## 4 HFAJAYA for the Multi-Constraint OPF Problems
### OPF Solutions, Including Stochastic Solar and Wind Power
#### 4.2.2 The Units of Wind Power  
An upcoming wind energy profile must be anticipated to create a working optimization approach for addressing OPF issues. The predictors in this work were generated using a Weibull probability distribution function [6; 7; 8]. Before establishing the optimal solution technique, the wind energy estimating work can be completed independently. The wind power generation model is often built using wind speed variables. In this part, the wind speed \(f_{v}(v)\) is provided and simulated mathematically using the Weibull probability distribution function basis of dimensionless form factor (\(K\)) and step size (\(c\)) [25]:  
\[f_{v}(v)=\frac{k}{c}\Big{(}\frac{v}{c}\Big{)}^{k-1}\times e^{-(\frac{v}{c})^{k}} \tag{34}\]  
As expressed in (35), \(\Gamma(x)\) is mainly accountable for the average of the Weibull probability distribution (\(M_{wbl}\)), as illustrated in Formula (36):  
\[\Gamma(x)=\int_{0}^{\infty}e^{-t}t^{x-1}dt \tag{35}\]  
\[M_{wbl}=c*\Gamma\Big{(}1+K^{-1}\Big{)} \tag{36}\]  
Generally, the wind turbine may transform the moving energy of the wind into electrical energy. Equation (37) shows the real production power of a wind turbine, \(P_{w}(v)\), as a function of wind speed:  
\[P_{w}(v)=\left\{ \begin{array}{ll}0;&\
u\leq\
u_{in}\text{ and }\
u>\
u_{out}\\ P_{w\tau}\Big{(}\frac{v-v_{in}}{v_{\tau}-v_{in}}\Big{)};&v_{in}<\
u\leq\
u_{r} \\ P_{\pi\pi};&\
u_{r}<\
u\leq\
u_{out}\end{array} \right. \tag{37}\]  
where \(P_{w\tau}\) is the wind turbine's rated power, the wind turbine's cut-in wind rate is \(v_{in}\), \(v_{out}\) is the cut-out wind rate, and \(v_{r}\) is the valued wind speed. The Weibull probability distribution was used in this study to include a stochastic procedure in the optimization approach. The Weibull probability distribution estimate offers the doubt findings for wind energy generation. The cost of wind power production units must be determined to reduce the power generation whole cost. The entire cost of a wind power generator is evaluated based on flow rate and real power (produced by wind turbines) to decrease the impact of unpredictability in wind power patterns. Equations (38)-(40) compute the direct, reserve, and penalty expenses in (\(USD\)/\(h\)), respectively. Equation (41) describes the overall cost of wind power generation in (\(USD\)/\(h\)), \(C_{W}^{T}\), which comprises three primary items: straight wind turbine cost, reserve or overestimation, and penalty costs [25; 44]. \[C_{w,j}\big{(}P_{ws,j}\big{)}=g_{j}P_{ws,j} \tag{38}\]  
\[ \begin{array}{l}C_{Rw,j}\big{(}P_{ws,j}-P_{wav,j}\big{)}=K_{Rw,j}\big{(}P_{ ws,j}-P_{wav,j}\big{)}\\ =K_{Rw,j}\int_{0}^{P_{ws,j}}\big{(}P_{ws,j}-P_{w,j}\big{)}f_{w}\big{(}P_{w,j} \big{)}dp_{w,j}\end{array}  \tag{39}\]  
\[ \begin{array}{l}C_{Pw,j}\big{(}P_{wav,j}-P_{ws,j}\big{)}=K_{Pw,j}\big{(}P_{ wav,j}-P_{ws,j}\big{)}\\ =K_{Pw,j}\int_{P_{ws,j}}\big{(}P_{w,j}-P_{ws,j}\big{)}f_{w}\big{(}P_{w,j} \big{)}dp_{w,j}\end{array}  \tag{40}\]  
\[C_{W}^{T}=\sum_{j=1}^{N_{W}}\big{[}C_{w,j}\big{(}P_{ws,j}\big{)}+C_{Rw,j} \big{(}P_{ws,j}-P_{wav,j}\big{)}+C_{Pw,j}\big{(}P_{wav,j}-P_{ws,j}\big{)} \big{]} \tag{41}\]  
where \(g_{j}\) is the direct cost coefficient associated with \(j\)th wind power plant, \(P_{ws,j}\) is the scheduled power from the same plant, \(K_{Rw,j}\) indicates the reserve cost coefficient pertaining to \(j\)th wind power plant, \(P_{waw,j}\) defines the actual available power from the same plant, \(K_{Pu,j}\) is the penalty cost coefficient for the \(j\)th wind power plant, and \(P_{aw,j}\) is rated output power from the same windfarm. Wind energy providers generally supply power network operators with an expected power generation profile. The network operator creates an operation plan for all producing units based on the expected wind power output to fulfill the demand. Suppose the actual wind power output is less than the predicted value. In that case, the reserve cost is applied to offset the anticipated value. A penalty fee is applied if the actual wind power consumption surpasses the predicted value.