## 2 Reactive Power Offers of a DG
Therefore, the QPF of DG \(i\) in these regions is expressed as follows:  
Figure 1:0  
Figure 1:1  
where \(Q2\) and \(Q3\) are in the following inequalities:  
Figure 1:2  
Figure 1:3  
_Region IV_ (\(Q_{A,i}\) to \(Q_{DG,i}^{max}\)): In this region, the DG has to reduce its initial planned active power production to generate the required reactive power. If the required reactive power from the DG is \(Q_{B}\), the operating point requires shifting back along the capability curve to point \((P_{B},Q_{B})\), in which \(P_{B}<P_{A}\). Thus, it deserves to receive an additional payment according to its LOC resulting from the reduced generation of active power, as well as the two other components. In the joint active and  
Figure 1:4 Capability curves of an inverter-based RES-DG for design \(pf\) = 0.95 and different grid voltages. (**a**) \(Vg\) = 0.95 p.u. ; (**b**) \(Vg\) =1.0 p.u. ; and (**c**) \(Vg\) = 1.05 p.u. reactive market, similar to decoupled reactive power market, a DG will be paid for the LOC if its active power output in the joint market is less than that of the energy-only market. In this study, LOC is paid according to an adjustment bid [38; 39; 40] offered by the DG. The adjustment bid is used by a generation unit to submit information about the maximum reduction in its initial planned active power obtained in the energy market that it admits, together with the price that it wants to receive for this change. In region IV, if the corresponding change in the active power which has a negative value is represented by \(\Delta P^{Adj}\) variable, the LOC can be calculated based on multiplying \(|\Delta P^{Adj}|\) by the adjustment price (\(\rho_{Adj}\)). Hence, the QPF of DG \(i\) in region IV is calculated as:  
Figure 1:5  
where \(\Delta P_{DG,i}^{Adj}\) is defined as follows:  
Figure 1:6  
Q4 is constrained by the following inequalities:  
Figure 1:7  
It should be mentioned that \(P_{A}\) is initial scheduled active power which will be obtained in the day ahead energy-only market for each DG. According to the above discussion, the following formulation is introduced for the QPF of DG \(i\)th:  
\[ \begin{split} QPF_{DG,i}=W_{1,i}&\cdot\rho_{0,i}-W_ {2,i}\cdot\rho_{1,i}\left(Q2_{DG,i}+Q_{mand,i}^{A}\right)+W_{3,i}\cdot\rho_{2, i}\left(Q3_{DG,i}-Q_{mand,i}^{A}\right)\\ &\hskip 14.226378 pt+W_{4,i}\left(\rho_{2,i}\left(Q4_{DG,i}-Q_{mand,i}^ {B}\right)+\rho_{Adj,i}|\Delta P_{DG,i}^{Adj}|\right)\end{split}  \tag{12}\]  
\(W_{1,i},\hskip 5.690551 ptW_{2,i},\hskip 5.690551 ptW_{3,i},\) and \(W_{4,i}\) are binary numbers. These numbers are employed to represent a limited region on which the DG operates. The two regions III and IV are merged in Equation (13) compared with (12), since both regions have the same operation payment:  
\[ \begin{split} QPF_{DG,i}=W_{1,i}&\cdot\rho_{0,i}-W_ {2,i}\cdot\rho_{1,i}\left(Q2_{DG,i}+Q_{mand,i}\right)\\ &\hskip 14.226378 pt+\left(W_{3,i}+W_{4,i}\right)\ \rho_{2,i}\left(Q3_{DG,i}-Q_{mand,i}\right)+W_{4,i}\left(\rho_{Adj,i}|\Delta P _{DG,i}^{Adj}|\right)\end{split}  \tag{13}\]  
where, the operation constraints of DG \(i\) are:  
\[W_{1,i},\hskip 5.690551 ptW_{2,i},\hskip 5.690551 ptW_{3,i}\hskip 5.690551 pt,W_{4,i}\in\{0,1\} \tag{14}\]  
\[-W_{1,i}Q_{mand,i}\leq Q1_{DG,i}\leq W_{1,i}Q_{mand,i} \tag{15}\]  
\[W_{2,i}\ Q_{DG,i}^{unit}\leq Q2_{DG,i}\leq-W_{2,i}Q_{mand,i} \tag{16}\]  
\[\left(W_{3,i}+W_{4,i}\right)\ Q_{mand,i}\leq Q3_{DG,i}\leq\left(W_{3,i}+W_{4,i }\right)Q_{cap,i} \tag{17}\]  
\[Q_{mand,i}=\left(P_{DG,i}\right)tan\left(cos^{-1}\left(pf_{mand}\right)\right) \tag{18}\]  
\[Q_{DG,i}=Q1_{DG,i}+Q2_{DG,i}+Q3_{DG,i} \tag{19}\]  
\[W_{2,i}+W_{3,i}+\ W_{4,i}\leq W_{1,i} \tag{20}\]  
In Equation (17), \(Q_{\text{cap,i}}\) is located on the capability curve border in accordance with \(P_{DG,i}\). Equation (20) ensures that the DG operates in only one of the four defined regions.