**Optimal Control of wind energy systems**

energia eólica,energia renovável

(Parte **3** de 5)

designed to values λA = 5 | 8. |

Fig. 2.2 Typical power coefficients of different rotor types over tip-speed ratio

Fig. 2.3 Typical torque coefficients of different rotor with hotizontal shaft

2.2.2 Forces and Torque

The main rotor properties follow from lifting force and drag force of a blade as described by aerofoil theory. Let an aerofoil element of depth t and width b be subjected to a wind speed v1, see Fig. 2.4. Dependent on the angle of attack α between wind direction and the blade profile cord, the lifting force FA and drag force FW are:

Note that these force components are directed perpendicular and parallel to the oncoming wind, respectively. Coefficients cA and cW are characteristic for a given blade profile; they depend on blade angle α. The example in Fig. 2.4 applies to real

proportional dependence of cA =( 5,1 | 5,8)·α is observed, while cW is compar- |

atively small in the considered interval of α. The ratio ε = cA/cW is called the glide ratio or lift/drag ratio.

When a wind rotor is rotating at an angular speed Ω, the circumferential speed of each blade at radius r is u(r)= Ω ·r. In the rotor plane the wind velocity is v2 in axial

2.2 Basics of Wind Energy Conversion 15

Fig. 2.4 Coefficients cA(α) of lift and cW(α) of drag over blade angle of specific profiles direction, which is, according to Betz theory, 2/3 of the upstream wind velocity v1. Both components added geometrically result in the speed c(r) which is directed under angle α relative to the rotor plane, see Fig. 2.5. Consequently increments dFA of lift force and dFW of drag force are acting on the area increment (t ·dr) of the blade. The force can be described by its components dFt in tangential and dFa in axial direction: [ dFt

dFa

[ cAsinα −cWcosα cAcosα +cWsinα

Integrating for a given profile, the torque can be obtained from the tangential forces, while the axial forces sum up to the drag force acting axially on the rotor.

At the tip of the blade, r = R, the tip speed is u(R)= Ω ·R. Note that the wind speed relative to the tip is:

An example of cp(λ) and associated cT(λ) curves for a rotor with fixed blade angle designed for optimum tip-speed ratio λA = 6,5 is shown in Fig. 2.6 [Gas07].

The basic characteristics of a wind rotor follow from the power coefficient cP(λ) and the torque coefficient cT(λ), (see 2.3, 2.4). Further a drag coefficient cS is

Fig. 2.5 Wind speeds and forces acting on the blade

Fig. 2.6 Curves of power coefficient cp(λ) and torque coeffient cT(λ) of a three-blade rotor defined which allows to calculate the axial drag force, see Fig. 2.7. With rigid blade this coefficient is small for low λ, and attains at no-load speed (large λ) values similar to a circular plane subjected to air flow in normal direction. Torque T,p ower P and axial drag force S may be expressed by a set of equations using the reference force FB which varies with the square of the wind speed and is proportional to the swept rotor area:

Fig. 2.7 Curve of drag (drag) coefficient cS(λ)

2.3 Wind Regime and Utilization 17 2.3 Wind Regime and Utilization

2.3.1 Wind Velocity Distribution

The wind regime is influenced by regional and local effects, and depends on seasonal and short-time variations. In a project to erect a wind energy system a realistic expertise to predict the wind velocity distribution and its average at the relevant place is of foremost importance. The actual wind regime can be determined by a measuring campaign, preferably at the height of the mast. However, data collations show significant variations when considering different years, months and days at a given location.

The wind velocity varies with the height above ground, influenced by the surface roughness. Assuming stable conditions, the dependence of velocity v on height z may be described by a logarithmic profile. The wind speed v2 at z2 is calculated from a reference speed v1 at z1 by:

are 0,03m for farmland, 0,1m for heath scattered shrubs and trees, 0,5 | 1,6m |

where z0 is the roughness length dependent on the country; conventional parameters for forest. Equation (2.9) is used when calculating the reference energy yield in the project stage, see 2.4.4.

When the regime of wind velocity v(t) is known at a specified height above ground, the distribution of power and energy yield can be evaluated by descriptive statistics. To this end the wind velocities are assigned to k equally distributed classes of

width Δv with centre values vi (i = 1 | k). Averages measured during a period |

of e.g. 10min are assigned to the k classes, so that each class i is covered by a time interval ti. The relative frequencies hi of wind velocity in the period under consideration T of e.g. 1 d are:

hi = tiT ti (2.10a)

The frequency distribution is represented in form of a histogram hi(vi). Figure 2.8a gives an example with class width Δv = 1m/s.

