## 3 Problem Formulation
### DR Optimization Model
The profit-seeking maximization model of the GO, considering the Jordanian power grid consumers, excluding the energy exports, can be expressed by:  
\[Max\sum_{d=1}^{D}\sum_{h=1}^{H}\left(\sum_{c=1}^{C}I_{c,h}(sp_{c,h},ds_{c,h})- \sum_{ps=1}^{PS}C_{ps,h}(bp_{ps,h},dp_{ps,h})\right)+\sum_{c=1}^{C}PCI_{C}(dp_{ c},dpp_{c}) \tag{1}\]  
Subject to:  
\[ds_{c,h}=ds_{c,h}^{0}\left[1+\epsilon_{h}\frac{sp_{c,h}-sp_{c,h}^{0}}{sp_{c,h}^ {0}}+\sum_{h=1}^{24}\epsilon_{h}^{\prime}\frac{sp_{c,h}-sp_{c,h}^{0}}{sp_{c,h}^ {0}}\right] \tag{2}\]  
\[spp_{c,h}...\limp_{c,h}\geq sp_{c,h}\geq sp_{c,h}... \tag{3}\]  
Figure 4: Proposed day-ahead demand response model for Jordan in this study. where \(l_{c,h}\) is the income at hour \(h\ \in\ \{1,2\ \ldots 24\}\) from the bulk consumer of type \(c\in\{1,2,\ldots 5\}\), which is a function of bulk supply price (\(sp_{c,h}\))and the demand sold (\(ds_{c,h}\)) at hour \(h\) for consumer \(c\). \(C_{ps,h}\) is the cost purchasing power from power supplier \(ps\in\{1,2,\ldots PS\}\) at hour \(h\ \in\ \{1,2\ \ldots 24\}\), which is a function of the buying price (\(bp_{ps,h}\)) estimated by the average prices shown in Table 2, and the demand purchased (\(dp_{ps,h}\)). At each hour \(h\), all the costs of purchasing demand from each \(ps\) depict the total hourly cost depending on which power plants were utilized, according to the unit commitment by the GO. \(PCI_{C}\) is the peak capacity income per consumer \(c\) which is a function of the highest demand peak (\(dp_{c}\)) for the respective consumer \(c\) and their demand peak price (\(dp_{c}\)). \(ps\) denotes the total power supply units utilized at the respective hour. \(spp_{c,h\cdot Min}\) and \(spp_{c,h\cdot Max}\) are the upper and lower ranges of the bulk supply price, as determined by the GO. \(\varepsilon_{h}\) and \(\varepsilon_{h}^{\prime}\) represent the self and cross-price elasticities of demand, respectively, where they capture the effect of electricity price change for customers on their electricity consumption. This relationship lies at the heart of DR systems to determine the price set by utilities to achieve economic benefits in the power market and technical benefits related to the operation of the power system [19]. To capture the price elasticity of demand on 24 h under the assumption that the rescheduling of the production does not go beyond a 24-h interval, the PEDM is formulated as [19; 20]:  
\[\left[ \begin{array}{c}\Delta ds_{c,1}/ds_{c,1}^{0}\\ \vdots\\ \Delta ds_{c,i}/ds_{c,i}^{0}\\ \vdots\\ \Delta ds_{c,24}/ds_{c,24}^{0}\end{array} \right]=\left[ \begin{array}{ccccc} \varepsilon_{1,1}&\cdots&\varepsilon_{1,j}&\cdots&\varepsilon_{1,24}\\ \vdots&&&\vdots\\ \varepsilon_{i,1}&\cdots&\varepsilon_{i,j}&\cdots&\varepsilon_{i,j24}\\ \vdots&&&\vdots\\ \varepsilon_{24,1}&\cdots&\varepsilon_{24,j}&\cdots&\varepsilon_{24,24}\end{array}  \right]*\left[ \begin{array}{c}\Delta sp_{c,1}/sp_{c,1}^{0}\\ \vdots\\ \Delta sp_{c,i}/sp_{c,i}^{0}\\ \vdots\\ \Delta sp_{c,24}/sp_{c,24}^{0}\end{array} \right] \tag{4}\]  
The PEMD relates the effect of the change of price in any hour of the day depicted by \(i\) to the change in demand of the hour itself as well as other hours depicted by \(j\). \(\varepsilon_{i,i}\) represents the self-elasticity, which relates the change of price in the period \(i\) to the change in demand in that period, whereas \(\varepsilon_{i,j}\) represents the cross-elasticity, relating the change of demand in hour \(i\) to the change in price in another period \(j\), which are shown as \(\varepsilon_{h}\) and \(\varepsilon_{h}^{\prime}\) in Equation (2), respectively. The proposed model was developed under the following assumptions:  
1. Daily environmental and residential demand data are available with an hourly sample rate. 2. The day-ahead generation electric power prices are available as a single value for each power plant. 3. The day-ahead selected power plants for dispatch by unit commitment are available for each day. 4.