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Subsynchronous resonance of two neighbouring generating units
Electric Power Systems Research, 10 (1986) 175 - 188
175
Subsynchronous Resonance of Two Neighbouring Generating Units G. D. JENNINGS and R. G. H A R L E Y
Department of Electrical Engineering, University of Natal, King George V Avenue, Durban (Republic of South Africa) D. C. LEVY
Department of Electronic Engineering, University of Natal, George V Avenue, Durban (Republic of South Africa) (Received December 18, 1985)
SUMMARY
This paper presents a generalized per-unit system for the analysis o f multi-pole, multiinertia turbogenerators. It then uses this per-unit system to investigate the effect on the subsynchronous stability o f the system o f an increase in the generating capacity when a second identical generating unit is added to an existing one. In carrying o u t the investigation, the increased generating capacity is represented by two models: a single equivalent generator o f increased capacity, and the actual two-machine system. A comparison o f the two models shows that for this type o f stability calculation there is no advantage gained in modelling the p o w e r station as two separate machines rather than a single equivalent machine. The results show further that previous stability calculations carried o u t by the authors and others for this system with one generating unit are not valid when the second unit comes on line and are in fact optimistic.
1. INTRODUCTION
Two 1072 MVA steam powered turbogenerators at a remote generating station are to be connected to a large grid some 1400 km away by a 400 kV transmission line. In order to increase the p o w e r transmission capability of such a long line, series capacitor compensation was introduced. However, previous investigations [ 1 ] have shown that this system is prone to subsynchronous resonance (SSR) for high levels of series compensation. 0378-7796/86/$3.50
Since the appearance of the first paper on SSR in this system, many other papers have followed dealing with topics such as system modelling, multi-machine SSR and possible countermeasures [ 1 - 7 ] . However, all these investigations only considered a single 1072 MVA turbogenerator unit, instead of two identical 1072 MVA units. The second benchmark model for computer simulation of SSR [8] considered two non-identical generators at a power station; however, it did not consider the effect which the introduction of the second generator had on the stability of the first generator. Conventional representation of a power station with more than one generating unit in a stability analysis usually entails 'lumping i all the units into a single equivalent unit with a rating equal to t h e sum of the individual units [9]. However, the effect of such a simplification on the modelling of the torsional characteristics of the shaft has not been fully established. Moreover, this simplification would not be possible if the individual units had different shaft torsional characteristics. This paper uses two models to reconsider this station's SSR problem with its increased generating capacity. The first model represents the two 1072 MVA units as a single equivalent unit of 2144 MVA so that the system may be represented by the singlemachine model of Fig. 1; this model will be referred to as the 'equivalent-machine model'. The second model represents the two generating units separately, as in Fig. 2, and will be referred to as the 'two-machine model'. The results calculated in this paper by an eigenvalue analysis agree with those obtained © Elsevier Sequoia/Printed in The Netherlands
176
Xc V~ Pt
TRANSFDRMER I STATION
Vb
I INFINITE
BUS
BUS
Fig. 1. Schematic diagram for single-machine and equivalent-machine system.
RI
Xt
TRANSFORMER
R
v~P~ R~
X~
Xc
V~
d
INFINITE BUS
TRANSFORMER
V~P~
X~
STATION BUS
Fig. 2. Schematic diagram for t w o - m a c h i n e system.
by frequency scanning techniques [10] where it was shown t h a t an increase in the number of generators at a station would most often result in a relatively less stable system. The importance of per-unit systems in the study of electrical machines and power systems is well known. The usefulness of a particular per-unit system depends on its simplicity and on the understanding of the system. Thus, before considering the problem of representing two machines at a power station, a generalized per-unit system for the study of multi-pole, multi-inertia generators in power systems is presented.
2. P E R - U N I T S Y S T E M
Many different per-unit systems are available for the analysis of electrical power systems and each has its own inherent advantages and disadvantages [11]. This paper proposes a generalized per-unit system which is based on one of the existing systems [12] and which can be used to analyse multi-pole and multiinertia turbogenerators. One of the major distinctions between different per-unit systems is the choice of the base time, normally taken as either one second or the reciprocal of the nominal speed. The former is employed in this paper as this results in the per-unit values of power and torque being numerically equal at nominal speed. In order for the machine equations in the per-unit system to be independent of the number of pole pairs, any multi-pole machine
is converted to an equivalent two-pole machine of the same pwer. This effectively requires two sets of base values, one for the electrical parameters and one for the mechanical parameters. 2.1. Derivation o f per-unit s y s t e m For this per-unit system five base values are chosen independently, namely the base armature power pb, the base armature voltage V~, the base field current I b, the base time t b and the base electrical angle 0 b. The rest of the base values are calculated from these five as follows. Electrical base values Base armature power pb = total three-phase rating Base armature voltage Vb = line voltage Base armature current I~ = Pb / v ~ v b Base armature impedance Z b = (V b / X/~) lab Base field power pb = 1.5 P ~ / 3 Base field voltage V~b = p b /Ib Base field current I b = X~o [12] reciprocal Base field impedance Z b = vD /I~ Base time t b = 1 s Base electrical angle 0 b = 1 electrical radian (rad ~) Base electrical speed cob = ob/tb, rad e s 1 Base electrical acceleration ~b = cob/tb, rad e S-2
Base torque is defined as that torque which produces base power at nominal speed eden. Thus, Base electrical torque T b = Pb /Cden = Pb/co0 The per-unit value of torque in the twoaxis theory is calculated from the per-unit values of d- and q-axis current and flux linkage as follows: Te = ~Pdiq - - '~qid Normalizing yields T u = To/Tt
= (~dq -- ~qid)/T~
= O.)O(~di q - -
~qid)/Pba
Since Pab = x/~vblba, I~ = 1 . 5 I~
Vt l T u = ¢O0(~diq -- ~ q Q ) / 2 V ~ I b U "U U "U = O~o(~dZq - - ~ q ~ d ) 1 2
and
Vbo =
177
The superscript u indicates that a parameter is being expressed as a per-unit value. In order for the mechanical angles, speed, acceleration and torque to be equal to the corresponding electrical quantities when expressed as per-unit values (for an equivalent two-pole machine) the following mechanical base values are chosen with n as the number of pole pairs. M e c h a n i c a l base values Base mechanical angle 0 b = Obe/n, mechanical radians (rad m) Base mechanical speed w b = 6~be/n, rad m s-1 Base mechanical acceleration o~n = o~be/n, rad m
Base inertia Jb Base stiffness
Base mechanical nP~/w0 Thus
torque
Tb
= P b / wmn
e = K b = n 2 T~/O b b
H = stored
kinetic energy speed/base power
Now
n2Tb -
n2P2
Thus
H =
60e = nCOm,
J ~ ( a~o/n ) 2 b 2O~e) b 2 P ab( n 2O~e/n ~b6o0
ab
= 0.5 J ~ 0 / a ~
Thus J~ = 2 H ~ / ¢ o 0
= T~O~m
= 2H/wo
and thus nTe = Tm
The physical units of mechanical parameters of inertia and shaft stiffness are normalized into the per-unit values by considering the second-order differential equation of motion: T m = J m ~ a + KmOm or
n T e = JmO~e/n + K m O e / n
and normalizing yields Te
JmOle
KmO e
Tbe
n2Tbe
n2T b
JmO~eO~e b
KmOeObe
-2~b~,b
--2~b~b
where
03 o
ae = nO~n
and, since the electrical power Pe and the mechanical p o w e r Pm must be equal at steady state,
= ju~
synchronous
2P2
= 0.5 Jm n : p ba
TU
at
J_mO.)mn2
In physical units the mechanical and electrical parameters are related as follows:
Te~
b b T~lOm
=
Tm=nTe
Oe = nora,
n~T~ /~be = Tbm/~bm
The machine's inertia is often expressed in terms of the inertia constant H in seconds, so it is useful to find the per-unit value of J in terms of H. By definition
jb-
S-2
=
+ K.OU
where H and COoare in per-unit. The equation o f motion in per-unit is then given b y T u = ( 2 H / w o ~ u + KuO u 2.2. P e r - u n i t single e q u i v a l e n t m a c h i n e
The conventional approach of representing a power system which contains more than one machine is to have a single equivalent machine with a rating equal to the sum of the ratings of the individual machines. This is a valid simplification if the stability study is concerned only with the transient oscillations of energy between the electrical network and the rotational systems of the generators. It may n o t be valid if more detail is required a b o u t the transient oscillations of energy among the various individual masses of the generator shafts. The two-machine system o f Fig. 2 can be simplified to the single-machine system of
178 Fig. 1 where the system in Fig. 2 is normalized to a base of 2144 MVA, twice that of Fig. 1. The base values for a single machine (1072 MVA) and those for the equivalent machine (2144 MVA) are given in Appendix A-1. The per-unit values of the electrical parameters in the equivalent system of Fig. 1 are given by the per-unit values in the twomachine system of Fig. 2 multiplied by the ratio of equivalent-machine base power to single-machine base power, that is by a factor of two. However, since the machine and transformer in Fig. 1 represent the parallel combination of machines and transformers in Fig. 2, their parameters in physical units are half those of a single machine and transformer in Fig. 2. Thus, in per-unit, the machine and transformer of Fig. 1 all have the same electrical parameters as those in Fig. 2. For the equivalent-machine's mechanical system to be equivalent to that of the twomachine system, its shaft's total inertia must be equal to the sum of the total inertias of the two machines in order to have the same quantity of stored rotational energy at synchronous speed. However, since the base inertia for the equivalent machine is twice that for a single machine, the per-unit values of inertia will be the same for both systems. Similarly, to carry through the torsional characteristics of the shaft to the equivalent machine, the physical values of shaft stiffnesses of the equivalent machine must be twice those of the single-machine's shaft. As in the case with the inertia, owing to the doubling of the base value of stiffness for the equivalent machine, the per-unit values of shaft stiffness will be the same in both cases. The inertia constant H in seconds is the same for all machines since both the stored kinetic energy at synchronous speed and
the base power increase by a factor of two for the equivalent machine. The equivalent system is summarized below in Table 1 where the elements in the Table give the ratio of equivalent-machine system values to the two-machine system values.