It is known that distributions measured in practice may be approximated by a

Weibull-function. Distributions obtained in inland Europe follow, with good accuracy, a Weibull-function with form factor k = 2, i.e. a Rayleigh-distribution:

vav vav (2.10b)

Figure 2.8b shows Weibull distributions with k as parameter; note the preference for k =2 in approximation which is the Raileigh distribution widely used in practice.

Fig. 2.8 Representation of wind velocity distribution. (a) example histogram; (b) approximation by Weibull-functions (Raileigh-function for k = 2)

2.3.2 Power Distribution and Energy

A power value Pi(vi) may be attributed to each class i of the distribution hi(vi) according to (2.2). It is advisable to refer Pi to the swept area A, to obtain a specific power value:

Note that a power limitation may take place at higher wind velocities. From the specific power follows a normalized energy distribution:

ei = pi ·hiE

Fig. 2.9 Histograms of wind velocity distribution and normalized energy yield

2.3 Wind Regime and Utilization 19

Due to the cubic relationship between power and wind velocity the maximum of ei(vi) is seen at significantly larger values vi than the maximum of hi(vi). Figure 2.9 gives an example, using a class width of Δv = 0.5m/s.

2.3.3 Power and Torque Characteristics

The power delivered by a wind turbine is a function of tip speed ratio (see Fig. 2.3) and hence depends on wind velocity and rotational speed. Figure 2.10 shows a normalized representation of power and torque of a rotor with fixed blade position over speed, with wind velocity values as parameter. In the example the rated wind velocity is 12m/s. The design is such that at approximately v = 8m/s the tip speed ratio is optimal, λ = λopt. The power maxima are indicated by the cubic function of v.

From the power characteristic it is seen that P/PN = 1a t n/nN = 1. Measures for power limitation at higher wind speeds are not considered in the graphs.

The graphs are analytically derived from a power coefficient curve, see Fig. 2.3.

When cp(λ) is given as an empirical function, e.g. in form of a look-up table, then the curves shown in Fig. 2.8 can be calculated by the following algorithm for a specific design. Let

– λA the design tip speed ratio and cpA = cp(λA) the optimum power coefficient of the rotor,

– λN the tip speed ratio for rated condition, with cp,N = cp(λN), – vN the wind speed for rated condition.

The ratio λN/λA can be chosen. In case of a constant speed system for operation at n/nN = 1 the wind speed at which cp is optimum is calculated by:

vA = λN λA vN (2.13)

To calculate power and torque curves, selected parameter wind speeds vi are chosen. The following equations are normalized to give P,T and n referred to rated values.

PN = cp cp,N

TN = cp/cp,N λ/λN nN = λ λN · vi vN (2.14)

The base quantities for rated condition are as follows:

) λNvN where ρ, A are as in (2.2), and cT as in (2.4); n is in min−1 .

The example of Fig. 2.10 is based on the cp(λ) curve of Fig. 2.6, from which λA = 6,5 and cp,A = 0,52. The rated wind speed is chosen vN = 12m/s, and the optimum cp is assigned to vA =8m/s. According to (2.13) the tip speed ratio at rated condition should be λN = λA(vA/vN); it was chosen λN = 4,5, with cp,N = 0,448.

2.4 Power Characteristics and Energy Yield

2.4.1 Control and Power Limitation

2.4.1.1 Pitching Mechanism

Variation of the blade angle is a means to control the rotor torque and power from the wind side, and at the same time provide power and speed limitation at high wind velocities. Normally the pitch mechanism is powered by a hydraulic or electric drive. Pitch controlled rotors prevail in all larger systems.

In rotors equipped with a pitch mechanism the blade angle is adjusted subject to a relevant controller output. Pitching is also used for power limitation at tip speed ratios above a predesigned value by turning the blades out of the wind. With increasing pitch angle the maximum power and torque coefficients are reduced, and the maxima are shifted to lower λ values. The no-load tip speed ratio is reduced, while the torque coefficient shows increased values at starting. The drag coefficient is significantly reduced. Principal curves of a design, based on the reference Fig. 2.6 are shown in Fig. 2.1.

2.4.1.2 Stall Mechanism

For reasons of safety and to avoid overload, limitation of power above a preset rotational speed is required. This can be accomplished by different methods. A simple way is to turn the rotor out of the wind direction, which is done in the Western Mill. For rotors without pitch angle variation the stall-effect can be utilized, where due to a shift from laminar to turbulent air flow a braking effect is created. Figure 2.12

2.4 Power Characteristics and Energy Yield 21

Fig. 2.1 Power, torque and drag coefficients over tip speed ratio with pitch angle as parameter

Fig. 2.12 Sketch of a blade with laminar and turbulent air flow illustrates the effect, where α is the angle between blade plane and oncoming wind direction (as in Fig. 2.4) and ϑ the angle between rotor and blade planes.

