Burton T. (et. al.) Wind energy Handbook

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(1) Low solidity produces a broad, ﬂat curve which means that the C

will change

very little over a wide tip speed ratio range but the maximum C

is low because

the drag losses are high (drag losses are roughly proportional to the cube of the

tip speed ratio).

(2) High solidity produces a narrow performance curve with a sharp peak making

the turbine very sensitive to tip speed ratio changes and, if the solidity is too

high, has a relatively low maximum C

. The reduction in C

max

is caused by stall

losses.

(3) An optimum solidity appears to be achieved with three blades, but two blades

might be an acceptable alternative because although the maximum C

is a little

lower the spread of the peak is wider and that might result in a larger energy

capture.

It might be argued that a good solution would be to have a large number of

blades of small individual solidity but this greatly increases production costs and

results in blades which are structurally weak and very ﬂexible.

There are applications which require turbines of relatively high solidity, one is

the directly driven water pump and the other is the very small turbine used for

battery charging. In both cases it is the high starting torque (high torque at very low

tip speed ratios) which is of importance and this also allows small amounts of

power to be developed at very low wind speeds, ideal for trickle charging batteries.

0.5

0.4

0.3

0.2

0.1

One blade

Two blades

Three blades

Four blades

Five blades

0 5 10 15

Figure 4.3 Effect of Changing Solidity

THE PERFORMANCE CURVES 175

4.1.3 The C

– º curve

The torque coefﬁcient is derived from the power coefﬁcient simply by dividing by

the tip speed ratio and so it does not give any additional information about the

turbine’s performance. The principal use of the C

–º curve is for torque assessment

purposes when the rotor is connected to a gear box and generator.

Figure 4.4 shows how the torque developed by a turbine rises with increasing

solidity. For modern high-speed turbines designed for electricity generation as low

a torque as possible is desirable in order to reduce gearbox costs. On the other hand

the multi-bladed, high-solidity turbine, developed in the nineteenth century for

water pumping, rotates slowly and has a very high starting torque coefﬁcient

necessary for overcoming the torque required to start a positive displacement

pump.

The peak of the torque curve occurs at a lower tip speed ratio than the peak of the

power curve. For the highest solidity shown in Figure 4.4 the peak of the curve

occurs while the blade is stalled.

4.1.4 The C

– º curve

The thrust force on the rotor is directly applied to the tower on which the rotor is

supported and so considerably inﬂuences the structural design of the tower.

Generally, the thrust on the rotor increases with increasing solidity (Figure 4.5).

0.12

0.1

0.08

0.06

0.04

0.02

0 5 10 15

One blade

Two blades

Three blades

Four blades

Five blades

Figure 4.4 The Effect of Solidity on Torque

176

WIND-TURBINE PERFORMANCE

4.2 Constant Rotational Speed Operation

The majority of wind turbines currently installed generate electricity. Whether or

not these turbines are grid connected they need to produce an electricity supply

which is of constant frequency or else many common appliances will not function

properly. Consequently, the most common mode of operation for a wind turbine is

constant rotational speed. Connected to the grid a constant speed turbine is

automatically controlled whereas a stand-alone machine needs to have speed

control and a means of dumping excess power.

4.2.1 The K

–1=º curve

An alternative performance curve can be produced for a turbine controlled at

constant speed. The C

– º curve shows, non-dimensionally, how the power would

vary with rotational speed if the wind speed was held constant. The K

–1=º curve

describes, again non-dimensionally, how the power would change with wind speed

when constant rotational speed is enforced. K

is deﬁned as

0.12

0.1

0.08

0.06

0.04

0.02

0 5 10 15 20

One blade

Two blades

Three blades

Four blades

Five blades

Figure 4.5 The Effect of Solidity on Thrust

CONSTANT ROTATIONAL SPEED OPERATION 177

Power

r(R)

(4:1)

The C

– º and K

–1=º curves for a typical ﬁxed-pitch wind turbine are shown

in Figure 4.6. The K

–1=º curve, as stated above, has the same form as the power

– wind speed characteristic of the turbine. The efﬁciency of the turbine (given by

the C

– º curve) varies greatly with wind speed, a disadvantage of constant speed

operation, but it should be designed such that the maximum efﬁciencies are

achieved at wind speeds where there is the most energy available.

4.2.2 Stall regulation

An important feature of this K

–1=º curve is that the power, initially, falls off once

stall has occurred and then gradually increases with wind speed. This feature

provides an element of passive power output regulation, ensuring that the gen-

erator is not overloaded as the wind speed increases. Ideally, the power should rise

with wind speed to the maximum value and then remain constant regardless of the

increase in wind speed; this is called perfect stall regulation. However, stall

regulated turbines do not exhibit the ideal, passive stall behaviour.

Stall regulation provides the simplest means of controlling the maximum power

generated by a turbine to suit the sizes of the installed generator and gearbox and

until recently, at the time of writing, is the most commonly adopted control method.

