Ieee Press - Power System Stability- Edward Wilson Kimbark 3f5ub

If'

~

~

0.7

o

I , I I I J

J

~

1...~

1...oIlI~

= 0.65.

~

II

~

i-

v- ~r-

4,....

~

sin 80

5.0

4.0

3.0

~WI >-~ ~ ~ Ort' I :r-I/O"l

tr-- -

J

t I

FIGS.

.,

~

o

4.0

,, I

I I 'D,I-

~()~ ~~

.C'\ ___ ....~,~

5.0

= 0.70.

33, 34, 35. Curves for determination of clearing time (copied from Ref. 3 by permission).

CURVES DETERMINING CRITICAL CLEARING TIME 1.0

IIi}

~Wd'l 0'Yo ~ ~~ I~t~ 0 t::i O'~tfll O' o 0' ~ ~~Xi~

u: :!J ~~I-~

~O

~~~

0.9

11

J J

'I' I/'

.,.

~

'I

~

II

}

~

~ ~

"

~

jllIII"

//'1 //~:t.

.. I

-

~~

v

I

o ~

~

T

~

~~ I-~

'h...-I......""

~ l..o""""

io"'"

'-

I

T T 11~ ~~ 1-1-

If'

.....

.""

L.oo~

0:'

.,

,<,

~" ~

l....olI~

'"

~

~

lIi""

"

~

1-1-

175

~~

_I-'"

...

1..- .....

0.8 0.7

o

1.0 FIG.

r'rr =

4.0

3.0

2.0

5.0

36. sin 00 = 0.75.

030

= O. 0

'1

1.0

~

1/

.

~l 17 ,~r~J \)'b\<)~, 'rCi.1: ~~. ~~ ~~a a:t a:l,;~ ~B:-t\iP ~ ~o'f.~ ~\ffi1<j~ ~o.~~ Iio' I/t ~ ~I- '- 'h~~- t\~,,~ 1.t~.,,~~o~'z;;..-e 1'\I"'~~~- "~~..-""'~

t~ 11 -I~""I-f-<~ ~

" Jt-,"" ~

0.9 0.8

oJ

;L

= . 0.40 l

11'1 ~ r:r/: ~~

tt"

I.' ~

~~

o

- '--

... .......

1.0

... "" 1--..... ~

1.1I ......

I..-

1....1"'"

l..o.IlIl--

~"

.""

..."'"'"

2.0

--

1- .......

;!Ii

•

~\-,,"'-~-~

-"

L.-

.... 11I""

1.-:- ~iI-"'"

3.0

4.0

5.0

Tc

FIG.

1.00 o

rt> - co'~'>~ co'> v o· Alo,,, oep " W ~!~~ ~~O~""I~~ ~~I ~............~J -"I" ~ ,jf- ..... <)I.~ ~ <~~ <.t;~~~ 0 n,

-

F-f-

I-~'

~...

r

~"'J~

1/

J

j

I

0.90 0.85

IJ

JI

I..;

~ ~

1....1

I;l'"

l'

J

v

1....1

I.l'"

"

I.l'"

~

i...ll

~

--

1.0

....... "'"'"

II""

ijll'"

~'-'

t-

..... jllIII"

~,....

L....o .....

I...-

~

I....

l.olII

o

- '(,?

l-

I;'

II IJ

//

rlf- f-I-

4......J"-'r- ....

I-

v

f-f-

1/

I-

0.95

37. sin 00 = 0.80.

2.0

3.0

4.0

5.0

Tc

FIG.

1.00

...~ 0.95

....

~

" -)- "...; II

..."

,

I

I.." I

T

o

"

1/

-- - --"'"

~"'1 4..' v: ~ --"

L.... II 0 ~I- lli~'j~I~(). O~~~.,.~~ D~09..... ~().t!.~~ 1\ ~ '(~~L.,- - f - ~'\....-I~ - '- t~~~ h~ -'" ..... ~

.....

1.0

~""" "'""',~

... ...... """'" ~.....

",

..... ~

l0- ll""

_"'" ~~

Iiiiiii

-~

2.0 FIG.

FIGS.

t1~$~:1Ci

ol-~~*~Y/)4:<§le::; O· O· + 0' ~ I

0.90

38. sin 00 = 0.85.

39. sin

3.0 80

4.0

5.0

= 0.90.

36, 37, 38, 39. Curves for determination of clearing time (copied from Ref. 3 by permission).

THE TWO-MACHINE SYSTEM

176

It is convenient to have an additional curve for determining the stability limits for instantaneous clearing and for sustained faults. To obtain the equation of such a curve refer to Fig. 12, which shows the application of the equal-area criterion to the case of instantaneous 1.0

Ii J

v 1/

0.9

iJ II' 1/

0.8

J

V II

0.7

/ I~ ~

0.6 ~

c 0.5

v

.~

V WI'" I;"

,., v:

0.4

0.3 I~

,., ,.,

0.2

~

l.'

~

I'

0.1 l;l"

'-'

I~

o I#"

o

FIG.

0.1

0.2

0.3

0.4

0.5 r J or r2

0.6

0.7

0.8

0.9

1.0

40. Curve for determining the stability limit for a sustained fault or for a fault cleared instantaneously (copied from Ref. 3 by permission).

clearing, and equate the area of the rectangle under the Pi line between ao and 6m to the area under the post-fault power-angle curve between the same limits. Area of rectangle = Pi(8m - 60) = Pm(6m - 60) sin 60

1

sin ~

T2(COS

ao -

Area under power-angle curve = T2Pm

Equating the two areas, we get (am - ao) sin ~o =

6m

as = T2Pm (cos ~o cos ~m)

cos 8m)

[28]

CURVES DETERMINING CRITICAL CLEARING TIME

177

where 8m is as given by eq. 26. Equation 28 expresses implicitly a relation between T2 and sin 80 which is plotted in Fig. 40. To find the stability limit for instantaneous clearing, enter this curve with T2 and read sin 80 • Then the stability limit is Pm sin 80 • The same curve can be used to find the stability limit for a sustained fault by entering it with P the value of TI instead of T2; this becomes evident when Fig. 13 is compared with Fig. 12. It has already been mentioned that, as the transmitted power (Pi == Pm sin 80 ) varies, the internal voltages E 1 and E 2 usually vary, and hence Pm = E 1E2/X12 also varies. For the present purpose the most useful way a 0------------"'· to show these relations is to plot Pi sin 00 and Pm against sin 80 , as shown in F10. 41. Typica · 1 curves 0 f P, an d Fig. 41. r; versus sin ~o. The procedure for obtaining a curve of power limit as a function of fault duration by Byrd and Pritchard's method will now be summarized. 1. Reduce the reactance network to an equivalent ~ between the two machines and neutral for each of the three conditions:

a. Before the fault. b. During the fault. c. After clearing the fault. Only the reactances X 12 between the two machines are used in what follows. 2. Compute the equivalent inertia constant M by eq. 2. 3. Calculate and plot curves (like those of Fig. 41) of

a. Pm = E IE2/XI 2 versus sin 80 b. Initial power Pi versus sin 80 maintaining the bus voltages at the values which would be held in actual operation. 4. Compute TI and T2 by eqs. 20 and 21. 5. a. Enter the curve of Fig. 40 with Tl and read sin 80 • From Fig. 41 read Pi corresponding to this value of sin 80 • This is the stability limit for a sustained fault.

b. Repeat, using T2 instead of TI. The value of Pi thus found is the stability limit for instantaneous clearing.

178


6. t Select values of sin 00 which are multiples of 0.05 and which are between the values found in steps 5a and 5b. For each such value of sin 00 find the proper family of curves from Figs. 23 to 39; find the curve for the value of Tl; enter this curve with T2 and read T e•

7. For each value of sin 00 used in step 6 read the corresponding values of Pi and Pm from the curves of step 3 (like Fig. 41). 8. t For each value of Tc found in step 6 compute the clearing time te by eq. 27, using the proper value of Pm determined in step 7. 9. Plot stability limit Pi as a function of clearing time te• This curve will look like the one in Fig. 11. The method which has been described fails if Tl = 0, since for this caser = 0 and t is indeterminate from eq. 27. To avoid this difficulty a new modified time p is now introduced, related to the actual time t by

t

= p J180M = p 7rPm

I GH

~7rfPm

. [29]

and differing from Tin eq. 27 in that Pm, the amplitude of the pre-fault power-angle curve, is used instead of T1Pm , the amplitude of the fault power-angle curve. Hence p = T/Vi). The swing equation now becomes [30]

with 8 in electrical radians, and it has the solution p=

2(0 - 00)

.sin 00

[31]

which will be recognized as eq. 15, transformed by the substitution of eq. 29 for t, and with 0 - 00 expressed in electrical radians. Given Tl = 0 and the values of T2 and sin 00, Pc may be found by using eqs. 25 and 26 to get oc, and then eq. 31 to get Pc. From the results of such calculations Pc has been plotted in Fig. 42 against sin 00 for various values of T2. If Tl = 0, steps 6 and 8 of the procedure are replaced by steps 6A and 8A, respectively, which are as follows: 6A. Select values of sin 00 which are between zero and the value found in step 5b. Find the curve in Fig. 42 corresponding to the [If rl = 0, substitute steps 6A and 8A, described on this page and page 180, for steps 6 and 8, respectively.

'(ii

c:

'0

0

o

2

0.4

0.6

0.8

1.0

o

0.5

FIG. 42.

1.0 2.0

2.5 3.0 Pc

3.5 4.0

4.5

5.0

5.5

6.0

Curves for determining clearing time -if rl = 0 (copied from Ref. 3 by permission).

1.5

6.5

7.0

~

co

~

~

s: l%j

~

~

o

~Z

&;

a

~

:j

§

o

z

~

z

a=

~

t;d

~

t:1

ax

~

a


180

value of r2. Enter it with the selected values of sin corresponding values of Pc.

ao and read the

8A. For each value of Pc found in step 6A compute the clearing time tc by eq. 29, using the proper value of Pm determined in step 7. EXAMPLE

4

Plot a curve of stability limit as a function of fault duration .for a threephase fault at the middle of one of the 132-kv. transmission lines of the power system of Fig. 43. The two hydroelectric generators i and k are to be ra. 0.175

~

33.0/13.2 O.l75~

33.0/13.2 0.30

0.35

0-111:

~30

6.6/132

0.35 )(

Fault

.6.'A0.140

~ 33.0/13.2

0.140

f----®

33.0/13.2 43. One-line diagram of power system, with reactances of lines and transformers in per unit on a lOO-Mva. base (Example 4). Data on generators and loads are given in Table 2. (From Ref. 3 by permisslon.)

FIG.

garded as one machine of a two-machine system, and the steam turbogenerators, a, b, g, and h, together with loads c, d, e, and J, as the other machine. Each load is assumed to consist of three equal parts, one of resistance, one of synchronous motors operating at half load, and one of induction motors at 1/2.75 load. The reactances and inertia constants of the generators and of the composite loads are given in Table 2, expressed on a base power equal to the rating of the individual generator or load. In Fig. 43 the reactances of the lines and transformers are given on a base of 100 Mva. and nominal voltage. The system frequency is 60 c.p.s, Solution. The procedure described on pp. 177-8 will be followed. 1. Network reduction. All reactances will be expressed on a system base of 100 Mva, The reactances in Fig. 43 are already on this base. The


181

machine reactances in Table 2, given on the machine base, are converted to the system base by multiplying them by (100 Mva.)/(machine rating in megavolt-am peres) .

TABLE 2 MACHINE REACTANCES AND INERTIA CONSTANTS (EXAMPLE

,

Rating Kind

Machine

b c

Machine System Machine System Base Base Base Base

0.8

0.18

0.45

6.0

"

"

"

"

"

"

Composite load 55

" " "

" "u

e g h i k

P.F.

Turbogenerator 40

d

f

H

Xd

~

Mva.

a

0.9

30

u

"

" "

55 Turbogenerator 50

"

"

0.8

"

0.56

" " " 0.18

"

1.01 1.86

"

1.01 0.36

"

1.95

" "

"

6.0

"

Hydro generator 29. 4

0.85

0.30

1.02

3.0

u

"

"

"

"

"

4)

"

2.40

"

1.07 0.58 u

1.07 3.00

"

0.88 u

The network is so nearly symmetrical about a horizontal axis that the reduction may be facilitated with very little sacrifice in accuracy by assuming that the two sections of the long receiving bus are tied together and also that the load busses d and e are tied together. In addition, machines i and k are assumed to be tied together behind their transient reactances; machines a, b, c, d, e, j, g, and h are assumed to be tied together similarly. Then, after simplifying the hydroelectric-station circuits and the receiving system by series and parallel combinations, the circuit of Fig. 44a results. The reactances of the three parts which are separated from one another by transformers are given on a common power base of 100 Mva, and on voltage bases equal to the nominal voltages of 6.6, 132.0, and 33.0 kv, It will be noticed that the ratio of the receiving-end transformers, 125.4 kv./33.0 kv., is not equal to the ratio of base voltages, 132.0 kv./33.0 kv, Therefore the base voltage for the receiving system should be changed from 33.0 kv. to 33.0 X 132.0/125.4 = 34.7 kv.; the reactance of that system on the new base is 0.13 X

(33'0)2 34.7 = 0.12 per unit

The new value is shown in Fig. 44b. Parallel and series combinations give the result shown in Fig. 44c for the pre-fault network. Xu = 1.10 per unit. The faulted network is obtained by grounding the midpoint X of one line in Fig. 44b. Then a ~-Y conversion, series combinations, and a Y-~ conversion lead to the ~ network of Fig. 44d. X 12 = 7.20 per unit. The post-fault network is obtained by omitting one of the parallel lines of


182

Fig. 44b. Then a series combination leads to Fig. 44e. X 12 = 1.28 per unit. 2. Inertia constant. The inertia constants in Table 2, given on the machine base, are converted to the system base by multiplying them by (rna-

~~~(a) 6.6/132 kv.

0.35

125.4/33.0 kv.

O.~15

2

~~3~(b) 051

0.35

~D

/>.

~ vvv ~

(c)

(d)

~vvv~

(e)

FIG. 44. Reduction of the network of Fig. 43 (Example 4). The reduced networks are given by c for the pre-fault condition, by d for the fault condition, and by e for the post-fault condition.

chine rating)/(100 Mva.). HI

=

Hi+ H k

=

Then 1.76

H2 = Ha + Hb+ H c + Hd+ He + H/+ H, H

=

+ Hh = 14.1

H 1H2 = 1.76 X 14.1 = 1.56 HI H2 15.9

+

3. Curves of P« and Pi versus sin 00. Assume that 132 kv. (1.00 per unit) is maintained on the high-voltage side of the step-up transformers, and 33.0 kv. (0.95 per unit) is maintained on the low-voltage side of the step-down transformers. The voltages behind transient reactance, E 1 and E 21 as functions of the initial angle 00 between them, can be found from the vector diagram of Fig. 45 by asg arbitrary values to the current I and solving,


183

either graphically or by trigonometric computation, the resulting triangles for Ell E 2, and 00. Then P« and Pi are calculated from the relations

P _ E1E2 X l2

m -

_

E 1E2

-

1.10

and

Pi = Pm sin 00 0.661

FIG.

0.121

----~~

45. Vector diagram for determining E 1 and E 2 as functions of ~o (Example 4). TABLE 3 CALCULATION OF

Pm AND Pi

sin

VERSUS

P,

EI

E2

~o

0.125 0.250 0.375 0.500 0.750 1.000

1.106 1.121 1.145 1.177 1.267 1.373

0.933 0.934 0.936 0.939 0.950 0.972

7.7 15.3 22.7 30.0 43.6 55.9

ao

sin

The results of the calculations are given in Table 3. functions of sin 00 are plotted in Fig. 46. 4. Computation of fl and f2.

(EXAMPLE ~o

0.134 0.264 0.386 0.499 0.689 0.828

4)

Pm

0.935 0.948 0.971 1.001

1.090 1.221

Curves of Pm and Pi as

By eqs. 20 and 21, 1.10

fl= - =

7.20

0.15

3

1.10 1.28

r2= - = 0.86 5. Stability limits for sustained fault and [or instantaneous clearing. a. Sustainedfault. Entering the curve of Fig. 40 with fl = 0.153, we read


184

sin 80 = 0.117. Entering the Pi curve of Fig. 46 with this value of sin 00, we read Pi = 0.110. b. Instantaneous clearing. Entering the curve of Fig. 40 with = 0.86,

'2

I

1.0 1--

---

Pm ~ / '

~

~

J

0.5

/ /1>;

oV

v

/ ~V

1/

/

J

V

/

o

05

ID

sin 00 FIG. 46.

Curves of Pm and Pi as functions of sin 80 (Example 4).

TABLE 4 CALCULATION OF POWER LIMIT Pi AS A FUNCTION OF CLEARING TIME t e (EXAMPLE 4)

1

2

3

4

5

6 te

sin 80

Pi

Pm

Tc/tc

Te

(sec.)

0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15

0.933 0.843 0.763 0.690 0.623 0.560 0.500 0.443 0.390 0.336 0.285 0.236 0.188 0.141

1.18 1.14 1.10 1.07 1.04 1.02 1.00 0.986 0.973 0.963 0.954 0.946 0.940 0.936

4.68 4.57 4.50 4.44 4.37 4.33 4.29 4.26 4.23 4.21 4.19 4.17 4.16 4.15

0.13 0.24 0.35 0.45 0.55 0.67 0.78 0.90 1.07 1.26 1.50 1.79 2.29 3.31

0.03 0.05 0.08 0.10 0.13 0.15 0.18 0.22 0.25 0.30 0.36 0.43 0.55 0.80

~

SUMMARY OF CALCULATING TRANSIENT STABILITY

185

we read sin 00 = 0.825. Entering the Pi curve of Fig. 46 with this value of sin 00, we read Pi = 0.995. 6, 7, and 8. Stability limii« [orfinite clearing times. These three steps are carried out in Table 4. In col. 1 are entered values of sin 00 from 0.15 to 0.80 at intervals of 0:05. In cola. 2 and 3 are the corresponding values of Pi and

1.0

,

1\ \

~

\ 'i\

",

'"r-,

o o

............ ............

~

r--.. ::::-

0.5

1.0

Clearing time t, (seconds)

FIG. 47. Curve of stability limit as a function of clearing time, three-phase short circuit at the middle of one of the 132-kv. lines of the system of Fig. 43 (Example 4).

PM' read from the curves of Fig. 46. In col. 4 is the value of'rcltc, which by eq, 27 is 1r

X 60 X 0.153 1 X 1.56

In col. 5 is the value of T e read from the curves. dividing Te from col. 5 by T c/t cfrom col. 4.

v'p = 4.3 vp m

m

In col. 6 is fe, obtained by

9. Curve oj stability limit as a Junction oj clearing time is plotted in Fig. 47.

Summary of methods of calculating transient stability. Byrd and Pritchard's method, which has just been illustrated, is probably the best way of finding the critical switching time corresponding to a given transmitted power, or of finding the transient stability limit for a given switching time, for a two-machine pure-reactance system with only one

instant of switching after inception of the fault. If the network is not purely reactive (for example, if line resistance

or shunt loads are to be taken into ), this method is not applicable. However, the equal-area method may be used in conjunction with pre-calculated swing curves as described on pp. 149-57 of this chapter. If the system has three or more machines, their swing curves must be


186

TABLE 5 RECOMMENDED METHODS OF CALCULATING TRANSIENT STABILITY OF A

Two-

MACHINE SYSTEM

Recommended Method of Calculating

Number of Instants of Switching or Circuit Change

Examples

1

Sustained fault Fault cleared instantly Switching out a sound line

B A Critical Switching Time Stability Limit CorreCorresponding to a sponding to a Given Given Power Switching Time Use equal-area criterion or Byrd and Pritchard's method (steps 1 to 5).

(Pure-reactance network) Plot part of curve by Byrd and Pritchard's method.

2

3

Fault cleared in finite (Linear impedance network) time by simultaneUse equal-area method Cut and try by assumous opening of all for finding critical ing power and probreakers switching angle, then ceeding as in A. pre-calculated swing curve for finding switching time. Fault cleared by se., quential opening of 2 breakers Fault cleared by si- Dse a combination of swing curves, equal-area multaneous opening criterion, and successive trials. of all breakers, followed by simultaneous reclosing

calculated to a value of time when one can determine whether the system is stable for the assumed power and switching time. The procedure is then repeated for different values of power or of switching time. If there is more than one instant of switching after inception of the fault on a two-machine system (for instance, if a fault is cleared by successive opening of two or more circuit breakers), it is still possible to use the equal-area criterion as an adjunct to swing curves, making continuation of the swing curves beyond the instant of the last switching operation unnecessary. Suppose, for example, that a fault is cleared by successive opening of two breakers at known times and that

CERTAIN FACTORS AFFECTING STABILITY

187

the power limit is sought. A value of power is assumed, and the swing curve is calculated till the time of opening of the second breaker. The equal-area criterion for stability is then used. The procedure is repeated for other values of power until one is found which makes the positive and negative areas equal. This value is the stability limit. A plot of net area against power enables one to find the stability limit by interpolation after two or three trials. Table 5 lists the foregoing methods of calculation for two-machine systems with recommendations concerning which to use in each of various situations. For estimating purposes two sets of curves which are not reproduced here will be found useful. One of them" gives, for a two-machine reactance system, curves of critical clearing time tc against sin 00 for constant rl and r2. Four families of such curves are given, each for a value of rl corresponding to a different type of short circuit at the sending end of a transmission line and for several values .of r2. The equivalent inertia constant, H = H1H2/(H1 + H2 ) , is assumed to be 1.5. If it has a different .value, the switching time read from the curves must be multiplied by a correction factor V H /1.5. The other set of curves" gives the critical clearing time of a fault on a metropolitan power system having generators of H = 8 operating at full load and 0.85 power factor, based on the assumption that the generator or group of generators nearest the fault swings with respect to the remaining generators. Certain factors affecting stability. With a knowledge of the methods of analyzing the stability of the two-machine system that have been described in the preceding pages, we can proceed to draw a number of general conclusions regarding the effect on stability of certain features of apparatus design, system layout, and operation. The effect of each feature must be considered under all three conditions-before the fault, during the fault, and after clearing the fault. Some features of layout or design promote stability during all three conditions, whereas others are beneficial during one condition but detrimental during another. In the equal-area criterion for stability, the power-angle curves for each of the three conditions are used. (Refer to Fig. 14.) The factors determining the relative sizes of the two compared areas Al and A 2 are (1) the clearing angle Oc and (2) the amplitudes of the three powerangle curves, Pm, rIPm, r2Pm» relative to the height of the input (initial power) line Pi. Stability is aided by decreasing area A 1 and by increasing area A 2 • For a given Pi this may be accomplished, from the geometrical viewpoint, chiefly by decreasing the clearing angle Oc

188


and by increasing the amplitudes of the fault and post-fault curves, T1Pm and T2Pm. It is also helpful to decrease the amplitude of the pre-fault power-angle curve if such reduction is possible without at the same time decreasing the amplitudes of the other two power-angle curves, because so doing increases the initial angle 00- If any feature changes the amplitudes of all three curves proportionally, however, it is beneficial to increase rather than to decrease the amplitudes. The clearing angle s, depends on the clearing time and on the equivalent inertia constant (eq. 2). The importance of rapid fault clearing, obtainable by the use of high-speed circuit breakers and fast relaying, has already been stressed. For a given clearing time the clearing angle is decreased by increasing the inertia constant. If the inertia constants of the two machines are far from equal, the equivalent inertia constant of the system is very nearly that of the lighter machine; hence it is more effective to increase the inertia of the light machine than of the heavy one. Seldom, however, has it proved economical to increase the inertia of a generator beyond its normal value. t But a consideration of the role of inertia sheds some light on the relative critical clearing times for hydroelectric generators (average H = 3) and steam turbogenerators (average H = 6) if the circuit reactances are about the same. Since the time varies as the square root of the inertia constant Ceq. 27), a fault on a hydroelectric system must be cleared in about 70% of the critical clearing time of the corresponding steam system. The amplitude of the power-angle curves is E 1E2/X12 , where E 1 and E 2 are the internal voltages of the two synchronous machines and X 12 is the reactance between these voltages, which is, in general, different for each of the three circuit conditions. Increasing the internal voltages increases the amplitude of all three power-angle curves and aids stability. An increase in internal voltages usually accompanies an increase of load on the machines, but this does not necessarily mean an increase of the equivalent input Pi. If both generators have local loads which are increased in the proper ratio, Pi is not affected. (See eq.3.) The amplitude of all the power-angle curves is increased by decreasing the reactance X 12 between the machines. This reactance consists principally of the reactances of the two synchronous machines, transformer reactance, and line reactance. A large part of it is in the machines. The transient reactance of each class of large synchronous machines (steam turbogenerator, water-wheel generator, condenser, etc.) has a characteristic value and does not vary much in normal tA notable exception is the Boulder Dam station.

