## 2 Wind Power Characterisation  
Wind power is becoming more important in energy generation. In deregulated electrical networks, the distribution market will vary the most, rendering wind-generated power more expensive for commercial and residential users than industrial customers. Electric power generally flows from the substation to the ends of feeders in distribution networks throughout operation and planning. However, the incorporation of WTGS may result in reverse power flow in the distribution lines. There are various approaches for load flow using WTGS for distribution systems described in the literature. In this study, a matrix-based approach is used to integrate power output from WTGS into distribution systems [37]. Induction generators are explored as WTGS power conversion devices; they primarily function as variable-reactive power generators. For the reliable inclusion of wind energy into the grid, the impacts of wind power, which vary with wind speed, have to be analysed using appropriate models. The wind statistics of wind power are thoroughly illustrated in [38] and are derived as follows:  
\[f(V_{w})=\frac{k}{c}\bigg{(}\frac{V_{w}}{c}\bigg{)}^{k-1}\exp\bigg{[}-\bigg{(} \frac{V_{w}}{c}\bigg{)}^{k}\bigg{]} \tag{1}\]  
where \(V_{w}\) represents wind speed, c is the scaling factor, and k is the shape parameter, which modifies the characteristics of the probability density function (PDF). The Weibull continuous PDF is defined as the Rayleigh PDF computed by Equation (2) for k = 2 and is the most commonly used distribution function for estimating wind speeds because it has intervals of low and strong wind speeds. The characteristic feature of such a PDF is the region between any two wind speeds, which represents the possibility that the wind is somewhere between those speeds. The Weibull PDF is the basic foundation for analysing wind speed statistics.  
\[f(V_{w})=\frac{2V_{w}}{c^{2}}\exp\bigg{[}-\bigg{(}\frac{V_{w}}{c}\bigg{)}^{2 }\bigg{]} \tag{2}\]Power output at any specified instant is estimated using the power curve for a specific model of wind turbine. As can be seen in Figure 1, the mechanical power produced by a wind turbine is directly proportional to the wind speed [38].  
The wind turbine power curve given by the manufacturer is used to determine the active power output and the reactive power required is expressed based on the inverter ratings in this study as follows:  
\[Q_{w}=\begin{cases}0.3*Q_{max},&V_{i}\leq V_{min}\\ \theta_{1}*(V_{i}-1),&V_{min}<V_{i}\leq 1\\ -\theta_{2}*(1-V_{i}),&1<V_{i}\leq V_{max}\\ -0.3*Q_{max},&V_{i}>V_{max}\end{cases} \tag{3}\]  
where,  
\[\theta_{1}=\frac{Q_{max}}{V_{min}},\ \ \ \ \&\theta_{2}=\frac{Q_{max}}{V_{max}}\]  
\(Q_{max}\) is the maximum limit the inverter can support for reactive power. To account for voltage changes at the buses, the inverter's dynamic characteristics entail adjusting the slope gradients. The inverter is operable in four major zones. If the voltage exceeds the upper or lower limits, the inverter injects or absorbs 30% of \(Q_{max}\) in reactive power at the bus terminals. Consequently, the remaining two regions use the droop slope gradients (\(\theta_{1},\theta_{2}\)) to evaluate the output corresponding to the \(Q_{max}\) set.