produce improved reduced order models of interconnected multimachine power systems. The method is illustrated with two fundamental examples. The first ...
IEEE Transactions on Power Systems, Vol. PWRS-2, No. 2, May 1987
310
REDUCED ORDER MODELING OF INTERCONNECTED MULTIMACHINE POWER SYSTEMS USING TIME-SCALE DECOMPOSITION
P. W. Sauer
M. A. Pai
S. Ahmed-Zaid
D. J. LaGesse
Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign 1406 West Green St. Urbana, IL 61801
2.
Abstract t Time-scale decomposition is used to systematically produce improved reduced order models of interconnected multimachine power systems. The method is illustrated with two fundamental examples. The first shows that the I = YV interface equation used for studying multimachine electromechanical transients is a zero order As representation of the fast network transients. such, the time-scale theory can be used to produce improved electromechanical models without adding the network transient equations. The second example shows how linear models for small change stability analysis can be improved to virtually any degree of accuracy. The advantages of the improved models are shown with both a two- and three-machine system. 1.
Two Time Scale Separation in Nonlinear Systems
As an introduction to the analysis of the interconnected multimachine equations, consider the following two-time-scale nonlinear system in explicit form [51
x(0)
=
g(x,z,e)
z(0)
=
x
(1)
z
(2)
where x [xl ... x I , z [z1 *.. Z and e is a goes to small positive parameter. In the limit as zero, the following reduced order model is obtained: t
-
systems involves
sients.
f(x,z,e)
dz -
Introduction
The analysis of interconnected multimachine power a wide spectrum of time-scale phenomena [1,21. It is this property which makes simulation very difficult without sufficiently accurate reduced It is also this property which makes order models. power system dynamics an excellent candidate for timeThe use of scale decomposition and aggregation. singular perturbation has already been shown to be a valuable technique for exploiting the natural timeIn this paper, the dynamic scale properties [3-7]. equations of an interconnected multimachine power system are examined as a two-time-scale system and presented in explicit singular perturbation form. The two-time scales essentially consist of the fast network 60 Hz transients and the slow machine mechanical plus field transients. It will be shown that the zero order (or simplest) approximation of the slow-time-scale reduced order model utilizes algebraic equations traditionally referred to as the I = YV interface equations. As a two-time-scale system, this is shown to be the simplest and least accurate reduced order model. To illustrate the approach to developing new improved reduced order models, two examples of steady state stability are examined. In addition to serving as an illustration, the results give an interesting case of steady state instabiity in multimachine systems induced by the fast network transients. The paper also shows how to predict the instability with an improved reduced order model that does not include the network tran-
=
dx dt
=
dx
=
f(x, z, 0)
0
=
g(X,z,
x(0)
=
(3)
x°
(4)
°)
The bar over the variables indicates that this is a system where e was set equal to zero. Under certain general conditions, the model approximates the true This model is often response of x to an order £ [5]. referred to as the quasi steady-state (qss) model. The algebraic relationship for z is not in general a good estimate of z near time zero. This is clear since the initial condition of z is not free to be taken as Although there are methods to recover accurate z . approximations for z [5,8], they will not be discussed at this time.
The reduced order model
can
in
the slow time scale
be written as:
dx
=
f(x
f(X
dt
¢(x),
0)
()I0
x(0)
x
(5)
()=x
where z
=
+(x) is the solution to: g(x,
z,
0)
=
(6)
0
When e is not sufficiently small, more accurate reduced order models can be obtained through a systematic procedure [51. In the following section, the multi-machine equations will be developed in the form of Equations (1) and (2). It will then be shown that algebraic YV Equation (6) is approximated by the standard I interface equation. =
A paper recommended and approved 86 WM 091-3 by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1986 Winter Meeting, New York, New 7, 1986. Manuscript submitted York, February 2 May 31, 1985; made available for printing November 7, 1985.
3.
The Interconnected Multimachine Power System Model
-
Models for interconnected multimachine power systems range in order from low (classical uncontrolled YV) to extremely high machines interconnected by I order (fully controlled high order machines interconnected through dynamic network transient equations).
