# AC-based Power Flow

#### Definition

Alternating Current (AC)-based Power Flow Importance, proposes an AC-based power flow element importance measure, while considering multi-element failures.

The importance of $i^{th}$ element is measured by

$$IM_i^{N-K}={PE_0-PE_i^{N-K}\over PE_0}$$

where $PE_0$ is the system performance under the default configuration, and $PE_i^{N-k}$ is the average system performance among all the scenarios in which element $i$ is removed (along with other $k-1$ elements). Here the system performance is the power demand that could be supplied in the system. $PE_i^{N-k}$ is computed via the following function,

$$PE_i^{N-K}={1\over C(N-1,K-1)} {\underset {j=1} {\overset{C(N,K)}{\sum}} PE_j.W_{i,j}}$$

where $PE_j$ is the system performance of the $j^{th}$ removal out of $C(N,k)$ possible removals $(N > 1)$. $W_{i,j}$ captures whether element $i$ is removed in the $j^{th}$ removal, which is computed by

$$W_{i,j}=\begin{cases}

1, & \text{$i\in R(j,:)$}\\

0, & \text{$i\notin R(j,:)^.$}

\end{cases}$$

With $Eq. (1)-(3)$, the absolute importance under $N-k$ of the $i^{th}$ element can be captured, and in the following we will first normalize the element importance, as follows:

were$IM^{N-K}=[IM_1^{N-K},IM_2^{N-K},...,IM_i^{N-K},...,IM_N^{N-K}]$ and $IM_{i,norm}^{N-K}$is the normalized element importance. The element importance considering $N-1$ to $N-K$ can be captured by

$$IM_i^{N-K'}=IM_{i,norm}^{N-1}+IM_{i,norm}^{N-2}+...+IM_{i,norm}^{N-K}$$

MATPOWER, an open-source MATLAB power system simulation package

Li, J., Dueñas-Osorio, L., Chen, C. and Shi, C., 2017. AC power flow importance measures considering multi-element failures. Reliability Engineering & System Safety, 160, pp.89-97.

The importance of $i^{th}$ element is measured by

$$IM_i^{N-K}={PE_0-PE_i^{N-K}\over PE_0}$$

where $PE_0$ is the system performance under the default configuration, and $PE_i^{N-k}$ is the average system performance among all the scenarios in which element $i$ is removed (along with other $k-1$ elements). Here the system performance is the power demand that could be supplied in the system. $PE_i^{N-k}$ is computed via the following function,

$$PE_i^{N-K}={1\over C(N-1,K-1)} {\underset {j=1} {\overset{C(N,K)}{\sum}} PE_j.W_{i,j}}$$

where $PE_j$ is the system performance of the $j^{th}$ removal out of $C(N,k)$ possible removals $(N > 1)$. $W_{i,j}$ captures whether element $i$ is removed in the $j^{th}$ removal, which is computed by

$$W_{i,j}=\begin{cases}

1, & \text{$i\in R(j,:)$}\\

0, & \text{$i\notin R(j,:)^.$}

\end{cases}$$

With $Eq. (1)-(3)$, the absolute importance under $N-k$ of the $i^{th}$ element can be captured, and in the following we will first normalize the element importance, as follows:

were$IM^{N-K}=[IM_1^{N-K},IM_2^{N-K},...,IM_i^{N-K},...,IM_N^{N-K}]$ and $IM_{i,norm}^{N-K}$is the normalized element importance. The element importance considering $N-1$ to $N-K$ can be captured by

$$IM_i^{N-K'}=IM_{i,norm}^{N-1}+IM_{i,norm}^{N-2}+...+IM_{i,norm}^{N-K}$$