Outline
1
Electrical Drive Systems 324
Synchronous Generators
Dr. P.J Randewijk
Stellenbosch University
Dep. of Electrical & Electronic Engineering
Stephan J. Chapman
Chapter 4 (5th Edition)
Chapter 4 – Synchronous Generator
4.1 – Synchronous Generator Construction
4.2 – The Speed of Rotation of a Synchronous
Generator
4.3 – The Internal Generated Voltage of a
Synchronous Generator
4.4 – The Equivalent Circuit of a Synchronous
Generator
4.5 – The Phasor Diagram of a Synchronous
Generator
4.6 – Power and Torque in Synchronous Generators
4.7 – Measuring Synchronous Generator Model
Parameters
4.8 – The Synchronous Generator Operating Alone
4.9 – Parallel Operation of Synchronous Generators
4.11 – Synchronous Generator Ratings
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4.1 Construction
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4.1 Construction (cont.)
Salient pole rotor machines
There are basically two types of synchronous
generators
for “low speed” applications (> 4 poles)
e.g. hydro-electrical turbines
Cylindrical or non-salient pole rotor machines
for “high speed” applications
e.g. steam turbines
Usually limited to 2 or 4 pole machines
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4.1 Construction (cont.)
4.2 The Speed of Rotation
Small machine may employ sliprings and brushes to
supply the field
From (3–34) in Chapter 3, it can be seen that in order to
generate at a certain frequency, a certain speed is
required
This is still better than using a DC machine with a
commutator and brushes – Why?
Large machines use “brushless” excitation
fse =
This implies no brushes and sliprings are used to supply
DC power to the field winding – see Fig. 4.3
(just for interest sake)
nsm poles
·
60
2
(3–34’)
Also from section 4.1, the higher the speed, the lower
the number of poles must be
Or, the lower the speed, the larger the number of poles
must be to generate a voltage at the required frequency
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4.3 The Internal Generated Voltage
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4.4. The Equivalent Circuit
The rotor’s magnetic field, BR (produced by a DC field
current on the rotor, IF ) produces EA
See section 3.4 again
Also due to hysteresis and the B-H characteristic, see
section 1.4, saturation occurs if the field current
becomes to high
The current in the stator winding, IA , produces a flux in
the stator, BS
The “magnetisation curve” (Fig. 4–7) looks exactly the
same as for a DC machine – Fig. 8–4
This flux also produces a voltage in the stator windings,
Estat
The terminal voltage of the machine,
Vφ = Enet = EA + Estat
(4–4’)
Bnet = BR + BS
(4–5)
and with
The difference in the angle between BR and Bnet
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4.4. The Equivalent Circuit (cont.)
4.4. The Equivalent Circuit (cont.)
and the difference in the angle between EA and Vφ
the equivalent circuit of the synchronous machine can
be deduced as shown in Fig. 4–9 and is almost similar
(an AC dual) of the equivalent circuit of the DC machine
as shown in Fig. 8–2(b)
will be the same and equal to δ, the torque – or power
angle of the machine
+ Remember P = ωm τind and ωm is constant for a
synchronous machine
With
Estat = −jX IA
(4–6)
Vφ = EA − jXS IA − RA IA
where X is equal to the magnetising reactance of the
machine – similar to a transformer
(4–11)
with the synchronous reactance given by
Thus from
Vφ = EA − jX IA
A more accurate/“full” equivalent circuit, will also have to
take into account the stator leakage reactance, XA , and
the stator resistance, RA , so that the KVL equation
becomes
(4–7)
XS = X + XA
(4–10)
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4.4. The Equivalent Circuit (cont.)
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4.5 The Phasor Diagram
The “full” equivalent circuit is shown in Fig. 4–10 for all
three phases
For generator operation, IA is defined as flowing out of
the machine
We only need to consider 1 phase (1φ) as shown in Fig.
4–12 and multiply the power or torque by three (×3)
Just apply KVL, i.e. e.q. (4–11) and remember the
phasors must add up graphically. . .
And that IA lags jXS IA by 90◦ (CIVIL) – see Fig. 4-13 &
4–14
+ Note: Synchronous machines are usually connected in
Y (also called “star” or “wye”) – Fig. 4–11 (a)
+ Note: We will always use Y connection. . .
