ENERGY CONVERSION MME9617a Eric Savory eng.uwo/people/esavory/mme9617a.htm Lecture 14 – Wind Energy Part 2: Wind turbine

1. ENERGY CONVERSION MME9617a Eric Savory www.eng.uwo.ca/people/esavory/mme9617a.htm Lecture 14 – Wind Energy Part 2: Wind turbines Department of Mechanical and Material Engineering University of Western Ontario

2. Contents Modern wind turbines and their key components Basic operation of a Horizontal Axis Wind Turbine (HAWT) Estimation of the wind resource Statistical analysis of wind data and its use to predict available power (using Rayleigh and Weibull distributions) 1-D momentum theory applied to an actuator disk model of the turbine, and the Betz limit Incorporation of wake rotation into the analysis Airfoil aerodynamics and blade design using momentum equation and blade element theory Blade optimization

3. Modern wind turbines • In contrast to a windmill, which converts wind power into mechanical power, a wind turbine converts wind power into electricity. • As an electricity generator a wind turbine is connected to an electrical network: • Battery charging circuit • Residential scale power units • Isolated or island networks • Large utility grids • Most are small (< 10 kW) but the total generating capacity is mostly from 0.5 - 2 MW machines.

4. Underlying features of conversion process Aerodynamic lift force on the blades  net positive torque on a rotating shaft  mechanical power  electrical power in a generator. No energy is stored – output is inherently fluctuating with the wind variability (though can limit output below what wind could produce at any given time). Any system turbine is connected to must be able to handle this variability.

5. Horizontal axis wind turbine (HAWT) Most common type of turbine. Rotor may be upwind or downwind of the tower.

6. Main components of a HAWT Drive train = shafts, gear box, coupling, mechanical brake, generator Electrical system on ground = cables, switchgear, transformers

7. Main options in wind turbine design - Number of blades (commonly two or three) - Rotor orientation: downwind or upwind of tower - Blade material, construction method, and profile - Hub design: rigid, teetering or hinged - Power control via aerodynamic control (stall control) or variable pitch blades (pitch control) - Fixed or variable rotor speed - Orientation by self aligning action (free yaw), or direct control (active yaw) - Synchronous or induction generator - Gearbox or direct drive generator

8. Power output prediction POWER CURVE (obtained from manufacturer, based on field tests using standard methods) RATED = point of max power output from generator CUT IN = Minimum speed at which machine delivers useful power CUT-OUT = max wind speed at which turbine is allowed to deliver power (limited by safety / engineering) Power output varies with wind speed, producing a typical chart like this of electrical power output as a function of the hub height wind speed. Chart allows prediction of turbine energy production without doing a component analysis.

9. Typical size, height, diameter and rated capacity of wind turbines

10. Estimation of the potential wind resource The mass flow rate dm/dt of air of density  and velocity U through a rotor disk of area A is: The kinetic energy per unit time, or power, of the flow is: The wind power per unit area, P/A or wind power density is: Note: density is generally taken as 1.225 kg/m3 (15oC at sea level). Actual power output is only about 45% of this available wind power for even the the best turbines

11. Power per unit area available from steady wind Maps of annual average wind speeds  maps of average wind power density. More accurate estimates can be made if hourly averages, Ui, are available for a year. The average wind power density, based on hourly averages is

12. where U is the annual average wind speed and Ke is called the energy pattern factor. The energy pattern factor is calculated from: where N = number of hours in a year = 8,760 Typical qualitative magnitude evaluations of the wind resource are: P / A < 100 W/m2 - poor P / A ~ 400 W/m2 - good P / A > 700 W/m2 - great

13. Direct methods of data analysis, resource characterization, and turbine productivity Given a series of N wind speed observations, Ui, each averaged over the time interval t, we obtain: (1) The long-term average wind speed, U, over the total period of data collection: (2) The standard deviation U of the individual wind speed averages:

14. (3) The average wind power density P / A is Similarly, the wind energy density per unit area for a given extended time period T = N t is: (4) The average machine wind power Pwis: where Pw ( Ui ) is the power output defined by a wind machine power curve.

15. (5) The energy from a wind machine, Ew , is: = Method of bins The method of bins also provides a way to summarize wind data and to determine expected turbine productivity. The data must be separated into the wind speed intervals or bins in which they occur. It is convenient to use the same size bins. Suppose that the data are separated into NB bins of width wiwith midpoints mj and with fj the number of occurrences in each bin or frequency, such that:

16. The wind speed data set can now be analyzed to give: A histogram (bar graph) showing the number of occurrences and bin widths is usually plotted when using this method.

