Download Generator Dynamic Models for Large Scale Grid Simulations

Generator Dynamic Models for Large Scale Grid Simulations
John Undrill
March 2016
1
Contents
1
General
3
1.1
Generator Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Modeling based on transient and subtransient behavior of stator current . . . . . . . . . . . . . . . .
3
1.3
Modeling based on electromagnetic inductance relationships . . . . . . . . . . . . . . . . . . . . . . .
4
1.3.1
The basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4
Electric circuit equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
Various forms of generator dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.6
The model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2
References
8
3
Figures
9
List of Tables
List of Figures
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Transfer function diagram corresponding to generator transfer function description (2), (3) and (4) . . . . . . .
9
2
Thevenin impedance form of generator dynamic model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3
Generator transfer function diagrams - neglecting saturation . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
4
Frequency response of generator model direct axis transfer functions . . . . . . . . . . . . . . . . . . . . . . .
11
5
Effect of stator current on flux density and saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
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1
1.1
General
Generator Behavior
Dynamic models of a synchronous machine are based on:
a. The observable variation of stator current when a generator is subjected to a short circuit at its terminals.
b. A description of the inductive couplings among electrical circuits on the stator and rotor of the machine.
.
1.2
Modeling based on transient and subtransient behavior of stator current
A good model of the electromagnetic dynamic behavior of a synchronous machine can be constructed entirely on
the basis of the observed behavior of its stator voltage and current.
The amplitude of the sinusoidal component of stator current when the generator is subjected to a short circuit is
deemed by most standards to have the form
i=E
1
Ls
+
1
1
−
L0
Ls
exp
t
T0
+
1
1
− 0
L”
L
exp
t
T”
(1)
The initial amplitude is 1/L”. The amplitude decays rapidly, with a time constant T” to a lesser amplitude, 1/L0 ,
and then decays slowly with a time constant, T 0 , to the value 1/Ls .
This profile of short circuit current can be observed in a test but translation of a test recording into ’measured
values’ of the parameters Ls , L0 , L”, T 0 , andT” is complicated by the fact that the time constant, T”, is little different
from the period of the sinusoidal current wave.
The variation of stator current seen in short circuit situations is attributable to variation of the flux linking the
stator circuits. This flux linkage is diminished by armature reaction when current flows out of the machine, with
the rate of diminishment being determined by the inductive time constants (i.e. L/R) of the magnetic circuits.
The magnetic dynamic behavior of the machine can be described in terms of flux linkage and stator current, and
excitation voltage by the transfer function relationship
vq (s) = Ψd (s) = G (s) E f d (s) − Ld (s) Id (s)
(2)
This relationshaip is frequently referred to as the operational impedance description of the generator; with L − d(s)
being deemed to be the operational impedance description of the direct axis of the machine. (Direct axis is yet to
be defined.)
The transfer functions Ld (s) and Lq (s) have the form


( L” − L )
( L0 − L ) 0
1 + ( L0 d− L l) T”do s
1 + ( Ld − Ll ) Tdo
s
l
d
l
d

 + Ll
Ld (s) = ( Ld − Ll ) 
0 s
1 + Tdo
1 + T”do s

G (s) =
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1
0
1 + sTdo
3
(3)