CERTAIN FACTORS AFFECTING STABILITY

189

designs. A lower value of reactance (in per unit) is obtained essentially by building a larger machine and under-rating it. This has seldom proved an economical way to aid stability.§ The reactances of transformers also have, for a given size and voltage, normal values below which it is difficult to go. The reactance of an overhead transmission line is only slightly affected by any change of spacing or conductor size which is practical from other standpoints than stability. A decrease in system frequency, of course, decreases the reactances of all parts of the system and thereby aids stability. Therefore a frequency of 25 or 50 c.p.s. is preferable to 60 c.p.s. from the standpoint of stability. Nevertheless, because of other advantages, 60 c.p.s. has become the standard power-system frequency in the United States, and a transmission project operating at a lower frequency would necessarily include frequency-changing equipment at one or both ends. This additional equipment might offset any advantage of the lower frequency in regard to stability. Up to the present time the stability limitations of 60-c.p.s. systems have not been serious enough to warrant lower frequencies. Series capacitors have been used on some transmission lines to partially compensate for the inductive reactance of the lines. However, they have not been used on any major project where stability was an important factor, although they have been seriously considered. 6 ,7,g The most important means of reducing the reactance are (1) to connect more lines in parallel and (2) to raise the transmission voltages. The ratio rl of the amplitude of the fault power-angle curve to the amplitude of the pre-fault power-angle curve depends upon the type and location of the fault. The effect of the type of fault will be discussed in Chapter VI. The effect of location will be considered briefly now. A fault on a bus or on a line close to a bus is more severe than a fault of the same type near the middle of a line. Most severe of all is a three-phase short circuit at some point (for example, fault b in Fig. 15) where it entirely blocks the transfer of power from one generator to the other; then rl = o. Although a fault near the end of a line and one near the middle of the line are equally probable, the most severe location is usually assumed in a stability study so that the results will be conservative. The ratio T2 of the amplitude of the post-fault curve to the amplitude of the pre-fault curve depends on the number and location of lines which are opened to clear the fault; therefore it depends upon the fault location and upon the relaying scheme. The most favorable §Again the Boulder Dam station is a notable exception.

190


value of T2 encountered in practice is 1. This value applies to a fault on an unloaded radial feeder cleared by disconnection of the feeder. It is also attained with quick-reclosing schemes which restore the faulted line to service after a brief time to allow for extinction of the arc. II A fault on a bus, although electrically equivalent before clearing to a fault on a line adjacent to a bus, is more severe in its effects after clearing than a line fault is because several lines must be opened to clear it. Sometimes a comparative study is made between a small number of high-voltage lines and a large number of low-voltage lines having the same parallel reactance in per unit. The large number of lines is preferable to the small number from the standpoint of stability, because to clear a fault requires the opening of one line in either case, 16 24

J

24 FIG.

16 16 48. System for which the effectof high-voltagebussingis considered in Prob, 4.

but one line is a small fraction of the entire number of lines when this number is great. For other reasons, however, a smaller number of lines may be preferable. When two or more parallel lines are used, the number of intermediate busses is a factor affecting stability. For example, in the system of Fig. 48 it is worth inquiring whether the addition of the two high-voltage busses shown in broken lines is beneficial or detrimental. If the high-voltage busses and circuit breakers are provided, a line fault can be cleared by switching out one line while leaving all the transformers connected, whereas, without the high-voltage busses, the transformers would be switched out with the line. Thus the busses increase the value of T2 and are beneficial after clearing of the fault. On the other hand, during the fault the shock to the system is increased by the busses; that is, Tt is lessened. Which effect predominates depends upon the speed of clearing. With fast clearing the busses are beneficial; with slow clearing they are detrimental. The same principle may determine, although less obviously, the effect upon stability of changes of layout in more complicated networks. The effect of a given change may be either beneficial or detrimental, depending upon the speed of fault clearing. Needless to say,_ many USee Chapter XI, Vol. II.

PROBLEMS

191

contemplated changes of connections are conceived from other motives than the improvement of stability. They may be intended, for example, to improve voltage regulation, to increase the reliability of supply to certain loads, or to relieve overloading of certain lines. Nevertheless the effect of such changes on stability conditions should be considered. Most of the conclusions which have been drawn from a study of the two-machine system apply equally well to a multimachine system. When a muItimachine system is unstable, as a rule, it is split into two groups of machines which go out of synchronism with each other, while the machines within each group stay in synchronism. The grouping may differ with the fault location. Still, for a given fault location, the general behavior of the multimachine system is similar to that of a two-machine system. It could be analyzed as a two-machine system except that the machines within a group may swing so far with respect to each other (yet without going out of step) as to have a marked effect on the relations between the two groups. In addition, the grouping for a given fault location is not always apparent in advance. REFERENCES

1. R. H. PARK and E. H. BANCKER, "System Stability as a Design Problem," A.l.E.E. Trans., vol. 48, pp. 170-93, January, 1929. 2. I. H. SUMMERS and J. B. MCCLURE, "Progress in the Study of System Stability," A.I.E.E. Trans., vol. 49, pp. 132-58, January, 1930. 3. H. L. BYRD and S. R. PRITCHARD, JR., "Solution of the Two-Machine Stability Problem," Gen. Elec. Rev., vol. 36, pp. 81-93, February, 1933. 4. R. D. EVANS and W. A. LEWIS, "Selecting Breaker Speeds for Stable Operation," Elec. Wid., vol. 95, pp. 336-40, Feb. 15, 1930. 5. S. B. GRISCOM, W. A. LEWIS, and W. R. ELLIS, "Generalized Stability Solution for Metropolitan Type Systems," A.I.E.E. Trans., vol. 51, pp, 363-72, June, 1932; disc., pp. 373-4. 6. E. C. STARR and R. D. EVANS, "Series Capacitors for Transmission Circuits," A. I. E. E. Trans., vol. 61, pp. 963-73, 1942; disc., pp. 1044-6. 7. R. B. BODINE, C. CONCORDIA, and GABRIEL KRON, "Self-Excited Oscillations of Capacitor-Compensated Long-Distance Transmission Lines," A.I.E.E. Trans., vol. 62, pp, 41-4, January, 1943; disc., pp. 371-2. 8. J. W. BUTLER, J. E. PAUL, and T. ·W. SCHROEDER, "Steady-State and Transient-Stability Analysis of Series Capacitors in Long Transmission Lines," A.I.E.E. Trans., vol. 62, pp. 58-65, February, 1943; disc., pp. 377-80. PROBLEMS ON CHAPTER V

1. Check the results of Example 3, part a, by Byrd and Pritchard's method. 2. Work Example 4 for a three-phase fault on a 132-kv. line near the hydroelectric-station high-voltage bus.

192


3. Work Example 4, assuming that the voltages behind transient reactance are independent of the initial power transmitted. 4. Find the effect of high-voltage bussing on the stability of the 6o-cycle system shown in Fig, 48 (consisting of a generator G1 feeding over a doublecircuit high-voltage line to the infinite bus G2) by plotting stability limit in per unit against fault duration in seconds for a three-phase fault at the sending end of one circuit (a) with no high-voltage bussing and (b) with highvoltage bussing at both ends, as indicated by the broken lines. The reactances of the circuit elements are given in per cent based on the rating of G1. Assume that the voltage of the infinite bus and the sending-end voltage of the high-voltage lines are 1.00 per unit for all values of initial power. Explain why the curves cross. 5. Find the effect of fault location on the system of Fig. 48 with highvoltage bussing, by plotting stability limit in per unit as a function of fault location as the three-phase fault moves from the sending end to the receiving end of the line. The clearing time is constant at 0.15 sec.

CHAPTER VI SOLUTION OF FAULTED THREE-PHASE NETWORKS

Symmetrical components. Most power systems are three-phase, but, as long ,as they are symmetrical, they may be represented on a single-phase line-to-neutral basis for purpose of calculations. Threephase systems are usually symmetrical or very nearly so during normal conditions and also during three-phase faults. During other types of fault, such as line-to-ground or line-to-line, they are unsymmetrical, and the simple single-phase representation no longer suffices. In such cases the method of symmetrical components! is generally used. In the method of symmetrical components an unsymmetrical set of vector currents or voltages is resolved into symmetrical sets of components. Before proving that such resolution is possible, let us first consider the inverse process, that of adding symmetrical sets of vectors to obtain an unsymmetrical set. Consider a symmetrical set of vectors (Vat, Vbl, Vel of Fig. la) representing the balanced voltages or currents of a three-phase circuit. The vectors are equal in magnitude and 1200 apart in phase. Their phase order, or phase sequence, is abc, which is called the positive sequence. Consider also another symmetrical set of vectors (Va2 , Vb2, Ve2 of Fig. Ib) of the opposite phase sequence, cba. These vectors are said to have negative phase sequence. If now corresponding vectors of the two sets are added, thus Va' =

Vb' =

v:

=

+ V a2 »« + Vb2 Vel + V e2

[1]

Val

[2] [3]

v:

as shown in Fig. Ic, the resultant vectors, Va', Ve', form an unsymmetrical set. They are neither equal in magnitude nor 1200 apart in phase. It has been shown that an unsymmetrical set of vectors results from the addition of corresponding vectors of two symmetrical sets of opposite phase sequence, By reversal of the process the resulting unsymmetrical set can be resolved into its symmetrical components. We may well ask, can any unsymmetrical set of three vectors be resolved into two symmetrical sets, one of positive phase sequence, the other of negative phase sequence? This question is answered if we 193

SOLUTION OF FAULTED NETWORKS

194

notice that, since the sum of the three vectors in each symmetrical set is zero, the sum of the three vectors in the resulting unsymmetrical set is also zero. Val Vbl Vel = 0 [4]

+ + V a2 + Vb2 + Ve2 = 0

[5]

Adding, we get (Val + Va2 )

+

(Vbl

+ V b2 ) + (Vel + V c2 ) = 0

[6]

or

[7]

(a)

\\\

~o VbO VeO (d)

1. Composition of unsymmetrical vectors from symmetrical components. (b) Negative-sequence vectors. (c) Addition of negative-sequence vectors to positive-sequence vectors. (d) Zero-sequence vectors. (e) Addition of the sum of positive-sequence and negative-sequence vectors to zerosequence vectors. FIG.

(a) Positive-sequence vectors.

In general, the sum of the three vectors in an unsymmetrical set is not zero but, instead, may have any value. Obviously, an unsymmetrical set whose sum is not zero cannot result from the addition of two sets, each having a sum of zero. Suppose, however, that we add to the positive-sequence and negativesequence sets a set of equal vectors, Van, VbO, VeO (shown in Fig. Id), known as a zero-sequence set. The resulting set of vectors (shown in

Fig. Ie) is:

v, = v: + Val = VaO + Val + Va2

[Sa]

+ V bl + Vb2

[8bl [Sc]

+ Vb' =

Vb =

VbO

v, =

VeO + V e'

V bO

= V eO + Vel

+ Ve2

SYMMETRICAL COMPONENTS

195

These vectors are unsymmetrical, and their sum is not zero. Can any unsymmetrical set of three vectors be resolved into three symmetrical sets-a positive-sequence set, a negative-sequence set, and a zerosequence set? It can if eqs. 8 can be solved for the symmetrical components in of the given vectors Va, Vb, Ve • In eqs. 8 there appear to be nine unknown components, but actually only three of them are independent because the three vectors of each set are determined by anyone of the three, say by the phase a vector, and by the symmetry of the set. [9a] Vbl = Va1!240° = a2Va l

= Val/120 = aVal Vb2 = Va2/120° = aVa2

[9b]

Ve2 = Va2/240° = a2Va 2

[9d]

VbO = VeO= VaO

[ge]

0

Vel

where

[ge]

a = /1200 = -0.500 + jO.866

[10]

a 2 = /240 0 = -0.500 - jO.866

[11]

a3

[12]

and hence also

=1 a4 = a

[13]

Substituting eqs. 9 into eqs. 8, one obtains

v. = VaO+ Val + Va2 Vb = VaO + a2Va1 + aVa2 v, = VaO+ aVal + a a2 2V

[14a] [14b] [14c]

Subscript a may now be dropped, giving

Va = Vo + Vt

[1Sa] + V2 [I5b] v, = v, + a t + aV2 [15e] v, = Vo + aVI + a2V2 It is desired to solve eqs. 15 for Vo, v; and V2 in of Va, Vb, and 2V

Ve• The simplest method of solution follows. To obtain Vo, add the three eqs. 15 together, obtaining

Va + Vb + Vc = 3Vo + (1 + a

+ a2 ) (Vl + V2 )


196

Note that

1+a

+a = 0 2

[16]

Hence

[17] To obtain VI, multiply the three eqs. 15 by 1, 8, and a2, respectively, obtaining = Vo VI V2

v;

aVb = 2

a Ye =

+ + aVo + VI + a Y a 2y o + VI + aV2 2

2

Then add, obtaining

Va

+ aVb + a2y c =

(1

+ a + a2 ) (y o + V2 ) + 3V1

whence

VI = i(Va

+ aVb + a2Ve )

[18]

To obtain V2 , multiply the three eqs. 15 by 1, a2, and a, respectively, add, and divide by 3, obtaining

V2

= i(Va + a2Vb + aYe)

The symmetrical components of any three given vectors Va, Vb, therefore Vo = i(Va + Vb + Ve )

[19]

v, are, [20a]

VI = i(Va

+ aVb + a Ve )

[20b]

V2 = l(Ya

+ a Vb + aYe)

[20c]

2

2

Any three given vectors can be resolved into symmetrical components of positive, negative, and zero phase sequence by using eqs. 20. The operations indicated by eqs. 20 may be performed either graphically or arithmetically. If the given vectors comprise a symmetrical set of a particular phase sequence, the other two components will vanish. If the sum of the given vectors is zero, they have no zero-sequence component. The vectors may represent any alternating quantities associated with a three-phase circuit or machine, such as line-to-neutral, line-to-ground, or line-to-line voltage; current; or flux. In this chapter we are concerned with voltage and current vectors. The actual or phase voltages are given in of their symmetrical components by eqs. 15, and the symmetrical components are given in of the actual voltages by eqs. 20. The corresponding equations for current are: [21a]


197

Ib = 10

[21b]

I, =

+ a211 + aI2 10 + all + a 212

[21c]

10 = J(la+lb+le}

[22a]

11 =

12 =

i n, + alb + a2I c ) l(la + a2Ib + ale)

[22b] [22c]

An alternative form of eqs. 15 and 20, obtained by using the rectangular form of operators a and a 2 (see eqs. 10 and 11), is as follows:

Va =

v; + (VI + V2 )

[23a]

Vb = Vo - 0.500(V1

+ V2 )

Ve = Vo - O.500(V 1

+ V2 ) + jO.866 (VI

-

jO.866(V I

V2 )

[23b]

- V2 )

[23c]

-

+ (Vb + Vc ) ] i[Va - O.500(V + Vc ) + jO.866 (Vb irva - 0.500(Vb + Vc ) - jO.866(Vb -

Vo = l[V a VI =

V2 =

b

[24a]

Ve ) ]

[24b]

Vc)]

[24cJ

This form is convenient if it is desired to compute with complex numbers all in rectangular form. Needless to say, similar equations hold for currents. EXAMPLE

1

Calculate the symmetrical components of the following unbalanced lineto-neutral voltages: Va = 100/90° volts

Vb

= 116LQ volts

Vc

=

71 /225° volts

Solution 1. The symmetrical components are given by eqs, 20.

v, = 100 /90° v, =

11612

+ jl00 116 + jO 0

Vc = 71/225° = -50 - j50 3Vo = Va + Vb + Vc = 66 + j50 =,83/37° Vo =

22

+

j17

= 28/37°

(Ans.)


198

s, =

100/90° =

0

+ j100

aVb = 116/120° = -58 + j100

a'lYe = 71/105° = -18 + j68 3VI

= VI + aVb + a'rYe = -76 + j268 = 279/106° VI = -25+ j89 Va = 100/90° =

0

=

93/106°

(Ans.)

+ jl00

a"Vb = 116/240° = -58 - jl00 aVe = 71/345° = +68 -

j18

+ aVe = +10 -

j18

3V2 = Vo + a"Vb

V2 = + 3 -

= 21/299°

j6 =

7/299°

(Ans.)

The symmetrical components are, therefore: Vo ==

22 + j17 = 28/37° volts

VI = -25 + j89 = 93/106° volts V2 = +3 -

j6

=

7/299° volts

Check 1. From the symmetrical components which have just been found, the phase voltages will be calculated by eqs. 15 as a check.

Vo = 28/37° =

a"VI = 93/346° = aV2 = 7/59° = Vb = Vo + a?:VI aV2 =

+

Vo = 28/37° = aVI = 93/2260

22

+ j17

90 - j23

4 + j6 116 jO (Check)

+.

22 + j17

= -65 -

j67

aZV2 = 7/179° = - 7 + jO Ve = Vo + aVl a?:V2 = -50 - j50 (Check)

+


Solution 2. Equations 24 will be used.

Vb= 116+

Vc Vb

O.500(Vb

= -50 -

+ Ve =

+V

c)

=

Va = 3Vo =

Va - O.500(Vb

+V

e)

j50

66 -

j50

33 -

j25

0+ jl00 66 + j50

= -33

Vb - V e =

+ j125

166+ j50

+ j43 = -43 + j143

O.866(Vb - Ve) = jO.866(Vb - Vc )

jO

143

3V1 = -76 + j268 3V2 =

10 -

jlS

The symmetrical components are

Vo =

22 + j17 volts

VI = -25 + j89 volts

V2 =

3-

j6 volts

agreeing with Solution 1. Check 2. Equations 23 will be used.