0885-8950/87/0500-0310$01.00©1987 IEEE
311 The accuracy of these models is normally judged by the order of the total number of state equations being considered. That is, the number of differential equations versus the number of algebraic equations. Since the algebraic equations can generally be shown to be quasi-steady-state (qss) time scale separation aproximations, the accuracy of the reduced order model can normally be improved without an increase in the number of differential equations. This section will develop one such set of equations in preparation for the illustration. The models used will be simplified models so that the method can be illustrated without the burden of excessive complexity. This in no way restricts the potential of the procedure from application to more complex problems. On the contrary, the advantages of the method become even greater when complexity is introduced.
di+ di -R i a+ As i di qi w0dt
I
d A5 S dt
1 0
dt
-
1
T'O,
E
-
+l
Lqi
~dididi +
i-
dtM
-D(w-
(
+
3
I
i+w
qi
(1+
i i
) 7c)
-L
i-l,
terminal constraints and
dt
-
iqi
-
R
cai
Ax + v di
idi' iqi" vdi, vqi, di
A i
di L
di
i=l,... ,m
qi
Aqi
qi
iiqi
m
(8)
m
(9)
where
im+l,...,b (13)
Aq5 qi
ism+l,...,b (14)
= L
i
iq
qi
sin(S) Adi
L2d -mdi Lf
m (10)
di
All quantities are in per unit except 6i I which are in electrical radians and seconds.
xi
and t
The network is assumed to be balanced three-phase three-wire R-L elements interconnected between machine terminals. Using standard notation, the mathematical model of a primitive three phase R-L branch is:
(6
Cos
(i
cC
dXd
=
(S
Co
) A
+
iinl,*
,m
(15)
m
(16)
i1,..,m
(17)
,...,m
(18)
qi
in ( ) A
i=
or, di Ai
sin (6i)i A8di - Cos (6i )
xqi
C08(
)
Aai
+ sin
(
A'qi
Ai)qi
i
Note that these relationships can be written in complex notation:
( +di J qi)
(Adi
+
jXqi) ei(6i -2)
(i
+
j 1iA
il,...,m (19)
or,
(Adi L'= L di di
A5di =L i i di
s
where the d-q quantities are written in the reference frame fixed in the rotor with rotor angle 8 = 6. + w t - r/2, E'iis proportional to the field ffux linkmust satisfy ages, and *di' the terminal constraints and
di A8
di
(7e) =
5s
Adi' Aqi idiq iqi,
(7d) +v
i-m+l,.i.,b (12)
The principle task in interfacing the equations is to obtain an independent set of network and machine variables. The first step is to express all d-q variables in the same reference frame. The synchronously rotating reference frame will be used. The machine flux linkages in the synchronously rotating reference frame are related to the machine flux linkages in their individual rotor reference frames by:
[Tmi - Di(wi-) + 3 (Eqici + (Lqi~Lqi)
di di --t-Ri R idi dt
qi
C'-7b)
fdi,,
idiiii)] 1
+v
The machine differential equations must be interfaced with the network differential equations to form a minimal number of differential equations.
.,m
1
di
(11)
lE i-l,..,
d 6i dwi dt
A5
i-m+1,...,b
where the superscript s indicates that these d-q quantities are referenced to the synchronously rotating reference frame with 0 = w t. The branch variables s ,ss 5 must satisfy the branch vdi, vqi
Using standard notation, the mathematical model of a synchronous machine without damper windings using motor convention is [14-15] (7a)
dE'I
--R is i qi
di
+ j Aqi )
)
ei\Ui
2) i=1,...,m (20)
Note that these relationships also hold if flux linkages are replaced by currents oi voltages. When expressed in terms of di Aqi di ii Equations (7d) and (7e) are:
1dAt d A8 1
w
0
q dt
'
~Ri idi+
=- R
is i qi
Aqi + vdi A
di
+ v
qi
i=.1,..,m
(21)
i1, .I
(22)
,m
312 Note that these are in the exact same form as Equat (11) and (12). The b equations can be combine vector form to be, Il
I,
u
dASd
is
dt
dt q
s
d
d
R
q
s
+
dt
(23)
d
(32)
Xt
dt
B14
- M D + Ml [Bt3 C +-3M dt(tM T tMl
isqd vs Xs + q
I8
-CtQ
t
ctQ 2c CtQ14C
lC
I I
I
i5, v5, vs are bxl vectors with where Xs, , id' i qp vdcv ponents I to m for the machines and m1 to b for work R-L branches and R is a diagonal resistance ma with identical ordering.
