EE532 Power System Dynamics and Transients

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EE532 Power System Dynamics and Transients
EUMP Distance Education Services

• Satish J Ranade
• Synchronous Generator Model
• Lecture 9

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Topics

• Modeling of synchronous generators
• Advanced stability studies
• Modeling machine in more detail
• Modeling controls
• Exciter/Voltage Regulator
• Results from more detailed modeling

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Synchronous Generator Model

• Effect of Field and Damper
• Machine transitions through
• Subtransient
• Transient
• Steady state
• Damping
• Field and Damper
• Excitation Control
• Voltage regulator
• Stabilizer

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• Fields Approach Coupled Coil Model
• Park/Kron/Blondel
• Transformation
• Two reaction theory Transient Studies
• Phasor Model
• Linearized Model
• Steady State Models Stability Studies

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Modeling of synchronous generators
Magnetic Circuits, Inductance and Circuit
Equations
Basic magnetic system (Transformer)
i1
i2
Main(Core flux) Largely Uniform
Leakage Flux
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Modeling of synchronous generators
Magnetic Circuits, Inductance and Circuit
Equations
Basic magnetic system (Transformer) Leakage
ignored
i1
i2
Amperes Law
H Magnetic Field Intesity(A/m) B µ H Flux
density (T) µ µr µo Permeability of core
material µrrelative permeability µo4p10-7
Permeability of free space
H L N1 i1 N2 i2
L length of magnetic path
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Modeling of synchronous generators
Magnetic Circuits, Inductance and Circuit
Equations
Basic magnetic system (Transformer) Leakage
ignored
i2
Flux
Flux Linkage
H (N1 i1 N2 i2)/L (A/m) B µ H Flux
density (T)
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Modeling of synchronous generators
Magnetic Circuits, Inductance and Circuit
Equations
Basic magnetic system (Transformer) Leakage
ignored
i2
Flux
Flux Linkage to Coil 1
f BA f (µ A/L) (N1 I1 N2 I2) Wb
?1 N1 f
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Modeling of synchronous generators
Magnetic Circuits, Inductance and Circuit
Equations
Basic magnetic system (Transformer) Leakage
ignored
i2
Equivalent Magnetic Circuit
Reluctance
Flux f (µ A/L) (N1 I1 N2 I2) Weber
R L/ µA
N2 I2
N1 I1
f
Magnetomotive Force(MMF)
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Modeling of synchronous generators
Magnetic Circuits, Inductance and Circuit
Equations
Basic magnetic system (Transformer) Leakage
ignored
i2
Self inductance Flux linkage Per Ampere in a coil
L21 L11 L22
Mutual inductance Flux linkage to a coil Per
Ampere in the other coil coil
With i2 0
?1N1f/i1 N12i1/R gt L11 N12/R
?2N2f/i1 N1N2i1/R gt L21 N1N2/R
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Modeling of synchronous generators
Magnetic Circuits, Inductance and Circuit
Equations
Basic magnetic system (Transformer) Leakage
ignored
L21 e1 e2 L11
L22
i2
i1
i1
i2
Circuit Equations form 1
e1 r1 i1 L11 di1/t L12 di2/dt e2 r2 i2
L22 di1/t L21 di2/dt
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Modeling of synchronous generators
Magnetic Circuits, Inductance and Circuit
Equations
Basic magnetic system (Transformer) Leakage
ignored
L21 e1 e2 L11
L22
i2
i1
i1
i2
Circuit Equations form 1
e1 r1 i1 d?1/dt ?1 L11 i1 L12 i2 e2 r2 i2
d?2/dt ?2 L21 i1 L12 i2
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Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
pd()/dt
? flux linkage ( Kundur Text)
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Modeling of synchronous generators
Construction Terminology
Field q Axis
Phase a Axis
Field q Axis
Phase a Axis
?
Field d Axis
Field d Axis
a
a is the position of an observer who measures
radial flux through air gap T is the position of
the rotor with respect to a fixed, vertical, axis
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Modeling of synchronous generators
Construction Terminology -- Developed View
d
a
q
a
q
q
d
0
p/2
-p/2
p
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Modeling of synchronous generators
Construction Terminology -- Flux density due
to field for rotor position ?0
d
a
q
q
d
q
d
d
Field mmf
A rotor field winding Distributed across the
Surface creates a stepped mmf Waveform intended
to approximate A sinusoid
0
p
p/2
a
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Modeling of synchronous generators
Construction Terminology -- Flux density due
to field for rotor position ?0
d
a
q
Field mmf
q
q
A rotor field winding Distributed across the
Surface creates a stepped mmf Waveform intended
to approximate A sinusoid
d
0
p
p/2
a
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Modeling of synchronous generators
Construction Terminology -- Flux density due
to field for rotor position ?0
d
Field MMF and Flux Density(B) is a function Of
rotor position ? and observer position a. Flux
density is essentially radial through air gap.
Since field current If is dc , the peak
amplitudes are constant in time and proportional
to If MMFf(a,?) MMFfmax cos(a-?) Bf(a,?)
Bfmax cos(a-?) If rotor rotates at constant
speed ?r, then ? ?o ?r t t time Observer
at a sees a flux density that varies sinusoidally
in time Bf(a,t) Bfmax cos( ?r t- a ?o ) The
peak amplitude is independent of time. The phase
is a, the observers position
a
q
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Modeling of synchronous generators
Construction Terminology -- Flux density due
to stator phase a
d
Phase a MMF and Flux Density(B) is a function of
observer position a. Flux density is essentially
radial through air gap The current in phase a is
a sinusoid with frequency ? so are mmf and flux
density MMFa(a,?) MMFamax cos(a)cos(?t) Observe
r at a sees a flux density that varies
sinusoidally in time Ba(a,?) Bamax cos(a)
cos(?t) Combined with the flux density from
phase b and c, the net stator flux density will
turn out to be a rotating field!
a
q
Stator phase a winding is also
actually Distributed across the Surface creates
a stepped mmf Waveform intended to approximate A
sinusoid
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