17. Velocity and power duration curves Can be useful when comparing the energy potential of candidate wind sites. The velocity duration curve is a graph with wind speed on the y axis and the number of hours in the year for which the speed equals or exceeds each particular value on the x axis. These plots give an indication of the nature of the wind regime at each site. The flatter the curve, the more constant are the wind speeds (e.g. characteristic of the trade-wind regions of the earth). The steeper the curve, the more irregular the wind regime.

18. Velocity duration curve example (Rohatgi J S and Nelson V, 1994, Alternative Energy Institute, Canyon, Texas)

19. A velocity duration curve can be converted to a power duration curve by cubing the ordinates, which are then proportional to the available wind power for a given rotor swept area. The difference between the energy potential of different sites is visually apparent, because the areas under the curves are proportional to the annual energy available from the wind. The following steps must be carried out to construct velocity and power duration curves from data: (1) Arrange the data in bins (2) Find the number of hours that a given velocity (or power per unit area) is exceeded (3) Plot the resulting curves

20. A machine productivity curve for a particular wind turbine at a given site may be constructed using the power duration curve in conjunction with a machine curve for a given turbine. Note that the losses in energy production with the use of a wind turbine at this site can be identified.

21. Statistical analysis of wind data This type of analysis relies on the use of the probability density function, p(U), of wind speed. One way to define the probability density function is that the probability of a wind speed occurring between Ua and Ub is given by: The total area under the probability distribution curve is given by: 0

22. If p(U) is known, the following parameters can be calculated: Mean wind speed, U 0 Standard deviation of wind speed, U 0 Mean available wind power density, P / A 0 It should be noted that the probability density function can be superimposed on a wind velocity histogram by scaling it to the area of the histogram.

23. Another important statistical parameter is the cumulative distribution function F(U) which represents the time fraction or probability that the wind speed is smaller than or equal to a given wind speed, U'. That is: F(U) = Probability (U'U ) where U' is a dummy variable. It can be shown that: 0 Also, the slope of the cumulative distribution function is equal to the probability density function:

24. Probability density function equations In general, either one of two probability distributions (or probability density functions) are used in wind data analysis: (1) Rayleigh and (2) Weibull (see Lecture 13 notes) The Rayleigh distribution uses one parameter, the mean wind speed. The Weibull distribution is based on two parameters and, thus, can better represent a wider variety of wind regimes. Both are 'skew' distributions (defined only for values > 0).

25. Rayleigh distribution Requires only a knowledge of the mean wind speed, U. The probability density function and the cumulative distribution function are given by:

26. Example of a Rayleigh distribution Note: a larger value of the mean wind speed gives a higher probability at higher wind speeds

27. Weibull distribution Determination of the Weibull probability density function requires a knowledge of two parameters: k, a shape factor and c, a scale factor. Both are a function of U and U . The Weibull probability density function and the cumulative distribution function may be given by: Note: methods for determining k and c from U and  U are given in an appendix at the end of these notes

28. Examples of Weibull distributions with different k for U = 8 m/s Note: as k increases the peak is sharper, indicating there is less wind speed variation

29. Wind Turbine Energy Production Estimates Using Statistical Techniques For a given wind regime probability distribution p(U) and a known machine power curve Pw(U), the average wind machine power Pw is given by: Pw(U) may be determined from the wind power, the rotor power coefficient Cp and the drive train efficiency ( = generator power / rotor power):

30. where Cp is also a function of tip speed ratio  defined as: where  is the rotor angular velocity and R is rotor radius. Hence, assuming constant,average wind machine power is also given by

31. Idealized machine productivity calculations using Rayleigh distribution Assuming: (1) Idealized wind turbine, no losses, machine power coefficient, Cp , equal to the Betz limit (Cp,Betz = 16/27 = the theoretical maximum possible power coefficient). (2) Wind speed probability distribution is given by a Rayleigh distribution. The average wind machine power equation becomes:

32. where Uc is a characteristic wind velocity given by For an ideal machine  = 1, Cp =Cp,Betz = 16/27, so Using x = U / Uc gives a simpler integral Over all wind speeds, the integral becomes so that

33. Substituting for the rotor disk area, A =  D2/4, and for the characteristic velocity Uc the equation for average power becomes simply: Example: What is the average annual energy production of an 18 m diameter Rayleigh-Betz machine at sea level in a 6m/s average annual wind velocity regime?

