### Related Work  
Comparative analysis of the performance of LU decomposition methods for sparse matrices, which occur in power system simulations, has been carried out in previous studies [3; 4]. Among others, the LU decomposition methods SuperLU, KLU and NICSLU have been benchmarked in these studies. In particular, it was shown in [3] that the application of NICSLU can yield improved refactorisation computation times, which is also a major motivation for employing NICSLU in this work. An advancement in this work compared to [3] is that refactorisation is sped up by means of partial refactorisation. LU decompositions for sparse systems are not only applied by MNA based real-time solvers, as described in the present work in Section 2.1. Another potential application is shown in [5], where LU decompositions were used as subroutines of differential-algebraic system of equations (DAE) solvers. This applies, for instance, to SUNDIALS/IDA, a general-purpose DAE solver that is used by the variable time step solver of Dyna\(\omega\)o, an offline power system simulation suite [6; 7]. Furthermore, Newton-Raphson (NR) method based solvers for nonlinear systems such as SUNDIALS/KINSOL, e.g., used by Dyna\(\omega\)o's fixed time step solver, also apply LU decompositions. Another application of such NR methods in power system simulation is the solution of load-flow problems based on Runge-Kutta methods [8], whose many Jacobian factorisations have motivated improvements of the applied LU factorisation methods. However, the computational improvement of LU decomposition methods used by outer NR methods is not the only way to improve the overall computational performance. Instead, the computational performance of the overall calculation can be improved by reducing the total LU factorisations required to reach the final solution. For instance, ref. [9] presents two modern methods of this kind. The first is based on the third-order Newton-like method of [PERSON], and the second is based on that of [PERSON]. Both methods are modified in a way to benefit from the seventh order of convergence. Although both new methods require more function evaluations and the first method also has one more Jacobian evaluation than its original counterpart, they enable computational savings higher than 20% and 30%, as was shown with the aid of three power system operation scenarios. However, in the present work, we focus on the improvement of the computational efficiency of the LU decomposition itself, particularly its refactorisation step. Other works have proposed approaches that target the speedup of the refactorisation step of LU decomposition methods. The authors in [10] suggested partition-based decompositions. These methods are based on the idea of partitioning matrices into sub-blocks, some of which contain numerically changing values. Subsequently, from this knowledge, a block-based updating algorithm can be developed for refactorisation. The drawback of this method is that it fully refactorises blocks that contain numerically changing values, without considering which parts of those that are refactorised need to be recomputed and which can be neglected. This leads to a potential unnecessary surplus of computations, especially if there are few numerically changing values. This motivates the improvement of the refactorisation approach by further reducing the number of computations, which is the main aspect of this work. The research work presented in [11] also aimed to speed up refactorisation. In [11], the sparse matrix solver under investigation was KLU. The authors proposed a pivot validity test to ensure numerical stability as well as a partial refactorisation approach to accelerate the refactorisation process. The partial refactorisation approach discussed in [11] is based on the idea that the factorisation algorithm successively constructs the LU columns, starting from the leftmost column. It uses the previously computed columns for the factorisation.