# Optimal Solution of Environmental Economic Dispatch Problems Using QPGPSO-\(\omega\)
## 2 Mathematical Formulations
### Formulations for the Consideration of Resource Variability  
RESs, due to their intermittent nature, require the consideration of over- and underestimations of the resources and corresponding power availability. Overestimation occurs when the estimated power and its corresponding costs are lower than the actual available resource and the resulting power and cost from wind sources. On the other hand, underestimation refers to a situation where the extracted power and its associated costs are higher than the predicted values. The mathematical representations for the overestimation, underestimation, and the direct generation cost of wind resources are provided in (2)-(4) respectively:  
\[f_{wij}\big{(}P_{wij}\big{)}=C_{pwij}\left(P_{ww,ij}-P_{wsh,ij}\right)+C_{rwij} \Big{(}P_{wsh,ij}-P_{wwz,ij}\Big{)}\times\big{(}k_{wij}\times P_{wwz,ij}\big{)} \tag{2}\]  
\[C_{rwij}\left(P_{wsh,ij}-P_{wwz,ij}\right)=k_{rwij}\times\int\limits_{0}^{P_{ wsh,ij}}(P_{iussh,ij}-w_{ij})f_{wij}(w_{ij})dw_{ij} \tag{3}\]  
\[C_{pwij}\left(P_{wwz,ij}-P_{wsh,ij}\right)=k_{pwij}\times\int\limits_{P_{wsh, ij}}^{P_{wz,ij}}(w_{ij}-P_{iussh,ij})f_{wij}(w_{ij})dw_{ij} \tag{4}\]  
where the power generated by the \(j^{\text{th}}\) wind turbine in region \(i\) is represented as \(P_{wij}\). Additionally, the coefficients representing the direct, penalty, and reserve costs of the \(j^{\text{th}}\) wind turbine in region \(i\) are denoted as \(k_{wij}\), \(k_{pwij}\), and \(k_{rwij}\), respectively. The cut-in, cut-out, and rated speeds of wind are represented as \(v_{cut-in}\), \(v_{cut-out}\), and \(v_{r}\), respectively [55]. Similarly, the penalties for over- (reserve) and underestimations, as well as the direct cost associated with PV-based generation, are presented in (5)-(7):  
\[f_{sij}\big{(}P_{sij}\big{)}=k_{sij}\times P_{swz,ij}+C_{psij}\big{(}P_{swz, ij}-P_{ssh,ij}\big{)}+C_{rsij}\big{(}P_{ssh,ij}-P_{swz,ij}\big{)} \tag{5}\]  
\[C_{rsij}\left(P_{ssh,ij}-P_{swz,ij}\right)=f_{s}\Big{(}P_{swz,ij}<P_{ssh,ij} \Big{)}\times\Big{[}E\Big{(}P_{swz,ij}<P_{ssh,ij}\Big{)}\Big{]}\times k_{rsij} \tag{6}\]  
\[C_{psij}\Big{(}P_{swz,ij}-P_{ssh,ij}\Big{)}=k_{psij}\times\Big{[}E\Big{(}P_{ swz,ij}>P_{ssh,ij}\Big{)}-P_{ssh,ij}\Big{]}\times f_{s}\Big{(}P_{swz,ij}>P_{ssh,ij} \Big{)} \tag{7}\]  
The power output extracted from the solar PV system, generated by the \(j^{\text{th}}\) solar plant in the designated \(i\) region, is represented as \(P_{sij}\). The coefficients associated with the costs for direct, penalty, and reserve expenses for the \(j^{\text{th}}\) PV plant in the \(I\) region are spec ified as \(k_{sij}\), \(k_{psij}\), and \(k_{rsij}\), respectively. The terms \(f_{s}\big{(}P_{sav,ij}>P_{ssh,ij}\big{)}\), \(f_{s}\big{(}P_{sav,ij}<P_{ssh,ij}\big{)}\) taken from Equations (6) and (7), respectively, represent the potential for the excess and deficiency of scheduled power from the PV source. Also, \(\left[E\big{(}P_{sav,ij}>P_{ssh,ij}\big{)}-P_{ssh,ij}\right]\), \(\left[E\big{(}P_{sav,ij}<P_{ssh,ij}\big{)}\right]\) indicate the anticipated PV power that exceeds or falls below the planned corresponding output [56].  
The third RES considered for power supply was biofuel. To determine the computational cost of power generation from biofuel, a quadratic cost function is employed, as described in (8) [34; 57]:  
\[f_{bij}\Big{(}P_{bij}\Big{)}=a\times P_{i}^{2}+b\times P_{i}+c \tag{8}\]  
where \(a\), \(b\), and \(c\) represent the coefficients for the quadratic cost function.