## 4 Reactive Power Incentive Method
### Contribution Factor  
DisCo pays incentives for requesting reactive power cooperation. Due to the decision of the incentive amount, DisCo has two processes to assess the contribution; (1) decide and notify \(Q_{i}^{*}(t)\) based on the day-ahead scheduling, (2) the contribution factor is calculated based on the difference between \(Q_{i}^{*}(t)\) and the provided reactive power \(Q_{i}(t)\).  
The proper value \(Q_{i}^{*}(t)\) is generated by solving Equation (16):  
\[Q_{i}^{*}(t)=\arg\min F_{con}(t). \tag{16}\]  
Then, consumers are requested to use the optimal reactive power amount allocated as the reference value \(Q_{i}^{*}(t)\) for the proposed DR event. Consumers can then make a contribution to reactive power if they can provide a value close to this reference value for the cooperative control. DisCo calculates the contribution rate according to the consumer's reactive power supply amount. In this work, linear and sigmoid functions have been developed to calculated the contribution factor.  
Linear contribution function:  
The contribution rate defined as a linear function is shown in Figure 4, in which the slope of the linear contribution function is generalized as follows:  
\[A_{lin,i}(t)=\begin{cases}\frac{1}{Q_{i}^{*}(t)},&\text{if }0\leq|Q_{i}(t)|\leq|Q_{i}^{*}(t)|\\ \frac{-1}{Q_{i}^{*}(t)}+2,&\text{if }|Q_{i}^{*}(t)|\leq|Q_{i}(t)|\\ 2\times\frac{1}{Q_{i}^{*}(t)},&\text{others}.\end{cases} \tag{17}\]  
Then, the contribution factor is determined as follows:  
\[\kappa_{lin,i}(t)=A_{lin,i}(t)\times Q_{i}(t) \tag{18}\]  
When the \(\kappa_{lin,i}(t)\) indicates a negative value, the participating customer will have a penalty applied.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Cost (JPY) \& Capacity (kVar)** \\ \hline \(C_{SVC}\) & 1.5 \(\times 10^{8}\) (PYY) \\ \(C_{SVR}\) & 3.0\(\times 10^{8}\) (PYY) \\ \(C_{Q}\) & 3.15\(\times 10^{9}\) (PYY) \\ \(S_{SVC}\) & 250 (kVar) \\ \(v_{q}\) & 4.8 or 3.4 (JPY/kVarh) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for the comparison case.  
Figure 4: Contribution factor of the reactive power incentive.  
Sigmoid function:  
The contribution factor is written by applying the sigmoid function as:  
\[\kappa_{sig}=2\times\left\{\frac{1}{1+exp(-A_{sig}*Q_{i}(t))}-0.5\right\}. \tag{19}\]  
where coefficient \(A_{sig}\) determines the shape of sigmoid function, whenever Equation (19) \(-1<\kappa_{sig}<1\). \(A_{sig,i}(t)\) is given by solving Equation (19) for each time step \(t\):  
\[A_{sig,i}(t)=\ln\left\{1-\frac{2}{1+\kappa_{ref}}\right\}\times\frac{1}{Q_{i}^ {*}(t)} \tag{20}\]  
Here, \(\kappa_{ref}\) is the contribution reference value when DisCo obtained \(Q_{i}(t)=Q_{i}^{*}(t)\). Furthermore, in order to determine the proper reactive power contribution, \(\kappa_{ref}=0.9\) is assigned in the work, since \(\kappa_{ref}=1(\because Q_{i}(t)=\infty)\). The sigmoid contribution factor is given by substituting \(A_{sig,i}(t)\) into Equation (20), yielding:  
\[\kappa_{sig,i}(t)=2\times\left\{\frac{1}{1+exp(-A_{sig,i}(t)*Q_{i}(t))}-0.5 \right\}\times 100[\%] \tag{21}\]  
The \(\kappa_{sig,i}(t)\) is plotted against several reference values \(Q_{i}^{*}(t)\) in Figure 5a. A feasible region comparison for linear and sigmoid contribution functions is illustrated in Figure 5b.