diXq'
1
+
ii
3
L
1
qJ[L
QmQ
dI
The second step in obtaining an independent of equations is identifying a network tree in eac the d-q circuits to be used in defining basic lo The number of basic (or independent) loops in eac the d-q circuits is:
I8
Lq FdA
~~dA
L
Q
0tQ 3I 3C J
m
=
(25)
b - n+1
where b is the total number of R-L branches plus machines and n is the total number of buses (incluiding the datum bus). The tree contains n-i elements in of the d-q circuits and the network contains b elem in each of the d-q circuits. Note that each of th q circuits contains a datum bus so that a hinged work is avoided. The topology of the d- and qcircuits is identical to the topology of the onediagram describing the three-phase system. The to ogy of the circuits can be described by the net topology matrices [9]. Define the d-and q-cir basic loop flux vectors as =
d
d A q
C
= C
t
t
d i
5
Xd
(26)
q
CI
S
(27)
A
q
q
L
AS
q
CtRC
I51
As1
where the loop flux linkages are related to the loop currents and other state variables by:
.t
A9
CtAC
C A
Lq j
L
IIct d
2
dI
CtA
3
(35)
I8
C
4
qJ
j
E' is mxl with E'(i) = El
L
2
Efd
is mxl with
wt is mxl with
Efd(i)
loop current in q-axis
=
i
loop flux linkage in d-axis
ith
loop flux linkage in q-axis
q
5 Ad5 is txl with A(i) circuit
A is
clrcuit
do 1
xl with A(i)= q
is mxm with T' (ii)
do '
is mxm with L (i,i) 1
M is mxm with
M(i,i)
=
Q
R(i,i)
is bxb with
Qkl (i
=
,
=
L
=
current in d-axis
T'
doi
di
-L' di
Mi
D is mxm with D(i,i) = D
R is bxb with (31)
i=1, *. .,m
i
L
q
Tmi
i
=
is txl with 1(i)
Tt
CBC][s2 + do'fE
- w
I
(30)
C gives the complete model for the interconnected system:
w
loop
ccrcuit
miltiplying both sides of Equations (23) and (24) by
=-
i
i
(29)
0
6i
i
=
5
=
=
Efdi
=
Id is txl with Id(i) d
(28)
q
wt(i)
T is mxl with T(i)
i
qi
q
d circuit
Tdo L [
I
(34)
dt
5
C t vdS =0 t v S d d
do q
Id IS
=_
d A'
6 is mxl with 6(i) =
Note that this topology matrix is defined differently in different texts [9-11]. The definition of reference [91 is used in this paper. Since Kirchhoff's voltage laws give
dt
JLqJ
-ctRC
q
d =
qIs
and the vector/matrix notation is:
S
where C is the bxQ branch-loop incidence matrix and superscript t denotes transposition. This matrix is more commonly known for its relation between loop currents and branch currents as: 5
Z
(33)
5I
I2
CQ 4 C
]
I
m
0
t
C
i
i=-l,.. ,m
R, i) = (Li -Lqi) sin6
cos6
i=k
313
Qk2 is bxb with Qk2(i,i)
Qk3 Qk4
is bxb with is bxb with
-cos 2 6 )
(Li-L 2)(sin
=
Qk3(i,i) =Qk2(i,i) Qk4(i,i) = -Qkl(i,i)
-
1Ct (D (I I+BT1 do
i=k i-k
=
A2 is bxb with A2(i,i)
=
Li
C
i-l,...,m
B1is
bxm with with
2
= B2(iti)-Cos)
i
El
-
il1, .. . ,m
-
El + T's do q
L1
I8 Id +
B 2 C]
[B1 C
T;1'do
Iaq
E
L J
fd
(36a)
(36b)
wt =MMT-
dt
Dwt + 3
BtC] B4C
B3C
_I_, tq
[d
(36c)
D
3
Pd 1
3
41
A1c
C
A3C
[CtQ1
C
CtQ12C
I
IICtQ
C
CtQ1C
Iai~
qj L13
14
5
dId
dIa
LdtJ
jC(R-A) C
-C AC11I
LtACCt(R+A2) ~~~ )C
cA
I
cCB
dA3
is
dt
t
di'- q)(sin6 cos 6 )wt
i=1,. ,m
(bxb) with D3(i,i) = D2(i,i)
dB1 (bim)
D5(i,i)
=
Cos( i)
dB
D6 isd$
(bxm) with D6(i,i)
-
sin(
-LCtBj
with
.ti
) u
, i31,.* * *m
and all nonspecified entries are zero. The IsYV Interface Equation
For c-0 in Equation (36), the following quasisteady state algebraic equations are obtained: (The superscript bar is omitted for clarity) Ct (R-A3)C
-CtA4C
CtA
Ct(R+A2)C
C
Ct
CtB2
d
d)C21 LIq
q
E LC
j
BlJ
(37)
8
These can be solved for Id8 and I as a function of the Rather than leaving the remaining state variables. equations in this form, add the first equation plus j=/-1 times the second equation:
t(R-A3+jA1 )CI q
CtA4C
i
D isdt 5
LqJ
FI81t [CtQlC CtQ2C [qi LCQ.3C CtQi4C JiLqs dtB
CA2C
C
(bxb) with D1(i,i)=2(L(i- Lqi)sin6icos6i
I
dt
(36d)
dA4 (bib) with D4(i,i) = - D1(i,i) D is dt
4.