VI = -25+ j89 V2 =

+V O.500(VI + V VI

2

2)

= -11

Va =

+V

2)

j6

= -22+ j83

Vo =

Vo - O.500(VI

3-

=

+

j42

22+ j17 0

+ jl00

33 -

(CMc1c)

j25

Vi - V2 = -28 + j95

O.866(V1 - V2 ) = -24

+

j82

= -82 Vb = 115 -

j24 jl

(Should be 116 - jO)

Vc = -49 -

j49

(Should be -50 - j50)

jO.866(V1 - V2)

199

200,


Sequence impedances. The most important property of symmetrical components of current and voltage is that, in a balanced threephase circuit or part of it, the three sequences are independent. A balanced three-phase circuit is one having similar connections in all three phases, equal self-impedances in all three phases, and equal mutual impedances between each pair of phases. The statement that the three sequences are independent means that currents of each phase sequence will produce voltage drops of the same phase sequence only. For example, if positive-sequence currents flow through a balanced circuit, the resulting voltage drops are exclusively of positive sequence, and their zero-sequence and negativesequence components are zero. The independence of the sequences in any FIG. 2. A balanced three- particular form of balanced circuit may be phase circuit element with shown algebraically. This will be done for self- and mutual impedances. the circuit shown in Fig. 2, consisting of equal series self-impedances Zs in the three phases and equal mutual impedances Zm between each pair of phases. The relation between phase currents la, I b, I, and phase voltage drops Va, Vb, Ve is given by the following equations:

z.r, + Zmlb + Zmle Vb = Zmla + Zslb + ZmIe Ve = Zmla + ZmIb + z.r,

·Va =

[25a] [25b] [25c]

It is now necessary to find relations between the symmetrical components of voltage and current similar in form to eqs. 25. To obtain these relations, substitute eqs. 25 into eqs. 20 and then use eqs. 22.

v; = lcva + v, + Ve ) == i CIa + r, + Ie) CZs + 2Zm ) = CZs

+ 2Zm)Io

VI = leVa + aVb + a2Ve ) = Zs!Cla + alb + a2l e) + Zm!CIb'+ I, 2 = ZsI1 + zmiC -Ia - alb - a Ie)

= (Z8 - Zm)I1 Similarly',

+ ala + ale + a21a + a2Ib)

SEQUENCE IMPEDANCES

201

These equations may be written as Vo = Zolo

[26a]

z.r,

[26b]

V2 = Z2I2

[26c]

Zo = Z, + 2Zm

[27a]

VI =

where

Zl == Z2

= Z, -

Zm

[27b]

Unlike eqs. 25, in which the value of each phase voltage depends upon all three phase currents, in eqs. 26 the voltage of each sequence depends only upon the current 'of the same sequence. The ratio of sequence voltage to sequence current may be called a sequence impedance; thus Zo is the zero-sequence impedances Zl, the posiiioe-sequence impedance; and Z2, the negative-sequence impedance. The positive- and negativesequence impedances of this particular circuit are equal; indeed, for any static three-phase circuit they are equal. For rotating machinery, however, the positive-sequence impedance usually differs from the negative-sequence impedance because of the mutual reactances between rotor and stator windings. The impedances of synchronous machines are considered further in Chapters XII and XIV, Vol. III. Equations 27 show that, if Z8 and Zm are of the same sign (or, more precisely, if their vectors lie approximately in the same direction, as they do for a transmission line or cable, where each of the three phases is considered to consist of a wire with ground return), then the zerosequence impedance is higher than the positive- or negative-sequence impedance. If, on the other hand, Zm is of opposite sign from Zs, as it is for the stator windings of a three-phase machine, then the zerosequence impedance is lower than the positive- or negative-sequence impedance. Let us show that Zm of the machine is a negative reactance. It is a familiar fact that, if two coils are placed initially with their axes parallel and if one coil is then turned with respect to the other, their mutual inductance diminishes, reaching zero at 90°; and, upon further turning of the coil, the mutual inductance becomes negative and is still so at 120°, which is the angular separation of the coils of a three-phase machine. The sequence impedances may be calculated from the phase selfand mutual impedances, as illustrated by eqs. 27 for a particular case; or they may be measured. For example, the positive-sequence impedance of an induction motor running at any given speed may be determined by impressing known positive-sequence voltages on its

202


terminals, measuring the resulting positive-sequence currents, and taking the ratio of voltage to current. The negative-sequence impedance may be measured in similar fashion by impressing negativesequence voltage on the terminals and driving the rotor at the same speed as before. For measuring the zerosequence impedance a single-phase voltage may be impressed between the neutral point and the three line terminals tied together. Kirchhoff's laws. In the preceding section it was shown that the symmetrical components of current and voltage obey Ohm's law, V = ZI. It will now be shown that they obey Kirchhoff's laws as well. FIG. 3. A three-phase Kirchhoff's law of currents states that the junction. sum of all the currents flowing into a junction (node) is zero. At a three-phase junction, as illustrated in Fig. 3, this law applies separately to each phase, thus:

+ i: + la" = 0

[28a]

+ Ib' + I b" = 0 Ie + Ie' + Ie" = 0

[28b]

la Ib

[28c]

If we add the three equations together and divide by 3, we obtain

!(la + I b + Ie) + !(Ia'

+ I b' + Ie') + i(Ia" + Ib" + Ie") = 0

or

10

+ 10' + 10"

=

0

[29]

Also, by multiplying eqs. 28a, 28b, and 28c by 1, a, and a 2, respectively, adding the resulting equations together, and dividing by 3, we obtain 11

+ 11' + II" = 0

[30]

By a similar procedure, with a and a2 interchanged, we get [31] Equations 29, 30, and 31 show that the sequence currents (symmetrical components of current) obey Kirchhoff's current law just as the phase currents do. Kirchhoff's voltage law states that the sum of all the voltage drops (or rises) around a closed path (loop or mesh) equals zero. In a three-

THE SEQUENCE NETWORKS

phase circuit the law holds for each phase. Fig. 4 we have

Va + Va' + Va"

203

Thus for the circuit of

=0

[32a)

+ Vb' + Vb" = 0 v, + V + v: == 0

[32b]

Vb

e'

FIG.

[32c]

4. A three-phase mesh.

These equations are of the same form as eqs. 28; therefore equations can be derived from them similar to eqs. 29, 30, and 31. They are

v; + Vo' + Vo" VI + VI' V2

+ V2 '

= 0

[33a]

+ VI" = 0 + V2" = 0

[33b) [33c]

Thus the sequence voltages obey Kirchhoff's voltage law. The sequence networks. It has been shown that in a balanced three-phase network the symmetrical components of voltage and current obey Ohm's law and Kirchhoff's laws, the same laws followed by the actual voltages and currents. There is thus good justification for the concept of sequence networks. The actual three-phase network, in which the actual voltages and currents exist, is replaced for purposes of analysis by three fictitious single-phase networks, in which the symmetrical components or sequence voltages and currents exist. They are the positive-sequence network, in which the positive-sequence voltage and current exist; the negative-sequence network, in which the negative-sequence voltage and current exist; and the zero-sequence network, in which the zero-sequence voltage and current exist. The

204


positive-sequence network is identical to the single-phase impedance diagram considered in Chapter III. The negative-sequence network is very much like the positive-sequence network but differs from it in the following respects: (1) ordinarily there are no negative-sequence generated electromotive forces; (2) the negative-sequence impedance of rotating machinery is different from the positive-sequence impedance; and (3) the phase displacement of transformer banks for negative sequence is of opposite sign to that for positive sequence. The zero-sequence network differs greatly from the other two in that (1) the impedance of transmission lines is higher than for positive sequence; and (2) the equivalent circuits of transformers are different. Consequently, the zero-sequence network is given further consideration later in this chapter. In so far as the three-phase network is balanced, the three sequence networks are independent; that is, they are not connected or coupled to one another. But wherever the three-phase network is unbalanced, there is a connection or coupling between t\VO or three of the sequence networks at the corresponding points of unbalance. The nature of the connection or coupling between the sequence networks will be determined in the next section for the types of unbalances most important in stability studies, that is, for short circuits. Consider the component parts of a three-phase power network to see whether they are balanced. Rotating three-phase machines, transformer banks consisting of three identical units, cables, and transposed overhead lines are balanced. Untransposed overhead lines are only slightly unbalanced. Loads, although often unbalanced when considered in small portions, are approximately balanced when the whole load of a substation is considered. Therefore, all the major parts of a three-phase power network are either exactly or very nearly balanced during normal conditions. During fault conditions the same statement holds for all parts of the network except for the fault itself if the fault is of some type other than three-phase. Therefore, the sequence networks representing the power system are entirely independent of one another under normal conditions and are connected only at the point of fault under unbalanced fault conditions. The usual three-phase power system has no generated electromotive forces except symmetrical positive-sequence ones. Therefore the sequence networks representing such a system have no generated electromotive forces except those in the positive-sequence network. The negative-sequence and zero-sequence networks, having no electromotive forces within themselves and not being connected to the positive...sequence network under balanced conditions, are "dead." There-

REPRESENTATION OF SHORT CIRCUITS

205

fore only the positive-sequence network need be considered under such conditions. Under unbalanced fault conditions, however, the negativesequence network and, for some types of faults, also the zero-sequence network are energized by their connection to the positive-sequence network at the point of fault. The nature of these connections will no,v be considered. Representation of short circuits by connections between the sequence networks. Short circuits on a three-phase system may be classified into the following types: one-line-to-ground, line-to-line, One- Llne-to- Ground

;~

Phase c

Phaseb

g,,:,

Phases band c

Line-to . Line Phases c and a

!=c :=E

Phases a and h

=t=

Iwo- Line-to· Ground Phases band c

Phases c and a

Phases a and b

:

~

;=:F =F =F g~

Three· Phase Involving ground Not involving ground

.~~

:=E

g-:r

FIG.

5. Types of short circuit on a three-phase network.

two-line-to-ground, and three-phase. Three-phase short circuits may be further classified into those involving ground and those not involving ground. The other types may be further classified by the phases involved. The different types of short circuit are shown diagrammatically in Fig. 5. The procedure for finding the connections of the sequence networks corresponding to a fault on a three-phase network consists of four steps: 1. Draw a circuit diagram of the fault on the three-phase network, labelling the pertinent phase voltages and currents by suitable letter symbols and by arrows denoting their positive directions. 2. Write three equations expressing relations between the phase

206


currents and voltages or particular values thereof imposed by the fault condition. 3. Convert these equations to three corresponding equations in sequence currents and voltages. This conversion is accomplished by using eqs. 15, 20, 21, and 22. 4. Find connections of the sequence networks satisfying the equations obtained in the last step. The procedure just outlined will now be carried out for each type of short circuit, commencing with the line-to-ground short circuit on phase a. Line-to-ground short circuits on phase b or phase c may be a c 1 I ., Va treated by renaming the phases so b .... Three· ., b~ phase as always to call the faulted phase, Vb Ie . . . C~ network " a. The relations of the symphase tVe g~ metrical components come out most simply when the fault is taken on (a) phase a, which was adopted as the reference phase. (That is, the components of the other phase vectors were expressed in of the Zero·sequence network components of the phase-a vector.) For the same reason line-to-line and two-line-to-ground faults will be taken on phases band c, thus Posmvesequence making the faults symmetrical with network respect to phase a. The three-phase network, whatever its form, may be symbolized Negativeby a box having four terminals at sequence the point of fault, namely, phases a, network b, and c and the ground, as shown (b) in Fig. 6a. Different types of short FIG. 6. (a) Schematic representacircuit may be applied to the nettion of a three-phase network with work by making the appropriate terminals at the point of fault. (b) connections between two, three, or Schematic representation of the corfour of these terminals. Each of responding sequence networks. the three sequence networks may be represented similarly by a box having two terminals at the point of fault (line terminal and neutral terminal) as shown in Fig. 6b. The connections between these terminals, corresponding to the connections between terminals of the three-phase network, will be derived. Let Va, Vb, and Vc denote the line-to-ground voltages at the point of 4 ....


207

fault, and let la, Ib, and I, denote the phase currents flowing into the fault, as marked on Fig. 6a. Let Vo, Vt, V2 be the symmetrical components of Va' Vb, Vc; and let 10 , It, 12 be the symmetrical components of la, Ib' r, Line-to-ground short circuit on phase a. (See Fig. 7a.) The line-toground voltage of the grounded phase is zero, and the currents flowing into the fault from the two unfaulted phases are zero. Hence the equations for phase voltages and currents are: [34]

[35] [36] From eqs. 34 and 15a

v, + VI + V2

= 0

[37]

= i1a

[38]

From eqs. 35, 36, and 22

10

= II = 12

Hence the relations between symmetrical components of fault voltage and current, corresponding to eqs. 34, 35, and 36 for actual fault voltages and currents, are

Vo + VI + V2 = 0 10

[39a]

= II = 12

[39b]

A series connection of the three sequence networks (Fig. 7b) satisfies these equations. Line-to-line short circuit on phases band c. (See Fig. 7c.) The equations for phase voltages and currents are

Vb

=

Va

[40]

I, = 0

[41]

Ib + t, = 0

[42]

Substituting eqs. 15 into eq. 40,

v; + a2V t

+ aV2 = Vo + aVI + a2V2

Subtracting V0 from each side and rearranging, we get

V1 (a2

-

a)

= V2 (a2

-

a)

whence

[43]

..... .,

12

VI

12

t+~

Neg.Jt~ -

POSe

11

o

Hv

FIG.

7.

Vo

Y2

Three· phase involvingground

(j)

_ Neg.

VI

t ~

_ Pas.

t ~

Zero

t Vea V"

t I 1f.1

(i)

g

C

3q, b

a

Connections between the sequence networks corresponding to various types of short circuit on a three-phase network.

(h)

Neg. ~t- V2

-t-

Zero

~

10

Three- phase not involvingground

V2

Pas.

,....

11

'r-Vo

(g)

t Ie ~ Vb Vo Va

(I)

~

t

Vl

t

..... 1 2

7'

.... 11

-t. . r

~o

c

g

Two •line•to- ground

o Neg.

POSe

-

.....

(e)

v, Vb

C::tmJ

g

Zero

3q,

(d)

V2

t

.... 12 '"

" tVI

o Zero

::Mtvo

(c)

-

c g

Line-to -line

o

tvo

.... 11

~

Va

~lar,

3q, 6

(b)

Neg.

POSe

o

Zero

.....

.... 10

b

ao-+--f>b

One-line -to ·-ground

-

o

-

(a)

-

gC -

I: +~

4~t tv:

3q, b

3q, b

a

v,

I CI Ib Ie

~

~

0

:d

J-3

t;rj

Z

tj

t;rj

t3

~

~

51

Z

0

§

m

~

co


209

Substituting eqs. 41 and 42 into the expression for 10 in eq.22a, we obtain 10 = 0 [44] Substituting eq. 44 into the expressions for Ib and Ie in eqs, 21 and substituting the results into eq. 42, we obtain (a2

+ a) (11 + 1

= 0

2)

or

[45] Hence the relations between symmetrical components of voltage and current for the line-to-line short circuit are: [46a]

VI = V2 11

+ 12 = 10

0

[46b]

=0

[46c]

These relations are realized by a parallel connection oj the positive- and negative-sequence networks, leaving the zero-sequence network opencircuited. (See Fig. 7d.) Two-line.. to-ground short circuit on phases band c. (See Fig. 7e.) The equations for phase voltages and currents are: [47a] [47b] [47c] These are like eqs. 34 to 36 for a one-line-to-ground short circuit, except that voltage and current are interchanged. Therefore the equations in symmetrical components will be like eqs. 39 except for an interchange of voltage with current. They are:

10 + 11

+ 12 = 0

Vo = VI = V2

[48a] [48b]

These equations are realized by a parallel connection oj the three sequence networks (Fig. 7f). Three-phase short circuit not involving ground. (See Fig. 7g.) The equations of phase voltages and currents are:

Va Ia

= v, = v,

+ Ib + I,

=

0

[49] [50]

210


Substituting eqs. 49 into eqs. 20, we get

VI = V2

= iVa(l + a + a2 ) = 0

Substituting eq. 50 into eq. 22a for 10 :

10

=0

Hence the equations of symmetrical components for this type of fault are: [5Ia] [51b]

[5Ic] They are satisfied by short-circuiting the positive- and negativesequence networks and leaving the zero-sequence network opencircuited (Fig. 7h). Three-phase short circuit involving ground. (See Fig. 7i.) The equations of phase quantities are:

v, == 0

[52a]

Vb == 0

[52b]

v, =

[52c]

0

The corresponding equations for symmetrical components are: Vo = 0

[53a]

VI = 0

[53b]

V2 == 0

[53c]

The equivalent circuit consists of a short circuit on each of the three sequence networks (Fig. 7;). Ordinarily no distinction need be made between the two types of three-phase short circuit, because the zero-sequence network is "dead" whether or not its terminals are short-circuited. The negativesequence network is also dead during a three-phase short circuit, and the zero-sequence network is dead during a line-to-line short circuit. These dead networks are shown in Fig. 7 for the sake of completeness, but they are disregarded in fault calculations except in the rare instances where (1) the three-phase network contains unsymmetrical generated electromotive forces, or (2) there is a second fault or other unbalance in addition to the one being considered. Solution of faulted three-phase networks by the method of symmetrical components. Any three-phase network which is symmetrical


211

throughout except for one unsymmetrical short circuit can be represented by its three sequence networks connected at the point of fault as shown in Fig. 7. When the appropriate connections have been made, the current or voltage anywhere in anyone of the sequence networks is equal to the corresponding component of current or voltage at the corresponding point of the three-phase network. If the values of the phase currents and voltages are desired, they can be computed by combining the sequence currents and voltages in accordance with eqs. 15 and 21 or eqs. 23. In stability studies, however, the symmetrical components usually suffice. This method of solving a faulted three-phase network is well suited to the calculating board. After the three sequence networks have been set up on the board, connections can be made readily for representing any type of fault at any location. The sequence currents and voltages can then be read by plugging in the measuring instruments. If the network is not too complicated, it is feasible to solve it by algebraic reduction of the sequence networks. ExAMPLE 2

A line-to-ground short circuit occurs on bus D of the power system of Fig. 8a. Find the symmetrical components of current and voltage and the phase currents and voltages throughout the network if the internal voltages of generators A and B are equal and in phase, both having the value jl.05 per unit. The sequence reactances in per unit are as marked on the sequence networks of Fig. 8b, and resistances are assumed to be negligible. Generator A is ungrounded; B has a solidly grounded neutral. Solution. The short circuit is assumed to be on phase a. It is represented by connecting the three sequence networks in series as shown by the broken lines in Fig. Sb. Note that zero-sequence current cannot flow in ungrounded generator A; hence the zero-sequence network is open at this point (between Co and 0 0, Fig. 8b). The positive-sequence network is reduced as shown in Fig. 9 to one e.m.f, in series with one reactance. During the process of reduction the identity of the e.m.f.'s of generators A and B is retained until step c, so that the results can be used later inExample 3. Then, as the e.m.f.'s are assumed to be equal in the present example, points Al and BI of Fig. 9c are ed to give Fig. 9d. In Figs. 10 and 11 the negative-sequence and zero-sequence networks, respectively, are reduced each to a single reactance. Then in Fig. 12 the three reduced networks are connected in series, as the original networks were in Fig. 8, to represent the line-to-ground short circuit; and the combined circuit is further reduced to an e.m.f, and a reactance. In Fig. 13 the current through the reactance is found by dividing the e.m.I, by the reactance, and the voltages across the sequence networks are found by multiplying the current by the respective reactances. The terminal voltages and cur-

212


(a)

Positive- Sequence Network

01

Negative· Sequence Network

O2 0.10

zero-Sequence Network

00

0-- - - - - - - ---' (b)

8. (a) One-linediagram of a powersystem used to illustrate fault calculations. (b) The sequence networks of the system with values of reactance in per unit

FIG.

marked thereon. The sequence networks are connected by broken lines so as to represent a line-to-ground short circuit at point D. (Example 2.)


Al

0.08

0.25

rv

D1

0.03

0.20

_0.13_

213

81

rv (a)

°1

(6)

(c)

(d)

0.293

(e)

0.216

01

01

FIG. 9. Reduction of positive-sequence network (Example 2).

214


0.08

D2 0.03

(a)

(6)

D2

D2

0.01 (d)

(c)

(e)

0.09

0.193

0.136

02 FIG.

0.08

02

10. Reduction of negative-sequence network (Example 2).


215

Do 0.054

0.14

(a)

0.054 (b)

Do (d)

00

0.066

00 FIG.

0-- - - - - ---..

::::::::J 0.046

0.02

(c)

11. Reduction of zero-sequence network (Example 2).

·105'

~1

D2 0.09 O2 Do0.066 00

r-VYV~--~--~

L

::-=-

..J

(0)

jl.05

ctt::J -+--

FIG.

(b)

12. (a) Connection of the reduced sequence networks to represent a lineto-ground fault. (b) Final reduction. (Example 2.)

..---N LV jO.57

3.62

+il.OS:J


216

.....

in

0

.~

~

il

in

sfrv .....

(0)

"ci

°1

3.62

~-

1.54

2.08

.~

(b)

(c)

FIG. 14. Determination of the positive-sequence currents and voltages by reexpansion of the positive-sequence network. See Fig. 9 for values of reactance. (Example 2.)


217

rents of the sequence networks thus found are the symmetrical components of the voltages at the point of fault and of the current in the fault. Each sequence network is now re-expanded by going through the steps of the network reduction backwards, finding all currents and voltages at each

~ ~

°2

(b)

I

·

(c)

°2

3.62 jO.03) I

I t jO.29

150

2.12

D2

I ( jO.06

jO.23 t

3.62

~

1.50

3.62

Jo.041 1.50

D2

~

3.62

-

3.62

(a)

D2

2.12

I t

2.12

jO.26 (d)

1.27

D2

2.35

- jO.OJ ~""'-
jO.33

°2

~ /0.26 w)

15. Determination of the negative-sequence currents and voltages by reexpansion of the negative-sequence network. See Fig. 10 for values of reactance. (Example 2.) FIG.

step. This re-expansion is illustrated in Figs. 14, 15, and 16. Then in Fig. 17 the sequence networks are redrawn with the values of the sequence currents and voltages marked on them. The phase currents and voltages are computed from their symmetrical components in Table 1 by use of eqs, 23 and are shown on the circuit diagram of Fig. 18. It should be noted that, since the generator e.m.f.'a were assumed to be imaginary (that is, 90° ahead of the reference phase), and since the impedances of the network were assumed to be purely reactive, all the sequence currents are real, and all the sequence voltages are imaginary. A similar statement holds for phase a currents and voltages, but not for phases band c.

218

SOLUTION OF FAULTED NETWORKS 3.62 3.62 Do 0----<]0-_--<10--.

!

jO.17

jO.24

~

I '0.07

(b)

i

J

°00---------' 3.08 (a)

(e)

0.54

i

Do

jO.08 jO.24

3.08

jO.17 (d)

jO.l6

jO.09

°0

FIG. 16. Determination of the zero-sequence currents and voltages by re-expansion of the zero-sequence network. See Fig. 11 for values of reactance. (Example 2.)

FIG. 17. Sequence currents and voltages in the power system of Fig. 8 (Example 2).

1.50

r,

!(11 + 12) 10 - !(Il + 12) 0.866(11 - 12)

0.04 3.04 0 3.04 1.52 -1.52 0.03

12

11 - 12 11 + 12 10

11

Gen.

A 1.54

r--

B 2.08 2.12 -0.04 4.20 3.62 7.82 2.10 1.52 -0.03

Gen. Line CD 1.29 1.27 0.02 2.56 0.54 3.10 1.28 -0.74 0.02

,.

Line CE 0.25 0.23 0.02 0.48 -0.54 -0.06 0.24 -0.78 0.02

Currents

0.52 -0.01

1.16

Line EDt 1.16 1.17 -0.01 2.33 1.68 4.01

,

(EXAIIPLE

Line EDS 1.16 1.17 -0.01 2.33 1.40 3.73 1.16 0.24 -0.01

CALCULATION OF PHASE CURRENTS AND VOLTAGES

TABLE 1

,

jO.7S

-jO.38

Bus C jO.67 -jO.23 jO.90 jO.44 -jO.16 jO.28 jO.22

2) ,.

Bus D jO.57 -jO.33 jO.90 jO.24 -jO.24 0 jO.12 -jO.36 jO.7S

Voltages

jO.38 -jO.07 jO.31 jO.19 -jO.26 jO.78

jO.90

Bus E jO.M -jO.26

"

co

.....

~

~

0

=a

t-3

Z t.'=j

tj

t.'=j

~

> d

~

~

0

Z

0

t-3 .....

tD

0 t"'4 d


220

BusD

BusE

4.01 + jO 0.52 +jO.ot O.52-jO.Ol \0\0

('t)('t)

0 00 .....................

