Power curves for wind turbines - Wind Power Generation and Wind Turbine Design - page 605

Figure 7: Illustration of the concept of fi xed points. For constant wind speed u ,
the power would relax to the stable value P s ( u ). The deterministic drift
D (1) ( P ), denoted by vertical arrows, drives the system towards this fi xed
point (see text). The sketch was taken from [17].

in such a state, this means that no deterministic drift will occur (see also Fig. 7).
(To separate stable from unstable fi xed points, also the slope of D (1) ( P ) has to be
considered [ 16 ].) D (1) ( P ) can be interpreted as an average slope of the power signal
P ( t ), depending on the power value. For the stable fi xed points this drift function
vanishes because for constant u the power would also be constant, and thus the
average slope of the power signal would be zero. A simple functional ansatz for
D (1) would be D (1) ( P ) = k [ P s ( u )

P ( t )], where P s ( u ) is a point on the ideal power
curve (see the explanation of P s below).
Using their defi nition [15], the drift and diffusion functions can be derived
directly from measurement data as conditional moments (called Kramers-Moyal
coeffi cients):

−

1
() im

n

( )

(9 )

D ()

P = Pt ( + t ) − Pt () Pt () = P ,

n t

t → 0

where n = 1 for the drift and n = 2 for the diffusion function. The average is performed

over t . The condition inside the brackets means that the difference [ P ( t + t )

〈 · 〉

P ( t )] is
only considered for those times for which P ( t ) = P . This ensures that averaging is
done separately not only for each wind speed bin u i but also for each level of the
power P . If one considers the state of the power conversion system as defi ned by
u and P , one could speak of a “state-based” averaging in contrast to the temporal
averaging performed in [3].
Using the mathematical framework of eqns (8) and (9), also uncertainty estima-
tions can be performed for the fi xed points. For details of the derivation, the reader
is kindly referred to [16, 17]. Figure 8 presents the dynamical power characteristic

−