+
f
Tao Efd
1/ow
Equation (34) to give
d6 dt
C
dA
Clearly these final equations can be written in closed form as a function of loop flux linkages As As or loop currents Id Iq The loop current formulation will be given by substituting Equation (35) into
Tldo
dA, dt
D is
i=1
sin6 i
with all nonspecified entries zero and e
-
-d1
q
=
is bxm binqiwith B4(i,i)
dEt dtt=
j~
2
LB
L:
B1(i,i) sin6i
B is bxm with B3(i,i) = Ei coss, 'a
pI1d 1Bt)C| 2
where the newly introduced matrices are:
i=m+l,... ,b
Li
L
do
Et
C,1d C (D6-B2Tdo)
(L i-L )sin6; cossi i=l..,m
=
(4+2
(D5-B1Tdo)
A is bxb with A(i,i))=5 12 cos26 +L sin2 4 4( 6i ~~** di i qi
A4(i ,i)
L1B~C
do
IL
i.+l ,. .. ,b
A3 is bxb with A3(i,i) = A2(i,i)
2
(DT+B 2 T doo
C
1 )C
i=k
inm A is bxb with A (i,i) = LI sin26 +L cos 6 i=l i qi i '' di 1 1
A(i ,i)
3
L
L B
or
+ C
(jA4+R+A2)CjI = C (B2 JB1)El (38)
314
Ct(R+JL')C (IS+jI5)
=
-C
E + +
Ct(JL'-
jA1+A3)C
Ct(JL'-jA
Is
+A2)Cj Is (39)
tion of the currents and thus the formulation loses its advantage of simplicity. Furthermore, if a q-axis damper winding is included, a similar formulation can be made [13].
5.
where
L'(i,i) d
L' is a bxb matrix with d
L'(i,i) d
=
=
L'
i1,
di
..
,m
iL,m+,. .. ,b
i
E is a bxl vector with E(i)
E(i)
=
E'
qi
e
i i=1,.*..m
i=m+l, ... ,b
0
The last two terms of Equation (39) are usually either included with E to make a current dependent voltage source or neglected under the assumption L' = L di qi [12] in which case A2 = A 3 =0 and Al1 = A4 = L'd to
yield Z
where
loop
I
loop
I
where
Y
loop
loop
The steady state stability of interconnected has received considerable multimachine power systems attention. Analysis of the linearized dynamic equations is the primary method for evaluating and enhancingsteady statestability [1,12,16,17,18]. The network transients are usually only included in studies involving small numbers of machines. When large scale systems are studied, the network transients are normally These modeled by the I = YV interface equations. algebraic equations were shown to be a zero order quasi steady state approximation in a two-time-scale system. This was done by expressing the nonlinear equations of unregulated machines in two-time-scale singular pertwo-time-scale s ystem, turbatio urbat ion epcit em, expli ci t ff orm. AAs a two-t ime-s ca le syst there are many higher order approximations which can be used when the lowest order model is insufficient. As an introduction to the analysis of the interconnected multimachine equations, consider the following two-time-scale linear system:
E
(40)
= (I8 + j; ) d q
(41)
dt
Ct(R+jLP)C
(42)
£dt
I
C
loop
Zo so
Linear Analysis
=Y =
C
loop
(43)
E
(44)
Z '
loop
dX
- AX =
+ BZ
X(O)
=
X
(50)
CX + DZ
Z(O)
=
Z
(51)
Z ] and e is a t.. where X = [X1 ... X ], Z = [z small positive parameter relative to the norms of A, B, C, and D. Suppose that a reduced order model of the following form is desired:
dx
dt
Now since the C matrix relates loop currents to branch
(52)
AX
=
currents (I), C
Iloop
=
I
multiplication of Equation (43) by C yields, I = C Y
loop
Ct
(46)
E
(47)
I =Y E
or
Note that in this equation, I is the vector of branch currents of the form,
-I(i)
=
where X should be a good approximation of the true solution X. Clearly if BZ is zero or small, the solution with A = A would be good. Whenr BZ is not small, a systematic procedure for finding A is required. The existence of £ in equation (51) makes such a procedure possible when D is nonsingular [5,6]. Equation (50) can be rewritten as:
dZ
dX dt
(A
-
(48)
id + jiq
where i=,... ,m are machine currents and i=m+l,... ,b are network R-L branch currents. Similarly, E is a vector of the form
