## 2 Synchrophasor Sequence Modal Models and Extraction of Modal Parameters  
### Synchrophasor Sequence Modal Models with Sub- or Super-Synchronous Components  
Firstly, the instantaneous signal model of the current or voltage of the power system is established, and it is assumed that the model remains fixed in the selected data window, which is denoted as \(x(t)\) and expressed as shown in Equation (1):  
\[x(t)=x_{0}\text{cos}(2\pi f_{0}t+\phi_{0})+x_{\text{sub}}\text{cos}(2\pi f_{ \text{sub}}+\phi_{\text{sub}})+x_{\text{sup}}\text{cos}(2\pi f_{\text{sup}}+ \phi_{\text{sup}}) \tag{1}\]  
where \(f\), \(x\) and \(\phi\) represent the frequency, amplitude and initial phase, respectively. The subscripts \"\(0,\text{sub},\text{sup}\)\" stand for the fundamental component, subsynchronous component and supersynchronous component, respectively. According to the standard IEEE std C37.118 [23], for the DFT, Equation (1) is used to obtain the synchrophasor sequence \(\dot{X}(k)\) corresponding to the instantaneous signal \(x(t)\). This expression is shown in Equation (2):  
\[\dot{X}(k)=\frac{2}{N}(e^{\text{j}k\tau})\sum_{n=0}^{N-1}x(t)e^{-\text{j}\frac {2\pi n}{N}}=\dot{X}_{\text{sup}}(k)+\dot{X}_{0}(k)+\dot{X}_{\text{sub}}(k) \tag{2}\]  
where \(N\) is the number of data points when calculating the synchrophasors according to the instantaneous value. [PERSON]'s formula is applied to Equation (2) to divide the synchrophasors corresponding to different components into positive frequency components and negative frequency components. Taking the supersynchronous component as an example, when the current sampling time is \(\frac{\text{it}}{N/N_{\text{s}}}+\frac{k}{f_{s}}\), its decomposition process is as follows:  
\[\dot{X}_{\text{sup}}(k) = \frac{2}{N}(e^{\text{j}k\tau})\sum_{n=0}^{N-1}x_{\text{sup}} \text{cos}(2\pi f_{\text{sup}}(\frac{n}{Nf_{\text{N}}}+\frac{k}{f_{s}})+\phi_ {\text{sup}})\text{e}^{-\text{j}\frac{2\pi}{N}n} \tag{3}\] \[= G(f_{\text{sup}},-1)x_{\text{sup}}e^{\text{j}\phi_{\text{sup}}} e^{\text{j}\frac{2\pi(\text{sup}-f_{\text{s}})k}{f_{s}}}+G^{*}f_{\text{sup}},+1)x_{ \text{sup}}\text{e}^{-\text{j}\phi_{\text{sup}}}e^{-\text{j}\frac{2\pi(f_{N} -\text{sup})k}{f_{s}}}\]  
In this formula, the superscript \"\(\ast\ast\ast\)\" indicates conjugation, \(f_{s}\) is the uploading frequency of the synchronous phasor data, and \(f_{s}=2f_{N}\). The function \(G(f,m)\) is introduced to represent the calculation process of synchrophasors, \(f\) can be the frequency of each component, and \(m\) is \(\pm 1\). Its expression is  
\[G(f,m)=\frac{1}{N}\sum_{n=0}^{N-1}e^{(\text{j}\frac{2\pi\tau}{N}+\text{j}\frac {2\pi}{N}m)n} \tag{4}\]  
The fundamental, subsynchronous and supersynchronous components all encompass frequency components which are both positive and negative in nature. Therefore, the expression of each component of the synchrophasor is as follows:  
\[ \begin{cases}\dot{X}_{0}^{+}(k)=G^{*}(f_{0},+1)x_{0}e^{-\text{j}\eta_{0}}e^{ \text{j}\omega_{0}k}\\ \dot{X}_{0}^{-}(k)=G(f_{0},-1)x_{0}e^{\text{j}\eta_{0}}e^{-\text{j}\omega_{0}k }\\ \end{cases} ; \begin{cases}\dot{X}_{\text{sub}}^{+}(k)=G^{*}(f_{\text{sub}},+1)x_{ \text{sub}}e^{-\text{j}\eta_{0}}e^{\text{j}\omega_{0}k}\\ \dot{X}_{\text{sub}}^{-}(k)=G(f_{\text{sub}},-1)x_{\text{sub}}e^{\text{j}\eta_{ 0}}e^{-\text{j}\omega_{0}k}\\ \end{cases} ; \begin{cases}\dot{X}_{\text{sup}}^{+}(k)=G(f_{\text{sup}},-1)x_{ \text{sup}}e^{\text{j}\omega_{0}k}\\ \dot{X}_{\text{sup}}^{-}(k)=G^{*}(f_{\text{sup}},+1)x_{\text{sup}}e^{-\text{j} \omega_{0}k}\\ \end{cases}  \tag{5}\]  
In this formula, the superscripts \"\(+\)\" and \"\(-\)\" indicate the positive frequency and negative frequency components of each component, respectively.