Towards uncovering the structure of power fluctuations of wind farms

The structure of the turbulence-driven power fluctuations in a wind farm is fundamentally described from basic concepts. A derived tuning-free model, supported with experiments, reveals the underlying spectral content of the power fluctuations of a wind farm. It contains two power-law trends and oscillations in the relatively low- and high-frequency ranges. The former is mostly due to the turbulent interaction between the flow and the turbine properties, whereas the latter is due to the advection between turbine pairs. The spectral wind-farm scale power fluctuations ${\mathrm{\Phi }}_P$ exhibit a power-law decay proportional to $f^{-5/3-2}$ in the region corresponding to the turbulence inertial subrange and at relatively large scales, ${\mathrm{\Phi }}_P\sim f^{-2}$. Due to the advection and turbulent diffusion of large-scale structures, a spectral oscillation exists with the product of a sinusoidal behavior and an exponential decay in the frequency domain.

Figure 1

(a) Photograph of the test section pointing upwind. (b) Mean velocity $U/U_{\mathrm{hub}}$. (c) Turbulence intensity $I_u={\sigma }_u/U_{\mathrm{hub}}$. Horizontal lines indicate the turbine-hub height.

Figure 2

Spectra of incoming turbulence ($\text{red}$), $\widehat{G}\left(f\right)$ ($\text{blue}$), and turbine output power ($\text{black}$) of (a) first and (b) fourth row, $S_x=7$; (c) fourth row, $S_x=10$.

Figure 3

Premultiplied power spectra of (a) the local incoming velocity and (b) the power output of the first, second, and fourth rows with $S_x=7$ and 10 (solid and dotted lines).

Figure 4

Measured and modeled spectra of hub-height velocity of incoming flow (subfigures) and output power inside the wind farm. (a) Fifth row, $S_x=7$; (b) fourth row, $S_x=10$.

Figure 5

Measured and modeled power variance of individual turbines in the wind farm with $S_x=7$ ($\circ$) and $S_x=10$ ($▵$).

Figure 6

(a) Mean velocity ($U/U_{\mathrm{hub}}$) and (b) turbulence intensity $I_u$ within the $S_x=7$, and (c) mean velocity ($U/U_{\text{hub}}$) and (d) turbulence intensity $I_u$ within the $S_x=10$ model wind farm at hub height.

Figure 7

Measured and modeled spectra of output power in the wind farm. (a) Five rows, $S_x=7$; (b) four rows, $S_x=10$.