+

I

I

OcoCO ,,-,,-

BusC

3.10+ jO -0.74 - jO.02 - 0.74 + jO.02

00 I

10.86+jO

-1.52 +jO.03

00

~ ~

\0\0

_NN

~~~ ....... I I

a ~~

0"":"":

0"-"-

....... I

+coco I

I

+ coco

00

-O.06+jO - 0.78- j 0.02 - 0.78 + jO.02

dd I

FIG. 18. Phase currents and voltages in the network of Fig. 8 with a line-toground short circuit on phase a of bus D. Values of current in arrow direction are marked on the horizontal lines; values of line-to-ground voltage, on the vertical lines (busses). (Example 2.)

Fault shunts. Since the internal electromotive forces of a threephase synchronous machine are of positive sequence, and since no power results from the combination of positive-sequence voltages with negative-sequence or zero-sequence currents, the generated power of a synchronous machine and the synchronizing power between the various synchronous machines of a power system are positive-sequence power. Therefore the positive-sequence network is of primary interest in a stability study, and the zero- and negative-sequence networks are only of secondary interest. In the positive-sequence network a short circuit can be represented by connecting a shunt impedance Zp at the point of fault. The value of ZF depends upon the type of fault and upon the impedances Zo and Z2 of the negative- and zero-sequence networks as viewed from the point of fault. The formula for the impedance of the fault shunt may be determined in either of two ways. The first way is by inspection of the connections between the sequence networks for representing the various types of short circuits (Fig. 7). The line-to-ground short circuit is represented by connecting in shunt with the positive-sequence network

FAULT SHUNTS

221

at the point of fault the series combination of the zero-sequence and negative-sequence networks; the line-to-line short circuit is represented by thus connecting the negative-sequence network only; and the two-line-to-ground short circuit is represented by the parallel combination of the zero- and negative-sequence networks. Hence the impedances of the fault shunts are as follows: Type of Short Circuit Line-to-ground Line-to-line

Impedance of Fault Shunt, Zl' Zo + Z2 Z2 ZoZ2 Two-line-to-ground z, +Z2 Three-phase o

[541 [55J [56] {57]

The second way of finding the impedance of the fault shunt is to take the two equations Vo = -ZoIo [58]

V2

-Z2I 2

=

[59]

which are independent of the type of fault, together with the three equations giving relations between, or special values of, the sequence currents and voltages at the fault, and to eliminate from the five equations the four quantities Vo, 10 , V2 , 12 , thereby obtaining one equation of the form [601 For example, for a line-to-ground short circuit on phase a, eqs. 39 are used. Substitution of.eqs. 58 and 59 into eq. 39a gives

-ZoIo

+ VI -

Z2I2

=0

VI = ZoIo + Z2I2 By eq. 39b 10 and 12 may be replaced by It, giving

VI = (Zo

+ Z2)II

[61J

By comparison of eqs. 60 and 61 the impedance of the fault shunt is ZF =

z, + Z2

[62]

agreeing with eq. 54. The same result would have been obtained if the short circuit had been taken on phase b or c instead of on a. It may be worth pointing out that, although the connections of Fig. 7 are valid even if the zero- or negative-sequence networks contain generated electromotive forces, the expressions given for the imped-


222

ances of fault shunts are restricted to the normal condition, in which such e.m.f.'s are lacking. The positive-sequence network with the fault shunt connected to it may be reduced in the usual way to the simplest form of network connecting the internal voltages of the synchronous machines. For a twomachine system the reduced network is, of course, a tJ. (or 1r). EXAMPLE

3

Find the fault shunt for representing a line-to-ground short circuit on bus D of the power system of Fig. 8, Example 2, and reduce the positive-sequence network (with shunt attached) to an equivalent 1f' between the two generator 0.293

(a)

(b)

(c)

FIG. 19. Reduction of positive-sequence network with attached fault shunt of reactance X, representing a line-to-ground short circuit on the power system of Fig. 8 (Example 3.)

e.m.f.'s and neutral. Repeat for other types of short circuit at the same location. Solution. In Example 2 the zero-sequence and negative-sequence networks were reduced to single impedances as viewed from the point of fault (terminals D and 0). These impedances were Zo = jO.066 and Z2 = jO.09 per unit. For a line-to-ground short circuit the impedance of the fault shunt is Zp = Zo Z2 = jO.066 jO.09 = jO.156 per unit

+

+

The positive-sequence network was reduced as shown in Fig. 9, and in part c of that figure it has been reduced as far as possible while still retaining the

EFFECT OF TYPE OF FAULT ON STABILITY

223

separate identities of generators A and B. In Fig. 19 the fault shunt is attached, and the network is reduced to an equivalent 1r. Since all the branches of this 1r are reactive, only the branch between A and B affects the power-angle equation. ground fault, is

X

= 0.293 AB

The reactance of that branch, for a line-to-

+ 0.216 + 0.293 X 0.216 = 0.010 + '0.156

0.509

+ 0.381

= 0.890 per unit

For any reactance X F of the fault shunt,

= 0.509 +

X AB

0.0633 0.010+ X,

Hence for a line-to-line short circuit X". = X 2 = 0.090

X AB = 0.509 + 0.0633 0.100

= 0.509 + 0.633 =

·

1.14 per unit

For a two-line-to-ground short circuit, X,. =

XoX2 = 0.066 X 0.090 = 0.038

x, + X 2

0.156

+ 1.32 = 1.83 per unit· X AB = 0.509 + 0.0633 0.048 = 0.509 For a three-phase short circuit

XF = 0 · X AB = 0.509 + 0.0633 0.010 = 0.509 + 6.33 := 6.84 per umt

The reactance between generators A and B is lowest for the line-to-ground fault, higher for the line-to-line fault, still higher for the two-line-to-ground fault, and highest of all for the three-phase fault.

Effect of type of fault on stability. The lower the impedance of the fault shunt, the less is the power exchanged between any two synchronous machines for a given angular displacement, and therefore the lower is the stability limit for a given fault duration. In a twomachine system, such as that of Example 3, it is clear that the lower the impedance of the fault shunt (connected in the shunt branch of the T network of Fig. 19a), the higher is the impedance of that branch of the reduced network which s the two generators (branch AlB!) Fig. 19c). The synchronizing power varies inversely as the impedance of branch AIB I. A similar situation exists on a system of more than two machines.


224

Comparison of the expressions for the impedances of the fault shunts for the several types of short circuit (eqs. 54 to 57) shows that this impedance is lowest-indeed, is zero-for the three-phase short circuit, higher for the two-line-to-ground short circuit, still higher for the lineto-line short circuit, and highest for the line-to-ground short circuit. It follows that the most severe type of fault, as ·regards power-system stability, is the three-phase short circuit, followed in order of decreasing severity by the two-line-to-ground, line-to-line, and one-line-to-ground

~

~ 0.5 J---+---+-++~~~~-+--i---+--f

~

e

0.5

0.8

00

Fault duration (seconds) FIG. 20. Curves of stability limit as a function of fault duration for four types of fault at the sending end of a line of the system of Fig. 15, Chapter V (Example 4).

short circuits. The relative severity of the various faults is shown to advantage by curves of power limit versus fault duration, such as the curves of Fig. 20. The difference in severity of the several types of fault becomes smaller as the fault duration is decreased, but, even with the fastest feasible clearing times, the difference is usually pronounced. For zero clearing time (which is not attainable in practice) the power limit is independent of the type of fault. In making a stability study some judgment is necessary in choosing the type of fault which is assumed to occur. The assumption of threephase faults (the most severe type) gives conservative results and the simplest computation. It is therefore a useful assumption when comparing the effects of different fault locations, different system layouts, bussing arrangements, load conditions, and similar factors.


225

However, three-phase faults are of infrequent occurrence, especially on high-voltage overhead lines on steel towers, and the power limits determined on this assumption may be unduly pessimistic. For this reason it has been common practice to assume two-line-to-ground faults as the most severe condition likely to be encountered. In some locations, such as on busses of isolated-phase construction, it would be reasonable to assume that only line-to-ground faults could occur. Line-to-ground faults are the type of most frequent occurrence, and three-phase faults, the least frequent. The relative and absolute frequency of occurrence of the several types differs on different power systems, depending largely on the type of line construction. Sporn and Muller 4 give the following figures on the number of faults of different types occurring on a group of transmission lines most of which are operated at 132 kv.: Line-to-ground Two-line-to-ground Three-phase Total

58

8 6

72

In deg a system or a modification of one to improve stability, it appears logical to make estimates both of the value of various degrees of reliability of service and of the cost of achieving them. Then it may be determined whether the expense of any particular improvement of reliability is warranted. If a certain system can be made stable during three-phase faults at reasonable cost, it may be worth doing; but, if the expense is excessive, one may have to be satisfied with a system which is stable during two-line-to-ground faults but not during three-phase faults. In another system for which the requirements of reliability are not so stringent, instability during two-line-to-ground faults may be tolerated provided that the system is stable during line-to-ground faults. If still lower standards of reliability prevail, interruptions of service can be expected from any type of fault. EXAMPLE

4

Determine and plot stability limit in per unit as a function of fault duration for each of the following types of short circuit at the sending end of one line of the two-machine system of Fig. 15, Example 3, Chapter V: (a) one-lineto-ground, (b) line-to-line, (c) two-line-to-ground. Positive-sequence reactances are given in Fig. 15 of Chapter V. Negativeand zero-sequence reactances in per unit are as follows:

Generators: Each transmission line: Receiving-end transformers:

X 2 = 0.24, K« = 0.06 X 2 = 0.40, X o = 0.65 X 2 = X o = 0.10


226

The transformers are connected Y-~ with neutral of primary windings solidly grounded. Solution. The negative-sequence and zero-sequence networks are shown in Figs. 21a and 22a, respectively. They are reduced to single reactances 0.24

0.30

(O)~

i°.l

(b)

FIG.

33

(c)

(d)

21. Reduction of the negative-sequence network, fault at sending end of line (Example 4). 0.65

r::-:=::J 0.06

(0)

0.06

0.325

0.425

0.10

~(b) FIG.

22.

(c)

(d)

Reduction of the zero-sequence network, fault at sending end of line (Example 4).

in parts b, c, and d of those figures. The values of the reactances are X 2 = 0.133 and K« = 0.053 per unit. The reactances of the fault shunts (X F ) are:

Line-to-ground: Line-to-line:

Xo

Two-line-to-ground:

X

+ X2 =

0.186 per unit

X 2 = 0.133 per unit

X OX2

•

+ X ~ = 0.038 per unit o

The positive-sequence network is shown in Fig. 16a of Chapter V. With the fault shunt attached to it at the proper point it becomes as shown in Fig. 23a, and, after a Y-d conversion, it is as shown in Fig. 23b. The reactance of the branch connecting the two generator e.m.L's is XAB

= 0.35

+ 0.30 + 0.35;F 0.30 =

0.65

+ 0~~5


227

Upon substituting the values of X F given above, we get the following values of XAB: Line-to-ground: 1.22 per unit Line-to-line: 1.44 per unit Two-line-to-ground: 3.41 per unit 0.65 + °i~5

~~~~~ (b)

(a) FIG.

23. Reduction of the positive-sequence network with attached fault shunt of reactance XI' (Example 4).

For the pre-fault condition (Chapter V, Fig. 16) X AB was 0.65 per unit, and for the post-fault condition (Chapter V, Fig. 17) it was 0.85 per unit. Henee n and r2 have the following values:

Line-to-ground.·

0.65 5 '1 = -= 0.3 1.22

Line-fo-line:

rl = 0.65 = 0.45

1.44

Two-line-to-ground.· rl = 0.65 = 0.19 3.41

All types ojJault.·

0.65 7 '2=-= 0.6 0.85

Byrd and Pritchard's curves will now be used. From Fig. 40 of Chapter V values of sin 80 are obtained, corresponding to the values of '1 and r2 above. The values of sin 00, multiplied by Pm = EAEB/XAB = 1.00 X 1.00/0.65 = 1.54 per unit, give the power limits for sustained fault and for instantaneous clearing. These values have been entered in Table 2 in the columns headed "sin 00" and Up i" on the lines for te = 00 and to = o. Values of sin 00 which are multiples of 0.05 and which lie between the values already found for te = 00 and for te = 0 are selected. The curves of Figs. 23 to 39 of Chapter V are entered with the proper values of sin 00, rl, and '2, and the values of Teare read from the curves and written in Table 2. Values of Te/t e are calculated from eq. 27 of Chapter V, as follows:

~ == ~1ff1'1P ... == ~1f X to

GH

60 X 1'1 X 1.54 == 1 X 3.00

~

and are written in Table 2. The critical clearing time to is then calculated from T e + Tc/t e•

228

SOLUTION OF FAULTED NETWORKS TABLE 2

CALCULATION OF POWER LIMIT

Pi

AS A FUNCTION OF CLEARING TIME t c

(EXAMPLE

Type of Fault Line-to-ground rl = 0.53 r2 = 0.76

Line-to-line Tl = 0.45 r2 = 0.76

Two-line-to-ground Tl = 0.19 T2 = 0.76

sin ~o 0.456 0.50 0.55 0.60 0.65 0.701

Pi

4)

= 1.54sin ~o 0.70 0.77 0.85 0.92 1.00 1.08

"cltc 7.15

t«

tc (sec.)

" "

3.65 2.45 1.70 1.15

0.51 0.34 0.24 0.16

It

u

00

"

0.378 0.40 0.45 0.50 0.55 0.60 0.65

0.58 0.62 0.69 0.77 0.85 0.92 1.00

6.59

0.147 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65

0.23 0.31 0.38 0.46 0.54 0.62 0.69 0.77 0.85 0.92 1.00

4.28

"

"

" " " It

It

" " " " "

" "u

"

0 00

4.40 2.90 2.20 1.70 1.25 0.85

0.67 0.44 0.33 0.26 0.19 0.13

2.95 2.25 1.75 1.40 1.20 1.00 0.80 0.65 0.5 0.32

0.69 0.53 0.41 0.33 0.28 0.23 0.19 0.15 0.12 0.07

00

The results of the calculations are shown by the curves of Fig. 20. The curve for a three-phase fault has been copied into this figure from Fig. 22 of Chapter V so that the effects of the four types of fault on stability may be compared.

Effect of fault impedance. In the derivation of the connections between the sequence networks for representing short circuits,. it was assumed that the fault itself had no impedance. Actually the fault may have some impedance, principally the resistance of the arc between conductors. On high-voltage circuits this resistance is usually negligible in comparison to the other impedances of the network, and no allowance is made for it in calculation. * In ground faults there may

be additional resistance in the path to ground through the tower foot*For valuesof arc resistance see footnote in sectionon reactance relays in Chapter IX, Vol. II.

EFFECT OF FAULT IMPEDANCE

229

ing or through the object causing the fault. The value of this resistance varies widely, with a median value of around 20 ohms. 5 This resistance also is usually neglected in stability studies, thereby simplifying calculation and giving results for the most severe condition. If, for any reason, it is desirable, fault impedance may readily be taken into , as shown below. Line-to-ground fault. Let F be the impedance of the fault between conductor a and ground. To facilitate the analysis suppose that at the point of fault there are attached to the three-phase network three equal impedances of value F, as shown in Fig. 24a. This artifice has the advantage of making the three-phase network symmetrical up to point M, where a line-to-ground fault of zero impedance may be applied. The added impedance F appears in each sequence network between the point of fault and point M. Terminals 0 and M of the sequence networks are then interconnected as shown in Fig. 24b to represent a line-to-ground short circuit. Conditions inside the sequence networks are not altered if an impedance 3F is connected in series with the networks as shown in Fig. 24c instead of three separate impedances of F each. Line-to-line fault. Let F be the impedance of the fault between conductors band c. Suppose that the fault condition is represented by connecting an impedance F /2 to each of the three conductors at the point of fault and then applying the line-to-line short circuit at the other end of these impedances (point M, Fig. 24d). Such a connection is equivalent to connecting impedance F/2 to the line terminal of each sequence network and then connecting the positive- and negativesequence networks in parallel as shown in Fig. 24e. Of course, the two impedances F/2 which are no,v in series are equivalent to a single impedance F. Two-line-to-ground fault. Let the impedance between conductors band c be F L , and let the impedance to ground be Fa. For generality Fa might be tapped onto any point of FL, but, as FL is usually much smaller than F G, it will be sufficiently accurate for the present purpose to assume that Fa is connected to the center point of F L , giving F L/2 between the junction and each faulted conductor, as shown in Fig. 24f. The impedance F G in the ground connection can be shown to be equivalent to an impedance 3FG in the zero-sequence network. The connections between the sequence networks are then as shown in Fig. 24g. Three-phase fault. For symmetrical fault impedances as shown in Fig. 24h the connections of the sequence networks are as shown in Fig. 24i.

a

°2

I

Mo

3F

a

POSe

F/2

~'/~

line -to -line fault

(e)

L!::J

0

F/2

Zeroo--f-!V'

-(d)

go

c

3c1> b

Zero

(I)

with fault impedance.

Fa

Three -phase fault

(i)

~ FL/2

I

I~

FL/2

Two-line -to -ground fault

(g)

(h)

;~

a~""-

3cb b

~J 19

F./2

g o.-.l.---J\ AA--I

'i'c~

- ll::J

F

a~~V!

3,1,. b

FIG. 24. Connections between the sequence networks corresponding to various types of fault

(c)

~

One• line •to -ground fault

(b)

(a)

g:r::vv':

3q,b~ c -

~

ee

t=rj

00

~

:ao

t-3

Z

t:1

~

~ ~

Z

o

t--04

~1-3

rn

<:)

UNSYMMETRICAL OPEN CIRCUITS

231

Fault shunts. The impedances of the fault shunts, connected across the positive-sequence network at the point of fault to represent the effect of faults having impedance, may be found from inspection of the connections in Fig. 24; they are as follows: Type of Fault (with Fault Impedance) Line-to-ground Line-to-line

Two-line-to-ground

Impedance of Fault Shunt, Z,.

z, + Z2 + 3F Z2 +F FL

2

+ (Zo +FL/2 + 3Fa ) (Z2 + FL/2) Zo + Z2 + FL + 3Fa FL

Three-phase

[63] [64] [65] [66)

2

Unsymmetrical open circuits and series impedances. Unsymmetrical open circuits may be caused by blown fuses or by single-pole switching. t The connections between the sequence networks for representing one or two open conductors or an impedance in series with one conductor are given in Fig. 25. The derivations of these connections are similar to those given for.ahort circuits earlier in this chapter. Before these connections are made, the line side of each sequence network is opened at the point of fault, and the line on each side of the opening is considered to be a terminal. (These are the two upper terminals shown on each block in Fig. 25.) The neutral terminal, used for representing short circuits, is not used for representing open circuits. The effect of an unsymmetrical open circuit or series impedance on positive-sequence quantities may be represented by connecting an impedance Zp in series with the positive-sequence network at the point of fault. The value of .Zp depends upon the impedances Zo of the zerosequence network and Z2 of the negative-sequence network, each measured or calculated from terminals at the point of fault in series, instead of in shunt, with the respective network. Type of Fault

Series Impedance z" Inserted in PositiveSequence Network ZoZ2

One conductor open

ZO

Two conductors open Three conductors open Impedance Z in series with one conductor

Zo

+ Z2

[67]

+ Z2

[68] [69]

co

(l/zo)

+

1 (l/z2)

+ (3fZ)

fSingle-pole switching is discussed in Chapter XI, Vol. II.

[701


232 a

a

-l!. b c g

b

c g

--!!.

.s.g

Zero

o Pos.

g

P~s.

0

o

g

z~o

0

o

Z a ~

-. -.

o

(

a b c

0

Neg.

Neg.

0

0

[5J

c

Z/3

'\Nv C)

()

Zero 0

()

0

Pos. 0

Neg.

o 0 Neg.

0'

0

(a)

(b)

(c)

(d)

(e)

Normal condition

One conduetor open

Two conductors open

Three conduetors open

Impedance Z in one conductor

FIG.

25. Connections between the sequence networks corresponding to various types of open circuit or series impedance.

Simultaneous faults and other double unbalances. Approximately 20% of the faults on double-circuit overhead transmission lines on the same towers involve both circuits. Occasionally two faults occur simultaneously at points which are separated geographically as well as electrically, particularly on systems which are grounded through high impedance. A combination of an unsymmetrical short circuit with an unsymmetrical open circuit may occur, as when the short circuit is partially cleared by the opening of a fuse or single-pole circuit breaker. The conditions described above may be called "double unbalances." In general, a double unbalance cannot be correctly represented by connecting the sequence networks at each of the two points of fault as they would be connected for single unbalances at those points, nor can the effect of a double unbalance on positive-sequence quantities be correctly represented by connecting into the positive-sequence network fault shunts or series impedances which have the same values that they would if the two faults had occurred separately. For methods of solving three-phase networks having double unbalances see Refs. 2, 3, 6, and 7.

LINES IN THE ZERO-SEQUENCE NETWORK

233

The zero-sequence network. The zero-sequence network plays an important role in determining the currents and voltages during ground faults. This network differs from the positive- and negative-sequence networks particularly with respect to the representation of lines and of transformers. Representation of lines in the zero-sequence network. In any three-phase circuit positive- and negative-sequence currents are confined to the line conductors. Zero-sequence currents, however, being in phase in all three line conductors, must find a return path elsewhere, usually either through the earth or through both the earth and overhead ground wires or cable sheaths. The spacing between the outgoing and return paths is therefore greater for zero-sequence than for positive-sequence currents. Accordingly, the zero-sequence series inductive reactance is greater; and the shunt capacitive susceptance is smaller, than the corresponding positive-sequence quantities. The zero-sequence inductive reactance of lines is greater than the positivesequence reactance approximately by the following factors: Single-circuit aerial lines without ground wires or with steel ground 3.5 wires. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. .. . Single-circuit aerial lines with copper or aluminum ground wires. . . . . . . 2 Double-circuit aerial lines without ground wires or with steel ground 5.5 wires " . . . . .. . . . . . Double-circuit aerial lines with copper or aluminum ground wires. . . . . . 3 Three-phase cables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 to 5 Single-phase cables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Although these factors are useful for estimating zero-sequence reactance of lines, more accurate values should be calculated if the conductor sizes and spacings and the earth resistivity are known. Methods of calculating zero-sequence resistance, reactance, and capacitance are given in Refs. 1, 2, 3, and 21 of Chapter III. There is a considerable zero-sequence mutual impedance between parallel aerial lines on the same towers or even on the same right-ofway. The methods of calculating mutual impedance are also given in the references. If t\VO lines having self-impedances Za and Zb and mutual impedance Zm are connected together at both ends, they may be replaced by a single impedance, Fig. 26a. If they are connected together at one end only, they can be represented by the equivalent Y circuit of Fig. 26b. If they are not connected at either end, they can be represented on an a-c. board by the equivalent circuit of Fig. 26c, which utilizes an insulating transformer of ratio 1: i.t For ordinary tWo A. Lewis, in discussing Ref. 8, attributes this circuit to C. F. Wagner.


