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Lec 3: Power System Components
Dr. Malabika Basu8/10/2009
Lesson plan 3 and L.O.
• Sequence analysis example ( detail fault analysis next sem)
• Transformer model recap, tap change and phase change, 3-phase
• Modeling of Synchronous Generator• Modeling of Induction motor
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Balanced Operation Of a Three-Phase Circuit
• In the language of Power Systems, a three-phase circuit is said to be balanced if the following conditions are true.
• If all the sources and loads are y-connected.
• There is no mutual inductance between the phases.
• All neutrals are at the same potential.
• As a consequence of the points (2) and (3) above, all phases aredecoupled.
• All network variables are balanced sets in the same sequence as the sources.
• If not…. decouple them through symmetrical component analysis
Symmetrical Component Analysis
=
c
b
a
VV
V
aaaa
VV
V
2
2
2
1
0
11
111
31
Vs = A-1Vp
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Symmetrical Component
=
2
1
0
2
2
1
1111
V
VV
aa
aa
V
VV
c
b
a
Vp = AVs
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Power invariance
• S = VaIa* + VbIb* + VcIc*
• S = [AVs]T[AIs]*
• S = 3[Va1Ia1* + Va2Ia2
* + Va0Ia0* ]
Example 8.1, 8.2 (Glover)
Transformer modeling
• Power balance• Voltage, current and impedance balance• Mmf and flux linkage balance ( matching primary
and secondary voltage and get rid of the isolation)• Power loss• Exact equivalent circuit ( phasor diagram)• Approximate equivalent circuit ( Power system
calculation)
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Equivalent circuit of a single-phase transformer
Simplified equivalent circuit of a single-phase transformer: (a) when referred to the primary side and (b) when referred to the secondary side.
500 MVA, 220/22 kV with a leakage reactance of 10%. Find out the impedance in each side.
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Transformer example• The VA base of the transformer is 500 MVAand the voltage bases in the
primary and secondary side are 200 kV and 22 kV respectively. Therefore the impedance bases of these two sides are
Ω Ω
Assume that the leakage reactance is referred to the primary side. Then for 10%, i.e., 0.1 per unit leakage reactance we have
The above reactance when referred to the secondary side is
Ω
Ω
Hence the per unit impedance in the secondary side is 0.0968/0.968 = 0.1.
Transformer
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A
C
B
a
b
c
n
Synchronous Generator Modeling
Ref: Chapter 4Ref: Chapter 4
Embedded GenerationEmbedded Generation
ByBy
N. Jenkins et alN. Jenkins et al
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Schematic diagram of a synchronous generator
The schematic diagram of a synchronous generator is shown above. This contains three stator windings that are spatially distributed. It is assumed that the windings are wye-connected. The winding currents are denoted by ia , ib and ic. The rotor contains the field winding the current through whichis denoted by if . The field winding is aligned with the so-called direct ( d ) axis. We also define a quadrature ( q ) axis that leads the d -axis by 90°. The angle between the d-axis and the a-phase of the stator winding is denoted by ?d.
Synch. Generator
• Consists of armature (stator)- 3 phase,• Field winding (rotor), dc• The armature develops mmf rotating at a speed
proportional to the frequency (f)• Field winding develops an mmf w.r.t. rotor• In normal operation, the rotor and hence the
field winding rotates synchronously with the mmfdeveloped by the stator with its relative angle, the load angle, determined by the torque applied to the shaft
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Synch. Generator
60*2PN
f =
f = electrical frequency in Hz
P = no of poles
N = rotor speed in RPM
fm = N/60 mechanical frequency in r.p.s
FIXED RELATIONSHIP BETWEEN SPEED AND FREQUENCY
Construction of Synchronous Generator
• Non-salient pole ( cylindrical rotor)Uniform airgap, big alternator
• Salient pole (low speed, hydro gen)
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Three-phase equivalent circuit of a synchronous generator
Single-phase equivalent circuit of a synchronous generator
S.S. operation•Magnetic ckt unsaturated•a/gap flux uniform, effect of saliency neglected•a/gap flux sinusoidal•Rstator negligible
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Loading Capability Curve
• MVA loading limit – maximum MVA, imposed by stator heating
• MW loading limit – maximum MW, imposed by turbine power rating
• Max. field limit – rotor current limit• Stability limit ( dynamic aspect), must
operate at a safe margin of d
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Loading Capability Curve
Governor Droop characteristics
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Excitation control droop characteristics
Consider the 50 Hz power system the single-line diagram of which is shown in Fig. The system contains three generators, three transformers and three transmission lines. The system ratings are
300 MVA, 220/22 kV, X = 10%Transformer T 3
Three single-phase units each rated 100 MVA, 130Y/25 kV, X = 10% Transformer T 2
300 MVA, 220Y/22 kV, Xd = 10%Transformer T 1
300 MVA, 20 kV, Xd = 20%Generator G3
300 MVA, 18 kV, Xd = 20%Generator G2
200 MVA, 20 kV, Xd = 15%Generator G1
Modeling of the P.S.
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The impedance diagram of the system in single line
Loads• Voltage and frequency dependence• Affect the behaviour of P.S.• Large no. of types of loads• Differ considerably in characteristics• Aggregation of load (lump together)
Types:
•Lighting – voltage dependence
•Heating – power and voltage relationship
•Ind. Motor – P-V, Q-V, p.f.
•Synch motor
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Induction Machine modeling
Next class (Lec 4)
• Discussion of Assignment 1• Further modeling concepts – on sync.
Machines – transient state• FACTS devices modeling• Power Electronic Converters with
generators
