## 4 Simulation Results and Convergence Analysis  
### Optimal Power Allocation of Smart Home Appliances  
In this section, we take a small system of six customers for case study. The network topology of the system is shown in the Figure 3. Each customer is connected with the rest of the customers through power lines and communication links. The electrical appliances used by each customer include air conditioners, water heaters, refrigerators and so on. In this paper, the on/off loads participating in the DLC are used as case studies, and they are sorted according to the priority order of different electrical appliances of each customer. According to the priority order of these home appliances, the load power of the higher priority is more likely to be activated. Figure 3: Communication topology of the six-customer system: neighbor interconnection. Figure 2: Flowchart of incremental cost consensus (ICC) algorithm. Assume that the power capacity parameters of the home appliances of the customers are as shown in Table 1. The priority order of appliances is Appliance 1 to Appliance 10. Assume that the customer compensation for participating in the DLC is randomly distributed between 0.1-1.5 \(\mathrm{\SIUnitSymbolMicro\text{V}}\)/kW according to the types of appliances. Considering the discrete response characteristics of the on/off loads, the priority ranking is implemented, and the customer cost function can be established by the fitting method. The parameters of six customer cost functions are shown in Table 2.  
Taking Customer 3 as an example, the cost function is drawn by the curve fitting method. The relationship between the customer costs and the load power is shown in Figure 4, and the fitting function is obtained as following:  
\[F_{3}(P_{R3})=0.0372{P_{R3}}^{2}+0.2163P_{R3}+0.3157. \tag{11}\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Appliances No.** & **Customer 1** & **Customer 2** & **Customer 3** & **Customer 4** & **Customer 5** & **Customer 6** \\ \hline Appliance 1/kW & 2.32 & 0.11 & 1.33 & 3.33 & 0.08 & 0.72 \\ Appliance 2/kW & 1.94 & 2.64 & 2.15 & 0.29 & 1.94 & 0.63 \\ Appliance 3/kW & 1.21 & 2.72 & 1.14 & 0.38 & 1.31 & 1.88 \\ Appliance 4/kW & 1.09 & 2.37 & 1.04 & 0.51 & 1.25 & 2.38 \\ Appliance 5/kW & 1.11 & 0.30 & 1.74 & 0.59 & 2.36 & 0.97 \\ Appliance 6/kW & 0.76 & 0.78 & 0.49 & 2.21 & 2.06 & 1.72 \\ Appliance 7/kW & 0.67 & 1.00 & 1.08 & 1.79 & 1.14 & 0.60 \\ Appliance 8/kW & 1.87 & 2.03 & 1.95 & 0.44 & 2.24 & 3.48 \\ Appliance 9/kW & 2.04 & 0.41 & 1.25 & 3.32 & 2.10 & 0.02 \\ Appliance 10/kW & 1.98 & 2.15 & 1.83 & 2.60 & 1.51 & 1.10 \\ Total capacity/kW & 15.00 & 14.50 & 14.00 & 15.50 & 16.00 & 13.50 \\ \hline \end{tabular}
\end{table} 
Table 1: The power capacity parameters of home appliances of six customers. \begin{table}
\begin{tabular}{c c c c} \hline
**Customer No.** & \(\mathbf{a_{i}}\)**(W/kW\({}^{2}\))** & \(\mathbf{b_{i}}\)**(W/kW)** & \(\mathbf{c_{i}}\)**(W)** \\ \hline
1 & 0.0347 & 0.2093 & 0.6321 \\
2 & 0.0331 & 0.1773 & 0.2518 \\
3 & 0.0372 & 0.2163 & 0.3157 \\
4 & 0.0459 & 0.1402 & 0.4869 \\
5 & 0.0273 & 0.1938 & 0.0132 \\
6 & 0.0256 & 0.2133 & 0.0098 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of six customer cost functions. Figure 4: The fitting function plot of Customer 3. In this example, it is assumed that the initial objective load power is 0. When the objective load power of the DLC is increased to 55 kW, the customers achieve optimal power allocation through the distributed control method. Customer 3 is selected as the dominant node with a fixed step size of 0.0002 s and a convergence coefficient of \(\varepsilon\) = 0.0005. As shown in Figure 5, all the customers are able to converge to the optimal IC asymptotically under the neighbor interconnection topology. This also proves that the distributed control strategy is able to make each customer get the global state information through the local information exchange. As shown in Figure 6, each customer in the system achieves the optimal power allocation according to the distributed control strategy.