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## EE532 Power System Dynamics and Transients

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### ? = flux linkage ( Kundur Text) 10/8/06. EE531 Lecture 9-11. Modeling of synchronous generators ... Using these principles develop a model for the generator ... – PowerPoint PPT presentation

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Title: EE532 Power System Dynamics and Transients

1
EE532 Power System Dynamics and Transients
EUMP Distance Education Services
• Synchronous Generator Model
• Lecture 12
• Modified for EE531

2
Topics
• Modeling of synchronous generators
• Modeling machine in more detail
• Results from more detailed modeling

3
Approaches
• Fields Approach Coupled Coil Model
• Park/Kron/Blondel
• Transformation
• Two reaction theory Transient Studies
• Phasor Model
• Linearized Model
• Steady State Models Stability Studies

4
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
pd()/dt
? flux linkage ( Kundur Text)
5
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Kundur notation Note fd, kd, kq subscripts Note
ia, ib, ic now come out of stator coils
6
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Ignore Dampers
Will fold negative sign on ia ib ic into R and
L Will suppress time variable. Remember
everything Is instantaneous value here
7
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Laa, etc. are functions of Rotor position ? ?
??dt so T and the inductance terms are
Functions of time
8
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Inductance Matrix Stator phase a Laa(?)
LalLgoLaa2 cos (2 ?) Round rotor Laa2 0
Leakage
Field q Axis
?
Field d Axis
a
9
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Phase a Axis
Field q Axis
Inductance Matrix Mutual Inductance
a Lab(?) -(1/2)Labo-Lab2 cos (2
?-2p/3) Round rotor Lab2 0
?
Field d Axis
a
Leakage
Field q Axis
?
Field d Axis
a
10
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Phase a Axis
Inductance Matrix Mutual Stator phase a-
Field Lafd(?) Lafd cos ( ?)
Field q Axis
a
Field d Axis
Field q Axis
Field d Axis
a
Maximum
Minimum
11
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Phase a Axis
Inductance Matrix Field Self Lffd(?) Lffd
Field q Axis
a
Field d Axis
Field q Axis
Field d Axis
a
Maximum
Minimum
12
Modeling of synchronous generators- Round Rotor
L(?)
13
Modeling of synchronous generators
L(?)
14
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Ignore Dampers
Will fold negative sign on ia ib ic into R and
L Will suppress time variable. Remember
everything Is instantaneous value here
15
Modeling of synchronous generators
dqo transformation
Field current ifd is not transformed Io
(iaibic)/3 is the usual zero sequence current
16
Modeling of synchronous generators
dqo transformation
Direct axis component of stator currents
Quadrature axis component of stator currents
17
Modeling of synchronous generators
dqo transformation interpretation of dq currents
? is an arbitrary variable ( reference
frame) If Stator currents are balanced three
phase positive sequence
Note unconventional convention
Im peak value of line current
18
Modeling of synchronous generators
dqo transformation interpretation of dq currents
d
d
d
a
a
a
q
q
q
t0
t2p/3?
t4p/3?
Three phase statorbal.pos.seq. currentgt Uniform
rotating field
19
Modeling of synchronous generators
d
d
d
a
a
a
q
q
q
dqo transformation interpretation of dq currents
A rotating magnetic field can be created by a
two-phase stator -- coils placed 90 deg
apart -- currents 90 deg out of phase --
simplifies analysis but also practical
20
Modeling of synchronous generators
dqo transformation interpretation
Replaces 3 phase stator by a 2 phase stator
zero sequence circuit
d
d
a
a
q
q
21
Modeling of synchronous generators
dqo transformation interpretation
Transformation to synchronously rotating
reference ??st ?o
d
?s
d
?s
?s
q
q
2 phase stator rotating At ?s dc
Sationary 3 phase stator balanced positive
sequence current at frequency ?s
dc!
22
Modeling of synchronous generators
dqo transformation
dqo inverse transformation
Same transformations for voltage and flux
23
Modeling of synchronous generators
dqo transformation transformed machine
equations
Similar Notation for voltages and flux
eabc R iabc p?abc ?abc L iabc
24
Modeling of synchronous generators
dqo transformation transformed machine
equations
eabc R iabc p?abc ?abc L iabc
T(?)eabc T(?) R T-1(?) T(?) iabc
T(?) pT-1(?) T(?) ?abc edqo T(?) R
T-1(?) idqo T(?) pT-1(?) ?dqo ?dqo T(?)
L T-1(?) idqo
25
Modeling of synchronous generators
dqo transformation transformed machine
equations
edqo T(?) R T-1(?) idqo T(?)
T-1(?) p?dqo T(?) pT-1(?) ?dqo edqo
R idqo p?dqo T(?) pT-1(?)
?dqo ?dqo T(?) L T-1(?) idqo
R is diagonal transformer voltage derivative
of flux Speed voltage
pT-1(?) ? dT-1(?)/d ?
26
Modeling of synchronous generators
dqo transformation transformed machine
equations
pT-1(?) ? T(?) L T-1(?) Ldqo ?
27
Modeling of synchronous generators
dqo transformation transformed machine
equations
T (?)pT-1(?) ?r T (?)d(T-1(?))/d ??
• Effect of rotation (generated voltage) captured
by voltage sources
• q axis flux linkage induces speed voltage in d
coil
• d axis flux linkage induces speed voltage in q
coil

