## 2 Statistical Modeling of Bidding Strategy Problem  
It is supposed that every power supplier (PS) is needed to send a bid to POOL as a non-decreasing linear supply feature in a single-sided POOL-based energy market, and the running cost function of any generating unit is given by Equation (1)  
\[PC_{m}(Pg_{m})=a_{m}Pg_{m}+b_{m}Pg_{m}^{2} \tag{1}\]  
where: \(m\) is the number of PS; bid constraints of the mth PS are \(a_{m}\) and \(b_{m}\); and \(Pg_{m}\) is the real power quantity of the mth PS.  
In a single-side bid model, the mth PS is submit the linear supply bid function which is non-decreasing according to Equation (2),  
\[CP_{m}(Pg_{m})=\pi_{m}+\phi_{m}Pg_{m},m=1,2,\ldots\ldots,\text{CPS} \tag{2}\]  
where \(\pi_{m}\) and \(\phi_{m}\) are bid parameters that are required to be non-negative.  
If the PS offers have been completed and submitted to ISO, the ISO compares the power supply with the overall demand of the system. After the matching, ISO decided the market clearing price (MCP) and cleared the marketplace. The bid function, power balance constraint, and power inequality constraint of the mth PS are given by Equations (3)-(5).  
\[\pi_{m}+\phi_{m}Pg_{m}=R \tag{3}\]  
\[\sum_{m=1}^{cps}Pg_{m}+\sum_{n=1}^{wg}Wg_{n}=Q(R) \tag{4}\]  
\[Pg_{\text{min,m}}\leq Pg_{m}\leq Pg_{\text{max,m}} \tag{5}\]where: \(Wg_{n}\) is the forecasted wind power generation output for bidding in MW, \(n\) is the number of wind power suppliers, \(Pg_{\text{min,m}}\) and \(Pg_{\text{max,m}}\) are the minimum and maximum active power generation by the mth power supplier, the MCP is \(R\), and the projected load by the market operator is \(Q(R)\). It is assumed that \(Q(R)\) is given by Equation (6)  
\[Q(R)=L_{c}-K*R \tag{6}\]  
where: \(L_{c}\) is constant; \(K\) = 0 is non-negative load price elasticity.  
For deciding MCP and calculation of the amount of bid power, ISO considered Equations (3) and (4) and ignored Equation (5). The MCP and amount of bid power are calculated by Equations (7) and (8), respectively.  
\[R=\frac{L_{c}-\sum\limits_{n=1}^{wg}Wg_{n}+\sum\limits_{m=1}^{cps}\frac{\pi_{ n}}{\phi_{m}}}{k+\sum\limits_{m=1}^{cps}\frac{1}{\phi_{m}}} \tag{7}\]  
\[Pg_{m}=\frac{R-\pi_{m}}{\phi_{m}} \tag{8}\]  
If the amount of bid power in Equation (8) exceeds its limits, it will be fixed by Equation (5).  
After the calculation of MCP and the amount of bid power, the profit of the mth PS can be calculated. Therefore, in this work, the main objective is to increase the earnings of the mth PS in the presence of renewable PS according to Equation (9)  
\[\text{Maximize}:\,F(\pi_{m},\phi_{m})=R\times Pg_{m}-PC_{m}(Pg_{m})+R\times Wg _{n}-IMC(Wg_{n}) \tag{9}\]  
Here, \(IMC(Wg_{n})\) is the imbalance cost related to wind power in USD/MW.  
The cost function of all thermal generating utilities is given by Equation (1). The underestimation and overestimation costs make up the running costs of an mth wind power generation utility. Due to its intermittent existence, wind sources are susceptible to both overestimation and underestimation. The expenditure task of wind power suppliers is given by Equation (10):  
\[IMC(Wg_{n})=O_{c}(w_{g})+U_{c}(w_{g}) \tag{10}\]  
where, \(O_{c}(w_{g})\) and \(U_{c}(w_{g})\) are the overestimation and underestimation cost related to wind power in USD.  
The penalty cost for overestimation is represented by the first term, and the penalty cost for underestimation is represented by the second term, which are given by Equations (11) and (12):  
\[O_{c}(w_{g})=K_{o}*\int\limits_{0}^{W_{g}}(W_{g}-W_{a})*f_{W_{a}}(W_{a})*dW_{a} \tag{11}\]  
where \(K_{o}\) is the penalty coefficient for overestimating power.  
\[U_{c}(w_{g})=K_{u}*\int\limits_{W_{g}}^{W_{\text{max}}}(W_{a}-W_{g})*f_{W_{a} }(W_{a})*dW_{a} \tag{12}\]  
where \(K_{u}\) is a fine for the lack of situational advantages per USD/kWh owing to underestimation of the capacity.