## 4 Case Studies
### Discrete Consensus  
On the basis of the above model and information topology, while taking the maximum and minimum adjustable capacity constraints of each charging station provided in Table 2 into consideration, the adjustment cost is considered as a consentaneous variable. Meanwhile, the convergence condition is \(|\Delta P_{\text{error}}|\leq 0.01\) and the error adjustment factor \(\gamma\) is set to 0.5. When \(\Delta P_{\text{EV}\,\Sigma}=10\) MW, the convergence process of AGC power allocation for 10 charging stations based on the cost consensus algorithm is shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Charging Stations**} & \multicolumn{2}{c}{**The Adjustment Cost**} & \multicolumn{1}{c}{**Maximum Regulation**} & \multicolumn{1}{c}{**Minimum Regulation**} \\  & **Coefficient** & **\(C_{i}\) **yuan (MW-h)** & **Capacity/MW** & **Capacity/MW** \\ \hline CS1 & 77.44 & 3 & 2.5 \\ CS2 & 65.12 & 4.2 & 3.6 \\ CS3 & 82.30 & 6.3 & 5.8 \\ CS4 & 78.08 & 3.2 & 2.6 \\ CS5 & 96.51 & 4.8 & 4.3 \\ CS6 & 83.27 & 1.8 & 1.4 \\ CS7 & 95.94 & 2.8 & 2.2 \\ CS8 & 92.02 & 1.5 & 1.2 \\ CS9 & 68.04 & 2.4 & 2.0 \\ CS10 & 70.93 & 5 & 4.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Regulation cost coefficient and total capacity power for regulation up and down in a period.  
Figure 5: Consensus convergence process of regulation cost; (**a**) Convergence process of adjusting cost; (**b**) Convergence process of power generation in charging stations.  
As demonstrated in Figure 5, one can readily observe that when the iteration number is 500, the regulation cost consensus variables of all charging power stations become identical, while the generation power of each charging power station converges at the same time, which is lower than 1.5 MW. Clearly, under the condition \(\Delta P_{\mathrm{EV}\,\Sigma}=10\) MW, the generation power of all charging stations does not reach the upper limit of regulation capacity. Under such circumstances, the convergence process of the actual consensus variables is identical to that of the virtual consensus variables, as demonstrated in Figure 5a.  
Figure 6 depicts the convergence process of adjusting the cost consensus algorithm when \(\Delta P_{\mathrm{EV}\,\Sigma}=25\) MW, upon which one can readily observe that, with the increase of iteration numbers, the virtual regulation cost consensus variables of all rechargeable power stations tend to be consentaneous, while the generation power of each charging station also converges to a certain value. Owing to the restriction of maximum regulation capacity, the generation power of some charging stations has reached the limit, such as CS6, CS8, and CS9 shown in Figure 6c. Meanwhile, the actual consensus variables of these three charging stations also reach the limit, as shown in Figure 6b. Hence, this indicates that, because the concepts of the virtual consensus variable and actual consensus variable are introduced, even when the actual consensus variable reaches the limit value, the virtual consensus variable of all charging stations can still reach a consensus. More specifically, as the interactive information between different charging stations, the virtual consensus variable is not limited by the capacity constraint and is only used for cost consensus calculation by Equation (9). Through enough interactions, all the charging stations can reach a consensus on the virtual cost, while the power balance constraint can be satisfied. Therefore, the convergence of such an algorithm can be guaranteed by the interaction of virtual consensus variables among agents.  
Figure 6: _Cont._