3. THEORY In order to investigate the non-linear power systems of Figs. 1 and 2 it is necessary to describe each system element by a mathematical model. The well-known [12] two-axis equations describing the generator, transmission system and mechanical system of Fig. 1 are summarized below for the sake of completeness.
3.1. The s y n c h r o n o u s m a c h i n e Subject to the usual assumptions [12] pertaining to the two-axis theory, the mathematical model of a synchronous generator, expressed in terms of currents and a rotor reference frame, takes the following statespace form [ 1 ] : p i = [L]-I{v -- ([R] + ¢o[V])i}
(1)
where 09 is the rotor speed. The vector of axis winding currents is given by i = [id,
i~d, ikd,
iq,
ikql,
ikq2]w
(2)
and that of voltages by v = [vd,
Vfd,
O, 0] T
O, V,,
(3)
In the above model the synchronous machine is represented by six coils with one damper circuit on the d-axis and two damper circuits on the q-axis. The axis currents from eqn. (1) are used to calculate the electrical torque Te as follows: Te = --iW[ G ]i090/2
(4)
TABLE 1 Ratio of equivalent-machine system parameters to two-machine system parameters Mechanical parameters p.u. 1
physical 2
Generator electrical and transformer parameters
Transmission line parameters
p.u.
physical
p.u.
physical
1
0.5
2
1
179 3.2. The transmission s y s t e m
The transmission system is assumed to consist of a lumped series combination of resistance, inductance and capacitance in each of the stator phases. The two-axis voltage vector at the machine terminals is given by =
_
(5)
where the terms on the right-hand side of eqn. (5) are infinite bus voltage and the voltage drops across the capacitance, inductance and resistance as given b y eqns. (6), (7) and (8) respectively: pVCd = i d / C - -
¢OVCq and
vL = L p i d + w L i q v~ = Rid
and
and
pv c
= iq/C + ~v C
(6) (7)
vL = L p i q + ~ o L i d
(8)
v R = Riq
[P]px = [ E ] x + [S] u
3.3. The mechanical s y s t e m
The mechanical system of a single generating unit is assumed to consist of four turbine stages, a generator and an exciter [1], all represented in Fig. 3 as six inertias interconnected by five torsional springs (the connecting shafts), and which can be described by a set of second-order differential equations of the form 0 = [j]p25 + [D]p6 + [K]6 + T
(9)
where [J] is a diagonal matrix of inertias, [D] is a diagonal matrix of damping coefficients, [K] is a symmetric matrix of shaft stiffnesses, T is a forcing torque vector (acting on each inertia) and 5 is an angular position vector. INFINITE r.
J~ 6,
~ 6~
J~ 6,
J, 6,
theory for which has been explained elsewhere [13]. The machines and network are represented in steady A B C variables for the initial load flow calculation and the entire system is represented in d,q variables for eigenvalue and transient response calculations. In a multi-machine system the d,q reference frames of individual generators rotate at different speeds and the stator variables of all machines are therefore transformed into a c o m m o n synchronously rotating D,Q reference frame. In the c o m m o n reference frame each generator is represented by a model which consists of a resistance, inductance and a voltage source which does n o t neglect stator transformer voltage and speed voltage terms. The state-space formulation of the total system can be summarized in the form [ 13]
~ 16~
BUS
4 6,
Fig. 3. Mechanical model of the turbogenerator shaft.
3.4. T w o - m a c h i n e s y s t e m
The t w o machines, closely coupled through the transformers as shown in Fig. 2, are analysed using multi-machine techniques, the
(10)
The state vector of the entire system is a composite vector consisting of the components
X= [/DE, iDg, /QE, /Qg, uC, V~, [tPdr], [~qr],
[fa]] T
(11)
and
U----[VbDQ, bvg, bQg, Vfd, Pt] T
(12)
where the subscript E denotes currents in non-machine inductors and the subscript g denotes quantities associated with the generators; [~d~] and [~qr] are the composite vectors made up of the ~dr and ~qr vectors of each generator; [fa] are the mechanical states of the generators (angles and speeds); VbDQ contains the D,Q-axis components of the infinite busbar voltage; bDg and bQg are the vectors of all the D,Q components of the generator voltages bg; V~d is the vector of generator field voltages and Pt is the vector of prime mover input powers. For the purpose of the eigenvalue calculations the generator equations, transmission system equations and the mechanical equations in single-machine and multi-machine form are linearized a b o u t some steady operating point by a truncated Taylor series expansion. The linearized equations are summarized in Appendix B.
180
and is the same station representation t h a t has been used in various studies [ 1 - 7 ] of this power station to date. The second set, Fig. 5, applies to both generating units in operation but considers them as a single equivalent machine so that the analysis may be carried out as a single-machine study. The third set, Fig. 6, considers both generating units in operation and the system is analysed as a multi-machine system. In all three cases the system stability as a function of the series capacitor compensation is investigated by calculating a locus of eigenvalues for the system as the compensation level is varied from about 14% (Xc = 0.08 ~2) to 105% (Xc = 0.57 ~2) where
4. R E S U L T S
The results in this section consider three different system configurations. The first set in Fig. 4 considers a single generating unit 600.
I
M~s
500.
400.
% compensation level = N
300.
Fig.
4(b)
100.
Ri" %
0. R2
-100. -1s.00
, -10.00
, 6.00
-s.'00
REAL
(a)
(13)
4.1. E f f e c t o f the second generating unit Figure 4(a) shows the eigenvalue loci in the direction of the arrows for a single generating unit; they are similar to those calculated in previous studies [1]. As the
z 2E
L
The eigenvalue loci are generated by repetitive calculation of the linearized statespace equations. In the case of complex conjugate pairs, only those loci with positive imaginary parts are drawn. The system parameters for the three cases are summarized in Appendix A. For all three systems the initial conditions, assumed are Vb = 1.0 p.u., Vt = 1.1 p.u. and P t = 0.6 p.u.
200.
see
= IOOXc/X
s.0~
I/S
120.0
1
M;~4
/
r
T
]o.u.
xc
~2)575
{N
=
56,9%)
xc
0,76 p.u.
(N
D)75
p.u,
(N
0,89
p.u.
(N -
88,1%)
0,96
p.u.
{N
95,1%)
100.0 O~
MK3
/
xc
-
75,3%) -
74,3%)
80.
MK2 xc
50. 0 Xc
40.0
=
=
MK1
Z ~E
(b)
2 0 . I~ _ -2. 00
REAL
_
~ -1.00
0. 0 0
1 1.00
---L 2. 0 0
£ 3. 0 0
4.00
l/S
Fig. 4. (a) g i g e n v a l u e loci for o n e g e n e r a t o r as X c is varied f r o m 0.14 p.u. t o 1.06 p.u. (1072 M V A base). (b) E x p a n d e d view o f m o d e s MK1 t o MK4 for o n e g e n e r a t o r as X c is varied f r o m 0.14 p.u. to 1.06 p.u. (1072 M V A base).
181
b~
?, 0" T
T
ii z
-
-
T
-
-
T
-
-
~
u
u
11
14
z
z
z
z
z
o ii xu
it
u
u
~
×u
"0 0
l
l '
"0 =
rz.1 1"
o
1" SIOVB
I
I
I
= b~ 0
=
IBVNIDVWI
I
I
i
;i
ul
|% "~
8
I: I
I
I
I
I
I
i
SIOV~
),BVNIOVHI
r-,
182
< > l
m 0 v
z
£
@ 0
r~
I
o
i
4
\/
o
0
i'
b~ o
i
0
I~¥NIgVNI
S/0¥~
r~ 1
o
Fj
_o
i
i
I
co eJ
SIOV~
~T i
X~VNIgV~I
183
compensation level is increased, the frequency of the supersynchronous electrical mode E1 increases, while that of the subsynchronous m o d e E2 decreases. Mechanical modes MK4 and MK5 are not affected by compensation level changes (since the generator mass is near a nodal shaft point [1] for those modes), but the damping of modes MK1, MK2 and MK3 is influenced. The frequency and damping of the hunting mode MK0 increases with increasing compensation owing to the stronger electrical coupling of the generator to the infinite bus as the compensation increases. Figure 4 ( b ) s h o w s an expanded view of mechanical modes 1 - 4. The system first becomes unstable at an Xc of a b o u t 0.575 p.u. {0.309 ~2) when mode MK3 moves into the right-hand portion. Mode MK2 moves into the right-hand portion when Xc = 0.75 p.u. (0.403 [2) and mode MK1 moves into it when Xc = 0.96 p.u. (0.516 [2). Thus the maximum permissible compensation level with one generating unit before the first occurrence of instability is 57% (X~ = 0.309 [2). The effect of an increase in the generating capacity of the power station by the addition of a second generating unit can be seen by comparing Fig. 5 with Fig. 4 since Fig. 5 is the locus calculated with the two units represented as a single equivalent unit. Table 2 contains a set of eigenvalues from each of these loci calculated with a compensation level of 25% (X¢ = 0.13 [2). The main differences between the loci of Figs. 4 and 5 can be explained b y noticing TABLE
that the effect of an additional unit, and hence a doubling of the power transferred between the generator and the bus, is the same as a per-unit equivalent single unit transferring the same power across a transmission line of twice the distance. Hence, since the transmission line resistance and reactance have doubled in per-unit whereas the machine and transformer parameters are the same, there is an increase in the per-unit values of R, XL and Xc between the internal voltage of the generator and the infinite bus of 96.5% in R, 69.4% in XL and 100% in Xc. This results in the following. (a) For a low value of compensation there is an effective higher value of reactance between the internal generator voltage and the infinite bus. Thus, since the generator is not modelled with a voltage regulator, the locus at M, which is associated with the magnetic stability of the machine [12], starts in the right-hand portion. Furthermore, this increased reactance means that there is a weaker electrical coupling between the generator and the infinite bus and this results in a decrease in the damping and frequency of the hunting mode, as can be seen in Table 2. (b) The damping of the electrical modes E1 and E2 has increased. This is due to the fact that the real part of the eigenvalues for an R L C circuit can be approximated to R / 2 L for R : / 4 L 2 < 1/LC and the per-unit resistance has increased by more than the per-unit inductance. The effect of this is that mode E2 does not interact as strongly with the generator mechanical modes and this can be seen b y the
