# Derivation of wind farm power

The interaction between wind flow and the rotating blades of a wind turbine generates turbulent wind flow, called a wake, resulting in non-uniform wind speed profile behind the wind turbine. A wake has been mathematically described in terms of both its domain (shape) and value (as the ratio of wind speed reduction). One of the most prevalent wake models is the Park wake model [6] and [7], which expresses the wind speed at the downstream-wake distance d and the radial-wake distance r in the wake formed behind the upstream wind turbine with an induction factor of α asequation(4)u(d,r)=(1-δu(d,r,α))Uu(d,r)=1-δud,r,αUwhere δu(d,r,α)δud,r,α is termed the wind speed deficit factor that TAE226 quantifies how much the wind speed at (d,r)(d,r) is reduced by the wake. As shown in Fig. 1, the Park wake model assumes microfilaments the radius of the wake increases linearly with the downstream distance d as R(d)=R0+κdRd=R0+κd, where R0R0 is the radius of the wind turbine rotor and κ is the wake expansion rate affected by the surface roughness of a site. Furthermore, for the Park wake model, the wind speed is assumed uniform inside the wake with the wind speed deficit factor δu(d,r,α)δud,r,α expressed as [6]:equation(5)δud,r,α=2αR0Rd2,ifr≤R(d)0,ifr>R(d)Note that the wind speed abruptly changes at the boundary of the wake region, corresponding to the boundary surface of a 3-D cone whose radius is R(d)=R0+κdR(d)=R0+κd.