Wind Energy

180 F. Böttcher et al.
symmetric around V , neglecting the terms O(v 3 ), and averaging (32.1) one
obtains:
〈P (u)〉 = Pr(V )+ 1 ∂
2
2P (V )
∂u2 · σ 2 , (32.2)
where σ 2 = � v 2 (t) � denotes the variance of fluctuations. Equation (32.2) shows
that the “real” power curve Pr(V ) (as it would be realized in laminar wind
flows) has to be modified. This modification is strong for large variance of
fluctuations. For negative curvatures the “real” power curve is underestimated
by 〈P (u)〉 while for positive curvatures it is overestimated.
Obviously (32.2) is only appropriate for small fluctuations, i.e., for small
turbulence intensities ζ = σ/V . In addition to that Pr(V ) is in general not
known in advance [2].
We thus propose an alternative approach (based on the theory of Langevin
processes) to determine the “real” power curve even for very noisy (turbulent)
wind conditions. To this end a simple power curve model will be analyzed.
32.1.1 Reconstruction of a Synthetic Power Curve
A Langevin equation generally describes the temporal evolution of a state
vector q(t) =(q1(t),q2(t), ..., qn(t)) and has the following form:
˙qi = D (1)
i (q)+ �
��
D (2) �
(q) Γj(t), (32.3)
j
where D (1) denotes the deterministic drift coefficient, D (2) the stochastic
diffusion coefficient [3], and Γ (t) Gaussian distributed white noise (with
〈Γi(t)Γj(t ′ )〉 = δijδ(t − t ′ )). The coefficients are obtained via the conditional
moments
M (1)
i (q,τ)=〈qi(t + τ) − qi(t)〉| q(t)=q,
M (2)
ij (q,τ)=〈[qi(t + τ) − qi(t)][qj(t + τ) − qj(t)]〉| q(t)=q (32.4)
by first dividing them by τ and then calculating the limit τ → 0. For a good
temporal resolution the relation between moments and coefficients can be
approximated according to
M (1)
i (q,τ) ≈ τ · D (1)
i (q); M (2)
ij
ij
(q,τ) ≈ τ · D(2) ij (q). (32.5)
In the following we will consider a simple approach identifying q(t) with
(u(t),P(t)):
˙u = −α [u(t) − V ]+ � β · Γ (t),
˙
P = κ [Pr(u(t)) − P (t)] . (32.6)
Thus the only terms entering (32.6) are D (1)
u which is linear in u, D (2)
uu as a
nonzero constant and D (1)
P which is linear in P . In this case the velocities are