2
Single-machine and equivalent-machine eigenvalues calculated for a compensation level of 2 5 % (X c = 0.13 ~) Single machine Mode E1 E2 MK5 MK4 MK3 MK2 MK1 MK0 M R1 R2 R3
Supersynch. flux Subsynch. flux Supersynch. mech. Subsynch. mech. Subsynch. mech. Subsynch. mech. Subsynch. mech. Inertial mode Magnetic stability Damper circuit Damper circuit Damper circuit
Equivalent machine
Real
Imaginary
Real
Imaginary
--9.755 --9.065 --0.7861 --1.049 --0.7773 --0.8163 --0.8246 --1.012 --0.1051 --1.028 --10.42 --17.33
+444.6 +183.5 -+581.8 +109.9 +99.97 +77.98 +42.49 +4.420 0.0 0.0 0.0 0.0
--11.16 --10.72 --0.7860 --1.049 --0.7754 --0.8142 --0.8149 --0.9601 0.0263 --0.8905 --9.807 --16.94
+455.6 -+172.5 +581.8 +109.9 +99.69 +77.83 +42.15 +2.586 0.0 0.0 0.0 0.0
184 f a c t t h a t M K 1 , M K 2 a n d M K 3 d o n o t m o v e as far i n t o t h e r i g h t - h a n d p o r t i o n . (c) F o r a n y c o m p e n s a t i o n level t h e freq u e n c y o f m o d e E1 has increased a n d t h a t o f m o d e E2 has d e c r e a s e d . This is d u e t o t h e f a c t t h a t t h e i m a g i n a r y p a r t s o f t h e eigenvalues o f an RLC circuit can be a p p r o x i m a t e d to 1/(LC) in f o r R2/4L2 < 1/LC and, since t h e increase in L is less t h a n t h e decrease in C (increase in Xc), t h e r e s o n a n t f r e q u e n c y o f t h e RLC circuit b e t w e e n t h e g e n e r a t o r i n t e r n a l v o l t a g e a n d t h e b u s voltage is higher. T h e c o n s e q u e n c e o f this is t h a t t h e s u b s y n chronous mode E2 interacts with the m e c h a n i c a l m o d e s at a l o w e r value o f Xc. T h u s at high c o m p e n s a t i o n levels t h e h u n t i n g m o d e M K 1 starts t o i n t e r a c t w i t h t h e s u b s y n c h r o n o u s m o d e E2 a n d changes d i r e c t i o n towards the right-hand portion. T h e e f f e c t s of {b) a n d (c) a b o v e o n t h e stability o f t h e m e c h a n i c a l m o d e s are o f o p p o s i t e sense; h o w e v e r , it can b e seen b y l o o k i n g at Fig. 5(b) t h a t t h e e f f e c t o f (c) p r e d o m i n a t e s a n d instability o c c u r s at a l o w e r c o m p e n s a t i o n level t h a n in Fig. 4(b). T h e c o m p e n s a t i o n levels at w h i c h t h e m e c h a n i c a l m o d e s go u n s t a b l e f o r o n e g e n e r a t i n g u n i t and t w o g e n e r a t i n g u n i t s are c o m p a r e d in T a b l e 3.
4.2. Two-machine station representation Figure 6 c o n t a i n s t h e loci c a l c u l a t e d f o r t h e two-machine multi-machine representation of t h e s t a t i o n a n d it is similar t o t h e singlee q u i v a l e n t - m a c h i n e loci o f Fig. 5. T a b l e 4
TABLE 3 Compensation X c in ohms giving instability of mechanical modes for one and two units respectively Mechanical mode
One unit Two units
MK1
MK2
MK3
0.516 0.448
0.403 0.367
0.309 0.278
c o n t a i n s a set o f eigenvalues c a l c u l a t e d f o r t h e two-machine system when the compensation level is 25% {Xc = 0 . 1 3 ~2). A c o m p a r i s o n o f Fig. 6 w i t h Figs 4 a n d 5 s h o w s t h e following. (a) T h e a d d i t i o n o f t h e s e c o n d m a c h i n e d o e s n o t p r o d u c e a d u p l i c a t e set of t h e six m e c h a n i c a l m o d e s and t h e f o u r r o t o r circuit m o d e s o f t h e single m a c h i n e . I n s t e a d , f o r e a c h m a c h i n e m o d e o f Figs. 4 a n d 5, t h e r e are t w o m o d e s in Fig. 6. T h e s e pairs o f m o d e s are t h e c o m m o n (or s y m m e t r i c a l ) a n d anti- (or antis y m m e t r i c a l ) m o d e s associated w i t h s y m m e t r i c a l r e s o n a n t s y s t e m s [14] (see A p p e n d i x C). F o r m e c h a n i c a l m o d e s M K 4 a n d M K 5 o n l y o n e m o d e is seen since in each case t h e c o m m o n m o d e a n d a n t i - m o d e are a l m o s t c o i n c i d e n t as these m o d e s are n o t e x c i t a b l e f r o m t h e electrical n e t w o r k . F o r t h e r o t o r circuit real m o d e s t h e a n t i - m o d e s are slightly m o r e d a m p e d , h o w e v e r this is n o t o b v i o u s as each c o m m o n - m o d e locus m o v e s t h r o u g h its a n t i - m o d e as t h e c o m p e n s a t i o n level is increased. F o r t h e c o m p l e x m e c h a n i c a l m o d e s
TABLE 4 Two-machine system eigenvalues calculated for a compensation level of 25% (Xc = 0.13 ~ ) Common mode Mode E1 E2 E3 MK5 MK4 MK3 MK2 MK1 MK0 M R1 R2 R3
Supersynch. flux Subsynch. flux Synch. flux Supersynch. mech. Subsynch. mech. Subsynch. mech. Subsynch. mech. Subsynch. mech. Inertial mode Magnetic stability Damper circuit Damper circuit Damper circuit
Anti-mode
Real
Imaginary
Real
Imaginary
--11.16 --10.72 --2.254 --0.7859 --1.049 --0.7754 --0.8142 --0.8148 --0.9600 0.0263 --0.8905 --9.807 --16.94
+455.6 -+172.5 -+314.1 -+581.8 -+109.9 -+99.69 -+77.83 -+42.15 -+2.586 0.0 0.0 0.0 0.0
--0.7858 --1.049 --0.8730 --0.8674 --0.9844 --1.730 --0.2714 --1.307 --12.59 --19.18
-+581.8 -+109.9 -+101.0 +78.51 -+43.66 -+7.433 0.0 0.0 0.0 0.0
185 the anti-mode (MK0', MKI', MK2' and MK3') is always slightly more damped and at a higher frequency than its corresponding c o m m o n mode. (b) The anti-mode loci of the two-machine case are only slightly affected by a change in compensation level. However, the commonmode loci are identical to the loci of the single~quivalent~machine case in Fig. 5. Thus the level of compensation permitted before the onset of instability can be calculated with either a two-machine multi-machine representation or a single-equivalent~machine representation. This is significant since the two-machine model requires an additional eighteen differential equations and so the use of the equivalent-single-machine model results in a large reduction of c o m p u t e r time. It must be noted that these results were calculated for two identical machines operating under identical conditions and do not account for any affects associated with variations in parameters between the two generators or differences in the electrical load on each generator. This will be the subject of a further study.
5. CONCLUSION
This paper has investigated the effect of increasing the generating capacity o f a p o w e r station by adding a second identical generating unit to an existing one. It has compared the effect of modelling the new system as a single equivalent generating unit or as a multi-machine system with two separate units. A per-unit system was developed for the analysis of multi-pole, multi-inertia turbogenerators. The following conclusions can be drawn from the results presented. (a) An increase in the generating capacity effectively results in a change in the interconnecting tie-line characteristics and so previous stability studies for the singlemachine case are not valid when the second unit comes on line. They are in fact optimistic. (b) The addition of the second unit will result in instability occurring at a lower value of compensation than was calculated for the single unit. In fact a per-unit value of Xc of a b o u t 10% lower than previous calculations will have to be used to maintain a stable MK3. Various different transmission systems
including shunt loads were analysed and the results (not presented) showed that the destabilization effect of the second generator does depend on the transmission system configuration although for most cases similar reductions in the capacitor compensation level were obtained; (c) The modelling of the station as t w o separate units has shown the introduction of anti-modes in the locus diagram. These anti-modes represent an out-of-phase resonance between the two machines. For the purpose of maximum permissible capacitor compensation calculations these anti-modes appear to have no effect and no advantage is gained by modelling the station as two separate generators rather than as an equivalent single generator. However, the nature of these anti-modes needs to be further investigated.
ACKNOWLEDGEMENTS
The authors acknowledge the assistance of R. C. S. Peplow and H. L. Nattrass in the Digital Processes Laboratory of the Department of Electronic Engineering, University of Natal. They are also grateful for financial support received from the CSIR and the University of Natal.
NOMENCLATURE
d,q D,Q [D] [G] i [J] [K] [L] n p Pe Pm [R] T Te v 0
d,q axis of machine rotating reference frame D,Q axis of synchronously rotating reference frame shaft damping matrix rotational voltage inductance unit vector of axis currents shaft inertia matrix shaft stiffness matrix machine inductance matrix number of pole pairs deriverative operator d/dt electric power mechanical shaft p o w e r machine resistance matrix forcing torque vector electric torque vector of axis voltages null vector
186 0/
6 A 0 09 o 5Or (-Den 5omn
angular acceleration rotor angle vector small change operator angular displacement flux linkage system frequency, rad e s I angular velocity of rotor, rad e s - 1 electrical nominal frequency mechanical nominal speed
Indices b base values e electrical quantities m mechanical quantities u per-unit values REFERENCES 1 D. J. N. L i m e b e e r , R. G. Harley a n d S. M. S c h u c k , S u b s y n c h r o n o u s r e s o n a n c e of t h e Koeberg t u r b o g e n e r a t o r s a n d o f a l a b o r a t o r y microa l t e r n a t o r s y s t e m , Trans. S. Aft. Inst. Electr. Eng., 70 (11) ( 1 9 7 9 ) 2 7 8 - 279. 2 R. G. Harley, G. D. J e n n i n g s , M. A. L a h o u d a n d M. F. H a d i n g h a m , Level of m a t h e m a t i c a l m o d e l r e q u i r e d t o p r e d i c t t h e r e s p o n s e of a t u r b o g e n e r a t o r s y s t e m w i t h a n d w i t h o u t series c a p a c i t o r s , Trans. S. Art. Inst. Electr. Eng., 75 (3) ( 1 9 8 4 ) 28 - 39. 3 D . J . N . L i m b e e r , R. G. Harley a n d M. A. L a h o u d , S u p p r e s s i n g s u b s y n c h r o n o u s r e s o n a n c e w i t h static filters, Proc. Inst. Electr. Eng., Part C, 128 ( 1 9 8 1 ) 33 - 44. 4 M. A. L a h o u d a n d R. G. Harley, A n o p t i m a l c o n t r o l l e r for t h e s u p p r e s s i o n o f s u b s y n c h r o n o u s r e s o n a n c e , Electr. Power Syst. Res., 6 ( 1 9 8 3 ) 203 - 216.
