Wind Energy

27
Phenomenological Response Theory
to Predict Power Output
Alexander Rauh, Edgar Anahua, Stephan Barth and Joachim Peinke
27.1 Introduction
This contribution is on power prediction of wind energy converters (WEC)
with emphasis on the effect of the delayed response of the WEC to fluctuating
winds. Let us consider the wind speed–power diagram, Fig. 27.1a, with a
typical power curve of a 2 MW turbine.
Suppose at time t0 the system is at working point {U0, L0} outside the
power curve with a wind speed U0 which is constant for a long time. Then, for
t>t0, the system will move toward the power curve, either from above or from
below. The power curve acts as an attractor [2]. What happens in a fluctuating
wind field? Let us follow a short-time series in the wind-speed power diagram,
see Fig. 27.1b. We start at some time t0 with a wind speed U(t0) and a
power output L(t0). At the next time step, one observes a jump to the point
{U(t1), L(t1)}, then to the point {U(t2), L(t2)}, and so on. In Fig. 27.1b,
consecutive points are connected by straight lines to form a trajectory. In the
ideal case of an instantaneous response of the turbine, and in the absence of
noise, all points would lie on the power curve. Actually, the turbine reacts with
a delayed response to the wind-speed fluctuations. The timely changes of the
wind speed, ˙ U, together with a finite response time hamper accurate power
prediction by means of the power curve alone. The application of a suitable
reponse theory may help to properly include the influence of turbulent wind
in the power assessment. In Fig. 27.2 we show a typical point cluster which is
broadly spread around the power curve.
In the following we will discuss in some detail the main idea of a previously
published phenomenological response theory [1]. We also will propose an
extremal principle to establish an empirical power curve from measurement
data [4]. The method is similar but not identical to the attractor principle
applied elsewhere [2]. In addition we present an elementary theorem on power
prediction in the case of a constant relaxation time of the WEC.