The principal advantage of stall control is simplicity but there are signiﬁcant

disadvantages. The power versus wind speed curve is ﬁxed by the aerodynamic

characteristics of the blades, in particular the stalling behaviour. The post stall

power output of a turbine varies very unsteadily and in a manner which, so far,

deﬁes prediction, see Figure 4.13, for example. The stalled blade also exhibits low

vibration damping because the ﬂow about the blade is unattached to the low

pressure surface and blade vibration velocity has little effect on the aerodynamic

forces. The low damping can give rise to large vibration displacement amplitudes

0.6

0.5

0.4

0.3

0.2

0.1

0.06

0.05

0.04

0.03

0.02

0.01

0.6

0.5

0.4

0.3

0.2

0.1

0 1 2 3 4 5 6 7 8 9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

l/λ

Figure 4.6 Non-dimensional Performance Curves for Constant Speed Operation

178

WIND-TURBINE PERFORMANCE

which will inevitably be accompanied by large bending moments and stresses,

causing fatigue damage. When parked in high, turbulent winds a rotor with ﬁxed

pitch blades may well be subject to large aerodynamic loads which cannot be

alleviated by adjusting (feathering) the blade pitch angle. Consequently, a ﬁxed-

pitch, stall-regulated turbine experiences more severe blade and tower loads than a

pitch regulated turbine.

4.2.3 Effect of rotational speed change

The power output of a turbine running at constant speed is strongly governed by

the chosen, operational rotational speed. If a low rotation speed is used the power

reaches a maximum at a low wind speed and consequently it is very low. To extract

energy at wind speeds higher than the stall peak the turbine must operate in a

stalled condition and so is very inefﬁcient. Conversely, a turbine operating at a high

speed will extract a great deal of power at high wind speeds but at moderate wind

speeds it will be operating inefﬁciently because of the high drag losses. Figure 4.7

demonstrates the sensitivity to rotation speed of the power output – a 33 percent

increase in r.p.m. from 45 to 60 results in a 150 percent increase in peak power,

reﬂecting the increased wind speed at which peak power occurs at 60 r.p.m.

At low wind speeds, on the other hand there is a marked fall in power with

increasing rotational speed as shown in Figure 4.8. In fact, the higher power

available at low wind speeds if a lower rotational speed is adopted has led to two-

speed turbines being built. Operating at one ﬁxed speed which maximizes energy

capture at wind speeds at, or above, the average level will result in a rather high

cut-in wind speed, the lowest wind speed at which generation is possible. Employ-

Electrical power (kW)

0 5 10 15 20 25

Wind speed (m/s)

45 rpm

50 rpm

55 rpm

60 rpm

Figure 4.7 Effect on Extracted Power of Rotational Speed

CONSTANT ROTATIONAL SPEED OPERATION 179

ing a lower rotational speed at low wind speeds reduces the cut-in wind speed and

increases energy capture. The increased energy capture is, of course, offset by the

cost of the extra machinery.

4.2.4 Effect of blade pitch angle change

Another parameter which affects the power output is the pitch setting angle of the

blades 

. Blade designs almost always involve twist but the blade can be set at the

root with an overall pitch angle. The effects of a few degrees of pitch are shown in

Figure 4.9.

Small changes in pitch setting angle can have a dramatic effect on the power

output. Positive pitch angle settings increase the design pitch angle and so decrease

the angle of incidence. Conversely, negative pitch angle settings increase the angle of

incidence and may cause stalling to occur as shown in Figure 4.9. A turbine rotor

designed to operate optimally at a given set of wind conditions can be suited to other

conditions by appropriate adjustments of blade pitch angle and rotational speed.

4.2.5 Pitch regulation

Many of the shortcomings of ﬁxed pitch/passive stall regulation can be overcome

by providing active pitch angle control. Figure 4.9 shows the sensitivity of power

output to pitch angle changes.

The most important application of pitch control is for power regulation but pitch

control has other advantages. By adopting a large positive pitch angle a large

Electrical power (kW)

0 5 6 7 8 9 10

Wind speed (m/s)

55 rpm

60 rpm

50rpm

45 rpm

Figure 4.8 Effect on Extracted Power of Rotational Speed at Low Wind Speeds

180

WIND-TURBINE PERFORMANCE

starting torque can be generated as a rotor begins to turn. A 908 pitch angle is

usually used when shutting down because this minimises the rotor idling speed at

which the parking brake is applied. At 908 of positive pitch the blade is said to be

‘feathered’. The principal disadvantages of pitch control are reliability and cost.

Power regulation can be achieved either by pitching to promote stalling or pitching

to feather which reduces the lift force on the blades by reducing the angle of attack.

4.2.6 Pitching to stall

Figure 4.9 shows the power curves for a turbine rated at 60 kW, which is achieved

at 12 m/s. At wind speeds below the rated level the blade pitch angle is kept at 08.

As rated power is reached only a small negative pitch angle, initially of about 28,is

necessary to promote stalling and so to limit the power to the rated level. As the

wind speed increases small adjustments in both the positive and negative directions

are all that are needed to maintain constant power. The small sizes of the pitch

angle adjustments make pitching to stall very attractive to designers but the blades

have the same damping and fatigue problems as ﬁxed pitch turbines.