28
Modeling of synchronous generators
dqo transformation transformed machine
equations
29
Modeling of synchronous generators
dqo transformation transformed machine
equations
The dqo transformation simplifies machine
equations by transforming to a rotating reference
frame on the rotor The three stator coils are
replaced by a pair of orthogonal d-q coils and a
zero sequence coil Result Inductances are no
longer time varying Coil- coil coupling
simplified - no coupling between d and q
30
Modeling of synchronous generators
dqo transformation transformed machine
equations
Next Understanding transformed model Using
model for simple transients Steady state
model Slowly varying phasor model for stability
31
EE532 Power System Dynamics and Transients
EUMP Distance Education Services
• Synchronous Generator Model
• Lecture 14

32
Modeling of synchronous generators
dqo transformation transformed machine
equations
Understanding transformed model Using model for
simple transients Steady state model Slowly
varying phasor model for stability
33
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Kundur notation Note fd, kd, kq subscripts Note
ia, ib, ic now come out of stator coils
34
Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
Ignore Dampers
Will fold negative sign on ia ib ic into R and
L Will suppress time variable. Remember
everything Is instantaneous value here
35
Modeling of synchronous generators
L(?)
36
Modeling of synchronous generators
Simulation of original model
37
Modeling of synchronous generators
dqo transformation Simulation of original model
38
Modeling of synchronous generators
dqo transformation Simulation of original model
39
Modeling of synchronous generators
dqo transformation Simulation of original model
40
Modeling of synchronous generators
dqo transformation Simulation of original model
Field Current
Time, S
Ia
41
Modeling of synchronous generators
dqo transformation
dqo inverse transformation
Same transformations for voltage and flux
42
Modeling of synchronous generators
dqo transformation transformed machine
equations
43
Modeling of synchronous generators
dqo transformation transformed machine
equations
The dqo transformation simplifies machine
equations by transforming to a rotating reference
frame on the rotor The three stator coils are
replaced by a pair of orthogonal d-q coils and a
zero sequence coil Result Inductances are no
longer time varying Coil- coil coupling
simplified - no coupling between d and q
44
Modeling of synchronous generators
dqo transformation transformed machine
equations
The dqo transformation simplifies machine
equations by transforming to a rotating reference
frame on the rotor The three stator coils are
replaced by a pair of orthogonal d-q coils and a
zero sequence coil Result Inductances are no
longer time varying Coil- coil coupling
simplified - no coupling between d and q
45
Modeling of synchronous generators
dqo model simulation
The dqo transformation simplifies machine
equations by transforming to a rotating reference
frame on the rotor The three stator coils are
replaced by a pair of orthogonal d-q coils and a
zero sequence coil Result Inductances are no
longer time varying Coil- coil coupling
simplified - no coupling between d and q
46
Modeling of synchronous generators
dqo transformation Simulation
47
Modeling of synchronous generators
dqo transformation Simulation
48
Modeling of synchronous generators
dqo transformation Simulation-Compare abc
solution
If
Id
Iq
49
Modeling of synchronous generators
dqo transformation Simulation
50
Modeling of synchronous generators
dqo transformation Simulation
Ia
Ib
Ic
51
EE532 Power System Dynamics and Transients
EUMP Distance Education Services
• Synchronous Generator Model
• Lecture 16

