## 4 Case Study and Proof of Concept
### Test System 1  
Table 4 shows the operating limits, cost, and emission coefficients of the CHP and fossil fueled units that deliver power to the load demand of distribution test system 1. The hourly load demand data for this test system are gathered from article [19]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{\
\begin{tabular}{c} **Power-Only** \\ **Generators**\(i\) \\ \end{tabular} } & **1** &
\begin{tabular}{c} **CHP-Based** \\ **Generators** \\ \end{tabular} & **1** & **2** & **3** \\ \hline \(a\) (USD/MW\({}^{3}\)) & \(2.55\times 10^{2}\) & \(a\) (USD/MW\({}^{3}\)) & \(1.25\times 10^{3}\) & \(2.65\times 10^{3}\) & \(1.57\times 10^{3}\) \\ \hline \(b\) (USD/MW\({}^{2}\)) & \(7.70\times 10^{0}\) & \(b\) (USD/MW\({}^{2}\)) & \(3.60\times 10^{1}\) & \(3.45\times 10^{1}\) & \(2.00\times 10^{1}\) \\ \hline \(c\) (USD/MW) & \(1.72\times 10^{-3}\) & \(c\) (USD/MW) & \(4.35\times 10^{-2}\) & \(1.04\times 10^{-1}\) & \(7.20\times 10^{-2}\) \\ \hline \(d\) (USD) & \(1.15\times 10^{-4}\) & \(d\) (USD) & \(6.00\times 10^{-1}\) & \(2.20\times 10^{0}\) & \(2.30\times 10^{0}\) \\ \hline \(\alpha\) (tons/MW\({}^{2}\)) & \(4.09\times 10^{-4}\) & \(\alpha\) (tons/MW\({}^{2}\)) & \(2.70\times 10^{-2}\) & \(2.50\times 10^{-2}\) & \(2.00\times 10^{-2}\) \\ \hline \(\beta\) (tons/MW\({}^{2}\)) & \(-0.0005554\) & \(\beta\) (tons/MW\({}^{2}\)) & \(1.10\times 10^{-2}\) & \(5.10\times 10^{-2}\) & \(4.00\times 10^{-2}\) \\ \hline \(\gamma\) (tons) & \(6.49\times 10^{-4}\) & \(\gamma\) (tons) & \(1.65\times 10^{-3}\) & \(2.20\times 10^{-3}\) & \(1.10\times 10^{-3}\) \\ \hline \(P_{\rm j,max}\) (MW) & \(1.35\times 10^{2}\) & & & & \\ \hline \(P_{\rm j,min}\) (MW) & \(3.50\times 10^{1}\) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Conventional and 3 CHP generator cost and emission coefficients. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{Function} & \multicolumn{4}{c}{F3} & \multicolumn{4}{c}{F4} \\ \cline{2-7}  & & CSA & AOA & CSAOA & CSA & AOA & CSAOA \\ \hline Best & Multimodal benchmark & \(-7320.13\) & \(-5886.74\) & \(-7915.45\) & \(5.42\) & \(4.17\) & \(0.01\) \\ \hline Worst & functions & \(-1967.51\) & \(-3502.98\) & \(-2871.64\) & \(9.53\) & \(20.55\) & \(0.12\) \\ \hline Mean & & \(-4414.72\) & \(-4454.94\) & \(-5513.72\) & \(6.76\) & \(8.88\) & \(0.04\) \\ \hline SD & & \(1691.85\) & \(536.98\) & \(1076.63\) & \(0.81\) & \(5.46\) & \(0.01\) \\ \hline Function & & \multicolumn{4}{c}{F5} & \multicolumn{4}{c}{F6} \\ \cline{2-7} Algorithm & & CSA & AOA & CSAOA & CSA & AOA & CSAOA \\ \hline Best & Fixed-dimension & \(-10.14\) & \(-9.81\) & \(-10.15\) & \(-10.53\) & \(-10.40\) & \(-10.53\) \\ \hline Worst & functions & \(-2.62\) & \(-2.37\) & \(-2.68\) & \(-2.41\) & \(-2.32\) & \(-2.42\) \\ \hline Mean & & \(-7.29\) & \(-6.03\) & \(-6.41\) & \(-8.52\) & \(-8.04\) & \(-6.23\) \\ \hline SD & & \(3.55\) & \(2.95\) & \(3.79\) & \(3.36\) & \(2.5 The load model for the first stage has been changed to account for 40% of the loads participating in the DSM approach. During the restructuring, the elastic loads are optimally transferred using the proposed technique. It is determined that a DSM participation level of 40% would result in the total demand, mean demand, peak demand, and load factor computed. Peak demand is lowered by up to 16% when the DSM technique is implemented, according to Table 5. Similarly, a change in the distribution of elastic loads lead to an increase in the load factor, which increases from 0.7489 to 0.8929. Including a DSM approach in the operation of a distribution system can bring various advantages. The total and average demand of the distribution system remains the same both with and without the presence of DSM, which is another important point to take into consideration. Figure 3 depicts the adjusted load model for different levels of DSM participation.