234

calculation they can be represented by the mesh circuit of Fig. 26d. 8 If a fault is to be represented on one of a pair of coupled lines, the lines on each side of the fault must be considered a separate section. Thus, ZeZb-Z",,'l

Za+ Z b - 2Z ""

Ca)

o------JVV\r---

c

I~---a

(bl

~---6

, (c)

CJ

----a

6'----6

a' (d)

a'----a 6'----b b' Zb

FIG. ?6. Equivalent circuits for two coupled lines of self-impedances Za and Zb and mutual impedance Zm. Lines connected together (a) at both ends, (b) at one end only, and (c) and (d) at neither end.

I

a

Fault )(

b

t=ml~

m(Zb-Zm)

FIG.

(l-m)(Zb-Zm)

27. Equivalent circuit of two coupled lines, one of which is faulted.

if there is a fault on one of two lines connected' at both ends, and the fault is distant from one end by a fraction m of the length of the lines, the equivalent circuit is as in Fig. 27. Equivalent circuits for three or more coupled lines have been devised" but are seldom needed. Two coupled circuits having appreciable shunt ittance can be represented by the equivalent or nominal circuits given in Ref. 9.

TRANSF9RMERS IN THE SEQUENCE NETWORKS

235

Representation of transformers in the sequence networks. As a rule, every important transformer bank on a three-phase power system consists either of three identical single-phase transformers having each set of windings connected in Y or in d or of a three-phase transformer having its windings connected likewise. Only 'such transformers having not more than three sets of windings are considered here. As was mentioned in Chapter III, the positive-sequence equivalent circuit of a two-circuit transformer bank, with exciting current neglected, consists of a series impedance and an ideal transformer of complex ratio. The angle of the complex ratio expresses the phase shift of positive-sequence voltage from one side of the transformer bank to the other. For D.-d or Y-Y connections the shift is either 0 or 180°. For !:J,.-Y or Y-!:J,. connections the phase shift, with the standard manner of naming the phases, is either +30° or -30°, depending upon the exact connections. By naming the phases in a different manner, the phase shift of +30 0 may become + 1500 or -90°, and the phase shift of -30° may become +90° or -150°. The nomenclature in which the phase shift is ±90° is the most convenient for computation and is recommended for that purpose.2, 10 If per-unit quantities are used, the magnitude of the complex ratio becomes 1, but the angle remains. In setting up the positive-sequence network, the phase shift is commonly neglected. The currents and voltages are first calculated as if there were no phase shift, and later the phase shift is taken into if it is of any moment in the use of the results. When phase shift is not represented, the positive-sequence equivalent circuit for per-unit quantities consists merely of a series impedance. The negative-sequence equivalent circuit of a transformer bank is like the positive-sequence circuit except that the angle of the complex ratio is of opposite sign. For example, if the phase shift for positivesequence current and voltage is +90°, the phase shift for negativesequence current and voltage is -90°. Here again the phase shift is usually disregarded in setting up the negative-sequence network, but it may be taken into later if desired. The zero-sequence equivalent circuit differs considerably from the positive- and negative-sequence circuits. For banks of single-phase transformers the value of impedance is the same, but, instead of this impedance being simply in series with the line, each end of the impedance may be either open or grounded, depending upon the connections of the bank. The proper connection may be deduced by considering the paths of zero-sequence currents. Zero-sequencecurrents are equal in the three phases. It is apparent that such currents cannot flow through a set of Y-connected windings if the neutral of the Y is isolated,


236

but they can flow if the neutral is grounded. Zero-sequence currents cannot flow from a three-phase line into a set of A-connected windings, but they can flow around the delta. If exciting current is neglected, zero-sequence currents can flow in one set of windings of a transformer bank only if zero-sequence currents can flo,v also in the other set of windings, The foregoing statements may be summarized as follows: Zero-sequence current cannot flow from a line into a transformer bank Three-phase circuit

Zero- sequence circuit (a)

Delta·delta

(b)

Ungrounded Yungrounded Y

(c) Grounded Ygrounded Y

(d)

Grounded Y delta

---VV\;---

_l

(e)

Ungrounded Y- delta

(I)

Ungrounded Y grounded Y FIG.

28. Zero-sequence equivalent circuits of two-circuit transformer banks.

unless the windings connected to the line are in Y with grounded neutral and unless the other windings are either in ~ or in Y with grounded neutral, The zero-sequence equivalent circuits of various transformer connections are shown in Fig. 28. Only connections c, grounded Y-grounded Y, and d, grounded Y-~, present a path for zero-sequence current. The other connections are equivalent to an open circuit in the zero-sequence network, The zero-sequence impedance of the grounded y-~ bank is equal to the short-circuit impedance of one transformer. Figure 29 may help

TRANSFORMERS IN THE SEQUENCE NETWORKS

237

to show why this is so. In part a of the figure the actual connections are shown. It is assumed that equal currents 10 are impressed on the primary windings, causing the secondary current 10 ' to circulate around the delta. Because of the symmetry of the delta, terminals A, B, and C are at the same potential and may be connected to one another, as indicated by broken lines in the figure, without effectively changing the circuit for zero sequence. Connecting terminals A, B, and C is equivalent to short-circuiting each secondary winding separately as in part b of the figure. It is apparent that the circuit of Fig.

(a)

10

#

~

10

~Iri

Vo 10

(b)

Vo 1

~Io'

0

-

tvo

FIG. 29. Diagram showing that the zero-sequence impedance of a transformer bank connected grounded Y-A is equal to the short-circuit impedance of the transformers.

29b presents between each primary line and ground an impedance equal to the short-circuit impedance of one of the transformers. As the line current is 10 and the line-to-ground voltage is Vo, the impedance VolIo is the zero-sequence impedance. It is equal to the positivesequence impedance because the latter is also the short-circuit impedance. Grounding impedance. The neutral of the Y-connected windings may be grounded either solidly or through a neutral impedance Zn. The current through Zn is 310, causing a voltage drop across it of 3ZnIo, which is added to all three line-to-neutral voltages and hence to the zero-sequence component of these voltages. In the zerosequence network the current is 10, and to obtain the correct zero-

238


sequence voltage drop the neutral impedance must appear in this network as an impedance 3Zn in series with the transformer impedance. Multicircuit transformers. As mentioned in Chapter III, the positive-sequence equivalent circuit of a three-circuit transformer, with exciting current, phase shift, and ratio neglected, consists of three impedances connected in Y between the terminals of the three circuits.

1

2

3

41£;

FIG.

(a)

30. Zero-sequence equivalent circuits of three-circuit transformer banks.

As long as phase shift is neglected, the negative-sequence circuit is identical to the positive-sequence circuit. The zero-sequence circuit consists of the same set of Y-connected impedances, but the three terminals of the, Y may not all be connected to the corresponding external circuits. Each of the three windings should be considered separately, and the corresponding terminal of the Y should be connected according to the same principles already established for twocircuit transformers, namely: (1) if the winding is ~-connected, the line is left open, and the corresponding impedance is grounded; (2) if

TRANSFORMERS IN THE SEQUENCE NETWORKS

239

the winding is Y-connected with grounded neutral, the line is connected to the impedance; and (3) if the winding is Y-connected with isolated neutral, both the line and the impedance are left open. In Fig. 30 these principles are applied to several different connections of three-circuit transformers. The same principles apply to transformers of four or more circuits. Three-phase transformers. Three-phase shell-type and three-phase five-legged core-type transformers resemble banks of three single-phase transformers in that they have three magnetic circuits which are almost independent of each other. Hence the equivalent circuits of these transformers are the same as those for banks of single-phase units. In three-phase three-legged core-type transformers, however, conditions are different. Application of Kirchhoff's law to the fluxesof three legs shows that there can be no zero-sequence flux in such a transformer except that leaking through air paths from top to bottom of the core. Therefore zero-sequence exciting currents produce no zero-sequence voltage except that induced by this leakage flux, which is small compared with the core flux that would be set up by positive-sequence currents of like magnitude but large compared with the leakage flux between primary and secondary windings. In other words, the zerosequence exciting impedance is much smaller than the positivesequence exciting impedance but is about five times as great as the leakage impedance, or equivalent impedance, between windings. Consequently, even a two-circuit transformer of this type should be represented by a T circuit. The shunt branch, representing the zerosequence exciting ittance, cannot be neglected, as it usually is in the positive-sequence circuit. The effect of the three-legged core is similar to that produced by adding a high-impedance il-connected winding, as is evidenced in the equivalent circuit by the third branch, the terminal of which is grounded. The connections at the other terminals, corresponding to the actual windings, depend upon the connections of those windings, as already discussed. Autotransformers. The commonest type of autotransformer on three-phase transmission systems has a set of Y-connected tapped windings with a grounded neutral and a separate A-connected stabilizing winding (formerly called a tertiary winding). It is, therefore, a three-circuit transformer connected Y- Y- il, and its equivalent circuit is the same as that of a transformer with three separate windings similarly connected, the impedances being given on the circuit basis. 2 The zero-sequence circuit is connected as shown in Fig. 30a without the grounding impedances. If the neutral of the autotransformer is grounded through impedance, this impedance is common to both

240


Y-connected circuits and, when expressed in actual ohms, is both the mutual impedance and a component of each self-impedance. When expressed in per unit or in ohms referred to a particular voltage, however, this impedance has three different values, one for the mutual impedance and two for the two self-impedances. Therefore, the single impedance must be represented by a Y circuit, as shown in Fig. 31. If the neutral of an autotransformer is ungrounded, zero-sequence currents can from one circuit to the other without transformation, the transformer acting merely as a series impedance. A

B

(a)

(b) FIG.

31. An impedance Z common to two circuits, A and B, must be represented

in the per-unit system by a Y circuit if different base voltages are used for the two circuits. N is the ratio of base voltage A to base voltage B, and Z' is the per-unit value of Z on base voltage A. In the zero-sequence network all impedances are multiplied by 3.

Other symmetrical transformer connections, including regulating transformers. See Ref. 12. Effect of grounding on stability. Methods of grounding a power system modify its zero-sequence impedance. This affects the impedance of the fault shunts for representing ground faults and thereby affects the severity of such faults. Most transmission systems are grounded through transformers, the high-voltage windings of which are connected in Y with grounded neutral. Such transformers are nearly always used at the points of supply and sometimes also at points of load. Occasionally transformers which do not carry load are used solely for the purpose of grounding. On high-voltage systems the usual practice is to ground the transformer neutrals solidly; on medium-voltage systems, to ground them either solidly or through resistors or reactors of low per-unit impedance. Some medium-voltage systems are nominally ungrounded but actually are grounded through the relatively high zero-sequence shunt capacitive reactance of the lines. If one or more transformer neutrals are grounded through reactors, the inductive reactance of which resonates with the capacitive reactance, the zero-sequence impedance may be made much higher than that of a nominally ungrounded system. Resonant grounding reactors, known as Petersen coils or ground-fault

EFFECT OF GROUNDING ON STABILITY

241

neutralizers, are used to a small but increasing extent. Transmission systems to which generators are connected directly, rather than through transformers, are grounded through the generators. The generator neutrals, like transformer neutrals, may be grounded either solidly or through resistors or reactors. The zero-sequence impedance of a transmission system, viewed from the point of fault, depends upon the number of grounding points, upon the distance of the grounding points from the point of fault, upon the zero-sequence impedance of the grounding transformers (or generators), and upon the impedance of the grounding resistor or reactor, if used. An increase in the zero-sequence impedance, viewed from the point of fault, results in a decrease in the severity of a one-line-to-ground or two-line-to-ground fault. The effect of a two-line-to-ground fault approaches that of a line-to-line fault at the same location, and the effect of a one-line-to-ground fault approaches that of no fault. Therefore, in its effect on system stability during ground faults, grounding through impedance is better than solid grounding. Either resistors or reactors may be used to decrease the severity of ground faults, but reactors are cheaper and commoner than resistors. At some locations, however, resistors are more effective than reactors. On a system consisting of several synchronous machines ed by low-resistance lines, the occurrence of a fault decreases' the electrical outputs of the generators and the electrical inputs of the actual or equivalent motors, causing the generators to be accelerated and the motors to be retarded. The closer the fault is to a particular machine, the greater is its effect on the electrical output or input of that machine, and the greater is the tendency for the machine to be accelerated or retarded, as the case may be. Stability is promoted by anything which increases the output of the generators during the fault, thereby lessening their acceleration, provided that the retardation of the motors is not increased correspondingly. A grounding resistor, by consuming power during a ground fault, exerts on a synchronous machine a braking effect which is greater the closer the fault is to the resistor and the closer the machine is to the fault. A grounding resistor located near a generator is therefore beneficial, as it exerts the greatest braking effect on the generator for faults close by, which would otherwise result in the greatest decrease of load and, consequently, in the greatest acceleration. Grounding resistors should not be used, however, near actual or equivalent synchronous motors or near synchronous condensers, for such machines already are retarded by faults. In a two-machine system resistance grounding may be advisable at the sending end and reactance grounding at the receiving end.

242


Neither grounding resistors nor grounding reactors have any effect during three-phase faults. The choice of a grounding system is influenced not only by consideration of stability but also by other factors, among which are: (a) limitation of current to prevent damage to generator windings from excessive mechanical forces; (b) limitation of neutral-to-ground and line-to-ground voltages of transformers to permit the use of graded insulation and lightning arresters of lower voltage rating, which give better protection; (c) limitation of inductive influenceon communication circuits during ground faults on the power system; and (d) obtaining suitable conditions for rapid and selective relay operation to clear ground faults. Since the conditions for limiting current conflict with those for limiting voltage, the choice of grounding method rests on compromise. The importance of the effect on stability of the method of grounding has been decreased in recent years by the trend toward higher speeds of fault clearing. EXAMPLE

5

Determine the effectiveness of grounding reactors or resistors in increasing the stability limit of the two-machine system of Fig. 32 during two-line-toground faults on the transmission lines. Assume the grounding reactance or resistance to be 10% on the basis of the kilovolt-ampere rating of. generating station A. Solution. A fault at either end of one of the transmission lines is more severe than a fault near the middle of a line; therefore, two-line-to-ground faults will be assumed, first at point C, then at point D, Fig. 32. The stability limit depends upon fault duration. For comparing the different methods of grounding it will be simplest to find the stability limit for sustained faults. This limit will be found by first obtaining the power-angle equation and then using the equal-area method, or, when the network is purely reactive, by using Fig. 40 of Chapter V, which was derived by the equal-area method. The grounding impedances, Zs at the sending end and ZR at the receiving end, appear in the zero-sequence network as 3Zs and 3ZB, respectively. The three sequence networks are connected in parallel at points C and 0 (as shown in Fig. 32b) to represent a two-line-to-ground fault at the sending end; they are connected similarly, but at D and 0, to represent a fault at the receiving end. The zero-sequence and negative-sequence networks are each reduced to a single impedance between terminals C and 0 (or D and 0), and then the positive-sequence network with fault shunt attached is reduced to a 7r circuit. Next the power-angle equation is found by the methods of Chapter IV. The details of calculation will not be shown, but the resulting equations are given in Table 3. The corresponding stability limits, found by the equal-area method, are presented in Table 4.

EFFECT OF GROUNDING ON STABILITY

243

30 30

(a)

32

10

15

10

11

1.0

Posltive- sequence network

°1 Zero- sequence network

3Z s

'--

3Z R ~~--

(b)

I I I I

-1

o

32. Two-machine system of Example 5. (a) One-line diagram. (b) Sequence networks interconnected to represent a two-line-to-ground fault at point C. Reactances are given in per cent on the rating of the generating station.

FIG.

TABLE 3 POWER-ANGLE EQUATIONS OF THE Two-MACHINE SYSTEM OF FIG.

VABIOUS

Grounding Impedance (per cent) ~~

Z8

o jl0 o

METHODS OF GROUNDING (EXAMPLE

SLG Fault at Sending End

0

P u = 34 sin 8 P u = 51 sin 8 P u = 36 sin 8 P« = 55 sin 8 P u = 5 60 sin (8 + 7°) P u = 10 + 56 sin (8 + 10°)

jl0 10 10

Any

Any

Any

Any

FOR

Power-Angle Equations (power in per cent of rating of station A)

Zs

0 jtO jtO j10 10

32

5)

+

LL Fault at Sending End Pu, = 77 sin 8 N orrnal Conditions P« = 154 sin 8

SLG Fault at Receiving End Pu, = 40 sin 8 Pu, = 42 sin 8 Pu, = 55 sin 8 Pu, = 59 sin 8 Pu, = 4 + 58 sin (8 + 2°) P u, = -5 + 55 sin (8 + 8°) LL Fault at Receiving End Pu, = 77 sin 8 One Line Disconnected Pu, = 129 sin 8


244

With solid grounding at both ends, the stability limit for a sustained two.. line-to-ground fault at the -sending end is 27% of the rated kilovolt-amperes of generating station A; and, for a similar fault at the receiving end, it is 32%, as given in the first line of Table 4. The addition of a 10% grounding reactor at either end increases the stability limit considerably for a fault at that end but only very slightly for a fault at the distant end. The use of reactors at both ends increases the stability limit for any fault location; the increase of stability limit over that obtained with solid grounding at both TABLE 4 STABILITY LIMITS OF THE Two-MACHINE SYSTEM OF FIG.

32

FOR

SUSTAINED FAULTS, AS AFFECTED BY GROUNDING REACTORS AND RESISTORS (EXAMPLE

Grounding Impedance (per cent) ~

Zs

o jto

o

jl0 10 10

Any

5)

Stability Limit for Sustained Fault (in per cent of rating of station A) A

~

2LG Fault at Sending End

2LG Fault at Receiving End

o o

27 42

32 34

jtO jtO jtO

29

45

ZR

10

Any

46 57 59 LL Fault at Sending End 66

48 52 41 LL Fault at Receiving End 66

Stability limit for any fault cleared instantaneously is 123%.

ends is considerable, but the increase over that obtained with a reactor at the end near the fault is slight. The substitution of a 10% resistor for the reactor at the sending end provides an additional increase in the stability limit, although the increase is greater for a fault at the sending end than for a fault at the receiving end. The further substitution of a resistor for the reactor at the receiving end results in a small additional increase in stability limit for a fault at the sending end, but the benefit is more than offset by a considerable decrease in stability limit for a fault at the receiving end. The best grounding scheme oj those investigated here is a resistor at the sending end and a reactor at the receiving end. Even when this scheme is used, the stability limit is lower for a two-line-to-ground fault than for a line-to-line fault. The improvement in stability limit obtained by the use of grounding impedance is smaller for a rapidly cleared fault than for a sustained fault, although the stability limit itself is greater. If grounding resistors are used at one or both ends, the stability limit depends upon the ratio of the inertia constant of the motor to that of the

TWO-PHASE COORDINATES

245

generator. The dependence of stability limit on the inertia constants may be understood by recalling that the equivalent power-angle curve of a twomachine system having resistance is a sinusoid displaced upward (according to eq. 28, Chapter IV) by an amount

Po = M2E12 Y ll cos811 - M1E22 Y 22 cos 8 22 M 1 + M2

where M l and M 2 are the inertia constants of the generator and of the motor, respectively; E 1 and E 2 are the internal voltages of the generator and of the motor, respectively; and Fu cos On and Y 22 cos 8 22 are the terminal selfconductances of the network between the internal voltages. If the two conductances are assumed to be equal, as they would be if the fault were electrically midway between the two internal voltages, and if, furthermore, the two internal voltages are assumed to be equal, Po would be zero if the inertia constants were equal. If, however, M 2 were greater than MI , Pc would be positive, indicating a raising of the power-angle curve by the amount of Po and a resultant raising of the stability limit by about the same amount. If, on the other hand, M 2 were smaller than MI , Pc would be negative, indicating a lowering of the curve and of the stability limit. In the present example it has been assumed that the generator and motor have equal values of stored energy per rated kilovolt-ampere but that the rating of the motor is three times that of the generator, giving M 2 = 3Ml • If the two-machine system consisted of a hydroelectric station supplying power to a metropolitan system of one or more steam stations, M 2 would be considerably larger than MI , both because of the greater capacity of the steam system and because of the greater stored energy per kilovolt-ampere of the steam turbogenerators; increasing M 2/M1 from 3 to 00, however, would increase the stability limit only a small amount. The stability limit is further increased if, as is usually true, the internal voltage of the generator is greater than that of the motor (E I > E 2 ) . In addition to the upward shift of the power-angle curve when M 2 > MI , there is a small shift to the left, which is beneficial. A thorough investigation of the methods of grounding would include a study of the effects of varying the magnitude of the grounding reactance or resistance on power limit and also on the line-to-neutral voltages. These matters, however, will not be considered here.

Two-phase coordlnates.P-l" Symmetrical components, which have been used in the preceding part of this chapter for analyzing unbalanced three-phase circuits, are a kind of substituted variables. Another kind of substituted variables, sometimes used for the same purpose, are two-phase coordinates. Two-phase components of current and voltage are defined as follows:§

I, =

i (2Ia -

Ib

-

Ie)

§In Ref. 14 a, {j, and 0 components are defined in a slightly different way.

