## 2 Cost of Reactive Power Production from DERs  
### Synchronous Machine-Based DGs  
An offer price framework used by a synchronous generator was developed in [19] for a reactive power market. In this section, a new framework is formulated to determine the cost of reactive power support for a synchronous machine-based DG (briefly speaking, DG). The capability curve of a synchronous generator is depicted in Figure 1. This diagram demonstrates the relationship between active and reactive power generated by this generator. In this study, it is assumed that there is a mandatory reactive power requirement approved by grid code for connectivity of a DG to the grid. We consider the grid code requirement to be such that the DG units should operate between a mandatory leading power factor ( \(pf_{mand}\) ) and lagging \(pf_{mand}\) at any operating point. Three operating regions for a DG on the reactive power generation (at an active power \(P_{A}\)) can be identified as follows:  
_Region I: \(-Q_{mand}^{A}\) to \(Q_{mand}^{A}\)_: Reactive power produced in this region is based on the grid code requirement. It is suggested that the DG is paid an _availability cost_ (with a price \(\rho_{0}\) in \(\$\)), which is a fixed component implying the portion of a supplier's capital cost that can be contributed to reactive power production.  
_Region II: ( \(Q_{mand}^{A}\) to \(Q_{A}\)) and (\(Q_{DG}^{min}\) to \(-Q_{mand}^{A}\))_: In this area, the DG is demanded by the DSO for additional reactive power provision beyond the amount of mandatory reactive power support, without requiring reduction of its generated active power. Due to increased losses in the windings, the DG can expect to receive a payment for its services. This payment consists of two components, referred to as the _availability component_ and the _cost of loss component_. In this region two loss payment components are defined as price \(\rho_{1}\) in \(\$\)/MVArh for operating in the under excitation mode and price \(\rho_{2}\) in \(\$\)/MVArh for operating in over excitation mode [19]. _Region III: (\(Q_{A}\) to \(Q_{DG}^{max}\))_: In this region the DG is requested by DSO to reduce its active power production so that the system reactive power requirements are fulfilled. Thus, together with the two other components, it deserves to receive an additional payment according to its _opportunity cost_ of  
Figure 1: Capability diagram of a synchronous generator. reduced active power generation. In this study, an adjustment bid is used for opportunity cost payment [20]. The adjustment bids are employed by the generation units to indicate information about the prices received for the reduction (change) in its initial active power scheduled by the Market Operator (MO). Also, along with this price, the maximum changes regarding the initial active power accepted are specified and sent to the DSO. In region III, if the corresponding change in the active power, which has a negative value, is represented by the \(\Delta P^{Adj}\) variable, the opportunity cost payment can be calculated by multiplying \(\left|\Delta P^{Adj}\right|\) by the adjustment price (\(p_{\text{Adj}}\)). Moreover, a change in the initial generation schedule of a DG may be due to the enforcement of operation constraints or to allow a rescheduling of initial active power output to compensate for active power reductions of other DGs operated in region III. This change can take both positive and negative values, which in both cases are compensated for by the adjustment price. Accordingly, the cost of reactive power production for each DG unit can be mathematically represented by the following equation, assuming that \(P_{A}=P_{0}\) represents the initial active power schedule:  
\[COST(Q_{DG})=