## 2 Proposed Energy Cost Model for Offshore Wind Farms
### Annual Energy Production
#### 2.2.2 Weibull Probability Density Distribution of Offshore Wind Statistics  
In this study, the Weibull probability density function \(f_{i}(v)\) was employed to represent the offshore wind statistics of the installation site, which depended on the Weibull scale and the shape parameters \(k_{i}\) and \(c_{i}\) that determined the shape and intensity of the wind during one year on a site [27].  
\[f_{i}(v)=(k_{i}/c_{i})(v/c_{i})^{k_{i}-1}e^{-(v/c_{i})^{k_{i}}}, \tag{12}\]  
where \(c_{i}\) and \(k_{i}\) denote the Weibull scale and the shape parameters in the location of wind turbine \(i\), respectively.  
Assuming that the shape factor \(k_{0}\) at the reference hub height \(H_{0}\) and the annual mean wind speed \(v_{mi}\) are known constants for the offshore wind farm, the scale factor \(c_{0}\) can be calculated as:  
\[c_{0}=v_{mi}/\Gamma(1+1/k_{0}), \tag{13}\]  
where \(\Gamma\) is the gamma function.  
The wind speed becomes stronger as the altitude, i.e., the distance from the ground, increases. Based on the relationship between the wind speed and the altitude, the shape factor \(k\) and scale factor \(c\) at the turbine \(i\) hub height H are obtained by:  
\[c_{i}=c_{0}(H/H_{0})^{\alpha}\text{ and } \tag{14}\]  
\[k_{i}=k_{0}[1-0.088\,ln(H_{0}/10)]/[1-0.088\,ln(H/10)], \tag{15}\]  
where, \(\alpha\) is the Hellmann exponent, which depends on surface properties of the wind field. The typical value of \(\alpha\) is 0.1 over water and 0.14 over land [27].  
When the Weibull probability distribution of a single wind turbine in its erected site is mainly determined by the wind characteristics of the wind farm, it is also influenced by the wake effects of the wind farm. Generally, the wake effect leads to a reduced wind speed that is faced by the downwind turbines. In this study, the [PERSON] wake loss model was adopted for its low computational cost. With the [PERSON] wake loss model, the wind velocity deficit behind an upstream wind turbine is calculated by the following equation:  
\[v_{i}=v_{m}[1-(1-\sqrt{1-C_{t}})(\frac{D}{D_{w}})^{2}], \tag{16}\]  
where \(v_{m}\) is the mean wind speed and \(C_{t}\) is the axial induction factor, which is set to 0.88 for simplify [28]. D and \(D_{w}\) denote the wind turbine rotor diameter and wake diameter, respectively. The equation of \(D_{w}\) is given by:  
\[D_{w}=D+2k_{w}x, \tag{17}\]  
where \(x\) is the downstream distance from the wind turbine and \(k_{w}\) presents the wake decay coefficient. In this study \(k_{w}\) was set at 0.04 [28].  
In the actual offshore wind farm, a downstream wind turbine \(i\) was affected by multiple wakes of upstream wind turbines. Supposing that the number of upstream wind turbines is \(M\), the wind velocity of wind turbine \(i\) can be represented as follows:  
\[v_{mi}=v_{m}[1-(1-\sqrt{1-C_{t}})\sqrt{\sum_{j=1}^{M}\frac{1}{(1+2k_{w}\frac{v_{ ij}}{D})}}], \tag{18}\]  
where \(x_{ij}\) denotes the horizontal distance between upstream wind turbine \(j\) and downstream wind turbine \(i\).  
Based on Equations (12)-(18), the shape factor \(k_{i}(H,k_{0})\) can be expressed as a function of hub height and \(k_{0}\), and the scale factor \(c_{i}(H,k_{0},v_{m})\) depends on the hub height, \(k_{0}\), and mean wind speed \(v_{mi}\). From Reference (18), it was gathered that the wind speed extracted by turbine \(i\) is a function of the distance \(x\) between the installation sites of two turbines and the rotor diameter \(D\) or rotor radius, so the scale factor \(c_{i}\) can be expressed as \(c_{i}(H,k_{0},v_{m},R,x)\).  
Therefore, based on Equations (7)-(18), the AEP of the offshore turbines was expressed as a function of the wind statistics factors \(v_{m}\), \(k_{0}\), and \(\alpha\), the turbine characteristics parameters \(v_{c}\), \(v_{r}\), \(v_{f}\) and \(R\) and layout factor \(x\), formulated by  
\[AEP_{i}=f_{AEPi}(R,H,v_{r},v_{c},v_{f},v_{m},k_{0},\alpha,x) \tag{19}\]