2.2.2. LU Decomposition

For computing the power flow calculation by utilizing either the NR or FDPF method, the solution of a linear system of matrix equations is required, as in Equation (1). Notably, the *Ybus* matrix reflects the admittance information of a power system, so it presents a sparse quality in containing many zeros. As shown in Figure 2, we need to update *V* and *θ* with Δ *V* and Δ*θ* obtained from the power mismatch, i.e., Δ *P* and Δ *Q*, using Equation (7). Hence, we apply a mathematical technique such as sparsity oriented LU decomposition to calculate the inverse of the Jacobian coefficient matrix based on the admittance information [17].

$$A \cdot X = (L \cdot \mathcal{U}) \cdot X = L \cdot (\mathcal{U} \cdot X) = b \tag{8}$$

$$L \cdot y = b$$

$$y = \mathcal{U} \cdot X \tag{10}$$

LU decomposition is also known as triangular factorization or LU factorization. To obtain *X* by solving a linear equation, i.e., *A* · *X* = *b*, the LU decomposition first divides the coefficient matrix *A* into two lower and upper triangular matrices, *L* and *U*, as in Equation (8). Then, *y* is calculated by performing a forward substitution in Equation (9), and *X* is finally obtained by a backward substitution in Equation (10).

Numerical computation techniques based on such sparsity are popularly employed in procedures to solve the network equations of large power systems [20]. The time required for the matrix calculations requires a high proportion of the total computation time; for instance, it was found to take up to about 80% of the total computation time in [21]. Hence, many related studies have already been performed [22–26], as we can significantly reduce the total execution time of PF analysis by applying parallel processing to the matrix calculations.
