## 5 Case Studies
### All-Island Irish Transmission System
#### 5.1.3 Sparse Algorithms  
The implementation of the subspace iteration by SLEPc only finds the desired number of LM eigenvalues. However, in the \(s\)-domain, the relevant eigenvalues from the stability point of view are not the LM ones, but the ones with LRP or SM. Especially for the GEP, the LM eigenvalue is infinite and, hence, does not provide any meaningful information on the system dynamics. For this reason and for the needs of power system SSSA, the subspace method and, in general, any method that looks for LM eigenvalues must always be combined with a spectral transform. For the needs of this example, we apply the invert transform and pass to SLEPc the pencil of the dual system, i.e., \(z\mathbf{A}-\mathbf{E}\). Subsequently, the method looks for the 50 LM eigenvalues of the dual system, which correspond to the 50 SM eigenvalues of the prime system. With this setup, the eigenvalues found by the subspace iteration for the GEP are shown in Table 8. As can be seen, the pair \(-0.1322\pm\)/0.4353 is not captured, since its magnitude is larger than the magnitudes of the 50 SM eigenvalues. In order to also obtain this pair, one can customize the spectral transform or simply increase the number of the eigenvalues to be returned. However, the best setup is not known a priori and, thus, some heuristic parameter tuning is required. Finally, the method does not scale well, since the solution of the GEP is completed in 6807.24 s. \begin{table}
\begin{tabular}{l c} \hline
**Library** & **SLEPc** \\ \hline Method & Subspace \\ Spectrum & 50 LM \\ Transform & Invert \\ \hline Time [s] & 6807.24 \\ Found & 50 \\ \hline LRP eigs. & \(-0.0000\) \\  & \(-0.0869\) \\  & \(-0.1276\pm\)/0.1706 \\  & \(-0.1376\) \\  & \(-0.1382\) \\  & \(-0.1386\) \\  & \(-0.1390\) \\  & \(-0.1391\) \\  & \(-0.1393\) \\  & \(-0.1397\) \\ \hline \end{tabular}
\end{table} 
Table 8: AIITS: subspace iteration method, GEP. Figure 3: AIITS: Cayley transform image of the spectrum, \(\sigma=2\). The rightmost eigenvalues found with Krylov subspace methods for the LEP and GEP are shown in Tables 9 and 10, respectively. For the LEP, ARPACK is set up in order to find the 50 LRP eigenvalues. Although all of the eigenvalues shown in Table 9 for ARPACK are actual eigenvalues of the system, some of the LRP ones are missed. Furthermore, no correct eigenvalues were found for the GEP, which we attribute to the fact that a non-symmetric \(\mathbf{E}\) is not supported. In SLEPc methods, both for LEP and GEP and in order to obtain the eigenvalues with good accuracy, we use the option \"Target Real Part\" (TRP), which allows targeting eigenvalues with specified real part. In particular, we set the TRP parameter to \(-0.01\), and apply a shift & invert transform with \(\sigma=-0.01\). ERD-Arnoldi and Krylov-Schur methods are both able to accurately capture all rightmost eigenvalues. For the sake of completeness, we mention that the eigenvalues obtained with SLEPc, when compared to the ones that were found by LAPACK, appeared to be shifted by a constant offset \(-\sigma\), i.e., \(0.01\) was returned instead of \(0\), and so on. The results shown in Tables 9 and 10 take into account such a shift by adding \(\sigma\) to all output values that were returned by SLEPc. Finally, the Krylov subspace methods by SLEPc appear to be more efficient than ARPACK's IR-Arnoldi. When compared to Schur decomposition methods, at this scale, Krylov methods, although they require some tuning, appear to be by far more efficient for the GEP, but less efficient for the LEP. Table 11 presents the results that are produced by z-PARES' CI-RR method for the LEP and GEP. The method is set to look for solutions in the circle with center the point \(\gamma=(-0.01,4)\) and radius \(\rho=8\). In both cases, the eigenvalues found by z-PARES are actual eigenvalues of the system, although the eigenvalues found for the GEP include noticeable errors, when compared to the results that were obtained with LAPACK.