L” − Ll )
1 + ( L0d − Ll ) T”do s
d


1 + sT”do

(4)
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In these transfer function definitions the parameters Ld , L0d , L”d are descriptions of the components of sinusoidal
current that can be observed in the result of a short circuit test and described by equation (1). Thus, description of
the generator by equations (1), (2), (3), (4) is entirely emperical on the basis of test data and makes no reference to
the characterization of the machine in terms of Maxwellian electromagnetic relationships.
The transfer function description of a dynamic model based entirely on equations (2), (4), and (3) is shown in
figure 1.
1.3
1.3.1
Modeling based on electromagnetic inductance relationships
The basic equations
While a model of the magnetic dynamic behavior of the machine can be constructed without reference to a
description in terms of inductive couplings, it is helpful to use a description of the machine terms of inductances.
It is not only heopful, but practically essential to describe the machine in terms of inductances when magnetic
saturation is to be described.
Basic description of a three phase machine is written in terms of self inductances of individual windings and
mutual inductances between pairs of windings. The inductance coefficients relating to the windings vary as the
machine rotates and symbolic analysis is impractical. Accordingly, it is common practice to transform the
description of the machine into a two-axis reference frame by the use of a standard three-to-two axis
transformation, which is known in the electric machinery industry as Park’s transformation. The inductance
coefficients in the transformer inductance equations are constant with respect to angular rotation of the machine
and these equations are reasonably amenable to symbolic analysis. The description of three phase machines in
terms of the two axes via Park’s transformation is developed in reference 1.
The inductance coefficients in the transformed equations, while constant with respect to angular rotation, are
variable with respect to flux density and this will soon become important, as discussed in section ??.
The remainder of this discussion of the synchronous machine refers to the set of equations on the following page.
These equations are the ’transformed’ form of the equations describing the individual windings of the machine.
They relate currents and voltages that cannot be observed in reality but which can be calculated directly from
measured point-on-wave values of currents and voltages.
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
 
ψfd
Lf fd
 ψ  L
 kd   f kd

=
ψkq1  

 
ifd
La f d
  i  L
  kd   akd

−
Lkq12  ikq1  
L f kd
Lkkd
Lkq11
Lkq12
ψkq2
Lkq22
ikq2

" #
 i
 d

L akq1  iq
L akq2
(5)

"
#
"
ψd
L
= afd
ψq
#
L akd
L akq1
L akq2

ifd
"
i 
Ld
 kd 

−
ikq1 
#" #
id
Lq iq
(6)
ikq2



sψ f d
rfd



1  sψkd 


 = −
ω0 sψkq1 

rkd
rkq1
sψkq2
"
  