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4.6 Power and Torque
4.6 Power and Torque (cont.)
+ N.B. From Fig. 4–16,
The mechanical input power to the generator that is
converted to electrical power,
Pconv = τind ωm
= 3EA IA cos γ
(4–14)
(4–15)
Which is similar to a DC machine, with the only
difference that:
+ this is an AC circuit, hence the angle between EA and IA
needs to be taken into account
+ there are three (3) phases that contributes to the
delivery of power & torque
If we assume that RA ≈ 0 (which is true for very large
synchronous generators)
Pconv = Pout = 3Vφ IA cos θ
it can be proven that:
Pconv =
(4–17)
3Vφ EA
sin δ
XS
(4–20)
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4.6 Power and Torque (cont.)
4.7 Measuring the Model Parameters
With δ the torque angle of power angle of the machine,
i.e. the angle between Vφ and EA
This is the same angle as the angle between Bnet and
BR with Vφ ≈ Enet – see eq. (4–4’)
Also with P = ωm τind ,
τind
3Vφ EA
=
sin δ
ωm X S
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(4–22)
The open- circuit characteristic (OCC) curve, Fig.
4–17(a) is the same as the magnetisation curve, Fig.
4–7(b)
Saturation occurs if the DC field current, IF , becomes to
high
Without saturation, EA (which is equal to Vφ because
the machine is open-circuit, i.e. IA = 0) would have
followed the air-gap line
For the short-circuit characteristic (SCC) curve, the
relationship between IA and IF is linear.
This is due to the fact that with XA & RA X , BR ≈ BS
so that Bnet ≈ 0 and thus Enet ≈ 0 and thus the total
flux, φ ≈ 0 – see eq. (3–38)
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4.7 Measuring the Model Parameters (cont.)
4.7 Measuring the Model Parameters (cont.)
The unsaturated synchronous reactance, XS , assuming
RA XS can thus be calculated as
Vφ |OCC,air-gap line XS|unsat ≈
(4–26’)
IA |SCC
IF =x A
with EA |OC,air-gap line measured from the air-gap line
during the open-circuit test, IA |SC measured during the
short-circuit test, both measure at the same value of
IF , say x A
Similarly, the saturated synchronous reactance can be
calculated as
Vφ |OCC,actual curve XS|sat ≈
(4–26)
IA |SCC
IF =x A
The rated voltage value of the synchronous generator
will be used to determine the rated field current from the
OCC curve
The current value from the SCC curve for the same
rated field current will be used to determine XS|sat
+ XS|sat will always be smaller than XS|unsat – see Fig. 4–19
(next slide)
+ See Example 4–1 also with regard to the DC test in
order to determine RA
+ Note: We will always work with XS|sat
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4.7 Measuring the Model Parameters (cont.)
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4.8 Operating Alone
In a nutshell:
EA is determined from the OCC and depends on IF , the
DC field current
IA is determined by the impedance of the load, and the
value of EA ,
IA =
EA 0◦
RA + jXS + Zload
+ Note: You can choose any angle for EA – but 0◦ is nice –
as the angle of IA will always be with respect to the angle
chosen for EA
Vφ is determined by the phase/load current,
Vφ = IA Zload
The Short-Circuit Ration – Skip
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4.8 Operating Alone (cont.)
4.9 Parallel Operation
When asked to draw phasor diagram, similar to that
shown in Fig. 4–22 – which are with respect to
Vφ = Vφ 0◦ , all the phasor are just rotated by the angle
of Vφ as calculated above
In this section we can clearly see the effect a unity,
lagging or leading power factor have on the voltage
regulation of the terminal voltage
+ Note: From Fig. 4–25 we can see that with a fixed
excitation (i.e. fixed field current), the terminal voltage of
the synchronous generator when feeding a load with a
leading power factor, actually increases with an increase
in line current – which is somewhat counter intuitive. . .
A synchronous generator needs to be synchronised with
the infinite bus before it can be connected to the infinite
bus
This implies that:
the line (or phase) voltage of both must be the same
the phase sequence for both must be the same
the phase angles of both phase voltage must be the
same
the frequency of both must be the same
+ both ⇐ the generator and the infinite bus
Frequency – Power and Voltage – Reactive Power
Characteristics of a Synchronous Generator – skip –
together with the speed droop (SD) characteristics and
all the house diagrams
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4.9 Parallel Operation (cont.)
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4.9 Parallel Operation (cont.)
