# Synthetical Modal Parameters Identification Method of Damped Oscillation Signals in Power System
## 2 Synthetical Modal Parameters Identification (SMPI) Method
### Stochastic Subspace Identification (SSI) Approach  
The SSI is an identification method based on discrete state space equations of linear systems, which calculates partial modal parameters [14]. The SSI is split into covariance-driven SSI and data-driven SSI [15]. The covariance-driven SSI uses the covariance as a statistic to describe the data correlation [16], while the data-driven SSI projects the line space of the \"future\" onto the line space of the \"past\" to describe the data correlation [17; 18]. In this paper, we utilize the covariance-driven SSI for its strength in noise resistance and data processing accuracy [19], which is based on a discrete-time state space model, takes the covariance as the statistic, and uses the singular value decomposition (SVD) to obtain the state matrix and its eigenvalues, and, finally, gains the modal parameters of the system [20]. The subspace involved in this method refers to the state space model of a multi-degree-of-freedom system [19; 21]. The state model of SSI is [22]  
\[\left\{\begin{array}{l}\mathbf{x}_{s(k+1)}=\mathbf{A}\mathbf{x}_{s(k)}+\mathbf{w}_{k}\\ \mathbf{y}_{k}=\mathbf{C}\mathbf{x}_{s(k)}+\mathbf{v}_{k}\end{array}\right. \tag{5}\]  
where \(k\) is the time instant, i.e., the sampling point number of the discrete signal; vector \(\mathbf{x}_{s}\) is the state vector of the discrete-time system; vector \(\mathbf{y}_{k}\) is the system output vector at time instant \(k\); matrix \(\mathbf{A}\) is the state matrix of the discrete-time state space equation; matrix \(\mathbf{C}\) is the system output matrix, also called the observation matrix; and \(\mathbf{w}_{k}\) is the process noise and \(\mathbf{v}_{k}\) is the measurement noise at time instant \(k\) with zero mean.  
The measured data \(\mathbf{y}_{k}\), i.e., the system output vector, are constructed into the Hankel matrix, as shown in the following equation [14]:  
\[\mathbf{H}=\mathbf{Y}_{0/2M-1}=\frac{1}{\sqrt{N}}\left(\begin{array}{cccc}y_{0}&y_{ 1}&\cdots&y_{N-1}\\ y_{1}&y_{2}&\cdots&y_{N}\\ \cdots&\cdots&\cdots&\cdots\\ y_{M-1}&y_{M}&\cdots&y_{M+N-2}\\ y_{M}&y_{M+1}&\cdots&y_{M+N-1}\\ y_{M+1}&y_{M+2}&\cdots&y_{M+N}\\ \cdots&\cdots&\cdots&\cdots\\ y_{2M-1}&y_{2M}&\cdots&y_{2M+N-2}\end{array}\right)=\frac{\mathbf{Y}_{0/M-where the number of row blocks is \(2M\); the number of columns is \(N\) which is the number of measured data as well; and \(Y_{p}=Y_{0/M-1}\) and \(Y_{f}=Y_{M/2M-1}\) denote the past and future parts, respectively, of the block Hankel matrix. Since the number of rows in each row block \(y_{k}\) is \(l\) (the number of measured points), and the number of columns is \(1\), the matrix \(Y_{0/2M-1}\) consists of \(2M\times l\) rows and \(N\) columns.  
The amount of measured data is limited, i.e., \(N\) is not infinite in actual measurement conditions. According to the Hankel matrix obtained above, the covariance of the output vector can be constructed as [19]  
\[R_{M}=\frac{1}{N}\sum_{k=0}^{N-1}y_{k+My}y_{k}^{T} \tag{7}\]  
The Hankel matrix in Equation (6) can be converted to the Toeplitz matrix as [23].  
\[\mathbf{T}_{1/M}=\mathbf{Y}_{f}\mathbf{Y}_{p}^{T}=\left(\begin{array}{cccc}R_{M}&R_{M-1}& \cdots&R_{1}\\ R_{M+1}&R_{M}&\cdots&R_{2}\\ \cdots&\cdots&\cdots&\cdots\\ R_{2M-1}&R_{  
The eigenvalues of the system \(\lambda_{m}\) can be calculated from the eigenvalues of the state matrix \(A\):  
\[\lambda_{m}{=}\frac{\ln\!\mu_{m}}{\Delta t} \tag{15}\]  
where \(\Delta t\) is the time interval.  
According to the above principle, the modal frequency \(f_{s}\) and damping ratio \(\xi_{s}\) corresponding to each eigenvalue can be calculated by  
\[\left\{\begin{array}{l}f_{s}{=}\frac{\sqrt{|\lambda_{m}|}}{2\pi_{s}}\\ \xi_{s}{=}-\frac{\lambda_{m}+\lambda_{m}^{*}}{2\sqrt{\lambda_{m}\lambda_{m}^{* }}}\end{array}\right. \tag{16}\]