ECE 576 – Power System Dynamics and Stability

ECE 576 – Power System
Dynamics and Stability
Lecture 11: Synchronous Machines Models
and Exciters
Prof. Tom Overbye
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
overbye@illinois.edu
1
Announcements
•
•
•
Homework 3 is on the website and is due on Feb 27
Read Chapters 6 and 4
Midterm exam is on March 13 in class
– Closed book, closed notes
– You may bring one 8.5 by 11" note sheet
– Simple calculators allowed
2
Summary of Five Book Models
a) Full model with stator transients
b) Sub-transient model
c) Two-axis model
d) One-axis model
 1


 0 
s

Tqo  Tdo  0
Tqo  0
e) Classical model (const. E behind X d )
3
Two-Axis vs Flux Decay
•
For 4 bus system, figure compares gen 4 rotor angle for
bus 3 fault, cleared at t=1.1 seconds
4
Industrial Models
•
•
There are just a handful of synchronous machine
models used in North America
– GENSAL
• Salient pole model
– GENROU
• Round rotor model that has X"d = X"q
– GENTPF
• Round or salient pole model that allows X"d <> X"q
– GENTPJ
• Just a slight variation on GENTPF
We'll briefly cover each one
5
Subtransient Models
•
•
•
•
The two-axis model is a transient model
Essentially all commercial studies now use subtransient
models
First models considered are GENSAL and GENROU,
which require X"d=X"q
This allows the internal, subtransient voltage to be
represented as
E   V  ( Rs  jX ) I
Ed  jEq    q  j d  
6
Subtransient Models
•
Usually represented by a Norton Injection with
•
May also be shown as
Ed  jEq   q  j d  
I d  jI q 

Rs  jX 
Rs  jX 
 j  I d  jI q   I q  jI d 
 j   q  j d  
Rs  jX 
   j   


d
q
Rs  jX 
In steady-state  = 1.0
7
GENSAL
•
The GENSAL model has been widely used to model
salient pole synchronous generators
– In the 2010 WECC cases about 1/3 of machine models were
•
GENSAL; in 2013 essentially none are, being replaced by
GENTPF or GENTPJ
In salient pole models saturation is only assumed to
affect the d-axis
8
GENSAL Block Diagram (PSLF)
A quadratic saturation function is used. For
initialization it only impacts the Efd value
9
GENSAL Initialization
•
To initialize this model
1.
2.
Use S(1.0) and S(1.2) to solve for the saturation coefficients
Determine the initial value of d with
E d  V   Rs  jX q  I
3.
4.
Transform current into dq reference frame, giving id and iq
Calculate the internal subtransient voltage as
E   V  ( Rs  jX ) I
5.
6.
Convert to dq reference, giving P"d+jP"q=  "d+  "q
Determine remaining elements from block diagram by
recognizing in steady-state input to integrators must be zero
10
GENSAL Example
•
•
•
Assume same system as before, but with the generator
parameters as H=3.0, D=0, Ra = 0.01, Xd = 1.1, Xq =
0.82, X'd = 0.5, X"d=X"q=0.28, Xl = 0.13, T'do = 8.2, T"do
= 0.073, T"qo =0.07, S(1.0) = 0.05, and S(1.2) = 0.2.
Same terminal conditions as before
•
Current of 1.0-j0.3286 and generator terminal voltage of
1.072+j0.22 = 1.0946 11.59 
Use same equation to get initial d
E d  V   Rs  jX q  I
 1.072  j 0.22  (0.01  j 0.82)(1.0  j 0.3286)
 1.35  j1.037  1.7037.5
11
GENSAL Example
•
Then
 I d   sin d  cos d   I r 
I   
 I 
cos
d
sin
d
 i
 q 
0.609 0.793  1.0  0.869 






0.793
0.609

0.3286
0.593


 

And
V  ( Rs  jX ) I
 1.072  j 0.22  (0.01  j 0.28)(1.0  j 0.3286)
 1.174  j 0.497
12
GENSAL Example
•
Giving the initial fluxes (with  = 1.0)
  q  0.609 0.793 1.174  0.321
      0.793 0.609  0.497   1.233

 

 d  
•
To get the remaining variables set the differential
equations equal to zero, e.g.,
 q    X q  X q  I q    0.82  0.28  0.593  0.321
Eq  1.425,  d  1.104
Solving the d-axis requires solving two linear
equations for two unknowns
13
GENSAL Example
•
Once E'q has been determined, the initial field current
(and hence field voltage) are easily determined by
recognizing in steady-state the derivative of E'q is zero
E fd  Eq 1  Sat ( Eq )    X d  X d  I D
Saturation
coefficients
2
 1.425 1  B Eq  A  1.1  0.5  (0.869) were
determined
2
 1.425 1  1.25 1.425  0.8   0.521  2.64 from the two
initial values