ifd
efd
i   0 
  kd   

+ 
 ikq1   0 

rkq2
"
#
" #
"
#
id
vd
1 0
1 sψd
−R
+
=
ω0 sψq
ω0 ω
iq
vq
ikq2
−ω
0
(7)
0
#"
ψd
ψq
#
(8)
Te = ψd iq − ψq id
(9)
Pe = vd id + vq iq
(10)
v2t = v2d + v2q
(11)
In these equations the subscripts refer to parts of the machine as follows:
d
q
s
f
k
k1
k2
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refers to the direct axis (which is aligned with the magnetic centerline of the machine)
refers to the quadrature axis
refers to the stator
refers to the field winding on the direct axis
refers to the effective path of rotor body or amortisseur winding currents on the direct axis
refers to the first effective rotor current path on the quadrature axis
refers to the second effective rotor current path on the quadrature axis
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1.4
Electric circuit equivalent
The objective of the dynamic model of the machine is to represent its behavior when the machine is connected to
an electric transmission network. The linkage of the generator model to the network model requires that the
machine be represented by an electrical equivalent; the Thevenin equivalent form is convenient here.
The basic equations, (5) to (8) can be organized into very straightforward dyanmic simulation model of the
electromagnetic behavior of the synchronous machine. This organization is described in reference 2.
It is convenient to write the first four of the basic equations in symbolic matrix form as follows
ψr = Lrr ir − Lrs is
(12)
ψs = Lsr Ir − Lss is
(13)
1 dψr
= −rrr ir + Er
ω0 dt
(14)
1 dψs
− rss is + Jψs
ω0 dt
vs =
(15)
Equations (5) and (13) to yield
−1
ψ”s = Lsr Lrr
ψr
(16)
Then, using this form the above equations the generator can be represented as a Thevenin source whose
amplitude and phase position are variable and a constant impedance, L”.
The form of the Thevenin source is shown in figure 2. The voltage source is given by
(v”d + jv”q ) = −ψ”sq + jψ”sd
(17)
The Thevenin impedance, L”, is related to the inductance coefficients appearing in equations (5) and (6) by
−1
L” = Lss − Lsr Lrr
Lrs
1.5
(18)
Various forms of generator dynamic model
All of the generator dynamic models used in PSS/E and PSLF are Thenenin equivalents based on
implementations of equation (17) and all use a constant inductive impedance, (r a + j(ω/ωo ) L”) as the Thevenin
impedance. The models are developed and implemented by:
making assumptions as to the values of the inductance coefficients in equations (5) and (6)
working through the matrix operation shown in equation (17) algebraically to yield an expression for the
two components. v”d and v”q of the Thevenin voltage source.
The algebraic manipulation is tiresome and not very informative. The end results are what is significant.
Differing assumptions and different wanderings through the algebraic implemtation of (16) result in the models,
genrou, gentpf, gentpj, and gensal. The algebraic ’wanderings’ reflect the state of engineering computation over the
past 75 years:
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The gensal model emerged as electronic analog computers came into industrial use. Analog computers
required the coefficients in equations to be constant and were extremely limited in their capabilities for
multiplication and function generation. Even large analog computers were limited in the number and
complexity of the equations that they could handle; gensal involves many simplifications and compromises
made in respect of these limitations.
The genrou model emerged as computation made the trasition from analog computers to early digital
computers. Variation of coefficients in the generator model could be handled in principle but limited
memory put a premium on compactness of code and practices used in analog computation persisted. The
genrou model was used exetnsively by General Electric and was the early ’backbone’ model of the PTI
PSS/2 and PSS/E programs.
The gentpf model is traceable to Westinghouse and Arizona Public Service. It emerged somewhat after
genrou and was able to be more liberal in its use of digital computer memory. Gentpf is descibed in
reference 3.
The gentpj model is a variation of gentpf. gentpj was developed as accumulated data from testing of hydro
and high speed generators revealed that genrou and gentpf do not represent magnetic saturation accurately
when generators are loaded close to their ratings. The coding implementation of gentpf and gentpj is
descibed in reference 4.
1.6
The model parameters
With the exception of direct axis synchronous reactance, Ld , it is not practical, or even possible in many cases, to
measure the values of the parameters appearing in equations (5) and (6). All of the generator models listed above
are based on assumed values of these parameters that will result in simulated behavior that is a reasonable
approximation to the factual behavior of the machine. All of the models are based on it being deemed that
equation (1) states the factual behavior of the machine in a short circuit and that the corresponding transfer
function statement of machine behavior, (2), (4), (3) correctly to the factual behavior. The empericaly defined
0 , T” , which describe the
parameters, Ld , L0d , L”d which desribe the evolution of current amplitude in (1) and Tdo
do
timing of this evolution, are input parameters to the models. The values of the inductance coefficients, Lkd for
example, are assumed to take values such that the models give behavior corresponding to the emperical
information embodies in (1) = (4).
The difference assumptions regarding the coefficients in (5) and (6) lead to linear transfer function diagrams of
different forms. Figures ??, ??, ?? show transfer function diagrams describing the direct axis behavior of the
models, gensal, genrou, and gentpj. These correspond to the transfer function diagram shown in figure 1 which
implements the emperical dynamic characteristic (2) without using reference to the inductance relationships of
(5)-(8).
The relative characteristics of the models can be by comparing the frequency response calculated to by these
0 , T” , for all. This comparison is shown by figure 4.
models, using the same input parameters, Ld , L0d , L”d , Tdo
do
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2
References
1. "Synchronous Machines", C. Concordia, Wiley, 1951
2. "Structure in the Computation of Power System Dynamical Response", J. Undrill, IEEE Trans, Vol PAS88,
pp 1-6, 1969
3. "Digital Simulation of Synchronous Machine Transients", D. W. Olive, IEEE Trans, Vol PAS-87, pp1669-1675,
1968
4. "The gentpj Model", J. Undrill, https://www.wecc.biz/Reliability/gentpj-typej-definition.pdf.
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Figures
Figure 1: Transfer function diagram corresponding to generator transfer function description (2), (3) and (4)
Figure 2: Thevenin impedance form of generator dynamic model
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(a) genopz
(b) genrou
(c) gentpj
Figure 3: Generator transfer function diagrams - neglecting saturation
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2
1.8
1.6
Amplitude
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-4
10
10
-3
10
-2
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
-1
10
0
10
1
10
2
0
-10
Phase, deg
-20
-30
-40
-50
-60
-4
10
Frequency, Hz
Figure 4: Frequency response of generator model direct axis transfer functions
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Figure 5: Effect of stator current on flux density and saturation
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