(49)
i=l,... ,m
E' et6i qi = 0 =
im+l,... ,b
Thus it follows that the upper left mxm submatrix of Y is the traditional reduced admittance matrix including It is important to each machine impedance Ri + jL'di note that this formulation can also be made if E is modified to include the terms neglected by the assump-
BD -1 C)X +
BD
e
-I
dZ d(53) (3
d
When £ is small due to e, the following zero order model results in the limit as £ = 0, dX
E(i)
=
Z-(0)
=
=
=
(A - BD
IC)
-
X
-DI-(0) CX
(0)
=
X°
(54)
(55)
When the reduced order model given by Equation true value of e being too large, higher order approximations can be derived as follows. Consider the transformation
(54) is insufficient due to the
P
tion
L'
di
=
L
qi
although
the
E
vector
would
be
a
func-
E
=
X -s
BD
z
(56)
315 which
when
applied
(50)
to
(51)
and
yields
the
trans-
formed system:
it will be assumed that either the initial condition dZ is order for all time or that the is such that c
Al ' (A
cZ-
when
C)£ + c(A
BD
-
dt
CE+ (D
+
1)z
eCBD
Z in (57) is replaced by
C) BD Z
BD
-
(57) (58)
expression for Z from
an
c
approximation will be used over a time frame after £ dZ has reduced to order e. This does not detract from the fact that the approximation of Equation (65) should be better than Equation (54), rather it notes that there is potential for even further improvement in the approximation.
(58)
dt
(A-BD C) (I-cBD
=
2d
+
c
1C)BD
(A-BD
1-
(D+eCBD-1)-l
1
6.
1) C)t
(D+eCBD
dZ
(59)
Note that: (D+eCBD
+
D71
_
-1
-
eD7cBDID
(higher order terms in
-(I)
-1
-1 (A-BD
C)
eBD
(I
-lc
=
x
eBD
(D
+
cCBD
dt
_
q dt
c
'
X(1)
or
(I
-
cBD
ID
-
dt
-( £BD (I
D
X(0)
C) (A - BD
-1
dt
IT
-
1
do
-1
AE
fd
-
T0
-1 oT
+
11+e
-1
T°
-D C XO, this will be the case [5,6]. When the initial condition of Z is different from -D CX0, e dt may not be small (order e). Thus while the zero order approximation of Equation (54) is still a good (error bound to order e) approximation and the first order approximation of Equation (65) is still a better (error bound to order e ) approximation after a time when £ has reduced to e, the first order approximation may be only good (error bound to order c) near time zero. There is a systematic procedure for accounting for this boundary layer error in X, but it will not be addressed in this paper. Alternatively,
mxm
-
To To -1
8
(To
eT0
-I
T0
-1
T7
Aw
+ (T6 Tg
TdOTl)
AIL
-1
AT
(67d)
do
X0 Al L
+
q1j(0
-1
),Ad
T0 T 1M
6
is
3
T -1
-1
T0
(65)
T-1)
8Tdo)
-1
CT0 T0TdOE +TTo-
The above process can be repeated to include additional terms containing higher powers of e. In this procedure, it was tacitly assumed that £ dt is small (order e). When the initial condition on Z is
2L
+M
(67c)
-1
(64)
where
k-1T
-1 t
D At
(T6
-dt
(63)
-M -1D
AT
M4 T4 AE'q
dAI5L
C)X(
AIL
(67b)
The first order improved reduced order model is then found from Equations (61) and (64) as, dX
-1
T doT
t
greater
c) g
AI +
0
A
dAt
C)
D-1C-(1)
q
(67a)
dA6 dt
one, -(1) + eBD'
-1 E Tddo
-T-1 MA' +
~~Tdo
+
Again neglecting terms with powers of
(66)
X° + A
-
for all state variables. When this is substituted into Equations (31)-(35) and all products of small changes are neglected, the following model is obtained:
-(1)
(dZ
)
X
C)(
As before, the bar superscript (1) indicates this system which retains £ to the first power. When (56) is replaced by an expression for Z from (58)
power
The closed form nonlinear model was presented in Equations (36a-36d). For linearization, it is more convenient to begin with Equations (31)-(35). The linearization is made about an equilibrium point signified by a superscript "o". The linearization is standard utilizing the small change relation,
(60)
e.