A variant of the stall method is the so-called active-stall where the blades are automatically turned into the wind direction when a preset speed is reached. Figure 2.13 illustrates the specific properties. While in stall the air flow breakoff occurs with rigid blade position, in active-stall the blades perform a self acting angle variation into the wind to control the air flow breakaway; in pitch eventually the angle variation to decrease the active rotor area is performed without air flow breakaway.

Fig. 2.13 Illustration of stall, active-stall and pitch effects

2.4.1.3 Other Power Limitation Concepts

Small wind turbines in the kW power-range sometimes use other concepts such as passive pitch control, or a passive mechanism tilting the turbine in dependence of wind exerted axial force and decreasing the swept area from circular to elliptic (see 5.2).

Power and speed limitation can also be provided by concepts other than manipulating the wind turbine. This may be an electrical or mechanical braking system.

2.4.2 Wind Classes

Wind energy systems can be assigned to different wind classes. The classes standardized by IEC are commonly used (see Table 2.1). The classes reflect the design dependence on locations with strong or weak wind performance. Characteristic for WES in classes of larger number (lower wind velocities) are larger rotor diameters at same rated power, and often also a larger tower height. Reference values are the average wind speed in hub height and an extremum which statistically happens as 10min mean value only once in 50 years.

Note that in Germany there is also a classification in wind zones according to the Deutsches Institut fur Bautechnik (DIBT).

Table 2.1 IEC type classes IEC wind class I I II IV

2.4 Power Characteristics and Energy Yield 23 2.4.3 System Power Characteristics

Most important for any wind energy system capability is the power curve. Measured curves of the delivered power over wind speed are, together with the knowledge of average wind velocity and distribution properties (e.g. Rayleigh), indispensable for predicting the annual energy yield. Figure 2.14 shows typical power curves of pitchcontrolled and stall-controlled systems. Below a predesigned wind speed, normally the rated wind speed, the power curve is intended to follow a v3 function using optimum cp(λ). Note that useful power generation starts only at the cut-in wind speed, normally at v between 3 and 4m/s.

Power limitation at wind speeds above the rated value is effected by either one of the control systems:

– pitch control, where the power is controlled to rated power above a preset threshold wind speed (mostly the rated speed), – stall control, where a transient phenomenon with power overshoot is observed for wind speeds above rated value.

A number of characteristic wind velocity values are specified with the design for each wind turbine:

• Average velocity vav (in e.g. 10m or 30m above ground, or measured in hub height).

• Optimum velocity vopt at λopt (best point). • Velocity at maximum energy yield.

• Value of velocity at which the power limitation begins to work. This point is mostly called the rated wind speed. • Cut-in velocity, at which the turbine starts to supply power.

• Cut-off velocity, at which the turbine is brought to standstill for safety reasons.

• Survival-velocity, in view of an assumed “once-in-a-century storm”.

Regulations require a wind energy converter to have two independent braking systems, where the first serves as the main brake and the second as fixing brake. Shut down must be tripped at a maximum wind speed of usually 25m/s.

Fig. 2.14 Typical power curves for pitch-controlled and stall-controlled systems

Fig. 2.15 Sketch of a the measuring setup Fig. 2.16 Power curve of a system specified for 1800kW

2.4 Power Characteristics and Energy Yield 25

In practice the power curves are determined by test, normally executed in a recognized testing field. Measurements are taken and recorded between cut-in wind speed and cut-off, at least up to 18 m/s. A graph is drawn of the electrical active power P together with the power coefficient cp versus wind speed v. An example of a turbine rated 1800kW at 14m/s is shown in Fig. 2.16, in form of curves of power P(v) and power coefficient cp(v). The measurement set-up and quantities to be recorded are illustrated in Fig. 2.15 [DEWI].

Figure 2.16 is typical for a design where 14m/s is the specified rated wind velocity, with a v3 power dependence below and a limitation above. The power coefficient has its maximum at 8m/s in a region near the average speed.

A view on specific power values of currently available WES is given in the next two figures, based on data supplied by the manufacturers [BWE07], covering ratings of 850kW and above. They give an insight into the rotor diameters D and the rotor speeds (upper values in case of speed variable systems) nm versus rated power, Fig. 2.17. Further, Fig. 2.18 shows the specific power, i.e. the ratio of rated WES

in the regieon 900 | 10 (kWh/a)/m2 can be observed. Note that among the data |

power and swept area. According to the trend line the values aggregate between 0,35 and 0,45kW/m2, with a tendency to increase with increasing rating. The right part of Fig. 2.18 shows also, for a variety of systems for 2000kW, the specified specific reference annual energy yield versus hub hight. Here from the trend line values there are designs for upper as well as for lower average wind velocities.

Fig. 2.17 Rotor diameters and rotation speeds of systems of 850kW and above

(Parte **3** de 5)