Title: EE532 Power System Dynamics and Transients
1EE532 Power System Dynamics and Transients
EUMP Distance Education Services
- Satish J Ranade
- Synchronous Generator Model
- Lecture 12
- Modified for EE531
2Topics
- Modeling of synchronous generators
- Modeling machine in more detail
- Results from more detailed modeling
3Approaches
- Fields Approach Coupled Coil Model
- Park/Kron/Blondel
- Transformation
- Two reaction theory Transient Studies
- Phasor Model
- Linearized Model
-
- Steady State Models Stability Studies
4Modeling of synchronous generators
Circuits approach Using these principles
develop a model for the generator
pd()/dt
? flux linkage ( Kundur Text)
5Modeling 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
6Modeling 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
7Modeling 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
8Modeling 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
9Modeling 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
10Modeling 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
11Modeling 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
12Modeling of synchronous generators- Round Rotor
L(?)
13Modeling of synchronous generators
L(?)
14Modeling 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
15Modeling of synchronous generators
dqo transformation
Field current ifd is not transformed Io
(iaibic)/3 is the usual zero sequence current
16Modeling of synchronous generators
dqo transformation
Direct axis component of stator currents
Quadrature axis component of stator currents
17Modeling 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
18Modeling 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
19Modeling 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
20Modeling of synchronous generators
dqo transformation interpretation
Replaces 3 phase stator by a 2 phase stator
zero sequence circuit
d
d
a
a
q
q
21Modeling 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!
22Modeling of synchronous generators
dqo transformation
dqo inverse transformation
Same transformations for voltage and flux
23Modeling of synchronous generators
dqo transformation transformed machine
equations
Similar Notation for voltages and flux
eabc R iabc p?abc ?abc L iabc
24Modeling 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
25Modeling 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 ?
26Modeling of synchronous generators
dqo transformation transformed machine
equations
pT-1(?) ? T(?) L T-1(?) Ldqo ?
27Modeling 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
28Modeling of synchronous generators
dqo transformation transformed machine
equations
29Modeling 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
30Modeling 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
31EE532 Power System Dynamics and Transients
EUMP Distance Education Services
- Satish J Ranade
- Synchronous Generator Model
- Lecture 14
32Modeling 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
33Modeling 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
34Modeling 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
35Modeling of synchronous generators
L(?)
36Modeling of synchronous generators
Simulation of original model
37Modeling of synchronous generators
dqo transformation Simulation of original model
38Modeling of synchronous generators
dqo transformation Simulation of original model
39Modeling of synchronous generators
dqo transformation Simulation of original model
40Modeling of synchronous generators
dqo transformation Simulation of original model
Field Current
Time, S
Ia
41Modeling of synchronous generators
dqo transformation
dqo inverse transformation
Same transformations for voltage and flux
42Modeling of synchronous generators
dqo transformation transformed machine
equations
43Modeling 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
44Modeling 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
45Modeling 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
46Modeling of synchronous generators
dqo transformation Simulation
47Modeling of synchronous generators
dqo transformation Simulation
48Modeling of synchronous generators
dqo transformation Simulation-Compare abc
solution
If
Id
Iq
49Modeling of synchronous generators
dqo transformation Simulation
50Modeling of synchronous generators
dqo transformation Simulation
Ia
Ib
Ic
51EE532 Power System Dynamics and Transients
EUMP Distance Education Services
- Satish J Ranade
- Synchronous Generator Model
- Lecture 16
52Modeling 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
53Modeling of synchronous generators
dqo transformation transformed machine
equations
54Modeling 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
55Modeling 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
56Modeling 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
57Modeling 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
58Modeling 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
59Modeling 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
60Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Example Revisited ( Kundur Ex 3.2)
61Modeling of synchronous generators
Synchronous Steady state model, balanced operation
Example Revisited ( Kundur Ex 3.2)
Now find Id and Iq
62Modeling of synchronous generators
Synchronous Steady state model, balanced operation
63Modeling of synchronous generators
Synchronous Steady state model, balanced operation
64Modeling of synchronous generators
Synchronous Steady state model, balanced operation
65Modeling of synchronous generators
Synchronous Steady state model, balanced operation
66Modeling of synchronous generators
Synchronous Steady state model, balanced operation
67Modeling 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
68Modeling 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
69Modeling 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)