## 5 Case Studies
### European Network of Transmission System Operators for Electricity  
This example presents the simulation results for a dynamic model of the ENTSO-E. The system includes 21,177 buses (1212 off-line); 30,968 transmission lines and transformers (2352 off-line); 1144 zero-impedance connections (420 off-line); 4828 power plants represented by 6-th order and 2-nd order synchronous machine models; and, 15,756 loads (364 off-line), modeled as constant active and reactive power consumption. Synchronous machines that are represented by 6-th order models are also equipped with dynamic AVR and TG models. The system also includes 364 PSSs.  
The system has, in total, \(n=49,396\) state variables and \(m=\) 96,770 algebraic variables, as summarized in Table 12. The pencil \(s\mathbf{E}-\mathbf{A}\) has dimensions 146,166 \(\times\) 146,166 and the matrix \(\mathbf{A}\) has 654,950 non-zero elements, which represent the 0.003% of the total number of elements of the matrix.  
Neither the LEP nor GEP could be solved while using Schur decomposition methods. At this scale, the dense matrix representation that is required by LAPACK and MAGMA libraries leads to massive memory requirements, and a segmentation fault error is returned by the CPU. Among the algorithms that support sparse matrices, here we only test the contour-integration based methods, which, in fact, were the ones that were able to tackle this large eigenvalue problem on the available hardware.  
The effect of changing the search region of the z-PARES CI-RR method on the eigenvalue analysis of the ENTSO-E system is shown in Table 13. Interestingly, simulations showed that shrinking the defined contour may lead to a marginal increase of the computation time. Although not intuitive, this result indicates that the large size of the ENTSO-E system mainly determines the mass of the computational burden, and that, at this scale, smaller subspaces are not necessarily constructed faster by the CI-RR algorithm. Regarding the number of eigenvalues obtained, using a region that is too small leads, as expected, to missing an important number of critical eigenvalues.  
We test the impact of applying spectral transforms to the matrix pencil \(s\mathbf{E}-\mathbf{A}\), on the eigenvalue analysis of the ENTSO-E with z-PARES' CI-RR. In particular, we test the invert transform that yields the dual pencil \(z\mathbf{A}-\mathbf{E}\); and, the inverted Cayley transform, i.e., \(s=(z+1)/(\sigma z-\sigma)\), which yields the pencil \(z(\mathbf{E}-\sigma\mathbf{A})-(-\sigma\mathbf{A}-\mathbf{E})\). Table 14 shows the results. Passing the transformed matrices to z-PARES provides a marginal speedup to the eigenvalue computation. In addition, when considering either the prime system or the inverted Cayley transform with \(\sigma=-1\), results in finding the same number of eigenvalues, whereas, when the dual system is considered, a number of eigenvalues is missed.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Library** & \multicolumn{2}{c}{**z-PARES**} \\ \hline Problem & \multicolumn{2}{c}{GEP} \\ Method & \multicolumn{2}{c}{CI-RR} \\ \hline \(c\) & \((-0.01,4)\) & \((-0.01,3)\) & \((-0.01,3)\) \\ \(\rho\) & 8 & 4 & 2 \\ \hline Time [s] & 364.85 & 375.67 & 378.71 \\ Found & 349 eigs. & 350 eigs. & 110 eigs. \\ \hline \hline \end{tabular}
\end{table}
Table 13: ENTSO-E: impact of the search region of the CI-RR method.  
\begin{table}
\begin{tabular}{c c} \hline \hline \(n\) & 49,396 \\ \(m\) & 96,770 \\ Dimensions of \(\mathbf{A}\) & 146,166 \(\times\) 146,166 \\ Sparsity degree of \(\mathbf{A}\) [\%] & 99.997 \\ \hline \hline \end{tabular}
\end{table}
Table 12: ENTSO-E: statistics.