52
Modeling of synchronous generators
dqo transformation transformed machine
equations
Understanding transformed model Using model for
simple transients Steady state model Slowly
varying phasor model for stability
53
Modeling of synchronous generators
dqo transformation transformed machine
equations
54
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
io 0 ?r ?s gt d, q quantities are dc gt
derivative terms 0
ed -Ra id ?r ?q ?d -Ld id Lad ifd eq
-Ra iq ?r ?d ?q -Lq iq efd Rf if
55
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
jEqEt Ra (id j iq) -Xq iq j Xd id
Eq/dEt /0 Ra (id j iq) /d-90o jXq iq /d j
Xd id /d-90o
q
Im
id It sin (diF) iq It cos (diF)
Eq Eq/d
-Xq iq Ra id
Id id /d-90o Iq iq /d It Id Iq Id It
sin (diF) /d-90o Iq It sin (diF) /d
jXq Iq
d
Iq
Ra iq Xd id
d
Re
jXd Id
Et
It
Id
Ra It
56
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Final Model Eq/dEt /0 Ra Ia jXq Iq j Xd Id
q
Im
Eq Eq/d
jXq Iq
Iq
d
d
Id id /d-90o Iq iq /d
Re
jXd Id
f
Et
Ia
Id
Ra Ia
57
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Final Model Eq/dEt /0 Ra Ia jXq Iq j Xd Id
Example Revisited ( Kundur Ex 3.2) 555 MVA, 24
kV, 0.9pf operating at rated conditions Saturated
values of parameters ( includes leakage) in
pu Xad1.386 Xaq 1.344 Xd 1.536 Xq
1.494 Find all quantities
58
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Example Revisited ( Kundur Ex 3.2)
To find Eq we need Iq and Id this in turns
requires d
Eq/dEt /0 Ra Ia jXq Iq j Xd Id
59
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Example Revisited ( Kundur Ex 3.2)
To find Eq we need Iq and Id this in turns
requires d
Eq/dEt /0 Ra Ia jXq Iq j Xd Id
Trick, Define
Eq/d Et /0 Ra Ia jXq Iq j Xq Id Et /0
Ra Ia jXq Ia
q
Im
Eq Eq/d
Eq/d is a vector at angle d!
Eq/d
jXq Iq
Find Eq/d first
Iq
d
d
Re
f
Et
jXd Id jXqId
Ia
Id
Ra Ia
60
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Example Revisited ( Kundur Ex 3.2)
61
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Example Revisited ( Kundur Ex 3.2)
Now find Id and Iq
62
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
63
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
64
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
65
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
66
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
67
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Power and Torque
S Et It Electrical (edjeq)(id-jiq)
(edid eqiq) j(eqid-ediq) T ?d iq
?q id P (1/w) Ra It2 Mechanical
68
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Alternate power formula( machine to infinite bus)
It Id Iq Make q axis the referenceRa0 Eq
Etcos(d)IdXd 0 Et sin d-XqIq Id -j
(Eq-Etcos d )/Xd Iq Etsin d /Xq It
Etsin d /Xq -j (Eq-Etcos d )/Xd S Et It
q
Im
Eq Eq/d
-Xq iq Ra id
jXq Iq
d
Iq
Ra iq Xd id
d
Re
jXd Id
Et
It
Id
Ra It
69
Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Alternate power formula( machine to infinite bus)
It Etsin d /Xq -j (Eq-Etcos d )/Xd S Et
It P (EtEq /Xd)sin d (Et2/2)((1/Xq)-(1
/Xd) sin(2 d)