Chapter 8 Problems

8.1 Wind power has been used for centuries for productive purposes. One early yaw control system for windmills, invented by Meikle about 1750, used a fan tail and gear system to turn the rotor into the wind. The windmill turret supported the rotor that faced into the wind. The fan tail rotor, oriented at right angles to the power producing rotor, was used to turn the whole turret and rotor (see Figure B.8). The gear ratio between the fan tail rotor shaft rotation and the turret rotation was about 3000:1. Explain the operation of the yaw orientation system, including the feedback path that oriented the rotor into the wind.

Power shaft Fan tail rotor

Main rotor

Figure B.8 Meikle yaw control

SOLUTION

The torque from the rotor drove the mill and resulted in a net torque on the power shaft.

That torque would drive the turret, including the fan tail rotor out of the wind. As the yaw error increased, the aerodynamic torque on the fan tail rotor would produce a counter torque, to stabilize the yaw position. The system would operate with a yaw error that depended on the design and wind speed. The yaw error sensor (the fan tail) produced a torque that increased as the yaw error increased from the desired set point (no yaw error).

The 3000:1 gear reduction provided the force amplification to turn the turret.

8.2 A supervisory control system, as part of its tasks, is to monitor gearbox operation and the need for gearbox maintenance or repairs. What information should be collected by the supervisory controller and what information should be reported to the system operators?

SOLUTION

Numerous variations on input information and reporting might be acceptable, but important input sensor information should include operating temperatures (bearing or lubricant temperatures), gearbox vibration, lubricant level, and measures of lubricant circulation system operation (flow, pressure, or power, etc.). Operators should be appraised of total operating hours on the gearbox and lubrication systems, peak vibration and temperature levels, low lubricant levels, and hours since the last regular maintenance work. Additional information might include files of sensor data collected at frequent intervals, long term temperature trends, and correlations between other operating measures and gearbox operation (gearbox temperatures as a function of power, etc.).

8.3 A variable speed wind turbine control system uses blade pitch and speed variations to provide constant power to the grid. What are the tradeoffs between fluctuations in rotor speed and the time response of the pitch control system?

SOLUTION

In a variable speed wind turbine, the converter is set to provide constant power or torque as wind speed fluctuations cause rotor speed changes. The blade pitch is used to smooth out the rotor speed fluctuations. A slower time response of the pitch control system would allow the rotor speed to increase or decrease more than with a faster pitch control system.

This has the advantage of possibly reducing pitch system loading and wear, but may result in undesirable excursions in rotor speed. Faster pitch system response increases pitch motor power requirements and loads in the pitch system and blades.

8.4 An accelerometer on an experimental wind turbine is used to measure tower vibration.

Measurements indicate that the sensor is very sensitive to temperature variations. To complete a series of tests in cold weather the test engineers rig up a quick electrical heating element and controller to keep the sensor's temperature constant. The system includes 1) a small electronic chip that provides a millivolt output that varies with temperature, 2) an electronic circuit, 3) a transistor, 4) a resistance heater, and 5) a housing to encase the chip,

the heater, and the accelerometer. The electronic circuit provides a voltage output related to the difference between the voltages at the two inputs. A small current (milliamps) flows into the transistor that is a function of the circuit output voltage. The transistor uses that current to provide up to two amps of current to the heater. The heater maintains the temperature in the enclosure at 70 F.

a) What is the process that is being influenced by the control system?

b) What elements play the roles of the sensor(s), the controller, the power amplifier, and the actuator?

c) What if any are the disturbances in the system?

SOLUTION

a) The process that is being influenced by the control system is the energy balance between the accelerometer and components inside the enclosure and the environment on the outside of the enclosure. The introduction of the heating element and the controller adds a heat source inside the enclosure, increasing the temperature inside the enclosure and increasing the heat transfer from the enclosure.

b) The chip is the sensor that responds to temperature changes and the electronic circuit is the controller that responds to differences between the enclosure temperature and the desired enclosure temperature. The transistor is the power amplifier. It takes the small current from the circuit and controls a much larger current that is capable of providing heat.

The actuator is the resistance heating element.

c) The heat transfer from the outside of the enclosure is a disturbance to the system.

This is mainly a function of ambient temperature, but also of wind speed and possibly the presence of snow, ice, rain, or dirt.

8.5 An accelerometer on an experimental wind turbine is used to measure tower vibration.

Measurements indicate that the sensor is very sensitive to temperature variations. To complete a series of tests in cold weather the test engineers rig up a quick electrical heating element and controller to keep the sensor's temperature constant. The closed loop transfer function of the system is:

1 . 0 5 . 0

1 . 0

2+ +

=s s

T T

ref

Plot the step response of the system to a step increase of the reference temperature of one degree F.

Hint: The step response is formed by multiplying the transfer function by T_ref = 1/s.

The solution is determined by taking the inverse Laplace transform of the resulting equation:

⎟⎠

⎜ ⎞

⎝

⎛ +

= +

s s

T s 1

1 . 0 5 . 0

1 . 0

2

First it must be converted into a sum of terms, using the method of partial fraction

Then the inverse Laplace transform of each of the terms can be determined. The fraction with the second order denominator may need to be broken into two terms to find the inverse Laplace transform.

SOLUTION

The step response of the system is the response for:

T_ref = 1s

The inverse Laplace transform of this transfer function can be found using a partial fraction expansion: manipulations result in:

( )

^{2 2}

( )

^{2 2}

The inverse Laplace transfer of this function is:

)

The graph looks like:

1.2

Temperature, degrees F

30

Temperature controller step response

8.6 The transfer function for a pitch control system is:

⎟⎠

The closed-loop system response to a step command to pitch the blades 1 degree is:

(

^{s bs c}

)

^s

a) Calculate and plot the closed-loop time domain system step response for K = 0.5, the initial choice of the designer. Comment on the closed-loop system response, including damping, overshoot, and response time.

b) For no good reason, the control system designer decides to increase the gain of the system, K. Calculate the closed-loop time domain system step response for K = 3, the new choice of the designer.

c) What differences are evident between the time responses, each with a different gain?

Hint: perform a partial fraction expansion of the general form of the closed-loop response:

and find the inverse Laplace transform of each term, using the a, b, c, d variables. This symbolic form will be handy, as it will be used twice in the solution.

For parts a) and b), find the real root, a, of the denominator by graphing d

s s

s³+ ²+ + , using the appropriate K. The values of s at the zero crossing is –a. The second order root is found using long division:

(

^a

)

^s

(

^{a a}

)

^s

Thus, the roots of the denominator are s, (s + a) and

( ) (

²

)

2

2 bs c s 1 as 1 a a

s + + = + − + − +

Insert the solutions for a, b, c, d into the general solution and plot the result.

SOLUTION provides the other imaginary root, so the whole transfer function is:

(

^s

) (

^{s s}

)

^s Using a partial fraction expansion:

(

^s+a

) (

^s2d+bs+c

)

^{s= As +}

⁽

^s+Ba

⁾

⁺

(

^s2Cs+bs+D+c

)

and, therefore, equating coefficients on the left and right hand sides:

( )

In document Manwell Wind Energy Explained Solutions (Page 120-125)