5 D . J . N . L i m b e e r , R. G. Harley a n d M. A. L a h o u d , The suppression of subsynchronous resonance w i t h t h e aid o f an auxiliary e x c i t a t i o n c o n t r o l signal, Trans. S. Afr. Inst. Electr. Eng., 74 (8) ( 1 9 8 3 ) 198 - 209. 6 M. A. L a h o u d a n d R. G. Harley, M u l t i m a c h i n e s u b s y n c h r o n o u s r e s o n a n c e : Part I - - L i n e repres e n t a t i o n , Trans. S. Aft. Inst. Electr. Eng., 75 (2) ( 1 9 8 4 ) 3 - 13. 7 M. A. L a h o u d a n d R. G. Harley, M u l t i m a c h i n e s u b s y n c h r o n o u s r e s o n a n c e : Part I I - L o a d repres e n t a t i o n , Trans. S. Afr. Inst. Electr. Eng., 75 (2) ( 1 9 8 4 ) 14 - 27. 8 IEEE Subsynchronous Resonance Working G r o u p , S e c o n d b e n c h m a r k m o d e l for c o m p u t e r s i m u l a t i o n o f s u b s y n c h r o n o u s r e s o n a n c e , IEEE Trans., PAS-104 ( 1 9 8 5 ) 1057 - 1066. 9 E. W. K i m b a r k , Power System Stability, Vol. 1, Wiley, New York, 1 9 6 7 . 10 B. L. Agrawal a n d R. G. F a r m e r , Use of freq u e n c y s c a n n i n g t e c h n i q u e s for s u b s y n c h r o n o u s r e s o n a n c e analysis, IEEE Trans., PAS-98 ( 1 9 7 9 ) 341 - 349. 11 M. R. Harris, P. J. L a w r e n s o n a n d J. M. S t e p h e n son, Per-Unit Systems: with Special Reference to Electrical Machines., C a m b r i d g e University Press, L o n d o n , ISBN 521 0 7 8 5 7 1, 1 9 7 0 . 12 B. A d k i n s a n d R. G. Harley, The General Theory
o f Alternating Current Machines. Applications to Practical Problems, C h a p m a n a n d Hall, L o n d o n , ISBN 0 4 1 2 1 5 5 6 0 5, 1 9 7 5 . 13 M. A. L a h o u d a n d R. G. Harley, Analysis t e c h n i q u e s for t h e d y n a m i c b e h a v i o r o f a m u l t i g e n e r a t o r s y s t e m , Trans. S. Afr. Inst. Electr. Eng., 74 (4) ( 1 9 8 3 ) 53 - 63. 14 T. S. H u a n g a n d R. R. Parker, Network Theory: An Introductory Course, Addison-Wesley, R e a d i n g , MA, 1 9 7 1 . 15 M. A. L a h o u d , T h e analysis a n d c o n t r o l o f sync h r o n o u s a n d a s y n c h r o n o u s m a c h i n e s in p o w e r s y s t e m s , Ph.D. Thesis, University of Natal, 1 9 8 2 .
APPENDIX A
A-1. System base values
Base Base Base Base Base Base Base Base Base Base Base Base
time t b (s) armature power P~ (MVA) armature voltage V~ (kV) armature current I b (kA) armature impedance Z~ ( ~ ) field power pb (MVA) field voltage V b (kV) field current I~ (kA) field impedance Z b (~2) electrical angle 0~ (tad e) electrical speed 5o~ (tad e s -1) electrical acceleration ~ (ra& s-2)
Single generator 1 1072 24 (1-1) 25.79 0.537 536 188 2.85 65.94 1
Equivalent generator 1 2144 24 (1-1) 51.58 0.269 1072 188 5.70 37.97 1
1
1
1
1
187 Base Base Base Base Base Base Base
electrical torque T b (N m (rade) - ' ) mechanical angle 0 b (rad m) mechanical speed COb (tad m s-l) mechanical acceleration C~m(tad m s-2) mechanical torque Tbm (N m (radm) -t) inertia j b (kg m 2 (radm) -2) stiffness K b (N m (radm) -2)
A-2. Generator electrical parameters Xd = 2.466 p.u. Xkd = 0.13 p.u. Xq = 2.28 p.u. R k d = 0.015 p.u. X, = 0.22 p.u. Xkql = 0.59 p.u. Ra = 0.003 p.u. Rkq I = 0.005 45 p.u. Xfd = 0.19 p.u. Xkq2 = 0.06 p.u. R f d = 0.000 984 p.u. R k q 2 = 0.011 p.u. A-3. Mechanical parameters Values of the inertia constants: H1 = 0.158 s //4 = 1.625 s //2 = 1.598 s Hs = 0.674 s H 3 = 1.593 s H6 = 0.034 s Per-unit values of shaft stiffness: K12 = 10.63 p.u. K4s = 26.93 p.u. K23 = 25.34 p.u. Ks6 = 70.53 p.u. K34 = 23.42 p.u. For the purpose of this study, the per-unit values of damping Di were taken as 1% of Hi [1]. The turbine torque distribution is assumed to be HP LPI LP2 LP3 33.7% 21.6% 18.8% 25.9% A-4. Transmission system parameters Transformer: X~ = 0.15 p.u. R1 = 0.0001 p.u. Transmission line: XL = 1.0097 p.u. R = 0.0845 p.u. {1072 MVA base) XL = 2.0194 p.u. R = 0.169 p.u. (2144 MVA base)
APPENDIX B The linearized system equations The linearization of eqn. (1) yields the linearized generator axis currents as A p i = [ L ] - I { - - [ R ] + COo[G] A / - - [G]i 0 Aco
+ At}
(B-l)
where the subscript 0 indicates the steady
3.412 0.5 0.5 0.5 6.825 1.365 1.365
× 106
× 106 × 10 6 × 106
6.825 0.5 0.5 0.5 1.365 2.730 2.730
× 106
× 10 7 X 10 7 × 107
state values of the variable and A indicates a small change in this value. The linearized electrical torque is obtained from eqn. (4) as ATe = --COo([G] T + [G]}io A i / 2
(B-2)
while the linearization of the mechanical torque yields ATm = ~Um0 -- (Pmo/co0) ACO
(B-3)
where Pro0 is the steady state power transmitted along the turbine shaft. The two-axis equations of the transmission line have been stated in § 3.2. The voltage across the capacitor i s a state variable, hence a linearized form of its differential equations is required. From eqn. (6) A p v c = Aid/C -- coo AvC -- vCo ACO
(S-4) A p v c = Aiq/C + coo AVd c + vdC0ACO
The linearization of the generator's electrical torques and of the non-linear sub-matrices contained in [S] and [E] of eqn. (10)results in new matrices [P'], [S'] and [E'] such that [15] A p x = [ P ' ] - ' [ E ' ] Ax + [ P ' ] - I [ S ' ] Au
(B-5)
APPENDIX C Symmetrical systems In this Appendix we determine the mode shapes and mode frequencies of a symmetrical gystem. Figure C-l(a) shows a single degree of freedom (SDOF) system consisting of a mass m connected to a rigid frame by a spring of stiffness k. This system has a single natural frequency given by
co,
=
(c-1)
The two degrees of freedom symmetrical system shown in Fig. C-l(b) consists of two identical SDOF systems, of the kind in Fig. C-l(a) coupled together-by a spring of stiff-
188
IX
-
(k + k¢)/m
kc/m =0
det kc/m
X -
(k + k c ) / m
(C-4) or (a)
k 2 -- k2(k + kc)/m + (k + kc)2/rn 2
/NNNN/ZZ/HIHH/
k
-- k~2/m 2 = 0
(C-5)
and solving this for X gives kl = k / m
ks = (k + 2k~)/m
(C-6)
Thus, the natural frequencies of the coupled system are
×1
(.01 "~ v ~ / m
///HI,
or
and
co2 = x~(k + 2kc)im
(C-7)
The mode shape vectors can be found by solving the equations
"HN//,'HHH/
(b)
(C-8)
[ M ] l [ g ] u = ku
Fig. C-1. (a) Single degree of freedom system. (b) Symmetrical two degrees of freedom system.
for each k to yield 151 = [ 1 , 1] w and
ness kc. The equations describing the unforced m o t i o n of the two masses are written in matrix form as k +
--k c
[p2x2] (C-2) The natural frequencies of this symmetrical system can be found by solving for the eigenvalues of the system. The eigenvalues are given by the solution of the scalar equation d e t [ ~ [ I ] -- [ M ] - I [ K ] ] = 0
(C-3)
where [M] and [K] are the mass and stiffness matrices respectively of eqn. (C-2) and [I] is the identity matrix. Substituting for [M] and [K] from eqn. (C-2) we get
us = [ 1 , - - 1 ] T
(C-9)
Thus the two modes of the coupled system are n o t just the SDOF system mode repeated twice, but two modes shifted slightly apart. These modes are the so-called c o m m o n mode and anti-mode associated with symmetrical systems. The difference in frequency between the c o m m o n mode and anti-mode is dependent on the strength of the coupling between the two systems and increases with stronger coupling. The common-mode frequency is the same as that of the SDOF system and represents the two masses oscillating in phase. For this mode the coupling spring kc does n o t stretch at all and the masses move as though they were uncoupled. The anti-mode frequency is slightly higher than that of the c o m m o n mode and represents an out-of-phase movement of the masses.