[71a]

SOLUTION OF FAULTED NET\VORKS

246

III = -

1

va (Ie -

[7tb]

Ib)

I z = l(la+ Ib + Ie)

[71c]

Vz = !(2Va

[72a]

V1I =

Vb - Vc )

-

va1 (V

c -

v, = leVa +

Vb

Vb)

[72b]

+ Vc )

[72c]

Note that the expression for I, differs from that for V z by a factor 2. If eqs. 71 and 72 are solved for the phase quantities in of their two-phase components, the following expressions are obtained: la = Ix + !Iz 1

r, =

-"2 I x

I, =

-"2I z

1

-

[73a]

2V3 r, + '21 I z

[73b]

va III + 2"Iz

[73c]

+2

1

Va = Vx + Vz 1

[74a]

va

+

Vz

[74b]

v, = -7jVx + 2 Vy + v,

[74c]

Vb = -2 Vx 1

- - 2 Vy

va

Relation to symmetrical components. Two-phase components are related to symmetrical components as follows:

r, = 11 + 12

[75a]

v. =

Iy

[75b]

Vy = j(V1

[75c]

v, = v;

=

j(II - 12)

VI + V2 -

V2 )

[76a] [76b] [76c]

Note that the x current and voltage are equal to the sum of the positive- and negative-sequence components, andthe y current and voltage are equal to the difference of the positive- and negative-sequence com-

TWO-PHASE COORDINATES

247

ponents multiplied by j. The z voltage is the zero-sequence voltage, but the z current is twice the zero-sequence current. Consider a balanced, positive-sequence set of three-phase voltages. Their negative-sequence component is zero. Their two-phase components, byeqs. 76, are V x = VI and V y = jV1, which are balanced two-phase voltages of phase order y, x. Similarly, the two-phase components of negative-sequence three-phase voltages are two-phase voltages of phase order x, y. The substitute networks. The x voltages and currents may be imagined to exist in a single-phase network called the z network; the y voltages and currents, in another called the y network; and the z voltages and currents, in a third called the z network. These networks are analogous to the sequence networks used in the method of symmetrical components and may be given the more general name of "substitute networks." The substitute networks representing a balanced threephase network whose positive- and negative-sequence impedances are equal are independent of one another. The x and y impedances are equal to each ather and to the positive- and negative-sequence impedances. The z impedances are one-half the corresponding zero-sequence impedances. The positive- and negative-sequence impedances of rotating polyphase machinery are usually unequal. For many purposes, however, they may be assumed equal without serious error, thus making the use of two-phase coordinates practicable. If the three-phase network contains balanced, positive-sequence generated electromotive forces, then the x and y networks contain electromotive forces of equal magnitude, those in the y network leading those in the x network by 90°. Ordinarily, the z network contains no generated electromotive forces. The self- and mutual impedances of the substitute networks and the connections between the networks corresponding to any given impedances and connections of the three-phase network or any portion of it may be found by the process previously described for symmetrical components and illustrated by the derivation of equivalent circuits for representing several types of short circuit. Connections between the substitute networks for representing short circuits are shown in Fig. 33.13 A line-to-ground short circuit is represented by connecting the x and z networks in series and leaving the y network open. A line-to-line short circuit is represented by shortcircuiting the y network and leaving the x and z networks open. The z network is dead and need not be set up. However, the x network in this case and the y network in the case of a line-to-ground fault, al-

248


though not connected to anything at the point of fault, contain generated voltages and carry normal load currents.] A two-line-to-ground fault is represented by short-circuiting the y network and paralleling the x and z networks through a transformer of 2 : 1 ratio. If the impedances of the z network are given four times their normal value ao

3

~b go

[Z]

[]

o % -.

[J

-. o 0 _ _ .11

0Z

-

o

[]

%

I;~p I

(a)

(6)

(c)

(d)

One •line•to- ground

une-to•line

Two . line•to- ground

Three- phase

FIG.

33. Connections between the substitute networks (z, y, z) for representing various types of short circuit on a three-phase network (34)).

(twice, instead of half, the zero-sequence impedances), the transformer can be omitted. If the three-phase network is symmetrical except at the point of fault, and if the negative-sequence impedances are assumed everywhere equal to the positive-sequence impedances, the x and y networks are not coupled to each other and are identical except for a 900 phase difference in generated e.m.f.'s. Therefore it is unnecessary to have both these networks set up simultaneously on a calculating board. The positive-sequence network is used in turn for both the x and y networks. For example, in calculating a line-to-line fault (Fig. 33b), readings are taken in the positive-sequence network, first with nothing [Under the assumptions made in using a d-e. calculating board, all line-toneutral voltages in this network are equal, and all currents in it are zero.

REFERENCES

249

connected to it at the point of fault, and then again with a short circuit there. It is not even necessary to shift the generator voltages by 90° because it is a simple calculation to rotate the second set of readings forward through 90°, that is, to multiply them by j. Readings can be taken for all types of short circuit with only two networks set up on the board. Applications oj two-phase coordinates. In general, only the positivesequence network need be set up for a stability study, faults being represented by fault shunts. Sometimes, however, it is desired to study the probable operation of protective relays, and for this purpose the currents and voltages while the system is faulted must be found. The method of two-phase coordinates furnishes a convenient way of calculating these quantities from readings taken on a calculating board on which only the z network is set up in addition to the positivesequence network. . Two-phase coordinates also find application to the analysis of unbalances which are symmetrical with respect to one phase, for example, single-phase series or shunt impedances, banks of transformers connected in V or T, and untransposed transmission lines.14 ,15,16

REFERENCES 1. C. L. FORTESCUE, "Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks," A.I.E.E. Trans., vol. 37, pp. 1027-140, 1918. Historical reference. 2. C. F. WAGNER and R. D. Evans, Symmetrical Components as Applied to the Analysis of Unbalanced Electrical Circuits, New York, McGraw-Hill Book Co., 1933. 3. W. V. LYON, Applications of the Method of Symmetrical Components, New York, McGraw-Hill Book ce., 1937. 4. PHILIP SPORN and C. A. MULLER, "Five Years' Experience with UltrahighSpeed Reclosing of High-Voltage Transmission Lines," A.I.E.E. Troms., vol. 60, pp. 241-6, May, 1941. 5. C. L. GILKESON, P. A. JEANNE, and E. F. VAAGE, "Power System Faults to Ground: Part II: Fault Resistance," Elec. Eng., vol. 56, pp. 428-33, 474, April, 1937. 6. EDITH CLARKE, "Simultaneous Faults on Three-Phase Systems," A.I.E.E. Trans., vol. 50, pp. 919-41, September, 1931. 7. E. W. KIMBARK, "Experimental Analysis of Double Unbalances," Elec. Eng., vol. 54, pp. 159-65, February, 1935. 8. FRANK M. STARR, "Equivalent Circuits-I," A.I.E.E. Trans., vol. 51, PP. 287-98, June, 1932; disc., pp. 321-6. 9. J. C. BALSBAUGH, R. B. Gow, W. P. DOUGLASS, and A. H. LEAL, "Equivalent Circuits-2 Coupled Circuits," Elee. Eng., vol. 55, pp. 366-71, April, 1936; disc., pp, 1037-9, September, 1936. This paper gives an exactly equivalent five-terminal mesh circuit for two coupled lines not connected to each other at either end. The


250

discussion gives (1) an alternative circuit using a transformer, and (2) nominal or approximate values of the impedances of the equivalent circuits with numerical comparisons of the nominal values with the exact values. 10. BRYCE BRADY, "Symmetrical Notation of Three-Phase Circuits," A.I.E.E. Trans., vol. 60, pp. 955-7, November, 1941. 11. A. N. GARIN, "Zero-Phase-Sequence Characteristics of Transformers," Gen. Elec. Reo., vol. 43, pp. 131-6, 174-9, March and April, 1940. 12. J. E. HOBSON and W. A. LEWIS, "Equivalent Circuits for Power and Regulating Transformers," Elec. Jour. preprint, January, 1939; also published as Appendix, Table 7, Electrical Transmission and DistributionReference Book by Central Station Engineers of the Westinghouse Electric and Manufacturing Company, 1st edition, 1942. 13. EDITH L. CLARKE, "Determination of Voltages and Currents during Unbalanced Faults: Use of the Sum and Difference of Positive and Negative Sequence Symmetrical Components," Gen. Elec. Reu., vol. 40, pp. 511-3, November, 1937. 14. EDITH CLARKE, "Problems Solved by Modified Symmetrical Components," Gen. Elec. Reo., vol. 41, pp, 488-94, 545-9, November and December, 1938. 15. EDWARD W. KIMBARK, "Two-Phase Coordinates of a Three-Phase Circuit," A.I.E.E. Trans., vol. 58, pp. 894-904, 1939; disc., PP. 904-10. 16. EDITH CLARKE, Circuit Analysis of A-C Power Systems, vol. I, Symmetrical and RelatedComponents, New York, John Wiley & Sons, 1943. Chapter V, "Two Component Networks for Three-Phase Systems"; Chapter X, "Alpha, Beta, and Zero Components of Three-Phase Systems." PROBLEMS ON CHAPTER VI 1 ~ Graphically and algebraically find the symmetrical components of the following sets of phase voltages:

Va = 100/0

v, =

b 100LQ

v, = 100LQ

v,

v, =

100 /120°

v, =

a

= 100 /240°

v, =

100 /120°

v,

= 100 /240°

C

100LQ

Vc=100LQ

2. Find the symmetrical components of the following sets of phase currents: b c a Ia = I Ia = I I" = 0

Ib

=I

Ib

=

-!I

Ie = -!I

I, = -I

Ib I,

=0 =0

3. Find the symmetrical components of the following sets of voltages, and check the results by computing the given voltages from them: = 100/180°, = 100/90°, v, = 100/0. 4. Find the phase currents corresponding to the following symmetrical components: 10 = 0.15/31°, 11 = 0.97 /18~, 12 = 0.45/303°.

v,

v.

PROBLEMS

251

5. Work Example 2 with a two-line-to-ground short circuit on phases band c of bus D instead of a one-line-to-ground short circuit. 6. Work Example 2 with a line-to-line short circuit on phases band c of bus D instead of a one-line-to-ground short circuit. 7. Work Example 2 with the neutrals of both generators A and B, instead of generator B only, solidly grounded. 8. Compute and diagram the flow of reactive power of each sequence in the faulted three-phase network of Example 2. 9. Find the equations similar to eqs. 39 for a line-to-ground short circuit on phase b instead of on phase a. Show that the impedance of the fault shunt is the same for a fault on either phase. 10. Find the expression for the impedance of the fault shunt representing a two-line-to-ground short circuit, using the second method described in the text (p. 221). 11. Find an expression for the impedance of a shunt representing a fault consisting of a line-to-ground short circuit on phase a and a line-to-line short circuit on phases band c at the same point of a three-phase circuit. 12. Find the sequence currents in the system of Fig. 8 when the internal voltages of generators A and B differ in phase by 60°. The magnitude of each internal voltage is 1.05 per unit. Use the fault current as reference phase. 13. Find the reactances of shunts for representing each of the four types of short circuit at the middle of one of the lines of the system of Fig. 15, Chap.. ter V. (The negative- and zero-sequence reactances are given in Figs. 21 and 22 of Chapter VI.) 14. Calculate and plot stability limit as a function of fault duration for one-line-to-ground and two-line-to-ground short circuits at the middle of one of the lines of the two-machine system of Fig. 15, Chapter V. Also plot for comparison the stability limit for a three-phase short circuit at the same point. (See Fig. 22, Chapter V.) 15. Work Example 4 of Chapter V with a two-line-to-ground fault, instead of a three-phase fault, at the center of one line. Assume that the negativesequence reactance of the system is equal to the positive-sequence reactance when viewed from the point of fault, and that the zero-sequence reactance of the transmission lines is equal to 3.5 times their positive-sequence reactance, mutual reactance between the two transmission circuits being negligible, 16. Which fault on the system of Example 4, Chapter V, is more severe in its effect on stability, a three-phase short circuit at the middle of one line or a two-line-to-ground short circuit at the sending end of one line? 17. Which fault on the system of Fig. 8 is more severe in its effect on stability, a one-line-to-ground short circuit on line CD near bus D or a threephase short circuit on one of the parallel lines DE' near bus D? Does the answer depend upon the clearing time? 18. Prove that the connections of Fig. 24g are correct by writing equations for the relations between phase currents and voltages at the fault and then


252

deriving the corresponding equations for the relations between symmetrical components. 19. If, in Example 4, the line-to-ground fault had a 20-ohm resistance, by what percentage would the maximum synchronizing power be increased over that for a solid fault? System base is 50 Mva., 33 kv. 20. Prove that there is no net power in a three-phase circuit attributable to a voltage of one sequence in conjunction with a current of a different sequence. 21. Find the stability limit of the two-machine system of Example 5 for sustained one-line-to-ground faults. Consider the following methods of grounding: (a) Zs = ZR = 0; (b) Zs = jl0%, ZR = 0; (c) ZB = 0, . ZR = jl0%; (d) Zs = ZR = j10%. 22. In Probe 21 consider an additional grounding method, as follows: (e) Zs = 10%, ZR = jl0%. 23. In Example 5 find the line-to-ground voltages of the sending-end transformers during a two-line-to-ground fault at the sending end. Consider each of the following grounding methods: (a) Zs = ZR = j10%; (b) Zs = 10%,ZR = j10%. 24. In Example 5 find the power limits for two-line-to-ground faults cleared in 0.25 sec. with the methods of grounding listed in Probe 21. The frequency is 60 c.p.s. 25. In Example 5 find the power dissipated in the 10% grounding resistor at the sending end during one-line-to-ground and two-line-to-ground faults at the sending end. A 10% reactor is used at the receiving end. Assume E I = 120 and E 2 = 100 26. Calculate and plot power dissipated in the grounding resistor of the following simple system during a line-to-ground fault as a function of the resistance in per cent. The system consists of a generator for which the positive- and negative-sequence reactances are each 20%, a bank of transformers of 10% reactance, connected Ll- Y with neutral grounded through a resistor, and a fault at the secondary terminals. 27. Calculate and plot the higher line-to-neutral voltage (in per cent) of the unfaulted phases, during a line-to-ground fault on the system of Probe 26, as a function of the grounding resistance in per cent. 28. the equivalent circuits of Figs. 26b and c. 29. Derive an equivalent circuit for two coupled lines of different nominal voltages in which per-unit impedances are used. 30. Prove the relations shown in Fig. 31 for representing the ground impedance of an autotransformer. 31. Find the value of series impedance presented to zero-sequence current by an autotransformer with ungrounded neutral. 32. Solve Example 2 by two-phase coordinates. Assume X2 of generators to equal Xl.

1!1.

1!1.%.

CHAPTER VII

TYPICAL STABILITY STUDIES In the past twenty years a great number of stability studies have been made, usually with the aid of an a-c. calculating board, on existing or projected electric power systems. Through the courtesy of a number of operating and engineering companies we are enabled to present in this chapter s of several such studies made during the last ten years. Most of the companies have preferred that their names and the identity of their stations, substations, and lines be withheld; accordingly, letter symbols have been substituted. Let the reader be assured, however, that neither the stations nor the studies are fictitious. These typical stability studies have been included in order to illustrate some of the purposes for which stability studies are made, to show how such studies are carried out, how the types and locations of faults to be studied are chosen, how simplifications of the power system (such as the combining of several stations) are decided upon, and how limiting or critical conditions are selected for study, and also to give a glimpse at the progress made in the improvement of power-system reliability. Some of the stability studies here presented are incidental parts of more general studies that included also steady-state operation (loading of lines, maintenance of voltage, required capacity of synchronous condensers, and so forth). All these studies involved great masses of data and calculations which it was not feasible to reproduce here in full; for example, length, spacing, wire size, zero- and positive-sequence impedances of transmission lines; name-plate data and impedances of transformers, generators, and other machines; actual and estimated future loads on substations at different times of day and year; moments of inertia of machines; calculations of per-unit impedances on a selected system base; calculating-board connection diagrams and instrument readings; and calculations of swing curves. The reports on some studies fill one or more volumes. The condensed s of the studies which follow give an inadequate idea of the amount of labor actually spent on the studies. Nevertheless, we hope that they convey a clear conception of the methods and results. 253

254

TYPICAL STABILITY STUDIES STUDY 1*

The power systems of company A and company B, when operating in parallel, experienced difficulties due to transient instability. As an increased flow of power from company B to company A was contemplated, stable operation would become of greater importance. Study 1 was made to determine what factors entering into the interconnected operation of the two systems were important in causing instability and how stability might be improved. Description of systems. A map of the two systems, showing generating stations, substations, and transmission lines (with their voltages and lengths), appears in Fig. 1. Each station and substation is there identified by a two-letter combination, the first letter of which denotes the power company owning or operating it. The combined systems extended approximately 620 miles from station BO to station AV. Company A had two steam stations (AL and part of AP), several fairly large hydroelectric stations (AB, AC, and AG), and many small hydroelectric stations, some of which were neglected or were combined with larger nearby stations in the study. The principal load centers were near AL and AM. The transmission system of this company was composed of (1) a backbone of three 132-kv.lines in parallel, about 140 miles long, firmly tying together the principal generating stations and load centers; and (2) an extensive 44-kv. network which paralleled the 132-kv. lines, connecting smaller generating stations and loads.und which extended the system south of the terminus of the 132-kv. lines at station AE to station AP and to numerous substations. Company A had few instability problems in itself, as the loss of anyone line (either 132-kv. or 44-kv.) would not seriously weaken the ties between generating stations. Company B had a dozen or so small hydroelectric stations, divided into an eastern and a western group. Most of the stations of the western group (BB to BF, inclusive) were connected to each other and to station AD of company A by a single 132-kv. line. Stations BG and BH, also in the western group, were connected to the other stations of this group by two 44-kv. lines. The stations of the eastern group (RI to RO, inclusive) were ed to one another and to the western group chiefly by 66-kv. lines, with some 44-kv. lines. Thus, over most of the length of system B there was only one 132...kv. line, or *Information concerning this study was obtained through the courtesy of Ebasco Services, Incorporated, New York City, with the concurrence of the operating companies concerned.

STUDY I-DESCRIPTION OF SYSTEMS

255

two or three lower-voltage lines. This system undoubtedly would present problems in transient instability even if it were not interconnected with the system of company A. In addition to what might o AA

Company B

125

34 BM

38

Principal line interconnecting companies A and B

@]

Steam generating station

@

Hydro generating station

~

Combination generating station

@

o

Synchronous condenser substation Substation

•

Fault location

+H++

132·kv. transmission line 66·kv. transmission line

- - 44·kv. transmission line Figures onlines aretheirlengths in miles~ Two·letter combinations are station designations.

71

AV AO FIG.

1. Map of the power systems of companies A and B, Study 1.

be called an inherently unstable arrangement of the transmission system of company B, due principally to the nature of the territory served, several hydroelectric stations on this system had generators with characteristics which were not well suited to maintaining stability.

1 2

3 1 1

3

2

N N

0

2

1

2 1 1 1

1

2

3 2 1

3

2

2 2

2

2

1

2 1

2 1

1

ABl AB2 AB3

AC

AE

AFl AF2

AG

AH

AI

ALl ALe AU AL4

AN

N

2

2

1

11,111 7,500 1,250 1,000 18,750 2,500

6.6 6.6 4.0 1.1 6.9 2.3

Hydro

N Hydro

"

Hydro

2.3

2.3

It

1 1

4.4

4.6 0.5

Steam

1

Hydro

Hydro

Hydro

It

Hydro

Condo

It

It

2,500

1,000

8,000 25,000 750

1,250

600

12,222 8,200 8,333

6.6 6.6 7.0

Hydro

It

2.3

2,500 625

2.4 2.3 2.3

Hydro It

kva.

Rating kv.

Kind of Machine

0

1

2

1

N N

Nt

1

1

1 0

2

2 1

2

Number of Units Running in Load Condition

AAl AA2 AA3

Station*

Number of Like Units

TABLE 1

360

600

1,800 1,800 330

600

600

150

400

600

400

180

514 150 171.5

180 150 150

r.p.m,

LIST OF SYNCHRONOUS MACHINES (STUDY

1)

,

13.5

14.5

8.5 .19 5

9

13

35

15 6

18

15

13 40 37

24

30 25.5

(%)

Xd

80 114 130 27

5,760

755

3,664

1,295 2,180 1,949

375

Generator

3

30 98 3 97

1

209

402

81.9 158 3.3 10.3

3.0

12.4

32.0

27.8

... 1 3

28.4

79.5 11.3 14.3

3.0 2.0 1.0

(Mj.)

of Unit

Kinetic Energy

144

14 185 170

44

Turbine

WR 2 (kilo-lb-ft.")

~

t?:j 00

t:1 .....

~

00

t""4 ..... ~ t-<

~

to

;>

~

00

;> t""4

~ ..... o

~ ~

~

l'-' 01

1 4

5 1

1

1 1

2

2 1

BB

BC

BD

BE

BF

BG

"

"

"

"

2

AV

2

2 1 2

2

I

2

1

1

2

2

1 1

1

2 1

2

1

1

I

0 1

0

4

5 1

0

N

1 1 2 1 1 2

1

1

1

N

0 0

1

1 1 1

3 1

APl AP2 AP3 AP4 AP5 AP6 AP7 AP8

Station*

Number of Units Number of Running in Like Load Condition Units 1 2

"

Hydro

Hydro

u

Hydro

Hydro

"

Hydro

"

Hydro

Condo

" " " "

Steam Hydro

u

Hydro

Kind of Machine

2,400 590 1,250 2,500

6.6

2.3 6.9

6.9

2.2 6.6

6.9

2.3 2.3

2.3 6.6

2,400 7,860 18,750 875

2.3 2.4 13.8 6.6 6.6 2.3 6.6 4.0

1,250 7,500

9,000

12,000

600

9,375

1,400 3,000

4,375 7,500

600

kva.

kv,

Rating

TABLE 1 (Continued)

300

600

138.5

514 240

240

200 200

164 120

360 514

600 300 300

514 1,800

300

r.p.m,

,

25 35

30

15 27.5

32

14 36

16.3 65

15.5

10 24 22 13.5 9 10 10 13

(%)

Xd

27 480

4,000

24 2,000

1,160

360 350

1,120 2,670

37

383 254 68 18 64 383 34

Generator

t t

t

t

t

t t t t

2.5 10.9

19.4

1.6 29.2

16.9

3.6 3.2

7.6 9.7

3.3

...

t

8.0 15.9 120.8 1.7 1.4 8.0 1.1 1.0

(Mj.)

Kinetic Energy of Unit

5 8 95 3 3 3 3 1

Turbine

WR 2 (kilo-lb-ft.F)

"'""

~

CJ1

00.

s=

~ trj

to<

00.

o~

Z

...-4

~ o

...-4

~

~ 00 (1

t;

I

......

to<

§

00.

.-3

1

2 2 1

1

BL

BNl BN1&

BO

1

2 1 1

1

2 2 2 4

1 3 3

1

1 1

2

1

0

1 1 1

0

2 2 2 4

0 0 0

2 1 1 0

1&

Number of Units Running in Load Condition

Hydro

u

"

Hydro

Condo

" " "

Hydro

u

Condo Hydro

"

u

u

Hydro

Kind of Machine

6.6

6.6 0.5 2.2

2.3

2.2 2.2 2.2 2.2

4.1 0.4 2.3

6.6 6.6 2.3 2.3

kv.