Saved as case B4_GENSAL
14
GENROU
•
•
The GENROU model has been widely used to model
round rotor machines
Saturation is assumed to occur on both the d-axis and
the q-axis, making initialization slightly more difficult
15
GENROU Block Diagram (PSLF)
The d-axis is
similar to that
of the
GENSAL; the
q-axis is now
similar to the
d-axis. Note
saturation
affects both
axes
16
GENROU Initialization
•
•
Because saturation impacts both axes, the simple
approach will no longer work
Key insight for determining initial d is that the
magnitude of the saturation depends upon the
magnitude of ", which is independent of d
   V  ( Rs  jX ) I
•
Solving for d requires an iterative approach; first get a
guess of d using 3.229 from the book
E d  V   Rs  jX q  I
17
GENROU Initialization
•
•
Then solve five nonlinear equations from five
unknowns
– The five unknowns are d, E'q, E'd,  'q, and  'd
Five equations come from the terminal power flow
constraints (giving voltage and current) and from the
differential equations initially evaluating to zero
– Two differential equations for the q-axis, one for the d-axis
(the other equation is used to set the field voltage
18
GENROU Initialization
•
•
Use dq transform to express terminal current as
 I d   sin d
I   
 q  cos d
 cos d   I r 
sin d   I i 
These values will change during
the iteration as d changes
Get expressions for  "q and  "d in terms of the initial
terminal voltage and d
– Use dq transform to express terminal voltage as
Vd   sin d
V   
 q  cos d
– Then from
 cos d  Vr 
sin d  Vi 
Recall Xd"=Xq"=X"
and =1 (in steady-state)
 q  j d  Vd  jVq   ( Rs  jX )  I d  jI q 
 q  Vd  Rs I d  X I q Expressing complex
 d  Vq  Rs I a  X I d
equation as two real
equations
19
GENROU Initialization Example
•
•
Extend the two-axis example
–
For two-axis assume H = 3.0 per unit-seconds, Rs=0, Xd =
2.1, Xq = 2.0, X'd= 0.3, X'q = 0.5, T'do = 7.0, T'qo = 0.75 per
unit using the 100 MVA base.
– For subtransient fields assume X"d=X"q=0.28, Xl = 0.13,
T"do = 0.073, T"qo =0.07
– for comparison we'll initially assume no saturation
From two-axis get a guess of d
E  1.094611.59   j 2.0 1.052  18.2   2.81452.1
 d  52.1
20
GENROU Initialization Example
•
And the network current and voltage in dq reference
Vd  0.7889 0.6146  1.0723 0.7107 