34. Solution: Multiplying this by 8,760 hrs/yr gives an expected annual energy production of 334,000 kWhr Comparing this result to the simple approach in Slide 10 where: P = ½  A U 3 = ½  (¼  D 2 ) U 3 = ( 0.627 D) 2 U 3 shows that the simple method under-estimates the power by about 12% Pw

35. Productivity calculations for a real wind turbine using a Weibull distribution The average wind machine power equation based upon the probability distribution function p(U) may be re-cast in terms of the cumulative distribution F(U):  (1) 0 0 The Weibull distribution is Therefore, using (2) in (1) and replacing the integral with a summation over NB bins gives (2)

36. Pw = Note: the above equation is the statistical method’s equivalent to the earlier equation: where the relative frequency f / N corresponds to the term in brackets and the wind turbine power is calculated at the mid-point between Uj - 1 and Uj

37. 1-D Momentum Theory and the Betz Limit A simple model may be used to determine the power from an ideal turbine rotor, the thrust of the wind on the ideal rotor and the effect of the rotor operation on the local wind field. The analysis assumes a control volume, in which the boundaries are the surface of a stream tube and two cross-sections of the stream tube:

38. The only flow is across the ends of the stream tube. The turbine is represented by a uniform "actuator disk" which creates a discontinuity of pressure in the stream tube of air flowing through it. This approach is not limited to any particular type of wind turbine. The analysis uses the following assumptions: - Homogenous, incompressible, steady state flow - No frictional drag - An infinite number of blades - Uniform thrust over the disk or rotor area - A non-rotating wake - The static pressure far upstream and far downstream of the rotor is equal to the undisturbed ambient static pressure

39. The thrust T is equal to the change in momentum rate: But the mass flow rate is: So that: T is +ve so the velocity behind the rotor U4is less than the free stream velocity U1. No work is done on either side of the turbine rotor. Thus, the Bernoulli equation can be used in the two control volumes on either side of the actuator disk.

40. Streamtube upstream of disk: 2 Streamtube downstream of disk: It is assumed that the far upstream and far downstream pressures are equal ( p1 = p4 ) and that the velocity across the disk remains the same ( U2 = U3 ). The thrust can also be expressed as the net sum of the forces on each side of the actuator disc:

41. Using the Bernoulli equations to solve for ( p2 - p3 ) we obtain the thrust as: Equating this with our first two expressions for T and recognizing that the mass flow rate is  A2 U2we obtain for the wind velocity in the rotor plane simply: Defining the axial induction factor a as the fractional decrease in wind velocity between the free stream and the rotor plane:  and

42. The quantity U1 a is the “induced velocity” at the rotor, so the wind velocity there is a combination of the free stream velocity and the induced velocity. As the axial induction factor increases from 0, the wind speed behind the rotor reduces. If a = 1/2, the wind has slowed to zero velocity behind the rotor and this simple theory is no longer applicable. The power out P is equal to the thrust times the velocity at the disk: U2

43. Substituting in the expressions for U2and U4: gives: where A is the rotor area and U is the freestream velocity. The power coefficient CP represents the amount of available wind power extracted by the rotor: 

44. The maximum value of CP occurs when a = 1/3 so that: CP,max = 16 / 27 = 0.5926 For this case, the flow through the disk corresponds to a stream tube with an upstream cross-sectional area of 2/3 the disk area that expands to twice the disk area downstream. This result indicates that, if an ideal rotor were designed and operated such that the wind speed at the rotor were 2/3 of the freestream wind speed, then it would be operating at the point of maximum power production.

45. The thrust T and thrust coefficient CT can now be computed as Hence, the thrust coefficient for an ideal wind turbine is equal to 4a (1 - a). CT has a maximum of 1.0 when a = 0.5 and the downstream velocity is zero. At maximum power output (a = 1/3), CT has a value of 8/9.

46. (a) Operating parameters for a Betz turbine; U = velocity of undisturbed air; U4 = air velocity behind rotor CP = power coefficient, CT = thrust coefficient

47. The Betz limit, CP,max = 16/27, is the maximum theoretically possible rotor power coefficient. In practice 3 effects lead to a decrease in the maximum achievable power coefficient: - Rotation of the wake behind the rotor - Finite number of blades and their tip losses - Non-zero aerodynamic drag Note that the overall turbine efficiency is a function of both the rotor power coefficient and the mechanical (including electrical) efficiency of the wind turbine: 

48. Ideal HAWT with Wake Rotation In reality the generation of rotational KE in the wake results in less energy extraction by the rotor than would be expected without wake rotation. In general, the extra KE in the wind turbine wake will be higher if the generated torque is higher. Thus, as will be shown here, slow running wind turbines (with a low rotational speed and a high torque) experience more wake rotation losses than high-speed wind machines with low torque.

49. Geometry for rotor analysis: U = undisturbed wind velocity a = induction factor Area of annular streamtube of radius r and thickness dr is 2  r dr

50. Assuming angular velocity  imparted to flow is small compared to angular velocity  of the rotor  pressure in far wake = pressure in freestream. The pressure, wake rotation and induction factors are all assumed to be a function of radial position r. Using a CV that moves with angular velocity  the energy equations can be applied at sections before and after the blades to derive the pressure difference across them. Across the flow disk the angular velocity of the air relative to the blade increases from  to  + ,whilst the axial component of the velocity remains constant.