## 5 LA Bidding Model
### Objective Function  
The net profit of the LA can be calculated by the income value, the incentive expenditure value, and the compensation value, which can be obtained by Equations (7)-(12).  
\[\max V^{n}_{k,j}=\max(V^{S}_{k,j}-V^{e}_{k,j}-V^{p}_{k,j}) \tag{7}\]  
\[Q^{b}_{k,j}=\sum\limits_{i=1}^{n}{(Q^{\text{DR}}_{\text{HVAC},i,j}+Q^{\text{ DR}}_{\text{EWH},i,j}+Q^{\text{DR}}_{\text{EV},i,j})} \tag{8}\]  
\[V^{S}_{k,j}=\lambda_{k,j}Q^{b}_{k,j} \tag{9}\]  
\[V^{e}_{k,j}=\gamma_{k,j}Q^{b}_{k,j} \tag{10}\]  
\[V^{p}_{k,j}=\left\{\begin{array}{ll}\mu/\left(\overline{p}^{\min}_{k,j}- \overline{p}^{k}_{j}\right)Q^{b}_{k,j}&\overline{p}^{k}_{j}<\overline{p}^{ \min}_{k,j}\\ 0&\overline{p}^{k}_{j}\geq\overline{p}^{\min}_{k,j}\end{array}\right. \tag{11}\]  
\[\overline{p}_{k,j}=\frac{\sum\limits_{i=1}^{n}{p_{i,j}(Q^{\text{DR}}_{\text{ HVAC},i,j}+Q^{\text{DR}}_{\text{EWH},i,j}+Q^{\text{DR}}_{\text{EV},i,j})}}{Q^{b}_{k,j}} \tag{12}\]  
where \(V^{n}_{k,j}\) is the net profit obtained by the LA \(k\). \(V^{S}_{k,j}\), \(V^{e}_{k,j}\), \(V^{p}_{k,j}\) are the income value obtained by the LA \(k\), the incentive expenditure value of the compensation consumer, and the compensation value caused by the response exceeding the limit in period \(j\); \(Q^{b}_{k,j}\), \(\lambda_{k,j}\), \(\gamma\), \(\lambda_{k,j}\) are bidding capacity, bidding unit price, and incentive unit price of the LA \(k\) in period \(j\); \(p_{i,j}\) is the willingness of resident \(i\) to participate in DR during the period \(j\); \(\overline{p}_{k,j}\) is average participation willingness of \(n\) residents in period \(j\); \(\mu_{j}\) is the unit price of compensation in response to the out-of-limit part.  
The income value is paid by the PSO to the LA, which can be calculated by multiplying the bidding unit price and the bidding capacity. The incentive expenditure value is the LA's compensation for residents' participation in DR, which is the part of the LA's cost and can be calculated by multiplying  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Response Quantity**} & \multirow{2}{*}{**Awareness Level**} & \multicolumn{3}{c}{**Willingness**} \\ \cline{3-5}  & & & **External Incentive** \\ \cline{3-5}  & & L & M & H \\ \hline \multirow{3}{*}{L} & L & VL & VL & L \\  & M & VL & L & M \\  & H & L & M & H \\ \hline \multirow{3}{*}{H} & L & VL & L & M \\  & M & L & M & H \\ \cline{1-1}  & M & M & H & VH \\ \cline{1-1}  & H & H & VH & VH \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fuzzy inference rules of user willingness.  
the incentive unit price and the bidding capacity. In addition, if the LA fails to reach the response volume declared in the previous bidding, the PSO will impose a fine on the LA. The total amount of the fine is the LA's compensation value. Considering the uncertainty of the actual response of residents, the LA is allowed to have a response deviation margin in the bid winning capacity. Suppose the ratio of the lowest response amount to the winning bid capacity is \(\overline{p_{k,j}^{\min}}\), and the LA does not need to compensate within the deviation margin. The compensation value can be obtained by Equation (11).