# A Decentralized Multi-Agent-Based Approach for Low Voltage Microgrid Restoration
## 5 Simulation Results and Discussion  
In order to evaluate the dynamic behavior of an MG during restoration procedure, a benchmark LV MG network presented in Figure 10 is implemented in the simulation platform. The electrical data for this LV test system can be found in [48]. It is supposed that the MG is subjected to a total blackout. The studied MG includes nine local controllers, and each one is assigned to a specific agent. Figure 11 shows the two topologies for connection of the agents. From (6), it can be observed that the speed of ACA is independent of the initial information matrix and it depends on how the agents are connected and how the weight coefficients are determined. Figure 9: Flowchart of the decision making process. To verify the convergence speed of the ACA for the topologies of Figure 11, let define the initial values assigned to each agent as follows:  
\[c_{1}^{0}=1,c_{2}^{0}=-1,c_{3}^{0}=1,c_{4}^{0}=-1,c_{5}^{0}=1,c_{6}^{0}=-1,c_{7}^ {0}=1,c_{8}^{0}=-1,c_{9}^{0}=0. \tag{12}\]  
By using Equation (2), the equilibrium point for unlimited iterations will be:  
\[c_{1}^{\infty}=c_{2}^{\infty}=c_{3}^{\infty}=c_{4}^{\infty}=c_{5}^{\infty}=c_{6 }^{\infty}=c_{7}^{\infty}=c_{8}^{\infty}=c_{9}^{\infty}=\frac{1}{9}\sum_{i=1}^{9 }c_{i}^{0}=0. \tag{13}\]  
Figure 11: Different topologies for connection of the local controller agents: (a) topology a; (b) topology b.  
Figure 10: Benchmark LV MG network. It means that after using the consensus algorithm (Equation (1)), the number '0' exists in each agent. The metropolis method is used for determining the weight coefficient. Figure 12 shows the value in each agent for 40 iterations. Considering an error tolerance equal to 0.01, the agents have reached the consensus with 19 and 17 iterations for the topology (a) and (b), respectively. The time delay to reach the consensus can be estimated by:  
\[T=\frac{N_{\text{iteration}}\times N_{M}\times N_{b}}{C}, \tag{14}\]  
where \(N_{\text{iteration}}\) is the number of required iterations to reach the consensus, \(N_{M}\) is the size of the information matrix, \(N_{b}\) is the number of required bits to represent each element of the information matrix, and \(C\) is the communication link speed. For the topology (a) and (b), the system requires 19 and 17 iterations to converge, respectively. There are nine agents, \(N_{M}=9\times 4\), and if 16 bits are used for representing each element of the information matrix, for a network with 5 Mbit/s, time delay for reaching the consensus for the topology (a), (b) are 0.002189 s and 0.001958 s, respectively. This time delay is very small compared with the pre-defined interruption time used in the decision making process. In the simulations, the interruption time is considered equal to 4 s. Therefore, the time delay for reaching the consensus can be neglected. In order to provide the black start capability for the studied MG, the microturbine is equipped with the battery storage in the DC link, and its inverter operates in the grid-forming control mode. The inverters of the fuel cell and PVs systems operate in the grid following control mode. The wind turbine is connected directly to the grid through an induction generator. It should be noted that the secondary control is carried out locally by using a PI controller at microturbine control system aiming to restore the frequency and voltage to the nominal value after any restoration action. The restoration process is started with forming the initial information matrices of the agents. The initial information matrices are as follows:  
Figure 12: The convergence speed of the different topologies of Figure 11: (**a**) topology a; (**b**) topology b. \[\mathbf{M_{1}}=\left[ \begin{array}{cccc}0&1&-5&2\\ 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_{2}}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&2&-40&1\\ 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_{3}}=\left[