175
Subsynchronous Resonance of Two Neighbouring Generating Units G. D. JENNINGS and R. G. H A R L E Y
Department of Electrical Engineering, University of Natal, King George V Avenue, Durban (Republic of South Africa) D. C. LEVY
Department of Electronic Engineering, University of Natal, George V Avenue, Durban (Republic of South Africa) (Received December 18, 1985)
SUMMARY
This paper presents a generalized per-unit system for the analysis o f multi-pole, multiinertia turbogenerators. It then uses this per-unit system to investigate the effect on the subsynchronous stability o f the system o f an increase in the generating capacity when a second identical generating unit is added to an existing one. In carrying o u t the investigation, the increased generating capacity is represented by two models: a single equivalent generator o f increased capacity, and the actual two-machine system. A comparison o f the two models shows that for this type o f stability calculation there is no advantage gained in modelling the p o w e r station as two separate machines rather than a single equivalent machine. The results show further that previous stability calculations carried o u t by the authors and others for this system with one generating unit are not valid when the second unit comes on line and are in fact optimistic.
1. INTRODUCTION
Two 1072 MVA steam powered turbogenerators at a remote generating station are to be connected to a large grid some 1400 km away by a 400 kV transmission line. In order to increase the p o w e r transmission capability of such a long line, series capacitor compensation was introduced. However, previous investigations [ 1 ] have shown that this system is prone to subsynchronous resonance (SSR) for high levels of series compensation. 0378-7796/86/$3.50
Since the appearance of the first paper on SSR in this system, many other papers have followed dealing with topics such as system modelling, multi-machine SSR and possible countermeasures [ 1 - 7 ] . However, all these investigations only considered a single 1072 MVA turbogenerator unit, instead of two identical 1072 MVA units. The second benchmark model for computer simulation of SSR [8] considered two non-identical generators at a power station; however, it did not consider the effect which the introduction of the second generator had on the stability of the first generator. Conventional representation of a power station with more than one generating unit in a stability analysis usually entails 'lumping i all the units into a single equivalent unit with a rating equal to t h e sum of the individual units [9]. However, the effect of such a simplification on the modelling of the torsional characteristics of the shaft has not been fully established. Moreover, this simplification would not be possible if the individual units had different shaft torsional characteristics. This paper uses two models to reconsider this station's SSR problem with its increased generating capacity. The first model represents the two 1072 MVA units as a single equivalent unit of 2144 MVA so that the system may be represented by the singlemachine model of Fig. 1; this model will be referred to as the 'equivalent-machine model'. The second model represents the two generating units separately, as in Fig. 2, and will be referred to as the 'two-machine model'. The results calculated in this paper by an eigenvalue analysis agree with those obtained © Elsevier Sequoia/Printed in The Netherlands
176
Xc V~ Pt
TRANSFDRMER I STATION
Vb
I INFINITE
BUS
BUS
Fig. 1. Schematic diagram for single-machine and equivalent-machine system.
RI
Xt
TRANSFORMER
R
v~P~ R~
X~
Xc
V~
d
INFINITE BUS
TRANSFORMER
V~P~
X~
STATION BUS
Fig. 2. Schematic diagram for t w o - m a c h i n e system.
by frequency scanning techniques [10] where it was shown t h a t an increase in the number of generators at a station would most often result in a relatively less stable system. The importance of per-unit systems in the study of electrical machines and power systems is well known. The usefulness of a particular per-unit system depends on its simplicity and on the understanding of the system. Thus, before considering the problem of representing two machines at a power station, a generalized per-unit system for the study of multi-pole, multi-inertia generators in power systems is presented.
2. P E R - U N I T S Y S T E M
Many different per-unit systems are available for the analysis of electrical power systems and each has its own inherent advantages and disadvantages [11]. This paper proposes a generalized per-unit system which is based on one of the existing systems [12] and which can be used to analyse multi-pole and multiinertia turbogenerators. One of the major distinctions between different per-unit systems is the choice of the base time, normally taken as either one second or the reciprocal of the nominal speed. The former is employed in this paper as this results in the per-unit values of power and torque being numerically equal at nominal speed. In order for the machine equations in the per-unit system to be independent of the number of pole pairs, any multi-pole machine
is converted to an equivalent two-pole machine of the same pwer. This effectively requires two sets of base values, one for the electrical parameters and one for the mechanical parameters. 2.1. Derivation o f per-unit s y s t e m For this per-unit system five base values are chosen independently, namely the base armature power pb, the base armature voltage V~, the base field current I b, the base time t b and the base electrical angle 0 b. The rest of the base values are calculated from these five as follows. Electrical base values Base armature power pb = total three-phase rating Base armature voltage Vb = line voltage Base armature current I~ = Pb / v ~ v b Base armature impedance Z b = (V b / X/~) lab Base field power pb = 1.5 P ~ / 3 Base field voltage V~b = p b /Ib Base field current I b = X~o [12] reciprocal Base field impedance Z b = vD /I~ Base time t b = 1 s Base electrical angle 0 b = 1 electrical radian (rad ~) Base electrical speed cob = ob/tb, rad e s 1 Base electrical acceleration ~b = cob/tb, rad e S-2
Base torque is defined as that torque which produces base power at nominal speed eden. Thus, Base electrical torque T b = Pb /Cden = Pb/co0 The per-unit value of torque in the twoaxis theory is calculated from the per-unit values of d- and q-axis current and flux linkage as follows: Te = ~Pdiq - - '~qid Normalizing yields T u = To/Tt
= (~dq -- ~qid)/T~
= O.)O(~di q - -
~qid)/Pba
Since Pab = x/~vblba, I~ = 1 . 5 I~
Vt l T u = ¢O0(~diq -- ~ q Q ) / 2 V ~ I b U "U U "U = O~o(~dZq - - ~ q ~ d ) 1 2
and
Vbo =
177
The superscript u indicates that a parameter is being expressed as a per-unit value. In order for the mechanical angles, speed, acceleration and torque to be equal to the corresponding electrical quantities when expressed as per-unit values (for an equivalent two-pole machine) the following mechanical base values are chosen with n as the number of pole pairs. M e c h a n i c a l base values Base mechanical angle 0 b = Obe/n, mechanical radians (rad m) Base mechanical speed w b = 6~be/n, rad m s-1 Base mechanical acceleration o~n = o~be/n, rad m
Base inertia Jb Base stiffness
Base mechanical nP~/w0 Thus
torque
Tb
= P b / wmn
e = K b = n 2 T~/O b b
H = stored
kinetic energy speed/base power
Now
n2Tb -
n2P2
Thus
H =
60e = nCOm,
J ~ ( a~o/n ) 2 b 2O~e) b 2 P ab( n 2O~e/n ~b6o0
ab
= 0.5 J ~ 0 / a ~
Thus J~ = 2 H ~ / ¢ o 0
= T~O~m
= 2H/wo
and thus nTe = Tm
The physical units of mechanical parameters of inertia and shaft stiffness are normalized into the per-unit values by considering the second-order differential equation of motion: T m = J m ~ a + KmOm or
n T e = JmO~e/n + K m O e / n
and normalizing yields Te
JmOle
KmO e
Tbe
n2Tbe
n2T b
JmO~eO~e b
KmOeObe
-2~b~,b
--2~b~b
where
03 o
ae = nO~n
and, since the electrical power Pe and the mechanical p o w e r Pm must be equal at steady state,
= ju~
synchronous
2P2
= 0.5 Jm n : p ba
TU
at
J_mO.)mn2
In physical units the mechanical and electrical parameters are related as follows:
Te~
b b T~lOm
=
Tm=nTe
Oe = nora,
n~T~ /~be = Tbm/~bm
The machine's inertia is often expressed in terms of the inertia constant H in seconds, so it is useful to find the per-unit value of J in terms of H. By definition
jb-
S-2
=
+ K.OU
where H and COoare in per-unit. The equation o f motion in per-unit is then given b y T u = ( 2 H / w o ~ u + KuO u 2.2. P e r - u n i t single e q u i v a l e n t m a c h i n e
The conventional approach of representing a power system which contains more than one machine is to have a single equivalent machine with a rating equal to the sum of the ratings of the individual machines. This is a valid simplification if the stability study is concerned only with the transient oscillations of energy between the electrical network and the rotational systems of the generators. It may n o t be valid if more detail is required a b o u t the transient oscillations of energy among the various individual masses of the generator shafts. The two-machine system o f Fig. 2 can be simplified to the single-machine system of
178 Fig. 1 where the system in Fig. 2 is normalized to a base of 2144 MVA, twice that of Fig. 1. The base values for a single machine (1072 MVA) and those for the equivalent machine (2144 MVA) are given in Appendix A-1. The per-unit values of the electrical parameters in the equivalent system of Fig. 1 are given by the per-unit values in the twomachine system of Fig. 2 multiplied by the ratio of equivalent-machine base power to single-machine base power, that is by a factor of two. However, since the machine and transformer in Fig. 1 represent the parallel combination of machines and transformers in Fig. 2, their parameters in physical units are half those of a single machine and transformer in Fig. 2. Thus, in per-unit, the machine and transformer of Fig. 1 all have the same electrical parameters as those in Fig. 2. For the equivalent-machine's mechanical system to be equivalent to that of the twomachine system, its shaft's total inertia must be equal to the sum of the total inertias of the two machines in order to have the same quantity of stored rotational energy at synchronous speed. However, since the base inertia for the equivalent machine is twice that for a single machine, the per-unit values of inertia will be the same for both systems. Similarly, to carry through the torsional characteristics of the shaft to the equivalent machine, the physical values of shaft stiffnesses of the equivalent machine must be twice those of the single-machine's shaft. As in the case with the inertia, owing to the doubling of the base value of stiffness for the equivalent machine, the per-unit values of shaft stiffness will be the same in both cases. The inertia constant H in seconds is the same for all machines since both the stored kinetic energy at synchronous speed and
the base power increase by a factor of two for the equivalent machine. The equivalent system is summarized below in Table 1 where the elements in the Table give the ratio of equivalent-machine system values to the two-machine system values.
