Wind Energy

31 Characterisation of the Power Curve for Wind Turbines 175
where −αℓ(t) is the deterministic relaxation of the fluctuations, which growth
and decay exponentially, on the stationary power Lfix(u). The term g(L, t)Γ (t)
describes the influence of dynamical noise from the system, e.g. shutdown
states, pitch-angle control, yaw errors, etc. [5].
31.3 Langevin Method
To obtain the stationary power curve by means of fixed points, the stochastic
temporal state n-vector L(t) =(L(t1),L(t2), ..., L(tn)) is assumed to be stationary
for wind velocity intervals: ua ≤ u < ub with an time evolution τ which
is described by an one-dimensional Langevin-equation [3, 4]
d
dt L(t) = D(1) (L)+
��
D (2) (L)
�
· Γ (t),
where D (1) and D (2) are called drift and diffusion coefficients and describe
the deterministic and stochastic part, respectively. √ D (2) describes the amplitude
of the dynamical noise with δ-correlated Gaussian distributed white noise
〈Γ (t)〉 = 0. These coefficients can be separated and quantified from measured
data by the first (n = 1) and second (n = 2) conditional moments [4]
M (n) (L, τ) =〈[L(t + τ) − L(t)] n 〉| L(t)=L.
Under the condition of L(u) =L. The coefficients are calculated according
D (n) 1
(L) = lim
τ→0 τ M (n) (L, τ).
Thus, the fixed points of the power, Lfix(u) = min{φD}, where δφD
δL = −D(1) (L)
is the deterministics potential.
31.4 Data Analysis
The analysis was based on measured data of about 1.6 × 10 6 samples of elect.
power output and wind speed at hub hight of a WEC of 2 MW [5]. First,
D (1) (L) was evaluated in a width of the wind velocity bins of 0.5 m s −1 and
power bins of 40 kW. Next, the fixed points of the power were found by searching
the min{φD}. We show evidence that the power exhibited multiple fixed
points for 〈u〉 < 20 m s −1 where the wind generator was switched to other
rated speed change by means of a maximal power extraction (optimal operation),
e.g. Fig. 31.2. Finally, all fixed points of the power were reconstructed
and presented in a two-dimensional vector field analysis D (1) (L, u) ofthe
deterministics dynamics of power and wind velocity, respectively, Fig. 31.3.