## Power Systems in Wind Turbines

### Context

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### Basic Theory

The power in an electrical current is the product of its voltage and current. For a simple circuit consisting of a power source, e.g. a battery, and a resistor, the power is given as:

p = vrir

Where vr and ir are the voltage and current across the resistor, respectively.

#### Direct and Alternating Current

If the voltage and current associated with a power source are steady in time, this power is referred to as direct current or DC. If the voltage and current vary with time, this is alternating current, or AC.

(Click to expand.) A low-frequency AC type signal. The x-axis marks time, the y-axis the signal amplitude. The curve is described by the equation y = A sin (2 π f t), where A is the peak amplitude of the varying quantity, 1 unit in this case, f is the frequency of the signal, 2 Hz in this case, and t is the total time elapsed, 1 second in this case. the quantity 2 π f can also be written as ω and the quantity 2 π f t = ω t is an angle, expressed in radians.

Practical AC signals are always sinusoidal. AC voltage can be described by:

v = Vpeak sin(ωt)

Where Vpeak is the peak voltage value and ω = 2 π f, where f is the signal frequency in Hz.

The root mean square (RMS) voltage or current of an AC signal, vrms, or simply V, is generally used to characterise the signal. The RMS voltage is the equivalent DC voltage or current that would have produced the same heating in a resistor connected across the power source terminals.

V = (1/2)0.5 Vpeak

(Click to expand.) In this case, two additional signals, B and C, lead and lag the reference signal in the order given. The equations for B and C are yB = A sin (2 π f t + π/2) and yC = A sin (2 π f t - π/2). B leads the reference signal by 90 degrees; C lags the reference signal by 90 degrees.

(Click to expand.) In this case, a signal, B, has a higher amplitude and higher frequency than the reference signal. Specifically, the peak amplitude of B is 1.5 units and the frequency is 4 Hz.

#### Resistors

There are three basic type of component in electrical circuits: resistors, capacitors and inductors. Each component offers resistance to the flow of electricity in different ways.

Resistors impose the following condition on the flow:

i = v/R

where v and i are the voltage and current across the resistor. The resistance of the resistor causes a voltage drop of v. The energy associated with this voltage drop is released as heat in the resistor.

If we assume that the source voltage has an AC signal,

v = Vpeak sin(ωt)

We can write that the current across the resistor is then,

i = (Vpeak/R) sin(ωt)

The instantaneous power in the circuit is then,

p = vi

Re-arranging this,

P = (Vpeak2/R) sin2(ωt) = 2 (V2/R) sin2(ωt)

Finally,

P = (V2/R) (1 - cos(2ωt))

#### Inductors

Small inductors. (Image from Wikipedia, licensed under the Creative Commons Attribution-Share Alike 2.5 Generic.)

Inductors oppose changes in current to them by dropping a voltage that is directly proportional to the change in the current, i.e.

v = L di/dt

Where L is measured in henries. Since i = Ipeak sin(ωt) and if we assume that a current i flows through the inductor,

v = ω L Ipeak cos(ωt)

Since cos(ωt) is equivalent to sin(ωt), only offset by 90 degrees, we can write that

Vpeak = ω L Ipeak

Further, we can say that the voltage and the current are offset by 90 degrees. If we take the current, i, as the reference signal, we can say that the voltage leads the current by 90 degrees. (If you plot cos(x) and sin(x) together, starting at x = 0, the cosine signal will lead the sine signal.)

The instantaneous power associated with the inductor is,

p = vi = VpeakIpeak cos(ωt)sin(ωt)

Remembering that,

Vpeak = ω L Ipeak

This can be written as,

p = ω LI2peak cos(ωt)sin(ωt) = 2 ω LI2 cos(ωt)sin(ωt) = ω LI2sin(2ωt)

Where I = irms = (1/2)0.5Ipeak.

#### Capacitors

Small capacitors, types that can by found on electronic circuitry. Image from Wikipedia, released into the public domain.

Capacitors draw or supply current to oppose changes in voltage.

i = C dv/dt

v = Vpeak sin(ωt)

i = C ωVpeak cos(ωt)

Since cos(ωt) is equivalent to sin(ωt), only offset by 90 degrees, and since em>i = Ipeak sin(ωt). we can write that,

Ipeak = C ωVpeak

And,

i = Ipeak cos(ωt)

Or,

i = Ipeak sin(ωt + π/2)

With the voltage v = Vpeak sin(ωt) as the reference signal, the current leads the voltage by π/2, or 90 degrees.

### Transformers

Transformers are electrical devices that transfers energy between windings in two AC electrical circuits using a phenomenon called electromagnetic induction. Tranformers allow the stepping up or stepping down a voltage. A varying voltage in the primary winding induces a voltage in the secondary winding proportional the ratio between the number of turns in the two windings, i.e.

v2/v1 = N2/N1

Where v1 and v2 are the voltages in the primary and secondary windings and N1 and N2 are the number of turns in the primary and secondary windings.

The associated currents are calculated from p = vi: the power is constant in both the primary and secondary circuits.

Transformer operation. Image from Wikipedia, released under the GNU Free Documentation License.

### Generators

• Generators create current by moving conductors through a magnetic field. i.e. electromagnetic induction.
• A rotating magnet, a rotor, moves inside conductors wound about a stationary casing, the stator.
• Depending on its construction, the generator's output can be either direct current (DC) or alternating current (AC).

The operation of a generator can be described by i = BLU, where i is the induced current, B is the magnetic flux density of the magnetic field, L is the length of the conductor and U is the speed with which the conductor moves through the magnetic field.

#### Synchronous Machines

Synchronous generators use permanent magnets or a DC current flowing through a coil to create a magnetic field "fixed to" the rotor. The magnetic field is produced independently. As the rotor rotates, the magnetic field rotates as well. An AC voltage proportional to the speed of rotation is induced in the stator windings. In this sense, the output is synchronised. A schematic of a permanent-magnet synchronous generator, in cross-section. Starting at the centre, the layers are, in order: (i) rotor; (ii) magnet layer; (iii) air gap; (iv) stator windings.

Synchronous wind turbine generators:

• Permanent magnets made from alloys of rare earth elements are used in many modern small wind turbines.
• Neodymium magnets, commonly used, are made of neodymium, iron and boron.
• Good for low speed applications; no gearbox.
• Reliable and efficient.
• Not easily sychronised to grid. Wild, or variable, AC output, i.e. both voltage and amplitude vary with RPM. Requires inverter.
• Unsuitable for fixed speed applications.
• Most popular solution at present.

Internal rotor with permanent magnets clearly visible.

#### Asynchronous or Induction Machines

Induction generators differ from synchronous machines in that the magnetic field is not produced independently, i.e. permanent magnets or a DC excitation are not required. Instead, the stator windings are initially energised with an AC signal. This signal induces a voltage in windings on the rotor using the same mechanism that underpins transformer operation. The resulting magnetic field folows the rotor and, in turn, acts as a medium to transfers the energy from the rotor's prime mover, i.e. the wind turbine rotor, back into the stator windings. This generator type is asynchronous in that there is slip between the rotor speed and the rate of rotation of the magnetic field, i.e. they are not tightly coupled, unlike the synchronous generator.

The induction generator:

• Uses current from the grid to magnetise its field.
• Is easily sychronised to grid.
• Is good for both fixed and semi-variable speed wind turbine applications.
• Has a cost per unit of electricity that is generally lower than for a synchronous machine.
• Is very robust.