# Wind-Energy-Powered Electric Vehicle Charging Stations: Resource Availability Data Analysis
## 3 Methodology
### Wind Energy Calculation  
The 68 utilized wind turbines operate at different hub heights. The wind speed data for these hub heights are mostly not available. The wind speed observations of the hub heights 20 m and 26 m above ground for M2 and M4, respectively, were selected for wind turbine power analysis. The power law wind profile (Hellmann exponential law) was used to extrapolate the wind speed observations to higher hub heights, which is expressed by:  
\[v=v_{0}\left(\frac{z}{z_{0}}\right)^{\alpha} \tag{1}\]  
where \(v\) is the wind speed at height, \(z\), \(v_{0}\) is the measured speed, height \(z_{0}\) is known, and \(\alpha\) is the power law exponent (shear exponent or Hellman exponent) describing the terrain topology, varying with atmosphere stability (temperature changes), which is commonly set to 1/7 or 0.143 for open land [31]. However, in this study the exponent \(\alpha\) parameter was of less concern than the stability of the wind speed itself for charging EVs. Therefore, \(\alpha=0.143\) was chosen for this assessment. The instantaneous output power of a wind turbine \(P_{w}(t)\) as a function of the wind speed \(v\) at a turbine hub height \(z\), given the power curve \(P_{c}\), is described by:  
\[P_{w}(t)=\left\{ \begin{array}{ll}0,&\quad\quad\text{if }v\leq v_{cut_{in}}\\ P_{c},&\quad\text{if }v_{cut_{in}}<v<v_{r}\\ P_{r},&\quad\text{if }v_{r}\leq v<v_{cut_{out}}\\ 0,&\quad\text{if }v\geq v_{cut_{out}}\end{array} \right. \tag{2}\]  
where \(v_{cut_{in}}\), \(v_{cut_{out}}\), and \(v_{r}\) are, respectively, the cut-in, cut-out, and nominal (rated) speed of the wind turbine (see Figure 3). \(P_{r}\) represents the turbine output power at rated wind speed \(v_{r}\). However, the actual wind power output is usually more than the theoretically calculated power, as in Equation (2), which may produce an estimation error of about 3%. This is because the gained momentum by the rotating turbine blades sustains the continual rotation when a sudden decrease in wind speed occurs with no major change in the angular wind direction [32]. The output power \(P(t)\) quantifies  
Figure 3: Power curves of 68 wind turbines with different capacities (10 to 9500 kW). the variable turbine power at each time point \(t\) (in minutes). However, the charging time of each EV battery depends on the stability of the supplied power, the battery capacity \(B_{c}\), the battery SoC, and the charging point specifications. In order for the EV battery to be fully charged, the charging time \(t_{charge}\) can be obtained according to [13] by:  
\[t_{charge}=\frac{B_{c}\times(1-SoC)}{P_{charger}}. \tag{3}\]  
Fast charging is a technology that allows higher currents to be delivered to the battery before it reaches its peak voltage (usually 80% \(SoC\)). Each EV model has its predefined parameters, such as \(B_{c}\) and \(t_{charge}\) for different \(P_{charger}\) capacities. In this study, we assumed that the \(t_{charge}\) value is known for carrying out the wind power analysis. Due to the intermittency of wind energy over time, the isolated EV charging station was restricted to the stability of the supplied piecewise energy. As shown in Figure 2 and described in Algorithm 1, the wind energy fed to the EV station is subject to several constraints, as detailed in the rest of this subsection. The instantaneous power contained in each \(t_{charge}\) interval must ensure the stability of the power supply to EVs with minimum variability, i.e., the standard deviation \(\sigma\big{(}Pv\big{(}t_{charge}\big{)}\big{)}\) must not exceed 0.1 kW. The interval-based average output power is selected such that  
\[ \begin{split}\overline{P_{w}}=\frac{\sum_{t}P_{w}(t_{charge})}{ t_{charge}}\\ \text{subject to }o\big{(}Pv\big{(}t_{charge}\big{)}\big{)}<0.1\text{ kW}.\end{split}