# Calculation Method for Electricity Price and Rebate Level in Demand Response Programs
## 3 Calculation Methodology of Electricity Prices and Rebate Levels
### Definition of the Optimal Electricity Price and Its Calculation  
The power suppliers decide the electricity price in the price-based DR to control the power demand. The optimal electricity price is defined as:  
\[p_{t}^{\prime}=p_{t}^{*}+\Delta p_{t}, \tag{12}\]  
where \(\Delta p_{t}\) is the change of electricity price in the DR.  
According to Equations (2) and (3), the values of the utility of the power suppliers and the consumers in the DR are calculated as:  
\[U_{1,t}\big{(}d_{t}^{\prime}\big{)} =p_{t}^{\prime}d_{t}^{\prime}-E_{t}\big{(}d_{t}^{\prime}\big{)}, \tag{13}\] \[U_{2,t}\big{(}d_{t}^{\prime}\big{)} =F_{t}\big{(}d_{t}^{\prime}\big{)}-p_{t}^{\prime}d_{t}^{\prime}. \tag{14}\]  
In addition, the DR-originated changes in them are calculated as:  
\[\Delta U_{1,t} =U_{1,t}\big{(}d_{t}^{\prime}\big{)}-U_{1,t}\big{(}d_{t}^{*} \big{)}=\big{(}p_{t}^{\prime}d_{t}^{\prime}-p_{t}^{*}d_{t}^{*}\big{)}-\Delta E _{t}, \tag{15}\] \[\Delta U_{2,t} =U_{2,t}\big{(}d_{t}^{\prime}\big{)}-U_{2,t}\big{(}d_{t}^{*} \big{)}=\Delta F_{t}-\big{(}p_{t}^{\prime}d_{t}^{\prime}-p_{t}^{*}d_{t}^{*} \big{)}, \tag{16}\]  
where \(\Delta E_{t}\) is the change in the suppliers' utility by the DR and is written as: \(E_{t}(d_{t}^{\prime})-E_{t}(d_{t}^{*})\); \(\Delta F_{t}\) is the change in the consumers' utility by the DR and is described as: \(F_{t}(d_{t}^{\prime})-F_{t}(d_{t}^{*})\).  
In the DR, each of the power suppliers' utility and the consumers' one (or each sum of them during the target period) is greater than or equal to zero (\(U_{1,t}(d_{t}^{\prime})\geq 0\); \(U_{2,t}(d_{t}^{\prime})\geq 0\)). Since the power suppliers request the DR cooperation considering their economic efficiency, the changes in the utility of the power suppliers are also greater than or equal to zero (\(\Delta U_{1,t}\geq 0\)). By contrast, without any incentive, the changes in the utility of the consumers are negative (\(\Delta U_{2,t}<0\)), because their utility function is maximized (approximately maximized in the actual situations) at the standard electricity consumption. By assumption 1, we can represent the electricity price that maximizes the consumers' utility at \(d_{t}^{\prime}\) as:  
\[p_{t}=\left.\frac{\partial F_{t}}{\partial d_{t}}\right|_{d_{t}=d_{t}^{*}}. \tag{17}\]  
If the power suppliers set higher electricity price \(p_{t}^{+}\) than its standard, the consumers' utility decreases according to Equation (16), and thus, the electricity consumption is reduced (\(\Delta d_{t}<0\)). Meanwhile, the electricity consumption is encouraged by setting lower electricity price \(p_{t}^{-}\) (\(\Delta d_{t}>0\)). The acceptable conditions \(p_{t}^{+}\) and \(p_{t}^{-}\) to achieve the target of the price-based DR are separately derived as:  
\[\left.\frac{\partial F_{t}}{\partial d_{t}}\right|_{d_{t}=d_{t}^{*}}\leq p_{t }^{+}\leq\left.\frac{F_{t}(d_{t}^{\prime})}{d_{t}^{\prime}}, \tag{18}\]  
\[\left.\frac{p_{t}^{*}dcooperation cannot reach the target of DR request. With a view to minimizing the burden in the price-based DR, the authors defined the optimal electricity price as:  
\[p_{t}^{\prime}=\frac{\partial F_{t}}{\partial d_{t}}\bigg{|}_{d_{t}=d_{t}^{ \prime}}. \tag{20}\]