3. THEORY In order to investigate the non-linear power systems of Figs. 1 and 2 it is necessary to describe each system element by a mathematical model. The well-known [12] two-axis equations describing the generator, transmission system and mechanical system of Fig. 1 are summarized below for the sake of completeness.
3.1. The s y n c h r o n o u s m a c h i n e Subject to the usual assumptions [12] pertaining to the two-axis theory, the mathematical model of a synchronous generator, expressed in terms of currents and a rotor reference frame, takes the following statespace form [ 1 ] : p i = [L]-I{v -- ([R] + ¢o[V])i}
(1)
where 09 is the rotor speed. The vector of axis winding currents is given by i = [id,
i~d, ikd,
iq,
ikql,
ikq2]w
(2)
and that of voltages by v = [vd,
Vfd,
O, 0] T
O, V,,
(3)
In the above model the synchronous machine is represented by six coils with one damper circuit on the d-axis and two damper circuits on the q-axis. The axis currents from eqn. (1) are used to calculate the electrical torque Te as follows: Te = --iW[ G ]i090/2
(4)
TABLE 1 Ratio of equivalent-machine system parameters to two-machine system parameters Mechanical parameters p.u. 1
physical 2
Generator electrical and transformer parameters
Transmission line parameters
p.u.
physical
p.u.
physical
1
0.5
2
1
179 3.2. The transmission s y s t e m
The transmission system is assumed to consist of a lumped series combination of resistance, inductance and capacitance in each of the stator phases. The two-axis voltage vector at the machine terminals is given by =
_
(5)
where the terms on the right-hand side of eqn. (5) are infinite bus voltage and the voltage drops across the capacitance, inductance and resistance as given b y eqns. (6), (7) and (8) respectively: pVCd = i d / C - -
¢OVCq and
vL = L p i d + w L i q v~ = Rid
and
and
pv c
= iq/C + ~v C
(6) (7)
vL = L p i q + ~ o L i d
(8)
v R = Riq
[P]px = [ E ] x + [S] u
3.3. The mechanical s y s t e m
The mechanical system of a single generating unit is assumed to consist of four turbine stages, a generator and an exciter [1], all represented in Fig. 3 as six inertias interconnected by five torsional springs (the connecting shafts), and which can be described by a set of second-order differential equations of the form 0 = [j]p25 + [D]p6 + [K]6 + T
(9)
where [J] is a diagonal matrix of inertias, [D] is a diagonal matrix of damping coefficients, [K] is a symmetric matrix of shaft stiffnesses, T is a forcing torque vector (acting on each inertia) and 5 is an angular position vector. INFINITE r.
J~ 6,
~ 6~
J~ 6,
J, 6,
theory for which has been explained elsewhere [13]. The machines and network are represented in steady A B C variables for the initial load flow calculation and the entire system is represented in d,q variables for eigenvalue and transient response calculations. In a multi-machine system the d,q reference frames of individual generators rotate at different speeds and the stator variables of all machines are therefore transformed into a c o m m o n synchronously rotating D,Q reference frame. In the c o m m o n reference frame each generator is represented by a model which consists of a resistance, inductance and a voltage source which does n o t neglect stator transformer voltage and speed voltage terms. The state-space formulation of the total system can be summarized in the form [ 13]
~ 16~
BUS
4 6,
Fig. 3. Mechanical model of the turbogenerator shaft.
3.4. T w o - m a c h i n e s y s t e m
The t w o machines, closely coupled through the transformers as shown in Fig. 2, are analysed using multi-machine techniques, the
(10)
The state vector of the entire system is a composite vector consisting of the components
X= [/DE, iDg, /QE, /Qg, uC, V~, [tPdr], [~qr],
[fa]] T
(11)
and
U----[VbDQ, bvg, bQg, Vfd, Pt] T
(12)
where the subscript E denotes currents in non-machine inductors and the subscript g denotes quantities associated with the generators; [~d~] and [~qr] are the composite vectors made up of the ~dr and ~qr vectors of each generator; [fa] are the mechanical states of the generators (angles and speeds); VbDQ contains the D,Q-axis components of the infinite busbar voltage; bDg and bQg are the vectors of all the D,Q components of the generator voltages bg; V~d is the vector of generator field voltages and Pt is the vector of prime mover input powers. For the purpose of the eigenvalue calculations the generator equations, transmission system equations and the mechanical equations in single-machine and multi-machine form are linearized a b o u t some steady operating point by a truncated Taylor series expansion. The linearized equations are summarized in Appendix B.
180
and is the same station representation t h a t has been used in various studies [ 1 - 7 ] of this power station to date. The second set, Fig. 5, applies to both generating units in operation but considers them as a single equivalent machine so that the analysis may be carried out as a single-machine study. The third set, Fig. 6, considers both generating units in operation and the system is analysed as a multi-machine system. In all three cases the system stability as a function of the series capacitor compensation is investigated by calculating a locus of eigenvalues for the system as the compensation level is varied from about 14% (Xc = 0.08 ~2) to 105% (Xc = 0.57 ~2) where
4. R E S U L T S
The results in this section consider three different system configurations. The first set in Fig. 4 considers a single generating unit 600.
I
M~s
500.
400.
% compensation level = N
300.
Fig.
4(b)
100.
Ri" %
0. R2
-100. -1s.00
, -10.00
, 6.00
-s.'00
REAL
(a)
(13)
4.1. E f f e c t o f the second generating unit Figure 4(a) shows the eigenvalue loci in the direction of the arrows for a single generating unit; they are similar to those calculated in previous studies [1]. As the
z 2E
L
The eigenvalue loci are generated by repetitive calculation of the linearized statespace equations. In the case of complex conjugate pairs, only those loci with positive imaginary parts are drawn. The system parameters for the three cases are summarized in Appendix A. For all three systems the initial conditions, assumed are Vb = 1.0 p.u., Vt = 1.1 p.u. and P t = 0.6 p.u.
200.
see
= IOOXc/X
s.0~
I/S
120.0
1
M;~4
/
r
T
]o.u.
xc
~2)575
{N
=
56,9%)
xc
0,76 p.u.
(N
D)75
p.u,
(N
0,89
p.u.
(N -
88,1%)
0,96
p.u.
{N
95,1%)
100.0 O~
MK3
/
xc
-
75,3%) -
74,3%)
80.
MK2 xc
50. 0 Xc
40.0
=
=
MK1
Z ~E
(b)
2 0 . I~ _ -2. 00
REAL
_
~ -1.00
0. 0 0
1 1.00
---L 2. 0 0
£ 3. 0 0
4.00
l/S
Fig. 4. (a) g i g e n v a l u e loci for o n e g e n e r a t o r as X c is varied f r o m 0.14 p.u. t o 1.06 p.u. (1072 M V A base). (b) E x p a n d e d view o f m o d e s MK1 t o MK4 for o n e g e n e r a t o r as X c is varied f r o m 0.14 p.u. to 1.06 p.u. (1072 M V A base).
181
b~
?, 0" T
T
ii z
-
-
T
-
-
T
-
-
~
u
u
11
14
z
z
z
z
z
o ii xu
it
u
u
~
×u
"0 0
l
l '
"0 =
rz.1 1"
o
1" SIOVB
I
I
I
= b~ 0
=
IBVNIDVWI
I
I
i
;i
ul
|% "~
8
I: I
I
I
I
I
I
i
SIOV~
),BVNIOVHI
r-,
182
< > l
m 0 v
z
£
@ 0
r~
I
o
i
4
\/
o
0
i'
b~ o
i
0
I~¥NIgVNI
S/0¥~
r~ 1
o
Fj
_o
i
i
I
co eJ
SIOV~
~T i
X~VNIgV~I
183
compensation level is increased, the frequency of the supersynchronous electrical mode E1 increases, while that of the subsynchronous m o d e E2 decreases. Mechanical modes MK4 and MK5 are not affected by compensation level changes (since the generator mass is near a nodal shaft point [1] for those modes), but the damping of modes MK1, MK2 and MK3 is influenced. The frequency and damping of the hunting mode MK0 increases with increasing compensation owing to the stronger electrical coupling of the generator to the infinite bus as the compensation increases. Figure 4 ( b ) s h o w s an expanded view of mechanical modes 1 - 4. The system first becomes unstable at an Xc of a b o u t 0.575 p.u. {0.309 ~2) when mode MK3 moves into the right-hand portion. Mode MK2 moves into the right-hand portion when Xc = 0.75 p.u. (0.403 [2) and mode MK1 moves into it when Xc = 0.96 p.u. (0.516 [2). Thus the maximum permissible compensation level with one generating unit before the first occurrence of instability is 57% (X~ = 0.309 [2). The effect of an increase in the generating capacity of the power station by the addition of a second generating unit can be seen by comparing Fig. 5 with Fig. 4 since Fig. 5 is the locus calculated with the two units represented as a single equivalent unit. Table 2 contains a set of eigenvalues from each of these loci calculated with a compensation level of 25% (X¢ = 0.13 [2). The main differences between the loci of Figs. 4 and 5 can be explained b y noticing TABLE
that the effect of an additional unit, and hence a doubling of the power transferred between the generator and the bus, is the same as a per-unit equivalent single unit transferring the same power across a transmission line of twice the distance. Hence, since the transmission line resistance and reactance have doubled in per-unit whereas the machine and transformer parameters are the same, there is an increase in the per-unit values of R, XL and Xc between the internal voltage of the generator and the infinite bus of 96.5% in R, 69.4% in XL and 100% in Xc. This results in the following. (a) For a low value of compensation there is an effective higher value of reactance between the internal generator voltage and the infinite bus. Thus, since the generator is not modelled with a voltage regulator, the locus at M, which is associated with the magnetic stability of the machine [12], starts in the right-hand portion. Furthermore, this increased reactance means that there is a weaker electrical coupling between the generator and the infinite bus and this results in a decrease in the damping and frequency of the hunting mode, as can be seen in Table 2. (b) The damping of the electrical modes E1 and E2 has increased. This is due to the fact that the real part of the eigenvalues for an R L C circuit can be approximated to R / 2 L for R : / 4 L 2 < 1/LC and the per-unit resistance has increased by more than the per-unit inductance. The effect of this is that mode E2 does not interact as strongly with the generator mechanical modes and this can be seen b y the
2
Single-machine and equivalent-machine eigenvalues calculated for a compensation level of 2 5 % (X c = 0.13 ~) Single machine Mode E1 E2 MK5 MK4 MK3 MK2 MK1 MK0 M R1 R2 R3
Supersynch. flux Subsynch. flux Supersynch. mech. Subsynch. mech. Subsynch. mech. Subsynch. mech. Subsynch. mech. Inertial mode Magnetic stability Damper circuit Damper circuit Damper circuit
Equivalent machine
Real
Imaginary
Real
Imaginary
--9.755 --9.065 --0.7861 --1.049 --0.7773 --0.8163 --0.8246 --1.012 --0.1051 --1.028 --10.42 --17.33
+444.6 +183.5 -+581.8 +109.9 +99.97 +77.98 +42.49 +4.420 0.0 0.0 0.0 0.0
--11.16 --10.72 --0.7860 --1.049 --0.7754 --0.8142 --0.8149 --0.9601 0.0263 --0.8905 --9.807 --16.94
+455.6 -+172.5 +581.8 +109.9 +99.69 +77.83 +42.15 +2.586 0.0 0.0 0.0 0.0
184 f a c t t h a t M K 1 , M K 2 a n d M K 3 d o n o t m o v e as far i n t o t h e r i g h t - h a n d p o r t i o n . (c) F o r a n y c o m p e n s a t i o n level t h e freq u e n c y o f m o d e E1 has increased a n d t h a t o f m o d e E2 has d e c r e a s e d . This is d u e t o t h e f a c t t h a t t h e i m a g i n a r y p a r t s o f t h e eigenvalues o f an RLC circuit can be a p p r o x i m a t e d to 1/(LC) in f o r R2/4L2 < 1/LC and, since t h e increase in L is less t h a n t h e decrease in C (increase in Xc), t h e r e s o n a n t f r e q u e n c y o f t h e RLC circuit b e t w e e n t h e g e n e r a t o r i n t e r n a l v o l t a g e a n d t h e b u s voltage is higher. T h e c o n s e q u e n c e o f this is t h a t t h e s u b s y n chronous mode E2 interacts with the m e c h a n i c a l m o d e s at a l o w e r value o f Xc. T h u s at high c o m p e n s a t i o n levels t h e h u n t i n g m o d e M K 1 starts t o i n t e r a c t w i t h t h e s u b s y n c h r o n o u s m o d e E2 a n d changes d i r e c t i o n towards the right-hand portion. T h e e f f e c t s of {b) a n d (c) a b o v e o n t h e stability o f t h e m e c h a n i c a l m o d e s are o f o p p o s i t e sense; h o w e v e r , it can b e seen b y l o o k i n g at Fig. 5(b) t h a t t h e e f f e c t o f (c) p r e d o m i n a t e s a n d instability o c c u r s at a l o w e r c o m p e n s a t i o n level t h a n in Fig. 4(b). T h e c o m p e n s a t i o n levels at w h i c h t h e m e c h a n i c a l m o d e s go u n s t a b l e f o r o n e g e n e r a t i n g u n i t and t w o g e n e r a t i n g u n i t s are c o m p a r e d in T a b l e 3.