In order to obtain an improved estimate of X, only terms with powers of c greater than one are neglected to obtain:
dt
The linearized interconnected multimachine system model
' [IA d]8 I
t
AqI
with E (i,i) 1
and -
(L -L'i)(cos
(6i)
is
~didii
+sin (
6i
)
di
i
=
1,... ,m
near
d-Z
E
is
mxm
with
-Eqi(sin (6i)idi-
E2 (i,i)
E is mxm with E3(i,i) 3 3
-
Cos
(6i) is
di
+
sin
cos(
Si)iqi)
i
l,...,m
is (Si) ~~~~~~~~~~qi
316
E
is
4
with E (i,i)
mxm
=
4
L
(L' di
)[cos(26
qi
+ 2 sin(2 6 )i i i di
is bxm with E5(i,i) 3 is 5
E 5
di
cos(O
+
E
T T
E6(i,i)
with
i 2
is mx2R
[BtC
L
=
3
-
ii (L'
+
sin(S
i
Lqi)
-
non
cos
1,
=
i-i ,.. .
)E'
,m
qi
C
t Q13C+iq
t id
T
T5
is
4
I .xm
is 2t
T7
is 2t
T8
~3
is 2Q
x
x m
x
dXRdt
is
2Qxm
=
1,...
,m
are zero.
of the form:
are
X~
(68)
D0+ 'D1
1
Qm2C+iq Qm4
=
(A
= (A
-
BD0
R(
C0)
)
maintaining the identity of C and DI. the first order improved reduced order model can be achieved by substituting CO + sc1 and Do + eD in Equation (65) and dropping the £ terms to obtain: -(1 dX dt
D1C D1C s[(I-eBD1 0 0 )+BIBD4 [IB0 D0 0 )(A-BD1C 0 (D 1 0 0 -c 1
-(1) (69)
The following section illustrates the use of these various models and their differences.
=
[ctRC
0tR
-
rCtA =
C
C
C
rctE5 ctE
=
Table 1 Two Machine Parameters
Machine #1
CtB2
=
2t with
with
Several interesting single machine steady state stability problems have been reported previously [19]. These cases are especially interesting since the stability of the machine is different for various models. The following two machine parameters were used to simulate a single machine infinite bus configuration:
A2 CA
CtA3C
A single machine infinite bus example
7.
ctRC
B2
-
T9(i,j)
T6(i+k,J)
=
T9(i+Q,j) T10
t
]
L
T9 is 2Q
-T8(i,j) j
Z = [1AIs and A, B, CO, where X [AE' A 6 Awt Z L01Cli q D1 are as given in Equations (67a) to (67b). These equations are in explicit form as given by Equations (1) and (2). The matrices CO + ECI and D0 + could be rewritten as C' and D' to obtain the eD1 standard form of Equations (50) and (51) of section 5 above. Since the progression to higher accuracy reduced order models utilizes the small parameter epsilon, the identity of these terms will be retained for consistency. The zero order model is obtained by analyzing these equations in the limit as £ goes to zero to obtain Equation (54):
Q14C
3
2Q
x m
1,...,Q and
.
is 2L x 2t
T6
=
B
0
T
+ E
i
Do0
[BtC tC B3CB4C
[E
=
Co+ EC
5dt
+ 2
mxm
T8(i+,)
LA
dz
_
26
t
is
=
specified entries
dx dt
) sin 26
Lq
id
T
TII(i,)
with
The linearized equations
26
26
cos
qi
(Ldi
i
zm
T11(i+t,j) and all
BtCl
BI
=I
i smc2
L)
-
qi
-is
is 2
qii
)E'
=
T1
qi~
i=1,
L) sin
di
s
di
qi
(L'
isqi (L'di
i
)(is
T10(i,j)
=
T10(i+t,j )
=
i
=
1,...,Q
-T6(i,j)i
=
1,... ,2Q
T7(i+L,j) =
i
-T7(i,j) j
=
1,...
