# A Decentralized Multi-Agent-Based Approach for Low Voltage Microgrid Restoration
## 5 Simulation Results and Discussion
\begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&3&30&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{4}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{5}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{5}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{5}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{6}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{7}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{8}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{9}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right],\mathbf{M_{9}}=\left[\begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0\\ 0\[\mathbf{M_{conv. }}=\left[ \begin{array}{cccccccccc}0&0&1&0&0&0&0&0&0\\ \dfrac{1}{9}&\dfrac{2}{9}&0&\dfrac{4}{9}&\dfrac{5}{9}&\dfrac{6}{9}&\dfrac{7}{9}& \dfrac{8}{9}&\dfrac{9}{9}\\ -5&\dfrac{-10}{9}&0&\dfrac{20}{9}&\dfrac{10}{9}&\dfrac{-20}{9}&\dfrac{-4.8}{9}& \dfrac{-20}{9}&\dfrac{15}{9}\\ \dfrac{2}{9}&\dfrac{1}{9}&0&0&0&0&0&0&0\end{array} \right]^{\mathrm{T}}\]  
The above matrix shows that the microturbine has reached its capacity limit and there is a need for connection of another MC-agent to release the capacity of the microturbine. In this step, among the MC-agents, the MC-agent 4 (correspond to the wind turbine) that has a higher production capacity is chosen to be connected. In the next step, the final converged matrix is as follows:  
\[\mathbf{M_{conv. }}=\left[ \begin{array}{cccccccccc}0&0&1&1&0&0&0&0&0\\ \dfrac{1}{9}&\dfrac{2}{9}&0&0&\dfrac{5}{9}&\dfrac{6}{9}&\dfrac{7}{9}& \dfrac{8}{9}&\dfrac{9}{9}\\ -5&-\dfrac{-10}{9}&\dfrac{20}{9}&0&\dfrac{10}{9}&\dfrac{-20}{9}&\dfrac{-4.8}{9 }&\dfrac{-20}{9}&\dfrac{15}{9}\\ \dfrac{2}{9}&\dfrac{1}{9}&0&0&0&0&0&0&0\end{array} \right]^{\mathrm{T}}\]  
In this step, the microturbine has 20 kW free capacity. The remaining load of LC-agent 2 along with the LC-agent 1 and one of the residence groups correspond to the LC-agent 6 are connected. This process will continue until all of the MC-agents are connected, and the microturbine operates in its capacity limits. The remaining disconnected loads will be supplied when the MG is connected to the upstream network. Table 1 shows the sequence of restoration actions carried out based on the proposed decentralized multi-agent based scheme. After step 8, the final converged matrix is expressed as follows:  
\[\mathbf{M_{conv. }}=\left[ \begin{array}{cccccccccc}1&1&1&1&1&0&0&1\\ 0&0&0&0&0&\dfrac{7}{9}&\dfrac{8}{9}&0\\ 0&0&0&0&0&0&\dfrac{-4.8}{9}&\dfrac{-10}{9}&0\\ 0&0&0&0&0&0&0&0\end{array} \right]^{\mathrm{T}}\]