4.2. Two-machine station representation Figure 6 c o n t a i n s t h e loci c a l c u l a t e d f o r t h e two-machine multi-machine representation of t h e s t a t i o n a n d it is similar t o t h e singlee q u i v a l e n t - m a c h i n e loci o f Fig. 5. T a b l e 4
TABLE 3 Compensation X c in ohms giving instability of mechanical modes for one and two units respectively Mechanical mode
One unit Two units
MK1
MK2
MK3
0.516 0.448
0.403 0.367
0.309 0.278
c o n t a i n s a set o f eigenvalues c a l c u l a t e d f o r t h e two-machine system when the compensation level is 25% {Xc = 0 . 1 3 ~2). A c o m p a r i s o n o f Fig. 6 w i t h Figs 4 a n d 5 s h o w s t h e following. (a) T h e a d d i t i o n o f t h e s e c o n d m a c h i n e d o e s n o t p r o d u c e a d u p l i c a t e set of t h e six m e c h a n i c a l m o d e s and t h e f o u r r o t o r circuit m o d e s o f t h e single m a c h i n e . I n s t e a d , f o r e a c h m a c h i n e m o d e o f Figs. 4 a n d 5, t h e r e are t w o m o d e s in Fig. 6. T h e s e pairs o f m o d e s are t h e c o m m o n (or s y m m e t r i c a l ) a n d anti- (or antis y m m e t r i c a l ) m o d e s associated w i t h s y m m e t r i c a l r e s o n a n t s y s t e m s [14] (see A p p e n d i x C). F o r m e c h a n i c a l m o d e s M K 4 a n d M K 5 o n l y o n e m o d e is seen since in each case t h e c o m m o n m o d e a n d a n t i - m o d e are a l m o s t c o i n c i d e n t as these m o d e s are n o t e x c i t a b l e f r o m t h e electrical n e t w o r k . F o r t h e r o t o r circuit real m o d e s t h e a n t i - m o d e s are slightly m o r e d a m p e d , h o w e v e r this is n o t o b v i o u s as each c o m m o n - m o d e locus m o v e s t h r o u g h its a n t i - m o d e as t h e c o m p e n s a t i o n level is increased. F o r t h e c o m p l e x m e c h a n i c a l m o d e s
TABLE 4 Two-machine system eigenvalues calculated for a compensation level of 25% (Xc = 0.13 ~ ) Common mode Mode E1 E2 E3 MK5 MK4 MK3 MK2 MK1 MK0 M R1 R2 R3
Supersynch. flux Subsynch. flux Synch. flux Supersynch. mech. Subsynch. mech. Subsynch. mech. Subsynch. mech. Subsynch. mech. Inertial mode Magnetic stability Damper circuit Damper circuit Damper circuit
Anti-mode
Real
Imaginary
Real
Imaginary
--11.16 --10.72 --2.254 --0.7859 --1.049 --0.7754 --0.8142 --0.8148 --0.9600 0.0263 --0.8905 --9.807 --16.94
+455.6 -+172.5 -+314.1 -+581.8 -+109.9 -+99.69 -+77.83 -+42.15 -+2.586 0.0 0.0 0.0 0.0
--0.7858 --1.049 --0.8730 --0.8674 --0.9844 --1.730 --0.2714 --1.307 --12.59 --19.18
-+581.8 -+109.9 -+101.0 +78.51 -+43.66 -+7.433 0.0 0.0 0.0 0.0
185 the anti-mode (MK0', MKI', MK2' and MK3') is always slightly more damped and at a higher frequency than its corresponding c o m m o n mode. (b) The anti-mode loci of the two-machine case are only slightly affected by a change in compensation level. However, the commonmode loci are identical to the loci of the single~quivalent~machine case in Fig. 5. Thus the level of compensation permitted before the onset of instability can be calculated with either a two-machine multi-machine representation or a single-equivalent~machine representation. This is significant since the two-machine model requires an additional eighteen differential equations and so the use of the equivalent-single-machine model results in a large reduction of c o m p u t e r time. It must be noted that these results were calculated for two identical machines operating under identical conditions and do not account for any affects associated with variations in parameters between the two generators or differences in the electrical load on each generator. This will be the subject of a further study.
5. CONCLUSION
This paper has investigated the effect of increasing the generating capacity o f a p o w e r station by adding a second identical generating unit to an existing one. It has compared the effect of modelling the new system as a single equivalent generating unit or as a multi-machine system with two separate units. A per-unit system was developed for the analysis of multi-pole, multi-inertia turbogenerators. The following conclusions can be drawn from the results presented. (a) An increase in the generating capacity effectively results in a change in the interconnecting tie-line characteristics and so previous stability studies for the singlemachine case are not valid when the second unit comes on line. They are in fact optimistic. (b) The addition of the second unit will result in instability occurring at a lower value of compensation than was calculated for the single unit. In fact a per-unit value of Xc of a b o u t 10% lower than previous calculations will have to be used to maintain a stable MK3. Various different transmission systems
including shunt loads were analysed and the results (not presented) showed that the destabilization effect of the second generator does depend on the transmission system configuration although for most cases similar reductions in the capacitor compensation level were obtained; (c) The modelling of the station as t w o separate units has shown the introduction of anti-modes in the locus diagram. These anti-modes represent an out-of-phase resonance between the two machines. For the purpose of maximum permissible capacitor compensation calculations these anti-modes appear to have no effect and no advantage is gained by modelling the station as two separate generators rather than as an equivalent single generator. However, the nature of these anti-modes needs to be further investigated.
ACKNOWLEDGEMENTS
The authors acknowledge the assistance of R. C. S. Peplow and H. L. Nattrass in the Digital Processes Laboratory of the Department of Electronic Engineering, University of Natal. They are also grateful for financial support received from the CSIR and the University of Natal.
NOMENCLATURE
d,q D,Q [D] [G] i [J] [K] [L] n p Pe Pm [R] T Te v 0
d,q axis of machine rotating reference frame D,Q axis of synchronously rotating reference frame shaft damping matrix rotational voltage inductance unit vector of axis currents shaft inertia matrix shaft stiffness matrix machine inductance matrix number of pole pairs deriverative operator d/dt electric power mechanical shaft p o w e r machine resistance matrix forcing torque vector electric torque vector of axis voltages null vector
186 0/
6 A 0 09 o 5Or (-Den 5omn
angular acceleration rotor angle vector small change operator angular displacement flux linkage system frequency, rad e s I angular velocity of rotor, rad e s - 1 electrical nominal frequency mechanical nominal speed
Indices b base values e electrical quantities m mechanical quantities u per-unit values REFERENCES 1 D. J. N. L i m e b e e r , R. G. Harley a n d S. M. S c h u c k , S u b s y n c h r o n o u s r e s o n a n c e of t h e Koeberg t u r b o g e n e r a t o r s a n d o f a l a b o r a t o r y microa l t e r n a t o r s y s t e m , Trans. S. Aft. Inst. Electr. Eng., 70 (11) ( 1 9 7 9 ) 2 7 8 - 279. 2 R. G. Harley, G. D. J e n n i n g s , M. A. L a h o u d a n d M. F. H a d i n g h a m , Level of m a t h e m a t i c a l m o d e l r e q u i r e d t o p r e d i c t t h e r e s p o n s e of a t u r b o g e n e r a t o r s y s t e m w i t h a n d w i t h o u t series c a p a c i t o r s , Trans. S. Art. Inst. Electr. Eng., 75 (3) ( 1 9 8 4 ) 28 - 39. 3 D . J . N . L i m b e e r , R. G. Harley a n d M. A. L a h o u d , S u p p r e s s i n g s u b s y n c h r o n o u s r e s o n a n c e w i t h static filters, Proc. Inst. Electr. Eng., Part C, 128 ( 1 9 8 1 ) 33 - 44. 4 M. A. L a h o u d a n d R. G. Harley, A n o p t i m a l c o n t r o l l e r for t h e s u p p r e s s i o n o f s u b s y n c h r o n o u s r e s o n a n c e , Electr. Power Syst. Res., 6 ( 1 9 8 3 ) 203 - 216.