,m
M2
=
DI
D
=
0.
=
0.
=
0.
=
0.
=
0.
R
L L
a1
d1 ql
L'
dI
T'
109
M1= 0° 004189 and
and
=
Machine #2 (infinite bus)
=
.00022
=
.09452
R
=
1.425
L
=
1.140
L
=
0.5342
=
0.7407 sec.
dO-
a2
d2
q2 L' d'? T
do2
10
sec.
317
Table 4
where all parameters are in per unit on the base of the rated values for the 10 kW 3*, 230 VI-I machine #1. There is no line or transformer between the two machines. An equilibrium point similar to that described in reference [19] was used as shown in Table 2,
Two Machine Example Zero Order Matrix Eigenvalues -
Table 2
El
Machine #2
2.227
-
qI
61
El -1.732
q2
-8.621 degrees
377 rad/sec
1
is
d
is q1
=
w2=
377
rad/sec
0.2923
is
-0.2924
0.9563
is
-0.9563
d2
This zero order model approximates the true slow eigenvalues very well in magnitude. However, since the damping of the mechanical modes is of order e (as is often the case) the instability of the true system is not represented. It is not necessary to return to a full model with fast transients to improve the accuracy of the model. An improved reduced order model is obtained by neglecting only e terms as shown in Equation (69). This first order improved reduced order model still has only five slow states and gives the eigenvalues shown in Table 5. Table 5
q2
Two Machine Example First Order Improved Matrix Eigenvalues
where again the currents and voltages are in per unit on a base equal to the machine #1 rating. This operating point corresponds to machine #1 acting as a motor taking 0.18 pu real power from the infinite bus machine #2 and giving 0.7 pu reactive power to the infinite bus machine #2. As such, machine #1 is overexcited but well within its thermal rating.
While each set of machine equations contains five state variables, the interconnection of the two machines in series reduces the full state formulation to only eight. The full system matrix for the equations linear-
ized about this operating point has the eigenvalues shown in Table 3. These are the eigenvalues of the system equations which have the form of Equation (68)
with
E
-
1/wo
m
1/377.
+ .04464 -
Two Machine Example Full System Matrix Eigenvalues
49.05 ± J376.4 3.598 + 0.04486 ± 18.15 0.0
-
0.0 0.0
One system zero eigenvalue is due to the angle dependence while the other two zero eigenvalues are due to the simulated infinite bus (infinite field time constant plus infinite inertia). The near 60 Hz frequency eigenvalues are the network transients, the stable eigenvalue is the machine number one field, and the unstable complex pair is the mechanical oscillation mode. This unstable equilibrium has been discussed in previous work [19]. Since these linearized equations have the form of Equation (68), the proper method of constructing reduced order models is clear from the previous section. Assuming that the network transients are not of interest, they can be eliminated as fast variables resulting in the zero order model of Equation (54). Note that this requires setting C equal to zero in both the left and right sides of Equation (68). The resulting three by three matrix has the eigenvalues shown in Table 4.
3.598 0.0 0.0 0.0
±
J18.15
This first order improved model accurately reflects the unstable mode due to the recovery of the epsilon order damping. Thus the true stability property is preserved in the reduced order model without the addition of fast differential equations. 8.
Table 3
J18.15
±
0.0 0.0
Two Machine Operating Point Machine #1
0.01343 3.595 0.0
A multimachine
example
The above single machine infinite bus example showed how reduced order models can be improved without higher dimensions. This technique has been extended to any number of machines connected by an R-L network in Section 6. These multimachine results are illustrated in this section. While the unstable mode of the previous example has been presented previously [191, an analogous multimachine example has not. A three machine, eight bus example was used. The system and machine data are shown in Tables 6 and 7 with all The branches unlabeled units in standard per unit. from buses 4 and 5 to 0 are R-L loads. The operating point of Table 8 was used for the linearization. This operating point satisfies the load flow equations of the 8 bus system and is also an equilibrium point of the multimachine nonlinear differential equations. The resulting state variables needed for the linearization are shown in Table 9.
