On Control Strategies for Wind Turbine Systems

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Bridget Higgins
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1 On Control Strategies for Wind Turbine Systems Niall McMahon December 21, 2011 More notes to follow at: 1 Calculations for Peak Tip Speed Ratio Assuming that a preliminary assessment has concluded that a small wind turbine system should have a rotor diameter of 2.5 m, what can we say about this system? Well, the theoretical power available will be: P t = 1 2 ρau 3 (1) Assuming that for the generator selected, the system is rated at a (high but typical) wind speed of 10 m/s, P t = (π ) 10 3 (2) (Nov. 11th: a small typographical error in this equation - division by 4 - has been corrected.) Or P t = 2945 W (3) Assuming that the system is designed to run at a tip speed ratio of 4, i.e. peak power extraction happens when λ = 4. At the rated wind speed of 10 m/s, the speed of the blade tips will be 40 m/s. The tip speed of a rotor is determined almost entirely by the setting angle of its blades, the twist of its blades and the chord length of its blades. Tip speed depends on blade geometry. Using a blade-element/momentum (BEM) code, which we ll cover later in the course, we can determine that the peak aerodynamic power coefficient 1

2 of the resulting rotor design, C p, is 0.49; this is high. The peak power coefficient corresponds to the design tip speed ratio, i.e. it occurs when λ = 4 for this system. The actual power output that we can expect from the rotor when it operates at its peak tip speed ratio is, P t = = 1443 W (4) Knowing the rotor radius, i.e. the blade length, we can write that, ω RPS = 40 = 5.1 (5) 2π 1.25 Where ω RPS is the rate of rotation expressed as revolutions per second, and 2πR is the circumference of the rotor disc, where R = 1.5 m is the rotor radius. We can also write, ω RPM = = 306 (6) Where ω RPM is the rate of rotation expressed as revolutions per minute, and that, ω = 5.1 2π = 32 (7) Where ω is the rate of rotation expressed in radians per second. Unless explicitly defined otherwise, you should always assume that ω has the units of radians per second. Rotor power is related to rotor torque, T, as, P = T ω (8) This means that for a peak power of 1443 W at a rate of rotation of 32 radians per second (in a wind speed of 10 m/s), T = P ω = = 45.1 Nm (9) 2 Power and Torque Curves Using BEM, again you will learn about this in the future, a C p - λ curve can be determined for this rotor design. For now, you can take the C p - λ curve as a given for a particular rotor design. The peak C p of 0.49, corresponding to the design tip speed ratio of 4, was determined using this graph. 2

3 0.50 Cp-Lambda Power Coefficient (Cp) [ - ] Tip Speed Ratio (Lambda) [ - ] Figure 1: C p - λ for this rotor design. From the C p - λ curve, it is evident that when λ is low or high, C p is low. The C p - λ is a dimensionless representation of how the power output of the rotor changes with wind speed and rate of rotation. Using the C p - λ curve, a dimensional power vs. RPM curve can be drawn for each free-stream wind speed. These are the curves of most practical use for working engineers. For a free-stream wind speed of 10 m/s, the power output of the rotor varies with RPM. This is shown in Figure 2. The corresponding torque variation with rate of rotation, for a wind speed of 10 m/s, calculated from the power curve using Eqn. 8 is shown in Figure 3. The peak torque clearly occurs at a lower RPM than peak power extraction. A family of power vs. RPM curves can be plotted for various wind speeds. See Figure 4. As the wind speed climbs, so too does the output power of the rotor. Similarly, a family of torque curves can be created corresponding to different wind speeds. See Figure 5. It s clear that as the wind speed increases, the power and torque increase quickly. The challenge of control in a wind turbine is to maintain operation close to the peak C p while at the same time ensuring that: (i) the forces acting on 3

4 160 Power Vs. RPM Power [W] Figure 2: Power variation with RPM for a wind speed of 10 m/s Torque Vs. RPM Torque [Nm] Figure 3: Torque variation with RPM for a wind speed of 10 m/s. 4

5 Power [W] Power Vs. RPM m/s 4 m/s 6 m/s 8 m/s 10 m/s 12 m/s 14 m/s 16 m/s 18 m/s 20 m/s 22 m/s Figure 4: Family of curves showing how the power varies with RPM for various wind speeds. Torque [Nm] Torque Vs. RPM m/s 4 m/s 6 m/s 8 m/s 10 m/s 12 m/s 14 m/s 16 m/s 18 m/s 20 m/s 22 m/s Figure 5: Family of curves showing how the torque varies with RPM for various wind speeds. 5

6 the system do not increase beyond certain thresholds and (ii) the system can safely slow and stop as the wind speed increases. This is essentially a problem of limiting torque, i.e. the acceleration of the rotor. A particular challenge is the fact that the peak rotor torque occurs at a lower rate of rotation that the peak rotor power at a given wind speed. Control techniques, for example, pitch systems, stall systems and brake systems, all function to reduce net shaft torque and power at high wind speeds and rates of rotation. The two broad approaches to power and torque control are: (i) to reduce the power extracted from the airflow (and, as a result, the rotor torque); (ii) to dissipate the rotor power (and torque). Pitch and stall control systems are examples of the first kind of control; mechanical rotating brakes of various types are examples of the second kind. Pitch and stall control systems reduce the power intake of the rotor (the result is a depressed C p - λ curve and, as a result, reduced power and torque values at every wind speed); brakes dissipate the rotor power as heat. So pitch and stall control systems change the C p - λ, power and torque curves while brake systems don t. It is almost always better to reduce torque by reducing the power extracted from the airflow rather than applying a brake, i.e. to avoid accepting the energy flow in the first place rather than having to dissipate it somehow afterwards. Reducing power extraction from the airflow, e.g. using pitch control, is something like controlling the speed of a car using the accelerator; applying a brake to the turbine, without controlling the power extraction, is something like controlling the speed of a car by braking without adjusting the accelerator. For safety reasons, all large wind turbines use a combination of both approaches. Niall McMahon, October Revised November 11th

EE 742 Chap. 7: Wind Power Generation. Y. Baghzouz

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