## 5 Simulation Examples
### Examples with No Consideration of Uncertainties
#### 5.1.1 Example 1--Coverage of the Required Load and Its Changes  
The algorithm will be presented by example, comprising four generators and two renewable power plants. One will be a large solar power plant, and the other will be a wind power plant. Three types of generators will be introduced. First will be **Type I** represents coal-fired steam unit, then **Type II** is oil-fired steam unit and **Type III** denotes oil-fired steam unit. The generator parameters were adapted from [10] only with the difference that the power of the generators will be considered in kW. Their values are described in Table 1.  
The total demand will be set as \(P_{D}=1500\) kW. Local demands \(P_{D_{i}}\) will be set randomly between all agents, with the condition that their sum must be equal to \(P_{D}\). Table 2 lists the algorithm initialization parameters. Example network topology is depicted in Figure 5, where vertices 1 and 2 denote **Type I** generators, vertex 3 mean **Type II** generator and vertex 4 indicate **Type III** generator. Next, vertex 5 represents the solar power plant, and the wind power plant is denoted by vertex 6. Weight matrix \(W\) will be calculated according to (29). The blue highlighted line in \(W\) denotes the wind farm, and the yellow highlighted line represents the solar power plant. \begin{table}
\begin{tabular}{l l l l l} \hline \hline  & \multicolumn{4}{c}{**Generation Unit Parameters**} \\ \hline
**Generator Type** & **Type I** & **Type II** & **Type III** \\ \hline Range [kW] & \([150,600]\) & \([100,400]\) & \([50,200]\) \\ \(\alpha\) [USD/kW\({}^{2}\)h] & \(-2535.211268\) & \(-2023.195876\) & \(-826.7634855\) \\ \(\beta\) [USD/kWh] & \(352.1126761\) & \(257.7319588\) & \(103.7344398\) \\ \(\gamma\) [USD/h] & \(-8616.760563\) & \(-7631.043814\) & \(-3216.65249\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Generator parameters. A total of 2000 panels with a power of 250 W are considered for the solar power plant; this is how the variable \(M_{PV_{i}}\) is characterized. The outside temperature is \(T_{t_{i}}=25\ ^{\circ}\)C. Furthermore, for the needs of wind power plants, it is considered that their windswept area will be \(A_{i}=1000\ \text{m}^{2}\) and \(\rho_{Air}=1\ \text{kgm}^{3}\). This setting is retained for all examples. To confirm convergence, the power outputs of the solar and wind power plants underwent four changes. These changes occurred in the time steps of \(100,200,300\), and \(400\), affecting \(P_{5}\) as it transitioned from 0 to 75, then to 200, followed by 85, and finally reaching 0. Similarly, \(P_{6}\) went from 0 to 50, then to 100, followed by 45, and eventually back to 0. The simulation results for \(\lambda_{i}\) and \(P_{i}\) are depicted in Figure 6. The algorithm effectively responds to variations in power output from solar and wind sources, consistently meeting the required power demand \(P_{D}\). The plots reveal that the total incremental cost \(\lambda\) decreases as renewable power generation increases. The lowest \(\lambda\) value within the \(k=200\) to 300 range is 8.559 USD/kWh, compared to the initial 8.84 USD/kWh at the start of the simulation. **Remark 5**. : _The results were compared with results from centralized methods. Namely, these were the Lagrangian multipliers method and the Lambda-iteration method. These centralized methods are always calculated for given initial conditions and required loads. In case of change, the calculation must be performed again. Thus, these calculations were performed for each time point of change (\(k=0,100,200,300\) and \(400\)), and their results were compared with those provided by the  
Figure 5: Network topology for Example 1. Figure 6: Example 1 results for \(\lambda_{i}\) and \(P_{i}\). This example shows the response of the BESS to the evolution of the resource price change. A total of six price changes will be considered here over a period of 86,400 steps. All steps represent the time horizon of one day in seconds.