Page 239 - Fluid Power Engineering

206 Chapter Te n

and then converts DC to AC at grid frequency. Variable-speed gener-
ators are typically connected to direct-drive turbines with no gearbox
or a simple gearbox. The speed of rotors of turbine and generator de-
pend on wind speed and are not fixed. This will be described in more
detail later in the chapter.
The equation for total power in all three phases is:

P T = 3V T . ¯ i (10-25)

Working in the complex domain, total power is:

V T
P T = P + jQ = 3 (E g (90 − δ) − V T 90)
Z
(10-26)
V T
= 3 (E g cos(90 − δ) + E g j sin(90 − δ)) − V T j
Z
V T
Real power, P = 3 (E g sin(δ)) = P max sin δ (10-27)
Z
V T
Reactive power, Q = 3 (E g cos(δ) − V T ) (10-28)
Z
P P max sin δ V T
τ = = = 3 (E g sin(δ)) = τ max sin δ (10-29)
ω ω Zω
For the purposes of this simplified analysis, assume a fixed-speed
synchronous generator, therefore, the following are constants: f, E g ,
V T ,ω, Z. The grid mandates fixed values of terminal voltage and fre-
quency. Fixed frequency means fixed ω. Thus, the real power P and
torque τ are functions of the power angle δ. A plot of power versus
load angle is in Fig. 10-5. The maximum power that may be gener-
ated and the maximum torque that it can withstand are given by P max
and τ max , respectively. When the applied torque is greater than τ max ,
then the power output reduces, the rotor accelerates, and the genera-
tor pulls out of synchronism. Note the power and torque curves are
both sinusoidal. When δ< 90, larger torque is required to increase δ,
which makes the system stable. However, when δ ≥ 90, lesser torque
is required, meaning in this condition, the rotor will start accelerating
because the resisting torque is smaller than the applied torque and
will never to able to resist the additional torque. When this condition
is achieved, the generator will be destroyed. Therefore, in order to
avoid this condition, adequate protection must be put on the turbine
or drive train side, such as aerodynamic or mechanical brakes.
Note, in a synchronous generator the external torque is resisted
by the tangential force between poles of a magnet. The tangential
force exists as long as the there is angular displacement between the
poles. The same machine can, therefore, serve as both a generator and
a motor. When the angular displacement is negative, the electrical