# WAMs Based Eigenvalue Space Model for High Impedance Fault Detection
## 1 Introduction
### Subspace Estimation  
Multiple techniques can be used to estimate the eigenvalues of a system. These techniques include approximate realizations and subspace identification. A linear time invariant approximate realization aims to derive a minimum order model using the impulse response of the system through Markov Parameters as follows. Starting with a State Space model [23]:  
\[x(k+1)=Ax(k)+Bu(k) \tag{2}\]  
\[y(k)=Cx(k)+Du(k)\]  
which has a transfer function  
\[G(z)=\sum_{k=0}^{\infty}g(k)z^{-k} \tag{3}\]  
where \(g(k)\) is a known impulse response for  
\[k=0,1,\ldots,\infty\]  
The state matrices and the transfer function share the following relationship:  
\[G(z)=D+C(zI-A)^{-1}B \tag{4}\]  
The matrices can be computed from the impulse as follows,  
\[g(k)= \begin{cases}D&\text{for }k=0\\ CA^{k-1}B&\text{for }k\geq\ 1\end{cases}  \tag{5}\]  
The Ho-Kalman algorithm is applied to find matrices \(A\), \(B\), and \(C\) through the Hankel matrix \(H\)  
\[H_{nr,nc}= \begin{bmatrix}g(1)&g(2)&g(3)&\cdots&g(n_{c})\\ g(2)&g(3)&g(4)&\cdots&g(n_{c}+1)\\ g(3)&g(4)&g(5)&\cdots&g(n_{c}+2)\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ g(n_{r})&g(n_{r}+1)&g(n_{r}+2)&\cdots&g(n_{r}+n_{c}-1)\end{bmatrix}  \tag{6}\]  
The Hankel matrix can be represented in the following form:  
\[H_{nr,nc}= \begin{bmatrix}C\\ CA\\ \vdots\\ CA^{n_{r}-1}\end{bmatrix}\begin{bmatrix}B&AB&\cdots&A^{n_{c}-1}B\end{bmatrix}  \tag{7}\]  
The rank \(n\) of the Hankel matrix is equal to the rank of matrix A. Singular Value decomposition is used to determine the rank of the Hankel matrix as follows:  
\[H=U_{n}\Sigma_{n}V_{n}^{T} \tag{8}\]  
where \(U_{n}\) and \(V_{n}\) are unitary matrices  
\[U_{n}^{T}U_{n}=V_{n}^{T}V_{n}=I_{n} \tag{9}\]\(\Sigma_{n}\) is a diagonal matrix with positive singular values on the diagonal. The number of non-zero singular values across the diagonal of the matrix is equivalent to the rank of the Hankel matrix, which is also, as discussed previously, the rank of \(A\). The Hankel matrix can then be expressed as  
\[H=U_{n}\Sigma_{n}^{1/2}\,\Sigma_{n}^{1/2}V_{n}^{T} \tag{10}\]  
where \(\Sigma_{n}^{1/2}\) represents a diagonal matrix composed of the square roots of the singular values. Matrix \(B\) is the first column of \(\Sigma_{n}^{1/2}V_{n}^{T}\), while \(C\) is the first row of \(U_{n}\Sigma_{n}^{1/2}\). Finally, \(A\) can be defined as  
\[A=\Sigma_{n}^{-1/2}U_{n}^{T}\overleftrightarrow{H}\ V_{n}\Sigma_{n}^{-1/2} \tag{11}\]  
where \(\overleftrightarrow{H}\) represents the shifted Hankel matrix  
\[\overleftrightarrow{H}_{nr,nc}= \begin{bmatrix}g(1)&g(2)&g(3)&\cdots&g(n_{c}) \\ g(2)&g(3)&g(4)&\cdots&g(n_{c}+1)\\ g(3)&g(4)&g(5)&\cdots&g(n_{c}+2)\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ g(n_{r})&g(n_{r}+1)&g(n_{r}+2)&\cdots&g(n_{r}+n_{c}-1)\end{bmatrix}  \tag{12}\]  
A limitation of the Approximate Realization approach is that system parameters are estimated from the response of the system to an impulse. In real-life, at least in the context of power systems, this is not practical. As a result, alternative methods that use information from the system's inputs and outputs to achieve similar results have been developed. In subspace identification the State Space model of the system can be expressed in the innovation form as [23]:  
\[x_{k+1}=Ax_{k}+Bu_{k}+Ke_{k} \tag{13}\]  
\[y_{k}=Cx_{k}+Du_{k}+e_{k}\]  
where \(u_{k}\) represents the input vector, and is \(y_{k}\) the output. \(e_{k}\) represents a source of noise and \(K\) contains parameters of the noise model. The output can be defined as  
\[y_{k}=G(z)u_{k}+H(z)e_{k} \tag{14}\]  
where  
\[G(z)=C(zI-A)^{-1}B+D \tag{15}\]  
and  
\[H(z)=C(zI-A)^{-1}K+I \tag{16}\]  
A key part of this estimation procedure is the derivation of the Observability matrix:  
\[O= \begin{bmatrix}C\\ CA\\ \vdots\\ CA^{r-1}\end{bmatrix}  \tag{17}\]  
Once again Singular Value Decomposition can be used to find the rank of the Observability matrix, which is also the rank of the \(A\) matrix.