5 D . J . N . L i m b e e r , R. G. Harley a n d M. A. L a h o u d , The suppression of subsynchronous resonance w i t h t h e aid o f an auxiliary e x c i t a t i o n c o n t r o l signal, Trans. S. Afr. Inst. Electr. Eng., 74 (8) ( 1 9 8 3 ) 198 - 209. 6 M. A. L a h o u d a n d R. G. Harley, M u l t i m a c h i n e s u b s y n c h r o n o u s r e s o n a n c e : Part I - - L i n e repres e n t a t i o n , Trans. S. Aft. Inst. Electr. Eng., 75 (2) ( 1 9 8 4 ) 3 - 13. 7 M. A. L a h o u d a n d R. G. Harley, M u l t i m a c h i n e s u b s y n c h r o n o u s r e s o n a n c e : Part I I - L o a d repres e n t a t i o n , Trans. S. Afr. Inst. Electr. Eng., 75 (2) ( 1 9 8 4 ) 14 - 27. 8 IEEE Subsynchronous Resonance Working G r o u p , S e c o n d b e n c h m a r k m o d e l for c o m p u t e r s i m u l a t i o n o f s u b s y n c h r o n o u s r e s o n a n c e , IEEE Trans., PAS-104 ( 1 9 8 5 ) 1057 - 1066. 9 E. W. K i m b a r k , Power System Stability, Vol. 1, Wiley, New York, 1 9 6 7 . 10 B. L. Agrawal a n d R. G. F a r m e r , Use of freq u e n c y s c a n n i n g t e c h n i q u e s for s u b s y n c h r o n o u s r e s o n a n c e analysis, IEEE Trans., PAS-98 ( 1 9 7 9 ) 341 - 349. 11 M. R. Harris, P. J. L a w r e n s o n a n d J. M. S t e p h e n son, Per-Unit Systems: with Special Reference to Electrical Machines., C a m b r i d g e University Press, L o n d o n , ISBN 521 0 7 8 5 7 1, 1 9 7 0 . 12 B. A d k i n s a n d R. G. Harley, The General Theory
o f Alternating Current Machines. Applications to Practical Problems, C h a p m a n a n d Hall, L o n d o n , ISBN 0 4 1 2 1 5 5 6 0 5, 1 9 7 5 . 13 M. A. L a h o u d a n d R. G. Harley, Analysis t e c h n i q u e s for t h e d y n a m i c b e h a v i o r o f a m u l t i g e n e r a t o r s y s t e m , Trans. S. Afr. Inst. Electr. Eng., 74 (4) ( 1 9 8 3 ) 53 - 63. 14 T. S. H u a n g a n d R. R. Parker, Network Theory: An Introductory Course, Addison-Wesley, R e a d i n g , MA, 1 9 7 1 . 15 M. A. L a h o u d , T h e analysis a n d c o n t r o l o f sync h r o n o u s a n d a s y n c h r o n o u s m a c h i n e s in p o w e r s y s t e m s , Ph.D. Thesis, University of Natal, 1 9 8 2 .
APPENDIX A
A-1. System base values
Base Base Base Base Base Base Base Base Base Base Base Base
time t b (s) armature power P~ (MVA) armature voltage V~ (kV) armature current I b (kA) armature impedance Z~ ( ~ ) field power pb (MVA) field voltage V b (kV) field current I~ (kA) field impedance Z b (~2) electrical angle 0~ (tad e) electrical speed 5o~ (tad e s -1) electrical acceleration ~ (ra& s-2)
Single generator 1 1072 24 (1-1) 25.79 0.537 536 188 2.85 65.94 1
Equivalent generator 1 2144 24 (1-1) 51.58 0.269 1072 188 5.70 37.97 1
1
1
1
1
187 Base Base Base Base Base Base Base
electrical torque T b (N m (rade) - ' ) mechanical angle 0 b (rad m) mechanical speed COb (tad m s-l) mechanical acceleration C~m(tad m s-2) mechanical torque Tbm (N m (radm) -t) inertia j b (kg m 2 (radm) -2) stiffness K b (N m (radm) -2)
A-2. Generator electrical parameters Xd = 2.466 p.u. Xkd = 0.13 p.u. Xq = 2.28 p.u. R k d = 0.015 p.u. X, = 0.22 p.u. Xkql = 0.59 p.u. Ra = 0.003 p.u. Rkq I = 0.005 45 p.u. Xfd = 0.19 p.u. Xkq2 = 0.06 p.u. R f d = 0.000 984 p.u. R k q 2 = 0.011 p.u. A-3. Mechanical parameters Values of the inertia constants: H1 = 0.158 s //4 = 1.625 s //2 = 1.598 s Hs = 0.674 s H 3 = 1.593 s H6 = 0.034 s Per-unit values of shaft stiffness: K12 = 10.63 p.u. K4s = 26.93 p.u. K23 = 25.34 p.u. Ks6 = 70.53 p.u. K34 = 23.42 p.u. For the purpose of this study, the per-unit values of damping Di were taken as 1% of Hi [1]. The turbine torque distribution is assumed to be HP LPI LP2 LP3 33.7% 21.6% 18.8% 25.9% A-4. Transmission system parameters Transformer: X~ = 0.15 p.u. R1 = 0.0001 p.u. Transmission line: XL = 1.0097 p.u. R = 0.0845 p.u. {1072 MVA base) XL = 2.0194 p.u. R = 0.169 p.u. (2144 MVA base)
APPENDIX B The linearized system equations The linearization of eqn. (1) yields the linearized generator axis currents as A p i = [ L ] - I { - - [ R ] + COo[G] A / - - [G]i 0 Aco
+ At}
(B-l)
where the subscript 0 indicates the steady
3.412 0.5 0.5 0.5 6.825 1.365 1.365
× 106
× 106 × 10 6 × 106
6.825 0.5 0.5 0.5 1.365 2.730 2.730
× 106
× 10 7 X 10 7 × 107
state values of the variable and A indicates a small change in this value. The linearized electrical torque is obtained from eqn. (4) as ATe = --COo([G] T + [G]}io A i / 2
(B-2)
while the linearization of the mechanical torque yields ATm = ~Um0 -- (Pmo/co0) ACO
(B-3)
where Pro0 is the steady state power transmitted along the turbine shaft. The two-axis equations of the transmission line have been stated in § 3.2. The voltage across the capacitor i s a state variable, hence a linearized form of its differential equations is required. From eqn. (6) A p v c = Aid/C -- coo AvC -- vCo ACO
(S-4) A p v c = Aiq/C + coo AVd c + vdC0ACO
The linearization of the generator's electrical torques and of the non-linear sub-matrices contained in [S] and [E] of eqn. (10)results in new matrices [P'], [S'] and [E'] such that [15] A p x = [ P ' ] - ' [ E ' ] Ax + [ P ' ] - I [ S ' ] Au
(B-5)
APPENDIX C Symmetrical systems In this Appendix we determine the mode shapes and mode frequencies of a symmetrical gystem. Figure C-l(a) shows a single degree of freedom (SDOF) system consisting of a mass m connected to a rigid frame by a spring of stiffness k. This system has a single natural frequency given by
co,
=
(c-1)
The two degrees of freedom symmetrical system shown in Fig. C-l(b) consists of two identical SDOF systems, of the kind in Fig. C-l(a) coupled together-by a spring of stiff-
188
IX
-
(k + k¢)/m
kc/m =0
det kc/m
X -
(k + k c ) / m
(C-4) or (a)
k 2 -- k2(k + kc)/m + (k + kc)2/rn 2
/NNNN/ZZ/HIHH/
k
-- k~2/m 2 = 0
(C-5)
and solving this for X gives kl = k / m
ks = (k + 2k~)/m
(C-6)
Thus, the natural frequencies of the coupled system are
×1
(.01 "~ v ~ / m
///HI,
or
and
co2 = x~(k + 2kc)im
(C-7)
The mode shape vectors can be found by solving the equations
"HN//,'HHH/
(b)
(C-8)
[ M ] l [ g ] u = ku
Fig. C-1. (a) Single degree of freedom system. (b) Symmetrical two degrees of freedom system.
for each k to yield 151 = [ 1 , 1] w and
ness kc. The equations describing the unforced m o t i o n of the two masses are written in matrix form as k +
--k c
[p2x2] (C-2) The natural frequencies of this symmetrical system can be found by solving for the eigenvalues of the system. The eigenvalues are given by the solution of the scalar equation d e t [ ~ [ I ] -- [ M ] - I [ K ] ] = 0
(C-3)
where [M] and [K] are the mass and stiffness matrices respectively of eqn. (C-2) and [I] is the identity matrix. Substituting for [M] and [K] from eqn. (C-2) we get
us = [ 1 , - - 1 ] T
(C-9)
Thus the two modes of the coupled system are n o t just the SDOF system mode repeated twice, but two modes shifted slightly apart. These modes are the so-called c o m m o n mode and anti-mode associated with symmetrical systems. The difference in frequency between the c o m m o n mode and anti-mode is dependent on the strength of the coupling between the two systems and increases with stronger coupling. The common-mode frequency is the same as that of the SDOF system and represents the two masses oscillating in phase. For this mode the coupling spring kc does n o t stretch at all and the masses move as though they were uncoupled. The anti-mode frequency is slightly higher than that of the c o m m o n mode and represents an out-of-phase movement of the masses.