V   




0.6146
0.7889
0.220
0.8326
q






 
 I d  0.7889 0.6146   1.000  0.9909 

I   




 q  0.6146 0.7889   0.3287  0.3553
•
Which gives initial subtransient fluxes (with Rs=0),
    j     V
q
d
d
 jVq   ( Rs  jX )  I d  jI q 
 q   Vd  Rs I d  X I q  0.7107  0.28  0.3553  0.611
 d   Vq  Rs I a  X I d  0.8326  0.28  0.9909  1.110
21
GENROU Initialization Example
•
•
•
Without saturation this is the exact solution
Initial values are: d = 52.1 , E'q=1.1298, E'd=0.533,  'q
=0.6645, and  'd=0.9614
Efd=2.9133
Saved as case
B4_GENROU_NoSat
22
Two-Axis versus GENROU Response
Figure compares rotor angle for bus 3 fault, cleared at
t=1.1 seconds
23
GENROU with Saturation
•
•
•
•
Nonlinear approach is needed in common situation in
which there is saturation
Assume previous GENROU model with S(1.0) = 0.05,
and S(1.2) = 0.2.
Initial values are: d = 49.2 , E'q=1.1591, E'd=0.4646,  'q
=0.6146, and  'd=0.9940
Efd=3.2186
Saved as case
B4_GENROU_Sat
24
Two-Axis versus GENROU Response
25
GENTPF and GENTPJ Models
•
These models were introduced by PSLF in 2009 to
provide a better match between simulated and actual
system results for salient pole machines
– Desire was to duplicate functionality from old BPA TS code
– Allows for subtransient saliency (X"d <> X"q)
– Can also be used with round rotor, replacing GENSAL and
•
GENROU
Useful reference is available at below link; includes all
the equations, and saturation details
http://www.wecc.biz/library/WECC%20Documents/Docum
ents%20for%20Generators/Generator%20Testing%20Pro
gram/gentpj-typej-definition.pdf
26
GENSAL Results
Chief Joseph
disturbance
playback
GENSAL
BLUE = MODEL
RED = ACTUAL
Image source :https://www.wecc.biz/library/WECC%20Documents/Documents%20for
%20Generators/Generator%20Testing%20Program/gentpj%20and%20gensal%20morel.pdf
27
GENTPJ Results
Chief Joseph
disturbance
playback
GENTPJ
BLUE = MODEL
RED = ACTUAL
28
GENTPF and GENTPJ Models
•
•
GENTPF/J d-axis block diagram
GENTPJ allows saturation function to include a
component that depends on the stator current
Se = 1 + fsat( ag + Kis*It)
Most of
WECC
machine
models
are now
GENTPF
or
GENTPJ
If nonzero, Kis typically ranges from 0.02 to 0.12
29
Voltage and Speed Control
P, 
Q,V 
Exciters, Including AVR
•
•
•
Exciters are used to control the synchronous machine
field voltage and current
– Usually modeled with automatic voltage regulator included
A useful reference is IEEE Std 421.5-2005
– Covers the major types of exciters used in transient stability
simulations
– Continuation of standard designs started with "Computer
Representation of Excitation Systems," IEEE Trans. Power
App. and Syst., vol. pas-87, pp. 1460-1464, June 1968
Another reference is P. Kundur, Power System Stability
and Control, EPRI, McGraw-Hill, 1994
– Exciters are covered in Chapter 8 as are block diagram basics
31
Functional Block Diagram
Image source: Fig 8.1 of Kundur, Power System Stability and Control
32
Types of Exciters
•
•
•
•
None, which would be the case for a permanent magnet
generator
– primarily used with wind turbines with ac-dc-ac converters
DC: Utilize a dc generator as the source of the field
voltage through slip rings
AC: Use an ac generator on the generator shaft, with
output rectified to produce the dc field voltage;
brushless with a rotating rectifier system
Static: Exciter is static, with field current supplied
through slip rings
33
Brief Review of DC Machines
•
•
•
•
Prior to widespread use of machine drives, dc motors
had a important advantage of easy speed control
On the stator a dc machine has either a permanent
magnet or a single concentrated winding
Rotor (armature) currents are supplied through brushes
and commutator
The f subscript refers to the field, the
Equations are
a to the armature;  is the machine's
v f  if Rf  Lf
di f
dt
dia
va  ia Ra  La
 Gmi f
dt
speed, G is a constant. In a
permanent magnet machine the field
flux is constant, the field equation
goes away, and the field impact is
embedded in a equivalent constant
to Gif
Taken mostly from ECE 330 book, M.A. Pai, Power Circuits and Electromechanics
34
Types of DC Machines
•
If there is a field winding (i.e., not a permanent magnet
machine) then the machine can be connected in the
following ways
– Separately-excited: Field and armature windings are
connected to separate power sources
• For an exciter, control is provided by varying the field
current (which is stationary), which changes the armature
voltage
– Series-excited: Field and armature windings are in series
– Shunt-excited: Field and armature windings are in parallel
35
Separately Excited DC Exciter
(to sync
mach)
ein1  r f 1iin1  N f 1
a1 
1
1
 f1
d f 1
dt
1 is coefficient of dispersion,
modeling the flux leakage
36
Separately Excited DC Exciter
•
Relate the input voltage, ein1, to vfd
f 1
v fd  K a11a1  K a11
1
N f 1 1
N f 1 f 1 
v fd
K a11
d f 1 N f 1 1 dv fd
Nf1

dt
K a11 dt
N f 1 1 dv fd
ein  iin rf 1 
K a11 dt
1
Assuming a constant
speed 1
1
37
Separately Excited DC Exciter
•
If it was a linear magnetic circuit, then vfd would be
proportional to in1; for a real system we need to account
for saturation
v fd
iin1 
 f sat v fd v fd
K g1
Without saturation we
can write
Kg1
K a11

L f 1us
N f 1 1
Where L f 1us is the
unsaturated field inductance
38
Separately Excited DC Exciter
ein  r f 1iin1  N f 1
1
d f 1
dt
Can be written as
rf 1
L f 1us dv fd
ein 
v fd  r f 1 f sat v fd v fd 
K g1
K g1 dt
1
 
This equation is then scaled based on the synchronous
machine base values
X md
X md v fd
E fd 
V fd 
R fd
R fd VBFD
39
Separately Excited Scaled Values
KE

sep
rf 1
K g1
L f 1us
TE 
K g1
X md
VR 
ein1
R fd VBFD
 
 VBFD R fd

S E E fd  r f 1 f sat 
E fd 
 X

md


Thus we have
TE
dE fd
dt
 


  KE
 S E E fd  E fd  VR


sep


Vr is the scaled
output of the
voltage
regulator
amplifier
40
The Self-Excited Exciter
•
When the exciter is self-excited, the amplifier voltage
appears in series with the exciter field
TE
dE fd
dt
 


  KE
 S E E fd  E fd  VR  E fd


sep


Note the
additional
Efd term on
the end
41
Self and Separated Exciter Exciters
•
The same model can be used for both by just modifying
the value of KE
TE
dE fd
dt


  K E  S E  E fd  E fd  VR


KE
 KE
 1  typically K E
 .01


self
sep
self


42