## 4 Simulation Results and Convergence Analysis
### Convergence Analysis Based on ICC Algorithm  
In the following, the ICC-based distributed control method is tested. In this section, we focus on convergence analysis based on the ICC algorithm. There are several factors that can affect the  
as degree centrality and closeness centrality. We can use these centrality indices as indicators for the selecting the dominant node.  
The ICC algorithm can handle a wide range of communication networks, so the topology of the communication network and the location of the dominant node can affect the convergence rate. As mentioned in the previous section, the dominant node will increase or decrease the group IC based on the mismatch \(\Delta P\). Thus, the convergence rate can be controlled by adjusting the convergence coefficient \(\varepsilon\).  
Figures 10 and 11 show two different convergence rates using different values for \(\varepsilon\). Using the same step size, the system with \(\varepsilon=0.001\) converges faster than the system with \(\varepsilon=1/6000\). Thus, the larger \(\varepsilon\) is related to a faster convergence speed. However, we should also consider its influence on system stability in practical working conditions.  
Figure 8: The IC when choosing different dominant nodes.  
Figure 9: The total output power of customers when choosing different dominant nodes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Dominant Node No.** & **Numbers of Adjacent Nodes** & **Iteration Number** \\ \hline Customer 1 & 3 & 1455 \\ Customer 2 & 3 & 1467 \\ Customer 3 & 5 & 876 \\ Customer 4 & 5 & 874 \\ Customer 5 & 3 & 1481 \\ Customer 6 & 3 & 1483 \\ \hline \hline \end{tabular}
\end{table}
Table 5: System response time when choosing different dominant nodes.  
As the previous section mentioned, the convergence rate is based on the topology of the system's communication network. Consider the six-customer system with a chain connection, which is shown in Figure 12 and a neighbor interconnection, which shown in Figure 3.  
Figure 11: The power of each customer with \(\varepsilon=0.001\).  
Figure 12: Communication topology of the chain connection.  
Figure 10: The power of each customer with \(\varepsilon=1/6000\).  
The Laplacian graphs of neighbor interconnection topology and chain connection topology are obtained as following:  
\[L_{N}=\left[\begin{array}{cccccc}3&-1&-1&-1&0&0\\ -1&3&-1&-1&0&0\\ -1&-1&5&-1&-1&-1\\ -1&-1&-1&5&-1&-1\\ 0&0&-1&-1&3&-1\\ 0&0&-1&-1&-1&3\end{array}\right] \tag{12}\]  
\[L_{C}=\left[\begin{array}{cccccc}1&-1&0&0&0&0\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&0&0\\ 0&0&-1&2&-1&0\\ 0&0&0&-1&2&-1\\ 0&0&0&0&-1&1\end{array}\right] \tag{13}\]  
Figures 5, 6, 13 and 14 show the simulation results under different communication topologies when Customer 3 is selected as the dominant node, and chain connection topology has a faster convergence rate than neighbor interconnection topology. This result suggests that the topology of the system's communication network can affect the convergence rate of the ICC algorithm.  
In general, when the number of nodes increases, the ICC algorithm needs more iterations to reach a consensus. The convergence rate under a different number of nodes is a very important performance measure. We will discuss the performance of the ICC algorithm under a  
Figure 14: The IC of each customer with chain connection.  
Figure 13: The power of each customer with chain connection.  
community with 32 residential customers, as shown in the Figure 15 as an example. As shown in Figure 16, when the objective load power of the DLC increases from 0 to 300 kW, all customers are able to converge to the optimal IC asymptotically based on the ICC algorithm. The optimal power allocation of customers is realized as shown in Figure 17. The ICC algorithm can handle a larger scale of communication network. Hence, the ICC algorithm is able to handle a relatively large number of customers in a reasonable period of time.