# Synthetical Modal Parameters Identification Method of Damped Oscillation Signals in Power System
## 2 Synthetical Modal Parameters Identification (SMPI) Method
### Prony Algorithm  
The Prony algorithm is a mathematical model in which equally spaced sampled data is represented by a linear combination of complex exponential functions [24]. The LFO mode can also be represented by a complex exponential function, so Prony can be applied to LFO. The Prony algorithm is fast and reliable, which does not depend on the mathematical model of the system [25; 26].  
Given a complex-valued data sequence \(x(n)\), \(n=0\), \(1\),..., \(N-1\), based on \(N\) original data, the traditional Prony method fits an exponential model to the data in the least-squares sense, and estimates \(\hat{x}(n)\)[27; 28; 29], as  
\[\hat{x}(n)=\sum_{i=1}^{p}A_{i}e^{j\theta_{i}}e^{(\alpha_{i}+j2\pi f_{i}) \Delta t}=\sum_{i=1}^{p}b_{i}z_{i}^{n}\quad n=0,1,\cdots,N-1 \tag{17}\]  
where \(p\) is equal to the order of system; \(j\) is the imaginary unit; the subscript \(i\) denotes the \(i\)-th mode; \(A_{i},f_{i}\), \(\theta_{i}\) and \(\alpha_{i}\) signify amplitude, frequency, initial phase, and damping factor, respectively; \(b_{i}\) and \(z_{i}\) are the \(i\)-th residue and polynomial root, correspondingly; and \(\Delta t\) is the time interval.  
Considering the linear prediction model is  
\[\left[\begin{array}{c}\mathbf{x}_{(p)}\\ \mathbf{x}_{(p+1)}\\ \vdots\\ \mathbf{x}_{(N-2)}\end{array}\right]=\left[\begin{array}{cccc}\mathbf{x}_{(p-1)}& \mathbf{x}_{(p-2)}&\cdots&\mathbf{x}_{(0)}\\ \mathbf{x}_{(p)}&\mathbf{x}_{(p-1)}&\cdots&\mathbf{x}_{(1)}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{x}_{(N-2)}&\mathbf{x}_{(N-3)}&\cdots&\mathbf{x}_{(N-p-1)}\end{array}\right] \left[\begin{array}{c}a_{1}\\ a_{2}\\ \vdots\\ a_{p}\end{array}\right] \tag{18}\]  
which can be denoted as \(\mathbf{d}=\mathbf{D}\mathbf{a}\), where \(a\) is the vector of coefficients, and \(p\) is the polynomial order assumed to be known a priori.  
Estimation of \(\mathbf{a}\) is obtained by solving Equation (18) in the least-squares sense as  
\[\hat{a}=\left(D^{\mathrm{H}}D\right)^{-1}\!\!D^{\mathrm{H}}\mathbf{d} \tag{19}\]  
where \(D^{\mathrm{H}}\) is the complex-conjugate transpose of \(D\).  
Using the previously estimated coefficients, the roots of the characteristic polynomial expressed as  
\[z^{p}-\left(a_{1}z^{p-1}+a_{2}z^{p-2}+\ldots+a_{p}z^{0}\right)=0 \tag{20}\]  
where each root \(z^{i}\), \(i=1,\ldots,p\), is a discrete-time approximation of its respective continuous-time eigenvalue in the Z-domain.  
The \(i\)-th mode frequency is calculated as  
\[f_{p(i)}=\arctan[\mathrm{Im}(z_{i})/\mathrm{Re}(z_{i})]/(2\pi\Delta t)\quad i =1,\ldots p \tag{21}\]Using the polynomial roots, \(z_{i}\), the linear regression model given by  
\[\left[\begin{array}{c}x_{(0)}\\ x_{(1)}\\ \vdots\\ x_{(N-1)}\end{array}\right]=\left[\begin{array}{cccc}1&1&\cdots&1\\ z_{1}^{1}&z_{2}^{1}&\cdots&z_{p}^{1}\\ \vdots&\vdots&\ddots&\vdots\\ z_{1}^{N-1}&z_{2}^{N-1}&\cdots&z_{p}^{N-1}\end{array}\right]\left[\begin{array}[ ]{c}b_{1}\\ b_{2}\\ \vdots\\ b_{p}\end{array}\right] \tag{22}\]  
which can be denoted as \(\boldsymbol{x}=\boldsymbol{V}\boldsymbol{b}\), where \(\boldsymbol{V}\)_is_ a Vandermonde matrix.  
Using the least square method, vector \(\boldsymbol{b}\) is estimated as  
\[\boldsymbol{\overset{\wedge}{b}}=\left(\boldsymbol{V}^{\mathrm{H}}\boldsymbol{V }\right)^{-1}\boldsymbol{V}^{\mathrm{H}}\boldsymbol{x} \tag{23}\]  
The \(i\)-th mode frequency, amplitude, and initial phase are, respectively, calculated as  
\[\left\{\begin{array}{l}A_{\boldsymbol{p}(i)}=|b_{i}|\\ \boldsymbol{\theta}_{\boldsymbol{p}(i)}=\arctan[\mathrm{Im}(b_{i})/\mathrm{Re }(b_{i})]\\ f_{b(i)}=\arctan[\mathrm{Im}(\boldsymbol{z}_{i})/\mathrm{Re}(\boldsymbol{z}_{i} )]/(2\beta\mathrm{\Delta t})\end{array}\right.\quad i=1,\ldots p \tag{24}\]