4.2.7 Pitching to feather

By increasing the pitch angle as rated power is reached the angle of attack can be

reduced. A reduced angle of attack will reduce the lift force and the torque. The

5 10 15 20 25

Electrical power (kW)

120

100

Wind speed

(

m/s

)

-4

-2

Figure 4.9 Effect on Extracted Power of Blade Pitch Set Angle

CONSTANT ROTATIONAL SPEED OPERATION 181

ﬂow around the blade remains attached. Figure 4.10 is for the same turbine as

Figure 4.9 but only the zero degree power curve is shown below the rated level.

Above the rated level fragments of power curves for higher pitch angles are shown

as they cross the rated power line; the crossing points give the necessary pitch

angles to maintain rated power at the corresponding wind speeds. As can be seen

in Figure 4.10, the required pitch angles increase progressively with wind speed

and are generally much larger than is needed for the pitching to stall method. In

gusty conditions large pitch excursions are needed to maintain constant power and

the inertia of the blades will limit the speed of the control system’s response.

Because the blades remain unstalled if large gusts occur at wind speeds above the

rated level large changes of angle of attack will take place with associated large

changes in lift. Gust loads on the blades can therefore be more severe than for

stalled blades. The advantages of the pitching to feather method are that the ﬂow

around the blade remains attached, and so well-understood, and provides good,

positive damping. Feathered blade parking and assisted starting are also available.

Pitching to feather has been the preferred pitch control option mainly because the

blade loads can be predicted with more conﬁdence than for stalled blades.

4.3 Comparison of Measured with Theoretical

Performance

The turbine considered in this section is run at constant rotational speed, the most

common mode of operation, because this allows electricity to be generated at

constant frequency. More detail about this method of operation will be discussed in

the next section but the main feature is that there is, theoretically, a unique power

output for a given wind speed.

100

Wind s

eed

(

m/s

)

5 10 15 20 25

Electrical power (kW)

0 Power curve

+12

+18

+24

Figure 4.10 Pitching to Feather Power Regulation Requires Large Changes of Pitch Angle

182

WIND-TURBINE PERFORMANCE

When the turbine was under test the chosen rotational speed was 44 r.p.m.

Energy output and wind speed were measured over 1 min time intervals and the

average power and wind speed determined. The test was continued until a

sufﬁcient range of wind speeds had been covered. The results were then sorted in

‘bins’ 0.5 m/s of wind speed wide and a fairly smooth power versus wind-speed

curve was obtained as shown in Figure 4.11. The turbine has a diameter of 17 m and

would be expected to produce rather more power than shown above if operated at

a higher rotational speed.

From the data in Figure 4.11 the C

– º curve can be derived. The tip speed of the

blades is 44 3 =30 rad=s 3 8:5m ¼ 39:2m=s, the swept area is  3 8:5

¼ 227 m

and the air density was measured (from air pressure and temperature readings) at

1:19 kg=m

. Therefore

º ¼

39:2

Windspeed

and C

Power º

1:19 39:2

227

The mechanical and electrical losses were estimated at 5.62 kW and this value was

used to adjust the theoretical values of C

. The resulting comparison of measured

and theoretical results are shown in Figure 4.12.

This comparison looks reasonable and shows that the theory is reliable but the

quality of the theoretical predictions really relies upon the quality of the aerofoil

data. The blade and aerofoil design are the same as given in Section 3.9.

One last point should be made before classifying the theory as complete: it would

be as well to look at the raw, 1 min averages data, which was reduced down by the

binning process and is shown in Figure 4.12. In the post stall region there seems to

be a much more complex process taking place than the simple theory predicts and

this could be caused by unsteady aerodynamic effects.

Wind s

eed

(

m/s

)

0 5 10 15 20 25 30

Electrical power (kW)

Figure 4.11 Power versus Wind Speed Curve from the Binned Measurements of a Three-

blade Stall Regulated Turbine

COMPARISON OF MEASURED WITH THEORETICAL PERFORMANCE 183

4.4 Variable-speed Operation

If the speed of the rotor can be continuously adjusted such that the tip speed ratio

remains constant at the level which gives the maximum C

then the efﬁciency of

the turbine will be signiﬁcantly increased. Active pitch control is necessary to

maintain a constant tip speed ratio but only in the process of adjustment of

0 2 4 6 8 10

Tip speed ratio

Power coefficient

0 5 10 15 20 25 30

Wind speed (m/s)

Electrical power (kW)

Measurement

Theory

Measurement

Theory

0.4

0.3

0.2

0.1

Figure 4.12 Comparison of Measured and Theoretical Performance Curves

0 5 10 15 20 25 30

Wind s

eed

(

m/s

)

Electrical power (kW)

ESE

SSW

WSW

Wind direction

Figure 4.13 Measured Raw Results of a Three-blade Wind Turbine

184

WIND-TURBINE PERFORMANCE