## 4 Evaluation via Use Cases
### Refactorisation of Large-Scale Matrices  
In the following use case, for a thorough analysis of the refactorisation performance, we extended the scope of our study by applying the proposed NICSLU method to matrices with additional characteristics. We considered matrices that, on the one hand, are of very  
Figure 6: Refactorisation and solution computation time per simulation step. Figure 7: Matrix patterns with varying elements. (**a**) Original input matrix **A** with the numerically changing entries marked in red; (**b**) decomposition matrix **LU** with changing entries between subsequent factorisations marked in red. large size and, on the other hand, exhibit a variety of sparsity patterns that correspond to real networks. Furthermore, we investigated the impact of the number of numerically varying matrix entries on the performance of the partial refactorisation approach. The analyses indicate the performance improvements that can be achieved in simulations of large real networks and take the impact of the number of nonlinear component models into account. We chose a set of 10 sample matrices with the characteristics shown in Table 1. The analysis involves the consideration of large-scale matrices corresponding to networks with up to 7500 nodes and matrix sizes of up to 220,000. The matrices belong to real networks of the French transmission system. They were extracted from the offline simulation tool suite Dyna\(\omega\)o [6; 33], which builds the mathematical problem for time-domain simulation in the form of a DAE from C++ and Modelica models and is capable of conducting steady-state, short-circuit and a variety of network stability studies. Even though the underlying mathematical problem formulation of the matrices in this case differs from the MNA based formulation in the real-time simulation described in Section 2.1, the matrices can serve as a basis for a systematic analysis of the performance of NICSLU and the implemented partial refactorisation approach, because they still exhibit sparsity patterns as they can be found in power system simulations of real networks. In the analysis carried out, the matrices were not refactorised during actual simulations. Instead, the matrices were extracted from the offline simulation tool and then refactorised after systematically randomised numerical variations were introduced. In particular, this enables the systematic analysis of the impact of different ratios of varying matrix entries. In addition, in this analysis, we compared the performance of NICSLU with that of KLU, since previous studies have shown that the latter performs well, as outlined in Section 1.2. To compare KLU, NICSLU and PartialNICSLU, a measurement environment was set up independently from Eigen. For each solver, a driver (i.e., a program with a main routine utilising the solver) that measures the elapsed time of refactorisation was implemented. To impose a certain ratio of varying matrix entries, we randomly varied a subset of rows after the factorisation of those entries. The size of the varied subset is an input parameter set by the user and is indicated here by percentage values. We analysed the performance of the refactorisation procedures of KLU and NICSLU, as described in Section 2.2, as well as that of PartialNICSLU, as described in Section 3.2.1. First, we considered a fixed percentage of varying matrix entries of 0.1%. The measured computation times for the refactorisation of the 10 sample matrices are depicted in Figure 8. It can be seen that the computation times of NICSLU are lower than those of KLU for all matrices, with computation time speedups in NICSLU ranging from 2.64 to 8.55 compared to KLU. This is in line with previous findings, e.g., in [3]. Furthermore, it can be seen that a further speedup of the refactorisation times for all matrices is accomplished when