The only thing that is important to remember with regard
to frequency, is that if the frequency of the synchronous
machine is
+ Note: This will lead to a change in the power angle, δ –
see Fig. 4–36 (b)
slightly higher than the infinite bus – see Fig. 4–34, the
machine will immediately start operating as a generator
and start delivering power
slightly lower than the infinite bus – see Fig. 4–35, the
machine will immediately start operating as a motor and
consume power, not delivering power
To change the amount of real power, P, delivered to the
infinite bus, the input power from the prime mover, e.g.
stream turbine needs to be changed
To change the amount of reactive power, Q, delivered to
the infinite bus, the DC field current, IF , or the excitation
of the machine needs to be changed
+ Note: This will lead to a change in the internal
generated voltage, EA – see Fig. 4–37
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4.9 Parallel Operation (cont.)
4.9 Parallel Operation (cont.)
+ Note: With Vφ fixed – because it is connected to the
infinite bus – the net flux in the machine is fixed, thus
EA ∝ IF
and does not follow the OCC curve line, but has a
linear relationship to one another through the rated
operating point. . .
+ See Fitzgerald et al, Fig. 5–10 (on the next slide)
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4.9 Parallel Operation (cont.)
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4.11 Synchronous Generator Ratings
With the net flux fixed the synchronous generator will
operate on the Op–line through the rated operating
point with a linear relationship between EA and IF
Various ratings apply to a synchronous machine –
voltage, frequency, speed, apparent power, power
factor, field current, service factor
For synchronous machines, the frequency is determined
by the speed of operation and vice versa
Operation of Generators in Parallel with Other
Generators of the Same Size – skip
fse =
nm
· poles2
60
(3–34)
The peak flux in the machine is determined not only by
the frequency, but also by the supply voltage
Bmax =
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Emax
2πNC Agap f
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4.11 Synchronous Generator Ratings (cont.)
4.11 Synchronous Generator Ratings (cont.)
+ N.B. The power loss in the stator windings is
determined by the magnitude of the stator current, and
is not influenced by the power factor
This implies that there must be a constant
volts-per-hertz ratio and also why 480 V, 60 Hz
machines will operate at the same flux-density value
when connected to a 400 V, 50 Hz supply – albeit at a
different speed. . .
PSCL = 3IA2 RA
The maximum AC current the machine can supply is
limited by the stator copper wire diameter and the
maximum allowable heat (i.e. power) this windings can
dissipate
With the voltage of the machine set (always given as a
line-to-line value), the maximum stator current is
indirectly given by the apparent power rating (measured
in kVA or MVA) of the machine
√
Srated = 3VL,rated IL,rated
(4–37)
(4–38)
The maximum DC field current possible is limited by the
field copper wire diagram and the maximum heat (i.e.
power) this windings can dissipate
PPCL = IF2 RF
(4-39)
The maximum IF directly influence the maximum EA as
E A ∝ IF
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4.11 Synchronous Generator Ratings (cont.)
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4.11 Synchronous Generator Ratings (cont.)
The maximum value of IA and IF or more specifically EA ,
limits the phasor diagram as shown in Fig. 4-47
By noting that the length of the “circle” with its origin at
Vφ 0◦ is proportional to IA ∝ S with Vφ and XS constant
3V
We scale the phasor diagram of Fig. 4–48 (a) by Xsφ
with its origin at Vφ 0◦ , to yield a power diagram as
shown in Fig. 4–48 (b)
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4.11 Synchronous Generator Ratings (cont.)
4.11 Synchronous Generator Ratings (cont.)
By swapping the x– and y–axis around, we obtain a
“traditional” power triangle type diagram with the
operational limits as shown in Fig. 4–49
If the prime mover’s (e.g. steam turbine) maximum
output power is lower than that of the generator, that
limit can be added to the generator’s capability diagram
to reflex the real operational limits of the generator –
see Fig. 4–50 (next slide)
Short-Time Operation and Service Factor
Unfortunately the power diagram has reactive power, Q,
as its x–axis and real of active power, P, a its y–axis
All electrical machines (not just synchronous machines)
can operate above their rated current values for short
periods of time
The time the current can be above the rated current
value depends on the thermal capacity or thermal time
constant of the machine
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4.11 Synchronous Generator Ratings (cont.)
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4.11 Synchronous Generator Ratings (cont.)
The service factor of a machine indicates at what level
the machine can operate indefinitely
Per-Unit System Revisited – see also 2.6 & 2.10
S3φ,base = S3φ,rated
S3φ,base
S1φ,base =
3
VL,rated
V1φ,base = √
3
S1φ,base
I1φ,base =
V1φ,base
V1φ,base
Zbase =
I1φ,base
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