Power curves for wind turbines - Wind Power Generation and Wind Turbine Design - page 604

Figure 6: One hertz measurements of wind speed and electrical power output of
a multi-MW wind turbine. Wind speed u n is normalized to equal power
at standard conditions, and power is normalized to rated power P r , both
according to IEC 61400-12-1 [3]. For a description of the measurements,
see [ 11 , 12 ].

speed, see also [13, 14]. For the (hypothetical) case of a constant wind speed u , the
electrical power output would relax to a fi xed value P s ( u ). Mathematically, these
power values P s ( u ) are called stable fi xed points of the power conversion process.
It is possible to derive them even from strongly fl uctuating data as shown in
Fig. 6 . To this end the wind speed measurements are divided into bins u i of 0.5 m/s
width, as it is done in [3]. It is thus possible to account to some degree for the non-
stationary nature of the wind, and obtain quasi-stationary segments P i ( t ) for those
times t with

∈ . The following mathematical considerations will be restricted
to those segments P i ( t ). For simplicity, the subscript i will be omitted and the term
P ( t ) will refer to the quasi-stationary segments P i ( t ).
The power conversion process is now modeled by a fi rst order stochastic dif-
ferential equation, the Langevin equation (which is also the reason for the name
Langevin Power Curve):

ut () u

i

d

(8 )

(1) (2)

Pt () = D () P + D () ( . P ⋅ Γ t

dt

Using this model, the evolution of the power signal is described by two terms.
The fi rst one, D (1) ( P ), represents the deterministic relaxation of the turbine,
which leads the power towards the fi xed point of the system. According to this
effect, D (1) ( P ) is commonly denoted as drift function . The second term involving
D (2) ( P ), serves as a simplifi ed model of the turbulent wind which drives the system
out of its equilibrium. The function ( t ) denotes Gaussian distributed, uncorrelated

noise with variance 2 and mean value 0. D (2) ( P ) is commonly denoted as diffu-
sion function . More details on the Langevin equation can be found in [15, 16].
For the power curve, only the deterministic term D (1) ( P ) is of interest. The stable
fi xed points of the system are those values of P where D (1) ( P ) = 0. If the system is

Γ