3,600

5,000 500 500

3,000

850 1,250 1,562 1,000

7,500 300 625

2,000 3,600 5,500 3,125

kva.

Rating

225

225 225 225

720

120 90 90 109

900 257 180

225 450 400

200

r.p.m.

,

10

45 12 8.5

34

15 29 36 36

31.5 16 22

20

10

18

(%)

Xd

593

1,200 126 100

22

286 546 546 270

140

39 31

574 593 500

Generator

15.4 1.6 1.3

...

t t t t

7.6

2.6

t

7.3 0.5 1.2

5.8 7.6 25.6 5.7

Kinetic Energy of Unit (Mj.)

1.0 1.1 1.1 0.8

t t t

: :

...

+ +

t t

Turbine

W R 2 (kilo-Ib-ft. 2 )

*The numerals appended to the station letters denote separate stations or parts of stations which were combined in this Study. tN = Neglected. tWR2 of turbines of company B was assumed to be 10% of generator WR 2•

"

" " "

2 2 2 4

3

~

1

2 1 1 1

BK

Bll BI2 BI3

BH2 BH3

"

BHl

Station*

Number of Like Units

TABLE 1 (Cootinued)

t:-t

H

m

~

H

tj

~

~

~ ~

H

S

E:

a

~ ~

t-d H

~

259

STUDY I-DESCRIPTION· OF SYSTEMS

Interconnected operation of the two systems resulted in additional transient stability problems for both systems. As will appear from the swing curves obtained in this study, however, faults occurring near TABLE 2 LIST OF STATIONS

HAVING

SYNCHRONOUS MACHINES (SroDY

Load Condition 1

Station

Kind of Machines

AA AB AC

Hydro

AE

Condo Hydro

AF AG AH AI AL It

u

AN AP u

u

AV

BB Be BD

BE BF BG BH BI It It

BK BL BN BO

It

u

Kinetic Energy (Mj.)

Aggregate Mva. Running

Kinetic Energy (Mj.)

6.8 53.2 33.3 15.0 4.5

10.0 264.1 85.2 55.6

6.8 20.5 11.1

10.0 93.8 28.4

Off

Neglected Off

4.5 18.8 5.0 1.2 25cO 1.8 26.8 2.5 18.8 17.9 36.7

Off Neglected

It

Off

5.0 1.2 41.0 1.8 42.8 2.5 18.8 14.8 33.6 5.0 34.4 10.0 9.4 12.6 18.0 10.0 16.1 7.5 2.8 10.3 11.3 3.0 11.0 3.6

It

It

u II

Ie It

a

Condo Hydro Combined Hydro Cond. Hydro It

Load Condition 2

Aggregate Mva. Running

It

Steam Hydro Combined Hydro Steam Hydro Combined Cond. Hydro

1)

24.8 3.0 321.8 13.6 335.4 Neglected

120.8 36.4 157.2 Neglected

46.6 21.4 16.9 30.8 38.8 15.9 50.5 7.3 5.0 12.3 9.8 2.6 33.7

7.6

32.0 24.8 3.0 158.0 13.6 171.6 Neglected

120.8 39.5 160.3

Off Off

Off Off

3.0 9.4 12.6 18.0 10.0 13.1

3.2 16.9 30.8 38.8 15.9 44.8

Off

Off Off Off 9.8 Off 18.3

OtT Off 11.3 Off 6.0 Off

Off

station BG or east of it had little relation to the stability of interconnected operation, although such faults sometimes presented serious stability problems for the eastern part of the system of company B. The 125-mile, 44-kv. line from AA to BA had several small hydroelectric stations (grouped as station AA in the study) and several small

BK

BL

BM

6& ky.

4Cky.

~

~

~

== ~I

~T

BJ

~

+' ~-

gl~£

::1+'~ ::J.~

~

4Cky.

AI'

~

+

:;;

G'

~

~

o

+

~

~ AS

=:+

~ ;

AR

lJF

~l

e

~

~

AU

AT

~

;

-

~

~ ~

~

~

+

~

..,

+ -~~ . :i ·...1 -.... aD

AK

~

C"I

.,p

o

AH

AJ

+ +' £

~

C"I

sil

+' ~

~

132 ky.

:it

+'

C!

2

:;-

s:

~

+ s:

C7>

2

:+

C"I

~l

Study 1. Positive-sequence network as prepared for a-c. calculating board. Impedances are in ohms on board or in per cent on a 48.4-Mw. base; shunt capacitances are in microfarads on board.

66 ky.

~T

BL

BL

~T

~T

FIG. 2.

;::

;-

eo

.c

.t

~

~

:+

:e

C!

BE

00

t:rj

~

~

§

to<

~

~

t"'1

~

~ OJ

rn

o

~

~ > t"'1

~

~ 0)

o

STUDY I-LOADING CONDITIONS

261

loads. Although faults on this line would sever it from the interconnected system, they would have little effect on the rest of the system. Table 1 lists synchronous machines (generators, condensers, and large motors) on the systems of both companies with their ratings, transient reactances in per cent based on their own ratings, moments of inertia (WR2 ) in thousands of pound-feet", and amounts of kinetic energy in megajoules. Table 2 lists the synchronous-machine stations (or in some cases groups of neighboring stations) represented in this study, with their aggregate megavolt-ampere ratings and amounts of kinetic energy. Figure 2 is a diagram of the positive-sequence network, simplified to the extent required for setting it up on the calculating board. The zero-sequence diagram is not reproduced here. The oil circuit breakers and protective relays on these systems were of slow-speed types at the time this study was made. Loading conditions. Transient stability was studied for two different loading conditions: (1) estimated August day loads and (2) estimated October night loads. (The loading of the various circuits for the two load conditions is shown in Figs. 3 and 4, respectively.) The first condition was one of heavy loads, due largely to pumping for irrigation, and involved a considerable transfer of power from company B to company A (30 Mw. from station BB to station AD). The second condition was one of light loads but of even greater transfer of power (36 Mw.) from company B to company A. In load condition 2, all the station BB generators and five of the six station Be generators were shut down because of shortage of water; hence much of the power transferred to company A was necessarily transmitted from sources (chieflyfrom stations BE and BF) more distant than in load condition 1. The calculating board used in this study had ten power sources for representing generators. In order to check connections and settings of the board and also to help determine which generators could be grouped together for transient stability studies, readings for load condition 1 were taken, first with company B generators represented in considerable detail and then with company A generators represented in considerable detail, according to the columns of Table 3 headed Run 1 and Run 2. The readings thus obtained are shown in Fig. 3. The generators were then grouped as shown in the columns of Table 3 for runs 3, 4, 5, or 6, according to fault location, to permit their representation by ten power sources on the board. Individual line and generator loads were somewhat distorted by the grouping; the readings are not reproduced here. Readings for load condition 2 were taken with the generators grouped as shown in Table 3 for run 7, and these readings are shown in Fig. 4.

262

TYPICAL STABILITY STUDIES

STUDY i-LOADING CONDITIONS

~

...

-.~9·O

-"b~'£)

;~..----~ --.tn· 8S't J

.

~=

~~

; 0 -..(&rU ~ CQ .... 0...

~

(tn)

...

"CLt'r) ~~'l

263

TABLE

3

GROUPING OF SYNCHRONOUS MACHINES FOR REPRESENTATION BY TEN POWER SOURCES ON THE CALCULATING BOARD AND VALUE OF KINETIC ENERGY OF EACH GROUP IN MEGAJOULES (STUDY 1)

Power Source Numbers and Kinetic Energy (Mj.) Station

Run 1

Run 2

Runs 3 and 4-

RunS

Run 6

Run 7

1 22.5

Off

BO BL BI

Static*

BN

2

2 33.8

1

18.3

BK

3

3

2

9.8

3

44.8

4

15.9

5

30.8

6

55.7

7

3.2

BHs

Static

1

4

BH1 BG (66kv.) BG (44 kv.)

5

BE

7

BF BD BC

8

1

Off

2

3

9 10

1 2

BB (44 kv.)

3

AA AG

4 Off

AB

5

AC

6

AF

7

AN

Off

1

66.1

69.6

4 25.6

44.8

3

15.9

4

30.8

8 30.8

5

38.8

Off

9 16.9

4

16.9

6

16.9

5

21.4

7

21.4

6

46.6

8

56.6

7

10.0 Off

9.8

2 66.4

6

BB (6.6 kv.) Load box

AH

66.1

5 19.2 6 10.9 7

5.0

Off With AP

Off

10 8 376.2

9 376.2

8 154.2 Q()

(Infinite bus)

8

AI

9 199.4

AL

9

AE

Static

AP

10

AV

Static

9

391.0 10 548.1

10 157.1

10 170.2 Off

*uStatic" means represented by capacitor on calculating board.

264

STUDY I-SWING CURVES

265

Fault locations. A principal object of the study was to determine maximum allowable clearing times for faults at locations where they would affect the stability of the interconnection. It is apparent that a fault anywhere on the single-circuit 132-kv. lines from station AD to station BE would split the system into two parts unless high-speed reclosing were used. Since high-speed reclosing of high-voltage lines was still in the experimental stage at the time of this study, however, it was not considered. Hence faults on the single-circuit 132-kv. lines were not studied. Faults on anyone circuit of the three-circuit 132kv. connection between stations AB and AE would not divide the system, and the critical clearing time of such faults had to be determined. Since the impedance of two or three 132-kv. lines in parallel is very low, the exact location of a fault on these lines would not make much difference. But a fault nearstation AD on one of the three 13S-kv. lines to station AC was believed to have a somewhat greater tendency to produce loss of synchronism between the machines of company A and those of company B than a fault anywhere else. Consequently, this fault location was chosen for study. Faults on the 44-kv. transmission system also required consideration. The most severe effect on stability would be produced by a fault near the 44-kv. terminals of a large 132-to-44-kv. transformer bank. Two such locations were selected for study, one near each end of the 132-kv. system. One was near station AE on the 44-kv. line to station AS, the other near station BE on the 44-kv. line to station BG. The critical clearing time of faults elsewhere on the 44-kv. lines would be expected to be longer than for those at the selected locations. Faults on 44-kv. or 66-kv. lines at some distance from the 132-kv. system would not be expected to have much effect on the stability of the interconnection. A fault location nearstation BG on the #-kv. line to station BH was tried to see whether it would disturb the interconnection. The four fault locations chosen for study and described above are marked on the map, Fig. 1. Two-line-to-ground faults were considered in most cases, and threephase faults in some cases. Nearly always, simultaneous clearing at both ends of the faulted line was assumed. Swing curves. Eighteen transient-stability runs were made in the course of the study. The swing curves obtained are reproduced in Figs. 5 to 22 inclusive, and the conditions and results of each run are summarized in Table 4. In numbering the stability runs, the first number, such as 3 of 3-8-1, refers to the load.run made to obtain proper initial conditions for the

2LG

1

"

3-8-1

3-8-2

"u

3-S-6 3-8-7 3-8-8

"

"

cc

"

"

BE

"

AD

BG

"

AC

BH

" "

BG

" " " " " "

u

r

0.2

0.3

0.4

0.3

0.7 0.8 0.3 0.7 0.3 0.2 0.2

0.6

0.4

Same as at near end except where noted

0.4 0.4 0.3 0.2 0.4 0.5

Far

End

End

Near

....

Clearing Time (sec.)

,

•Load Condition 1: estimated August day loads; load condition 2: estimated October night loads.

44

132

a

cc

2

BG

" "

"

u

"

a

It

"

It

"

BE

44

"

" "

5-8-1

5-8-2 5-8-3 6-8-1 7-8-1 7-8-2 7-8-3

"

2LG

" "

It

"

II

3t/I

It

It

3-8-10 4-8-1 4-8-2

a

"

II

II

II

" " " " " " "

"

IC

2LG

II

II

3-8-5

AC

AD

"

u

132

"

II

"

AS

AE

II

On Line to

Near

Station

Line

Voltage (kv.) 44

3-S-3 3-S-4

" "

3t/I

Fault Type

Load Oondition"

Run

Fault Location

SUIOlABY OJ' STABILITY RUNS (STUDT 1)

TABLE 4

Stable Stable Unstable Stable Stable Stable Unstable Unstable Stable Unstable Stable Unstable Stable Stable Stable Unstable Stable Stable

Stable or Unstable

"

" " "

"

Series R at sta. BB Low:xti at sta, BB Shunt X at sta. BB

Remarks

t-4

00

t:J=j

tj

d

~

00

t<

~

eH

H

6;

~

00

a > e-

H

t-3

:;3

~ C) C)

STUDY i-SWING CURVES

267

320 r----oyo---r--.,.---r--r-~-__r_-_..._-....

280 ..--t---t--

2401---t---t--

40

9 (AE, AL)

0.2

0.4

0.6

ne (,seconds) FIG. 5. Swing curves, Study 1, stability run 3-8-3, load condition 1, three-phase fault near substation AD on one 132-ky. line to station AC, cleared in 0.3 sec. Unstable.


268

stability run, and 8-1 refers to the first transient-stability run made with these initial conditions, 8-2 refers to the second run, and so on. The usual simplifying assumptions were made (constant voltage behind transient reactance, constant input, and so forth). A time interval ~t = 0.1 sec. was used in point-by-point calculations. 240 r---r--r---r----r---r--~-'I'_-,

6(BB)

200

40

t---t---:--"""~

t----I---

-+---+----+---+-----f

~ -f.---f----+----t ~.:( cu·

U

.....

1-----I.-3+---+----+---+---+---+----i

~r

0 ......- .....-"'"-----.......- ......- - - -......- .....

o

0.2

0.4 Time (seconds)

0.6

0.8

6. Swing curves, Study 1, stability run 3-8-4, load condition 1, three-phase fault near substation AD on one 132-kv. line to station AC, cleared in 0.2 sec. Stable.

FIG.

In determining the allowableclearingtime of faults, the usual method was first to estimate this time and then to make a run using this estimated time. If conditions proved stable, another run was made using a clearing time 0.1 sec. longer than before. If the second run showed the systems to be unstable, then the clearing time of the first run gave

269


the answer desired. If the second condition was still stable, then a third run was made with 0.1 sec. longer clearing time. Runs 3-8-5,3-8-6, and 3-8-7 (Table 4) illustrate the method. For a two-phase-to-ground fault at station AD on one of the lines to station AC under load condition 1, run 3-8-5 showed the systems to be stable 200 _-_-or---r---__-.,..--..,...--..,...----...

O--_......_.Io-_""-_.a..-_..I..-_"'--......- - '

o

0.2

0.4 Time (seconds)

0.6

0.8

FIG. 7. Swing curves, Study 1, stability run 3-8-5, load condition 1, two-line-toground fault near substation AD on one 132-kv. line to station AC, cleared in 0.4 sec. Stable.

for 0.4-sec. clearing. Run 3-8-6 showed the systems to be still stable for 0.5-sec. clearing. However, run 3-8-7 showed them to be unstable for 0.6-sec. clearing time. Thus it was shown that 0.5 sec. was the maximum allowable clearing time for these particular conditions. In some cases it was possible to determine maximum allowable clearing time from a single run because the angular swing of particular machines was wide enough, and the trend of accelerating and decelerating forces on the calculating sheets was such that the run was readily seen to be a borderline case, and additional clearing time would

270


undoubtedly have caused instability. Sometimes comparison with runs previously made was helpful in such interpretations. A detailed discussion of the various swing curves follows. Stability during load condition 1; faults on 132-kv. system of company A. The generators were grouped on the power sources of the calculating board as shown in Table 3 for run 3. Initial conditions for 200 . - - o r - - - r - - - r - - - - r - - - r - - ' T " " " - - " - - - '

160 t---f----.......~~F-

4Ot-----ir---J---+---+---I--qL-..;::!~-+----I

0.2

0.4 Time (seconds)

0.6

0.8

FIG. 8. Swing curves, Study 1, stability run 3-8-6, load condition 1, two-line-toground fault near substation AD on one 132-kv. line to station AC, cleared in 0.5 sec. Stable.

the stability runs were those of load condition 1 (Fig. 3). Swing curves 3-8-3 and 3-8-4 (Figs. 5 and 6) show that the systems were unstable for O.3-sec. clearing but were stable for O.2-sec. clearing of a three-phase fault at station AD on one of the 132-kv. lines to station AC. Curves 3-8-5, 3-8-6, and 3-8-7 (Figs. 7,8, and 9) show that, with a two-line-to-ground fault at the same location and with the same load condition, the systems were stable for O.5-sec. clearing and unstable for O.6-sec. clearing. On all these curves it should be noted that the three main generating


271

280 - - - - - . - - - - - . . - - - - - - - - - - - -

80

1----+----I---+-~~lIr___+-__tf__-_+___t

40 I----+---+---+----+----+-~~-+-~

O.....

-~_..r.-

o

0.2

. . . _. . . . 0.4 Time (seconds)

'___""_____.;;:::I

0.6

0.8

FIG. 9. Swing curves, Study 1, stability run 3-8-7, load condition 1, two-line-toground fault near substation AD on one 132-kv. line to station AC. cleared in 0.6 sec. Unstable.


272

320 --.....--------------.---.....--.....---

280 1 - - - + - - - 0 + -

240 t - - + - - r - - -

80

I----a.--~~__r_-__+_---+--..._-+_____t

8(AB,AC,AF AH,AI,AN)

t

40 I--_...-_-+-

--.l~r--__+-~~-+____f

0'---............ .-_....._ ......._ ....._ ....._ ......_ ... o 0.2 0.4 0.6 0.8 Time (seconds) FIG. 10. Swing curves, Study 1, stability run 4-8-1, load condition 1, 15,OOo-kva. shunt reactors added on station BB low-voltage bus, three-phase fault near substation AD on one 132-kv. line to station AC, cleared in 0.3 sec. Unstable.

STUDY 1-PROPOSED CHANGES AT STATION BB

273

stations of company A swung almost exactly together, slowing down during the fault, thus acting like the equivalent motor of a two-machine system. Station AA and those stations of company B which were connected at 132 kv. (BB to BE, inclusive) swung fairly closely together-although not as closely as the stations of system A-and speeded up, acting like an equivalent generator. In every case station BB, the station closest to the fault 'of this group, speeded up more than the others (although it was closely followed by station AA) and was the first station of this group to pull out of step. Stations BG to BO, which were farther from the fault, were less affected, but tended to stay with the other company B stations. Study of proposed changes at station BB. As station BB was found to be the first station of company B to pull out of step during faults on the 132-kv. transmission system of company A, it was believed that any measures taken to help this station stay in step might materially improve the stability of the interconnected systems. Three different changes were studied. Use of shunt reactors. The design of the generators at station BB contemplated normal operation at 80% lagging power factor. Actually, however, these machines were usually operated near unity power factor. Such operation caused the machines to have internal voltage lower than normal and tended to make them unstable. Studies were made to show the effect of connecting shunt reactors to the low-tension busses at station BB in order to put about 15-Mvar lagging load on the machines and thereby increase their internal voltages. The initial load conditions with the reactors added were obtained in load run 4 and differed only slightly from the conditions of load run 3. The same generator grouping was used. Three-phase and twoline-to-ground faults were assumed at the same location as before, namely, near station AD on a 132-kv. line to station AC. Curve 4-8-1 (Fig. 10) shows that the system was unstable for 0.3-sec. clearing of a three-phase fault, whereas curve 3-S-4 (Fig. 6) shows that the system was stable for O.2-sec. clearing of such a fault without the use of shunt reactors at station BB. Therefore it is apparent that the addition of the reactors would not increase the maximum allowable clearing time for a three-phase fault by as much as 0.1 sec. Curve 4-8-2 (Fig. 11) shows that a clearing time of 0.7 sec. was permissible for a two-line-to..ground fault, and curve 3-S-6 (Fig. 8) shows a permissible clearing time of 0.5 sec. if no reactors are used. Therefore, a gain of at least 0.2 sec. in allowable clearing time of a two-line-toground fault would result from the use of the reactors. Use of series resistors. A method of increasing the stability of hydro-

274


electric generators by automatically cutting resistors in series with them during faults and thereby maintaining their load and decreasing their acceleration during faults was proposed by R. C. Bergvall.' As this method appeared to offer a possibility for stabilizing station BB, it was studied on the board. A resistance of 7.2 ohms per phase, connected in series with each of the four 7,500-kva. 6,600-volt generators, 200 ...---,.----r---~-....--..,..--.,__-_r_-_r_-_,._-_

0.2

0.4

0.6

1.0

Time (seconds) FIG. 11. Swing curves, Study 1, stability run 4-8-2, load condition 1, 16,OOO-kva. shunt reactors added on station BB low-voltage bus, two-line-to-ground fault near substation AD on one 132-kv. line to station AC, cleared in 0.7 sec. Stable.

was found to maintain practically fuIlload on the generators during the fault. Resistors of this value were assumed to be cut into the circuit 0.1 sec. after occurrence of the fault and short-circuited again 0.2 sec. after clearing of the fault. The result is shown in curve 3-8-8 (Fig. 12), in which a two-line-to-ground fault near station AD on a 132-kv. line to station AC caused instability when cleared in 0.7 sec. This appeared to be a borderline case, however, and it seemed safe to conclude that 0.6-sec. clearing of the fault would have made the system stable. Comparison of this result with curve 3-8-6 (Fig. 8) for the same condi-

STUDY I-PROPOSED CHANGES AT STATION BB

275

200 - - - - . . . . . - -......--'I"--..--..,----r----r-----r--r-----.

120 1--..;;;;::::::'l. . ._!:::I----a.~---+--~--+---~--+------"'~~I--~

40

~

~I

~.J----+---+-----+--1~-4--~H-~-+---+-~r------I

o

.ml S .:...I----+--.J----+---+----+----I-~~--I---_+_---i ~I~

~Iu bO ~ .5 \0 -I----+---i---+--__+_5

t

~

-40

1.

Q)

Cl)4----+----+--__+_---+---i-

.51 ~I

''tn+---+---I----4----+----11--en

~I

- 80 1...--.&-_..1.-_..1..._..1-_....1-_.....1-_....1-_-"--......- - - ' 0.2 o 0.8 0.4 0.6 1.0 Time (seconds)

__

FIG. 12. Swing curves, Study 1, stability run 3-S-8, load condition 1, two-line-toground fault near substation AD on one 132-kv. line to station AC, cleared in 0.7 sec.: 7.2-ohm resistors inserted in station BB generator leads 0.1 sec. after initiation of fault and removed 0.2 sec. after clearing of fault. Unstable.


