Wind Turbine Power Performance and Application to Monitoring

Abstract

The concept of power performance is introduced as the ability of a wind turbine to extract power from the wind. The general performance estimates such as the power coefficient or the theoretical power curve are introduced in laminar conditions. Following Betz’ limit, an upper limit for the power available in the wind is derived, as well the main sources of energy loss. This laminar theory is too simple to describe operating wind turbines, and turbulent and atmospheric effects call for statistical tools. An IEC norm defines the international standard to measure and analyze power performance. The resulting IEC power curve gives a first estimation, and can be used to evaluate the annual energy production. An alternative is proposed with the Langevin power curve, which quantifies the high-frequency dynamics of a wind turbine power production to changing wind speeds. This brings further insight on the overall performance, and allows for applications such as performance monitoring or power modeling.

Appendices

Appendix 1: Aerodynamics of Rotor Blades

The essential (mechanical) element of a wind turbine is the rotor, that transforms the power of the wind into a rotational or mechanical power. The ideal requirements are:

• the rotation should be steady and smooth;

• dynamical loads should be minimal;

• the regulation should be done without sudden jumps.

The number of blades, their profile and design should guarantee these features. Modern wind turbines rotate due to the lift forces $F_{L}$ acting on the airfoils. For an airfoil the effective area³¹ can be expressed in terms of the depth $t$ and the wingspan $b$ (normally equal to the rotor radius $R$), such that

$$ \begin{gathered} F_{D} = C_{D} (\alpha )\frac{1}{2}\rho c^{2} (t \cdot b) \hfill \\ F_{L} = C_{L} (\alpha )\frac{1}{2}\rho c^{2} (t \cdot b) , \hfill \\ \end{gathered} $$

(31)

where $\alpha$ is the angle of attack, as displayed in Fig. 21. The lift-drag ratio $F_{L} /F_{D}$ relates to the quality of the airfoil. The larger the ratio, the better the quality.

Fig. 21

Cut through an airfoil to illustrate the forces acting on it. The depth $t$ is given by the distance between the leading and the trailing edge. The wingspan $b$ is the length of the airfoil, here perpendicular to the illustrated plane

In Fig. 21, the velocity vector $\mathbf{c}$ gives the wind velocity in the frame of reference of the airfoil. The wind velocity is $\mathbf{u}_{2}$ in the frame of the ground, but the rotational motion of the rotor must be considered for the motion of the wind with respect to the blades. Hence, $\mathbf{c}$ is the superposition of the horizontal axial velocity $\mathbf{u}_{2}$ ³² and of the rotational velocity $v = \omega r$, such that

$$ c^{2} (r) = \left( {2u_{1} /3} \right)^{2} + \left( {\omega r} \right)^{2} . $$

(32)

The rotor feels the effective wind speed $\mathbf{c}$. This is illustrated in Fig. 22.

Fig. 22

Cut through a rotating airfoil. The rotational velocity $\omega r$ is perpendicular to the axial velocity vector ${\mathbf{u}}_{2}$. $\beta$ denotes the angle between the resulting velocity $\mathbf{c}$ and the rotational direction

Instead of integrating the lift and drag forces on the entire airfoil, one can estimate the local force on each infinitesimal element. Also, the total force is divided into its rotational component $F_{r}$ and its axial component $F_{a}$. Considering a cut $dr$ at $r$ in the polar plane of the rotor, the resulting force is

$$ \begin{aligned} F_{r} & = \frac{\rho }{2}c^{2} \cdot t \cdot dr\left[ {C_{L} \cos(\beta ) - C_{D} \sin(\beta )} \right] \hfill \\ F_{a} & = \frac{\rho }{2}c^{2} \cdot t \cdot dr\left[ {C_{L} \sin(\beta ) + C_{D} \cos(\beta )} \right]. \hfill \\ \end{aligned} $$

(33)

Also, on can estimate

$$ \tan(\beta ) = \frac{\omega r}{{u_{2} }} = \frac{\omega R}{{u_{1} }}\frac{r}{R}\frac{{u_{1} }}{{u_{2} }} = \frac{3}{2}\lambda \frac{r}{R} . $$

(34)

The idea is to construct the blades in such a way that for each infinitesimal radial annulus they extract the infinitesimal (Betz) optimal power out of the wind

$$ dP_{r,Betz} = \frac{16}{27} \cdot \frac{\rho }{2} \cdot u_{1}^{3} \cdot (2\pi rdr) . $$

(35)

