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 10

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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

3

• 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
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
Fields -- Flux density due to field for rotor
position ?0
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
Fields -- 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
Fields -- Flux density due to field for rotor
position ?0
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
d
a
q
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Modeling of synchronous generators
Construction Terminology -- Flux density due
to stator phase a
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) Observer
at a sees a flux density that varies sinusoidally
in time Ba(a) Bamax cos(a) cos(?t) Ba(a) K
Iamax 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!
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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Modeling of synchronous generators
Construction Terminology -- Flux density due
to stator phase a Salient Rotor
Phase a MMF pattern is the same as before Flux
density is essentially radial through air gap
but reluctance is a function of location
a reluctance is lower along d-axis and higher
along q Thus flux density is higher along d axis
MMFa(a,?) MMFamax cos(a)cos(?t)
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Modeling of synchronous generators
Fields -- Flux density due to stator phase a
Salient Rotor
Phase a MMF pattern is the same as
before MMFa(a,?) MMFamax cos(a)cos(?t) Resol
ve MMFa into space vectors MMFad and
MMFaq MMFadMMFa cos(?) MMFaq-MMFa sin(?)
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Modeling of synchronous generators
Fields -- Flux density due to stator phase a
Salient Rotor
Flux density components depend On reluctance
along d and q axes BadBad cos(?)cos(a) Baq-Baq
sin(?) cos(a-90) The magnitudes Bad Baq are
proportional to current ia BadKd Ia
cos(?)cos(a)cos( Baq-Baq sin(?) cos(a-90)
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Modeling of synchronous generators
Fields -- Flux density due to stator phase a
Salient Rotor
x
x
Flux density components depend On reluctance
along d and q axes BadBad cos(?)cos(a) Baq-Baq
sin(?) cos(a-90) The magnitudes Bad Baq are
proportional to current ia BadKd Iam
cos(?)cos(a)cos(?t) Baq-Kq Iam sin(?) cos(a-90)
cos(?t) Later we will call Iam cos(?) and Iam
sin(?) the d and q components of current Ia
?
a
MMFa
d
MMFaq
MMFad
q
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Modeling of synchronous generators
Net Stator Field Rotating Magnetic
Field- balanced positive sequence operation
d
a
d
a
Schematic of Net Stator Field Phases a,b,and c At
t0
q
q
Ba(a) (3/2)K Iamax cos(a)
Schematic Representation Of Phase a flux
density Ba(a) K Iamax cos(a) cos(?t)
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Modeling of synchronous generators
d
d
d
a
a
a
q
q
q
t0
t2p/3?
t4p/3?
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Modeling of synchronous generators
d
d
d
a
a
a
q
q
q
Bs(a,t) K Iamax cos(a) cos(?t) K
Iamax cos(a-2p/3) cos(?t -2p/3)
K Iamax cos(a2p/3) cos(?t 2p/3) Bs(a,t)
(3/2)K Iamax cos(a- ?t ) ROTATING FIELD
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Modeling of synchronous generators
d
d
d
a
a
a
q
q
q
ROTATING FIELD Bs(a,t) (3/2)K Iamax cos(?t-
a) Observer at a sees sinusoidal flux density in
time Frequency ? lags ia by a At any time
t flux density is sinusoidally distributed in
space Amplitude is time independent (3/2)K
Iamax Peak is at a ?/t
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Modeling of synchronous generators
d
a
Steady State Model from Fields Terminal voltage
under balance conditions
q

• The terminal voltage comprises of
• Voltage induced in phase ( a) by Field Flux
• Voltage induced by net stator flux
• Resistive voltage drop
• Inductive voltage drop in parts of the winding
  outside air gap (Overhang)
• Inductive voltage drop in individual phase from
  leakage flux

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Modeling of synchronous generators
Steady State Model from Fields-Terminal voltage
under balance conditions
a Vt - n
Ra
-Ear
Ll
Ia
Round Rotor
Ea -

• The terminal voltage comprises of
• Voltage induced in phase ( a) by Field Flux(Ea)
• Excitation, Induced Voltage, Generated Voltage,
  Open Circuit Voltage
• Voltage induced by net stator flux(Ear)
• -Armature Reaction
• Resistive voltage drop (Ra armature resistance)
• Inductive voltage drop in parts of the winding
  outside air gap (Overhang),Inductive voltage drop
  in individual phase from leakage flux (Ll leakage
  inductance)

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Modeling of synchronous generators
Steady State Model from Fields-Terminal voltage
under balance conditions

• Voltage induced in phase ( a) by Field Flux(Ea)
• Excitation, Induced Voltage, Generated Voltage,
  Open Circuit Voltage

Bf(a,?) Bfmax cos(a-?)
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Modeling of synchronous generators
Steady State Model from Fields-Terminal voltage
under balance conditions

• Voltage induced in phase ( a) by Net Stator
  Flux(Ear)
• Armature reaction

Bs(a,t) (3/2)K Iamax cos(a- ?t )
?
d
a
q
q
a
a
a
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Modeling of synchronous generators
Steady State Model from Fields-Terminal voltage
under balance conditions
Voltage induced in phase ( a) by Field Flux(Ea)
?
Ea?N? /?o
d
? rms value of flux , depends on field current
Ia
a
q
Ear?N(3K) Ia/-90 -j?LaIa
q
a
a
a
Vt Ea- j?LaIa- j?Ll Ia RaIa Vt Ea- -jXsIa-
RaIa
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Modeling of synchronous generators
Next More on steady state, round rotor
model Salient Pole Machine Transients and flux
freezing Coupled Circuit model