318 Table 6
equations, three field equations, and six mechanical equations give a 23rd order system. The full system eigenvaues are listed in Table 10.
Multimachine Line R-L Data From 1 2 3 4 4 7 6 4 6 5 4 5
To
Rbr
Lbr
6 7 8 6 5 8 7 7 8 8 0 0
.001 .001 .001 .003 .003 .003 .004 .004 .004 .004 .582 .576
.005 .005 .005 .015 .015 .015 .030 .030 .030 .030 .466 .461
Table 7 Multimachine Data 1
2
3
M
0.02
0.0026
0.02
D
0.010
0.00013
0.010
R a Ld
0.03
0.06
0.03
0.60
1.20
0.60
L
0.50
1.00
0.50
0.15
0.31
0.15
5.00 sec.
5.04 sec.
5.00 sec.
q
d
Tddo
Table 10
Multimachine Full System Matrix Eigenvalues -466.3 -325.2 -61.47 -58.78 -45.81 -48.77 -52.12
± ± ± ± ± ± ± +0.056 ± -0.314 ±
-0.201 -0.723 -0.560 -0.491 -0.001
j376.99 j373.28 j376.99 j376.98 j375.90 j376.10 j376.99 j24.33 j12.26
The seven 60 Hz eigenvalue pairs correspond to the seven d-q network loop currents, the one stable and one unstable low frequency pairs correspond to the two independent mechanical oscillations. The -.001 eigenvalue is the center of inertia angle which is not zero in this case due to round off in the equilibrium input. The -0.201 eigenvalue is the slow center of speed mode, and the remaining three correspond to the three field equations. This unstable equilibrium is a multi-machine extension of the condition illustrated in the last section. When the 14 network equations are eliminated in the zero order model using Equation (54), the resulting matrix eigenvalues are as shown in Table 11.
Table 8 Table 11
Multimachine Operating Point BUS 1 2
0.990 1.000 0.995 0.977 0.972 0.988 0.995 0.992
3 4 5 6 7 8
Multimachine Zero Order Slow Model
QIN
IN
0 0.0
-0.3190 0.7580 -0.7900 -0.9440 -0.150° -0.2140 0.313°
.5651 -.1500 1.600 -1.000 -1.000 0. 0. 0.
0.268 1.080 0.343 -0.800 -0.800 0. 0. 0.
Table 9
Multimachine Equilibrium Point 1
El
q
1.795
2
2.298
3
1.807
6
13.64 degrees
-6.24 degrees
33.90 degrees
X
377 rad/sec
377 rad/sec
377 rad/sec
id
-0.9886
0.2702
-2.793
iS
0.4687
1.868
-0.5601
q
The full system has 15 branches and 7 independent loops in each of the d and q axes. The fourteen network
-0.055 ± j24.34 -0.333 ± J12.26 -0.256
-0.725 -0.548 -0.455 -0.001
As in the single machine case, the zero order model approximates the eigenvalue lengths quite well, but does not show the instability. Thus this traditional reduced order multimachine model indicates a stable equilibrium whereas the actual full system equilibrium is unstable. The error is due to epsilon order damping in the mechanical modes. When the first order improved model is used employing Equation (69), the eigenvalues of Table 12 are obtained.
Table 12 Multimachine First Order Improved Model +0.055 ± j24.34 -0.314 t J12.26 -0.201 -0.723 -0.560 -0.491 -0.001
319 The first order improved model has clearly captured the instability of the full order system without the introduction of additional network equations. 9.
Discussion
Reduced order models of individual and multimachine power systems have traditionally been obtained through intimate knowledge of physical behavior and While this traditional historical observation. approach is clearly vaid and in fact quite ingenious, the method of time-scale decomposition can justify known results and extend them in a systematic way. It has been shown that using I YV for network transient representation is the simplest in a series of reduced order models which can be obtained. It was also shown that improved reduced order models can be obtained without resorting to solving the network differential This technique is not limited to power equations. system network transients. It can be applied whenever multi-time scale separations exist. Clearly this is a very large portion of the systems involved in power system dynamics. -
10.
11.
References
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Acknowledgements
The authors wish to thank Prof. Petar Kokotovic of the University of Illinois and Dr. Jim Winkelman of the General Electric Co. for their useful discussions. This research was supported in part by the General Electric Co. and the University of Illinois Power Affiliates Program.
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