This power also reads

$$ dP = z \times F_{r} \times \omega r , $$

(36)

where $z$ denotes the number of blades, $\omega r$ the velocity in rotational direction and $F_{r}$ the force in this direction.³³ Inserting Eq. (33) and combining Eqs. (35) and (36), the optimal value of the depth $t$ as a function of $r$ can be determined. Assuming that $C_{D} { \ll }C_{L}$ and a sufficiently large tip speed ratio (for details see [20]), the profile of the airfoil $t(r)$ reads

$$ \begin{aligned} t(r) & \approx \frac{16\pi }{9}\frac{{R^{2} }}{{zC_{L} r\lambda^{2} }} \hfill \\ & \propto z^{ - 1} \cdot C_{L}^{ - 1} \cdot r^{ - 1} \cdot \lambda^{ - 2} . \hfill \\ \end{aligned} $$

(37)

This has an important consequence on the design of rotor blades. The depth decreases with increasing number of blades, larger lift coefficient, increasing radius and especially increasing tip speed ratios. This explains why fast rotating wind turbines tend to have only two or three narrow blades while old western-mill machines have many, rather broad blades.

Appendix 2: A Relaxation Model for the Power Output

As introduced in Sect. 3.3.2, the power output of wind turbines is assumed to be solution of a Langevin equation

$$ \frac{d}{dt}P(t) = D^{(1)} (P) + \sqrt {D^{(2)} (P)} \times \varGamma (t) . $$

(38)

For reminder, the power value $P$ and the functions $D^{(1)}$ and $D^{(2)}$ are conditioned on the wind speed bins, as done in the main text. The subscript $_{i}$ indicating the wind speed bin was dropped for simplicity. $D^{(1)}$ represents the deterministic dynamics of the conversion process, that always push the power output towards the Langevin power curve $P_{L} (u)$. Additional random fluctuations are superposed as a simplified model for all the microscopic degrees of freedom acting on the conversion process.³⁴ A simple but rather realistic ansatz for $D^{(1)}$ would be

$$ D^{(1)} (P) = \alpha \left(P_{theo} (u(t)) - P(t)\right) , $$

(39)

where $D^{(1)}$ linearly drives the power output towards the instantaneous value of the theoretical power curve $P_{theo} (u(t))$, which might read

$$ P_{theo} (u) = \left( {\begin{array}{ll} {P_{r} \left( {\frac{u}{{u_{r} }}} \right)^{3} } &{{\text{for}}\;{\text{u}} \le{\text{u}}_{\text{r}} ,} \\ {P_{r} } & {{\text{for}}\;{\text{u}} \ge {\text{u}}_{\text{r}} .} \\ \end{array} } \right) $$

(40)

Assuming Eq. (39) and a constant diffusion function $D^{(2)} (P) = \beta$,³⁵ the Langevin equation becomes a relaxation model for the power output

$$ \frac{d}{dt}P(t) = \alpha \left( {P_{theo} (u(t)) - P(t)} \right) + \sqrt \beta \times \varGamma (t) . $$

(41)

Equation (41) is a phenomenological model for the power signal. This special case of the Langevin process is mathematically called an Ornstein–Uhlenbeck process [17].

Equation (41) is a simplified model for the power output, where the wind turbine design is described through the parameters $\alpha$ and $\beta$, as well as the power curve $P_{theo} (u)$. The parameter $\alpha$ is related to the reaction time of the model wind turbine,³⁶ while $\beta$ quantifies the strength of the stochastic noise. $\varGamma (t)$ is a Gaussian-distributed white noise with mean value 0 and variance 2, which can be generated easily from most mathematical softwares. Using a wind speed time series $u(t)$ as an input for the model equation,³⁷ a time series of power output $P(t)$ can be generated at the same sampling frequency. An example is provided in Fig. 23.

Fig. 23

a Power signal measured (red line). b Power signal modeled (blue line) following Eq. (41) with $\alpha = 0.005\, / \text{s}$, $\beta = 0.5\, {\text{W/s}}^{2}$, $u_{r} = 13\,\text{m/s}$ and $P_{r} = 1\, \text{W}$. The wind speed signal used was measured simultaneously as the measured power output

From Fig. 23, it can be seen that the relaxation model manages to estimate the power output of a wind turbine to a first approximation. Fluctuations and their statistics are more difficult to reproduce than long-time behavior, which is mostly driven by the changes in wind speed rather than by the stochastic fluctuations. More advanced methods are being developed, as introduced in [11], where $D^{(1)}$ and $D^{(2)}$ are not assumed but estimated from measurement data.