# Day-Ahead PV Generation Scheduling in Incentive Program for Accurate Renewable Forecasting
## 3 Modeling of the PV Generation Scheduling Problem
### Linearization of the PV Generation Scheduling Problem  
The generalized incentive unit price in Equation (15) has a nonlinear structure, making it impossible to solve the nonlinear constraint in the MILP model. Therefore, to formulate the scheduling problem as a MILP model, a new variable is introduced as follows:  
\[e_{t}^{s}=\sum_{k=1}^{K}e_{t,k}^{s} \tag{17}\]  
where \(e_{t,k}^{s}\) is a fragmented energy deviation of the \(k\)-th interval when dividing the \(e_{t}^{s}\) into \(K\) intervals. Equation (17) allows \(e_{t}^{s}\) to be divided into \(K\) intervals since \(e_{t,k}^{s}\) is accumulated to satisfy \(e_{t}^{s}\).  
\[e_{t,k}^{s}\leq(E_{k}-E_{k-1})\times w_{t,k}^{s} \tag{18}\]  
\[w_{t,k}^{s}\leq w_{t,k-1}^{s} \tag{19}\]  
where \(w_{t,k}^{s}\) is a binary variable indicating in which of the \(k\)-th energy deviation \(e_{t,k}^{s}\) is to be accumulated. Equation (18) restricting the value determination of \(e_{t,k}^{s}\) for each interval to within an interval size \(E_{k}-E_{k-1}\) using a binary variable \(w_{t,k}^{s}\). Equation (19) allows \(w_{t,k}^{s}\) to increase sequentially for \(k\), and through Equation (18), \(e_{t,k}^{s}\) can be increased sequentially for \(k\) accordingly.  
Ultimately, the incentive unit price \(i_{t}^{s}\) is determined based on the energy deviation as follows:  
\[i_{t}^{s}=\sum_{k=1}^{K}(u_{k}-u_{k-1})\times w_{t,k}^{s} \tag{20}\]  
Figure 3 shows an example of applying the linearization method to determine the incentive unit price considering the energy deviation. If the first interval is defined as ranging from 0% to 6%, the second interval as 6% to 8%, and the third interval as 8% to 100%, then if the percentage of forecast error is 7%, \(e_{t,1}^{s}\), \(e_{t,2}^{s}\), and \(e_{t,3}^{s}\) will have values of 6%, 1%, and 0%, respectively. In addition, to maximize \(i_{t}^{s}\), \(w_{t,1}^{s}\), \(w_{t,2}^{s}\), and \(w_{t,3}^{s}\) will have values of 1, 1, and 0, respectively.  
In order to linearize the calculation of incentive revenue, ESS operation, and PV curtailment are both set to zero within the portion of incentive revenue calculation. Therefore,  
Figure 3: A concept of linearization method for determining the incentive unit price.  
by multiplying the PV generation by the incentive unit price determined in Equation (20), Equation (16) can be modified in the mixed integer linear optimization model as follows:  
\[\underset{a_{t}}{maximize}\sum_{s=1}^{S}p^{s}\sum_{t=1}^{T}output_{t}^{s}\times \left(MP_{t}+\alpha_{pv}\times REC\right)+\left(PV_{t}^{s}\times\tilde{t}_{t}^{s}\right) \tag{21}\]  
A linearization process is required because Equation (14) contains a nonlinear structure due to the absolute value. Therefore, it can be expressed in the mixed integer linear optimization model as follows [20]:  
\[-e_{t}^{s}\leq\frac{output_{t}^{s}-a_{t}}{CAP}\times 100\% \tag{22}\]  
\[\frac{output_{t}^{s}-a_{t}}{CAP}\times 100\%\leq e_{t}^{s} \tag{23}\]  
The formalized scheduling problem has the objective function of Equation (21) and can be defined as a mixed integer linear optimization problem with Equations (4)-(13), (17)-(20) and (22)-(23) as constraints.