) are determined or resolved in accordance with Equation (21):  
\[\begin{bmatrix}R_{1}\\ R_{2}\\ R_{3}\\ R_{4}\end{bmatrix}=\begin{bmatrix}1&1&1&1\\ e^{\mathrm{i}\omega_{0}}&e^{-\mathrm{i}\omega_{0}}&e^{\mathrm{i}\omega_{\mathrm{ s}}}&e^{-\mathrm{i}\omega_{\mathrm{s}}}\\ \vdots&\vdots&\vdots&\vdots\\ e^{\mathrm{i}\omega_{0}(r-1)}&e^{-\mathrm{i}\omega_{0}(r-1)}&e^{\mathrm{i} \omega_{\mathrm{s}}(r-1)}&e^{-\mathrm{i}\omega_{\mathrm{s}}(r-1)}\end{bmatrix}^ {-1}\begin{bmatrix}\dot{X}(0)\\ \dot{X}(1)\\ \vdots\\ X(r-1)\end{bmatrix} \tag{21}\]  
Second, \(\dot{X}_{0}^{+}(k)\), \(\dot{X}_{0}^{-}(k)\), \(\dot{X}_{\mathrm{s}}^{+}(k)\) and \(\dot{X}_{\mathrm{s}}^{-}(k)\) are solved for using Equations (8) and (10) by utilizing \(\omega_{0}\), \(\omega_{\mathrm{s}}\) and \(R_{m}\):  
\[\begin{cases}\dot{X}_{0}^{+}(k)=R_{1}e^{\mathrm{i}\omega_{0}},&\dot{X}_{0}^{-} (k)=R_{2}e^{-\mathrm{i}\omega_{0}}\\ \dot{X}_{\mathrm{s}}^{+}(k)=R_{3}e^{\mathrm{i}\omega_{\mathrm{s}}},&\dot{X}_{ \mathrm{s}}^{-}(k)=R_{4}e^{-\mathrm{i}\omega_{\mathrm{s}}}\end{cases} \tag{22}\]  
Subsequently, by utilizing Equations (4) and (8) in conjunction, the fundamental component's frequency, amplitude and initial phase are ascertained, as demonstrated in Equation (23):  
\[f_{0}=\begin{cases}f_{N}-\frac{\omega_{0}}{2\pi}f_{\mathrm{s}}\,||\dot{X}_{0 }^{+}(k)||_{2}<||\dot{X}_{0}^{-}(k)||_{2}&\\ f_{N}+\frac{\omega_{0}}{2\pi}f_{\mathrm{s}}\,||\dot{X}_{0}^{+}(k)||_{2}>||\dot{X }_{0}^{-}(k)||_{2}&\\ \end{cases};\quad\begin{cases}G(f_{0},-1)x_{0}e^{-\mathrm{i}\phi_{0}}=\dot{X}_{0 }^{-}(k),||\dot{X}_{0}^{+}(k)||_{2}<||\dot{X}_{0}^{-}(k)||_{2}\\ G^{*}(f_{0},+1)x_{0}e^{-\mathrm{i}\phi_{0}}=\dot{X}_{0}^{+}(k),||\dot{X}_{0}^{+} (k)||_{2}\geq||\dot{X}_{0}^{-}(k)||_{2}\end{cases} \tag{23}\]  
Finally, by integrating Equations (4) and (8) and \(f_{\mathrm{sub}}+f_{\mathrm{sup}}=2f_{N}\), the frequencies, magnitudes and phases corresponding to both subsynchronous and supersynchronous oscillations are subsequently determined:  
\[\begin{cases}f_{\mathrm{sub}}=f_{N}-\frac{\omega_{\mathrm{s}}}{2\pi}f_{ \mathrm{s}}&\begin{bmatrix}G^{*}(f_{\mathrm{sub}},+1)&G(f_{\mathrm{sup}},-1)\\ G^{*}(f_{\mathrm{sub}},-1)&G(f_{\mathrm{sup}},+1)\end{bmatrix}\begin{bmatrix}x_{ \mathrm{sub}}e^{-\mathrm{i}\beta_{\mathrm{sub}}}\\ x_{\mathrm{sup}}e^{\mathrm{i}\beta_{\mathrm{sup}}}\end{bmatrix}=\begin{bmatrix} \dot{X}_{\mathrm{s}}^{+}(k)\\ \dot{X}_{\mathrm{s}}^{-*}(k)\end{bmatrix} \tag{24}\]  
Up to this point, the comprehensive calculations pertaining to the determination of the frequencies, magnitudes and initial phases of the fundamental, subsynchronous and super synchronous constituents of the synchrophasor have been thoroughly executed, marking the successful conclusion of the modal parameter characterization for the synchrophasor.