276

tions but without the series resistors showed an increase of permissible clearing time of only 0.1 sec. Decreased transient reactance. The four large generators at station BB had the abnormally high transient reactance of 65% by test. Un200 .---.,.--or---..,---..,.--...,.---.,--....,---.....----

i

120

~

f

-----

"'t:J ~

CD

bo c «J

~

80

~

g

(ij

c

'-

~

.E

40

t---+--+----t--+---r--~----:~--I---+--~

Ot---+---+---+---+---+---+----+---HI.3IiIl~--=t

- 40

o

- - - - -......-"""-----_...r.-_....._-'-_....._....I

0.2

0.4

0.6

0.8

1.0

Time (seconds) FIG. 13. Swing curves, Study 1, stability run 3-8-10, load condition 1, transient reactance of station BB generators reduced from 65 to 35%, two-line-to-ground fault near substation AD on one 132-kv. line to station AC, cleared in 0.8 sec. Stable.

doubtedly, this high value of reactance decreased the stability of station BB. In order to determine the effect of a lower value of transient reactance for these machines, curve 3-8-10 (Fig. 13) was taken with the reactance assumed as 35%. The results showed that O.S-sec. clearing of a two-line-to-ground fault would make the system stable. This represented an improvement of 0.3 sec. over the 0.5-sec.

STUDY I-FAULTS BETWEEN STATIONS BE AND BG

277

clearing which was necessary for the same fault conditions but with the high value of transient reactance. Faults on the 44-kv. system of company A. Three-phase and twoline-to-ground faults were tried near station AE on the 44-kv. line to station AS under load condition 1. Curves 3-8-1 and 3-8-2 (Figs. 14 200 r---r--.,--w=---_-.---~,.----,...___

160

9(AE,AL)

40 I--f.--+----I---+-~~~-I---~----t

o

1:-.-

9

---------~--------_ . .......-

......

0.2

0.4

0.6

......

0.8

Time (seconds) FIG. 14. Swing curves, Study 1, stability run 3-8-1, load condition 1, two-line-toground fault near substation AE on 44-kv. line to substation AS, cleared in 0.4 sec. Stable.

and 15) show that the systems remained stable for either type of fault 'cleared in 0.4 sec. The limiting case for a two-line-to-ground fault was not obtained, and O.5-sec. clearing was judged to be permissible. A fault on the low-voltage circuits of any of the large stations tapping the 132-kv. lines of company A would be of approximately the same severity in regard to stability. Faults on the 44-kv. line between stations BE and BG.' The effect of a two-line-to-ground fault near station BE on the 44-kv. line to station BG under load condition 1 is shown in curves 5-8-1,5-8-2, and 5-8-3 (Figs. 16, 17, and 18). The generators were regrouped as shown


278

200 r - - - r - - -_ _--,.--_---r--~-_-___,

en Q)

80

f

~

Cl>

"'C Cl)

00 c co

Cl>

40

~

.s '0

8 (AB, AC, AF,

AH, AI, AN)

>

ro

E

$

.s

0

- 40

t----+---+---+---t--+---t----+---"---i---t

- 80 t----t-----+----+-1---+---+----..--

3 +---+---+---+---f ~I

- 120""'-_-'--_......._ - - ' - _ - - A . _...... o 0.2 0.4 Time (seconds)

....... _

0.6

....._

...

0.8

FIG. 15. Swing curves, Study 1, stability run 3-8-2, load condition 1, three-phase fault near substation AE on 44-kv. line to substation AS, cleared in 0.4 sec. Stable.


279

320 --,...--..,.----r---rlr---r---r--,---,

280 1----f---t---f---1---t--t----t---t--,

240 1---I----+-~J+---+--_+-__t-__1-___1

~ GJ

200 1---4---'---#--+--+-~r---+--;--,

e

iP

-so "C

C1J

c: co

~ ~ ~

~--""'_-.I 8 (AA, BB) ....... ---+-__1~__1

160 1----#--~:::...-.-+-~.....-

7(BC)

(ij

E

~

120 ~~$!!!!!I!!!!!!!IIiI~~.f--_9 (AB, AC, AF, AH, AI, AN)

I I

10 (AE, AL, AP, AV)

80 l----+----"--~--+--__t_-__+-__1-____1

40 1---I---+---;+.Ji-~---t----t"----1---;

_....Ioo._....._ ....

O'--_J..-_.L-_..Io.-_..a..___.......

o

0.2

0.4

0.6

0.8

Time· ( seconds)

16. Swing curves, Study 1, stability run 5-8-1, load condition 1, two-line-toground fault near station BE on one 44-kv. line to station BG, cleared in 0.3 sec. Unstable.

FIG.

280


for run 5 in Table 3 to give more detailed representation near the fault and less detailed representation far away from it. Curve 5-8-1 shows the system to be unstable with O.3-sec. clearing. Station BE, which was closest to the fault, pulled ahead and out of 240 .--_r--~r---r--r----r---..---....--..

200 I - - t - - -...

lO(AE,AL,

AP,AV)

40 t - - - + - I - - - + - - ...., +---+--+--+---+---0+-----1

~I

o............-----'0.8 o .....-.-.--~-"--~0.2 0.4 0.6 Time (seconds) FIG. 17. Swing curves, Study 1, stability run 5-8-2, load condition 1, two-line-toground fault near station BE on one 44-kv. line to station BG, cleared in 0.2 sec. Stable.

step with the rest. Stations BD and BF also speeded up, and stations BG to BO slowed down. It should be noted that the initial flow of power was from the former to the latter group of stations, which may be regarded as the equivalent generator and motor, respectively, of a two-machine system. Stations BB and Be were but little affected} and the stations of company A were affected hardly at all. Curve 5-S-2 shows the system to be stable with O.2-sec. clearing at


281

240 r--......---,.--.,..--....--..,.---.----.----.

200 1 - - - + - - - ' 1 - - - + - - . . - - - + - - - + - - - + - - - 1

10 (AE, AL, AP, AV)

,

./1 (BI,BK, BL,BN,BO)

4(BE) breakers opened 12 CYCles'*1

40

o .....------...--.......-...-.------.......----~ o

0.2

0.4

0.6

0.8

Time (seconds) FIG. 18. Swing curves, Study 1, stability run 5-8-3, load condition 1, two-line-toground fault near station BE on one 44-kv. line to station BG, cleared in 0.2 'sec. at BE and in 0.4 sec. at BO. Stable.


282

both ends of the line. It was also stable, as shown by curve 5-8-3, with sequential clearing in 0.2 sec. at station BE and in 0.4 sec. at station BG. Faults on 44-kv..line between stations BG and BH. The effect of a two-line-to-ground fault near station BG on the 44-kv. line to station 240 ..---r----r--,r--~r---.,r----r.--~r--_

-;; 160 .....~~~----+~

I

-~

."

«S

~ ~

120 I---~----l~""-"~-+

5(B~1)

I

7 (BG 44 kv.) I

3 (BJC)

~

I

4(B82)

Ci ...c

~

9(BD) I 10 (BB, Be, and company A as infinite bus)

80

1---t---t--~L..-....lII~~-fo--.--t----1

40

J---+---t--

I---t---t--...,-I--+---f--+--+---f

:; ~I

O'---'-------.....-"--"--------~---.t o 0.2 0.4 0.6 0.8 Time(seconds) FIG. 19. Swing curves, Study 1, stability run 6-8-1, load condition 1, two-line-toground fault near station BG on one 44-kv. line to station BH, cleared in 0.3 sec. Stable.

BH was ascertained for load condition 1. The generators were regrouped, as shown in Table 3 for run 6, to give more detail in the neighborhood of the fault, including the separation of stations BG and BH into two parts each. The machines of company A, together with stations BB and Be, all of which were found to be only slightly affected by a fault between BE and BG, were represented in this run as an

STUDY I-STABILITY DURING LOAD CONDITION 2

283

infinite bus. Curve 6-8-1 (Fig. 19) shows the system to be stable for O.3-sec. clearing at both ends of the line. The general direction of power flow was eastward. During the fault stations west of the fault 360 r - - - - y - - r - - - , . - - . . - - - y o - - - - , n - - - - - r - - - - .

280 I---+---+----+-+--I--.......~H---+---+----I

160 1--'--+--~~-.f--7l'o~+--+---+----+---4

120 t--+---+---+--~r__~.-+--+---t

0.2

0.4

0.6

0.8

Time (seconds) FIG. 20. Swing curves, Study 1, stability run 7-8-1, load condition 2, two-line-toground fault near substation AD on one 132-kv. line to station AC, cleared in 0.4 sec. Unstable.

(equivalent generators) speeded up, and stations east of the fault (equivalent motors) slowed down. Stability during load condition 2. In this load condition, as already mentioned, station BB and most of the generators at station Be were


284

shut down, and more power was transmitted from company B to company A than in the previous load condition. The generators were grouped in a manner (shown in Table 3 for run 7) which was judged 360 .---".--....---..----.,.....-.,..--.,.---ro--_

280 t---t---t----.,..;~__II---._, ~

en CD

~

CD ~

240 1---t-----#~~.__

+-I-~lIt---+---t-_4

.!l co C

tV

Wo ~

~ 200 1-JIIliIr:---J1----#-+---~-+---J~+---+---t-_of tV

... C

10{AA, AP)

J!! .E

~9(AE,AH,

AI, AL, AN)

160 1---+--~~-+----4~--+--+--+-_of

120 ~-+---+---__- + - - + - - + - - + - - - - t

0.2

0.4

0.6

Time (seconds) FIG. 21. Swing curves, Study 1, stability run 7-S-2, load condition 2, two-line-toground fault near substation AD on one 132-kv. line to station AC, cleared in 0.3 sec. Stable.

to be suitable for faults at either of two locations-near AD on the 132-kv. line to AC or near BE on the 44-kv. line to BG. Curves 7-8-1 and 7-8-2 (Figs. 20 and 21) show that, for a two-lineto-ground fault at the former location, the systems were stable for

STUDY l-STABILITY DURING LOAD CONDITION 2

285

O.3-sec. clearing and unstable for O.4-sec. clearing. It will be noted that the permissible clearing time of 0.3 sec. for this load condition was considerably shorter than the permissible clearing time of a similar fault under load condition 1, which was found to be 0.5 sec. This can be attributed to three reasons: 320 r----~-......--_-..,._-_.__-....,._-...._-_..

120 I - - + . - - - " ' - - - - t - - + - - - + - - - t - - - r - - - ;

80 ,--_",--_.a..-_~_..Io.-_-'-

o

0.2

0.4

"""_'"

0.6

0.8

Time (seconds) FIG. 22 Swing curves, Study 1, stability run 7-8-3, load condition 2, two-lineto-ground fault near station BE on one 44-kv. line to station BG, cleared in 0.2 sec. Stable.

1. The amount of power transferred from company B to company A was greater under load condition 2 than under load condition 1. 2. With station BB shut down, much of the power which it formerly supplied had to be transmitted from a greater distance. The result was a greater angular displacement between the generators of the two systems in the steady state and, therefore, less stable operation in the transient state.

286


3. When station BB was on the line, regardless of the amount of power supplied by it, it served to the voltage near the middle of the interconnecting line and thus to improve both steady-state and transient stability. In load condition 2, station BB was shut down, and this effect was nonexistent. The permissible clearing time of a three-phase fault was not determined, but, as it was 0.2 sec. under load condition 1, it would be 0.2 sec. or less under load condition 2. Curve 7-8-3 (Fig. 22) for a two-line-to-ground fault near BE on the 44-kv. line to BG shows a stable condition for O.2-sec. clearing at both ends. It was thought probable that sequential clearing, 0.2 sec. at BE and 0.4 sec. at BG, would be stable under load condition 2, as it was under load condition 1. Summary of allowable clearing times. The slowest clearing times permissible with stable operation, as determined from the swing curves, are summarized in Table 5.

TABLE 5 SLOWEST PERMISSIBLE CLEARING TIMES OF FAULTS (STUDY 1)

Load condition 1 (estimated August day loads) Three phase fault near AD on 132-kv. line to AC: Two-line-to-ground fault near AD on 132-kv. line to AC: Three-phase fault near AE on 44-kv. line to AS: Two-line-to-ground fault near AE on 44-kv. line to AS: Two-line-to-ground fault near BE on 44-kv. line to BG: Two-line-to-ground fault near BG on 44-kv. line to BH: Load condition 2 (estimated October night loads) Three-phase fault near AD on 132-kv. line to AC: Two-line-to-ground fault near AD on 132-kv. line to AC: Two-line-to-ground fault near BE on 44-kv. line to BG:

0.2 sec. 0.5 sec. 0.4 sec. 0.5 see." 0.2 sec. 0.3 sec. 0.2 sec.* 0.3 sec. 0.2 sec.

*Estimated.

Conclusion and recommendations. The study showed that the characteristics of the transmission system, especially in regard to the speed of clearing faults, had a decidedly more important bearing on the problem of improving stability than did the characteristics of individual machines or stations. It was concluded that no adequate correction of transient instability could be obtained until both relays and oil

STUDY 1-CONCLUSION AND RECOMMENDATIONS

287

circuit breakers on most of the 132-kv. circuits, on many of the 44-kv. circuits, and at major generating stations were modernized to obtain high-speed clearing of faults. If relaying .and oil circuit breakers were modernized, no other changes would be needed in the system to obtain a practicable solution of the instability problems which were studied. Three proposals for increasing the stability of the generators at station BB by changes at that station were studied, but the usefulness of the proposed changes was limited by the fact that this plant was usually shut down, because of water conditions, at night during the fall and winter seasons when the greatest amount of power was usually transferred from company B to company A and when instability of the systems was aggravated by this heavy power flow. Furthermore, changes at station BB would have very little effect in improving stability under most fault conditions which occurred on the system of company B. Consideration was given to the possibility of making changes affecting generators at plants other than BB. In all cases studied, however, it was found that the machines in a particular area swung closely together as a group, almost without regard to characteristics of individual stations. In practically every case the machines nearest the fault showed the greatest tendency to lose synchronism. From the foregoing observations, it was concluded that any changes which might be made at any of the generating stations would be limited in their effect and relatively costly to attain and, therefore, would assume a position of minor importance which did not justify further study. It was recommended that a definite program of relay and oil circuitbreaker improvement be prepared, taking into consideration the amount of good to be accomplished by various changes and the cost of the changes. It was also advised to concentrate on those changes which would allow the systems to remain stable after a two-line-toground fault, as three-phase faults are infrequent on properly relayed high-voltage systems. Modernization of relay systems and of oil circuit breakers would not only increase stability but would also decrease damage to equipment and improve service to customers. It appeared that the most improvement at the least cost could be obtained at many points on the system by relay changes. Modernization of some oil circuit breakers might be accomplished at reasonable cost but would, in general, be more expensive than relay changes to attain the same improvement in stability. Some other breakers should be replaced rather than rebuilt.


288

lorain Toledo

S Chicago

(OhioP.S. Co.)

(Toledo Edison Co.)

...an C'f)

.""

Indiana and

Cincinnati

S

South Point 10

FIG.

23. Central 132-kv. transmission system of American

289

STUDY 2

Akron

Alliance

(Ohio Edison Co.)

(OhioP.S. Co.)

Massilon

(Ohio P.s. Co.)

..,....,........-5

41.8mf.

6.68+j19.0

Newcomerstown

5'--0

West Penn Power Co.

Windsor

(Power, W. Va., nearWheelinl)

®-

Steam - electric generatintplant

@- Synchronous condenser(s).

®-

Equivalent of power system.

~Load.

-- -

Future 132·kv. line.

- - Existing 132·kv. line.

Turner

(Institute, W. Va., nearCharleston)

Cabin Creek W. Va.

Impedances aregiven in per cent on l00·Mw. 132·kv. base.

en Parallel impedance of double circuitline. Figures on busses areshunt capacitive susceptances of connected linesin percent on same base.

Gas and Electric Company and interconnections (Study 2).

290

TYPICAL STABILITY. STUDIES STUDY 2t

In 1940 the Ohio Power Company was adding two new generating units (numbers 4 and 5, rated at 94.5 Mva. each) to its steam-electric generating station at Philo (on the Muskingum River near Zanesville, Ohio), thereby increasing the aggregate capacity of the station from 271.4 Mva. to 460.4 Mva. The generators at Philo were paralleled through 132-kv. busses. The 132-kv. oil circuit breakers had a rated interrupting capacity of 2,500 Mva, The addition of the new generators would have increased the interrupting duty of these breakers from approximately 2,250 Mva. to approximately 3,150 Mva., a value in excess of their rating, had not steps been taken to limit the shortcircuit currents. For this purpose it was decided to install reactors between two sections of the 132-kv. bus, which will be called sections A and B. In October, 1940, an a-c. calculating-board study was made by the American Gas and Electric Service Corporation of the central system of the American Gas and Electric Company and interconnected systems, including the expanded Philo station with the bus reactors. The study included short-circuit studies, load studies, and a few stability runs. The system studied is represented in Fig. 23. It extended from Indiana across Ohio to West Virginia with interconnections to Chicago, Cincinnati, western Pennsylvania,' Vir-ginia, and Tennessee. Steamelectric generating plants of the represented part of the American Gas and Electric Company's system were located at Twin Branch (Mishawaka, Indiana), at Philo, at Cabin Creek, West Virginia, at Logan, West Virginia, and at Windsor (near Wheeling), West Virginia. The generating plants of 'the interconnected systems also were predominantly steam stations. All the transmission lines represented in the study operated at 132 kv. Impedances of the lines and of the equivalent generators are given in per cent on a 100-Mva. base in Fig. 23; total shunt capacitive susceptances at each bus, resulting from nominal-e representation of the transmission lines, are also given in per cent on the same base. The impedances of the equivalent generators were obtained by simplifying the circuits of generating or synchronouscondenser stations, using the transient reactances of the machines. For example, Fig. 24 is a simplified diagram of Philo station with the machines and lines grouped on the bus sections in the manner assumed tData on this study were obtained from the American Gas and Electric Service Corporation, New York City, through the courtesy of Philip Sporn, Vice President in Charge of Engineering, Harry P. St. Clair, System Planning Engineer, and Charles A. Imburgia.

STUDY 2 Crooksville Zanesville Howard No. 1 No. 2 No. 1 No. 2

291 Torrey North· Newcom· Rutland No. 1 No. 2 east erstown No. 1

Generators

j35.3

e

j61.7 j25.6 j57.0 j57.0

j33.7

j35.3

j61.1 j25.6

Circuit breakers assumed to open to clear the fault

FIG. 24. Philo steam-electric generating station. Simplified one-line diagram of the connections assumed in Study 2. Impedances are in per cent on 100-Mva. base. Transient reactances are used for the generators.

(a)

(b)

FIG. 25. Philo plant reduced to two equivalent generators, A and B. Impedances are in per cent on l00-Mva. base. (a) Normal condition, (b) after clearing fault on bus section AI.

292


in the stability study. Data on the generators and transformers at this station are given in Table 6. Values of impedance on a 100-Mva. base, taken from this table, are marked on Fig. 24. By means of series and parallel combinations the circuit was further simplified as shown in Fig. 25. Similar simplifications were performed for the other generating stations and interconnected power systems shown in Fig. 23. TABLE 6 DATA

ON GENERATORS AND TRANSFORMERS AT PHILO PLANT (STUDY 2) GENERATORS Xd'

% on

Xd" , . . - -......

Unit

Number Mva. 42.1 1 42.1 2 62.4 3-1 62.4 3-2 62.4 3-3 4HP 50.7 43.8 4LP 50.7 5 lIP 43.8 5LP

r.p.m,

1800 1800 1800 1800 1800 3600 1800 3600 1800

% on

Rating 100 Mva, Rating 100 Mva,

24.0 24.0 22.0 21.0 22.0 13.0 27.0 13.0 27.0

57.0 57.0 35.3 33.7 35.3 25.6 61.7 25.6 61.7

15.0 15.0 13.0 14.0 13.0 10.0 16.0 10.0 16.0

33.6 33.6 20.8 22.4 20.8 19.7 36.6 19.7 36.6

Hon 100 Mva. 2.35 2.35 3.81 3.81 3.81 1.74 3.22 1.74 3.22

TRANSFORMERS x%on ~

Unit

Mva.

T-l T-2 T-3-1 T-3-2

45 45 63

T-3-3 T-4

T-5

63 63

120 120

Rating 100 Mva.

8.9 8.2 17.8 17.5 17.9 13.4: 13.4

19.8 18.2 28.3 27.8 28.4 11.2 11.2

The inertia constants of the equivalent generators are listed in Table 7. Most of these values are based upon detailed information, but a few, particularly those for future additional generating capacity, are estimated at the rate of 6 per 100 Mva. for old generators and 5 per 100 Mva. for new generators, the new ones being assumed to be hydrogen-cooled. The calculating-board set-up was similar to that of Fig. 23, except that Fort Wayne and the stations and systems west and south of it were combined and represented by an equivalent generator at Fort

STUDY 2

293

TABLE 7

VALUES OF INERTIA CONSTANT H OF SYNCHRONOUS MACHINES ON l00-MVA. BASE (STUDY 2) Machines Chicago, Indana, and Cincinnati'systems Twin .Branch generators 1, 2, 3, 172 Mva. Twin Branch generator 4, 94 Mva, Fort Wayne condenser, 30 Mva, Total, Fort .Wayne and West

Case 2 1 3 Off 61 76* 7.8 7.8 7.8 Off Off 5.0 1.2 1.2 1.2

9

70

90

Fostoria condensers, 40 Mva.

0.7

0.7 Negl.

Toledo Edison Co. system, approx, 100 Mva.

6

6

7.7*

Lorain generating station of Ohio P. S. Co., 36 Mva.

2.1

2.1

2.7*

Canton condensers (Torrey, Sunnyside, Northeast) Akron and Alliance systems, over 500 Mva.

2.1 8

2.1 20

2.1 36.5*

Total, Canton and interconnections Windsor generators (Beech Bottom Power Co.), 257 Mva. West Penn Power Co. system, approx. 1,000 Mva.

38.6 15.1 15.1 15.1 Off 71.2 71.2

Total, Windsor and East

86.3 86.3

Philo A (gens. 1, 2, 3-1, 4) before clearing fault, 241 kyat

13.5 13.5 13.5

Philo A (gens. 1, 2, 3-1, 4) after clearing fault, 199 kva,

11.1

Philo B (gens. 3-2, 3-3, 5), 219 kva,

12.6 12.6 12.6

11.1

11.1

Rutland, future installed capacity of Appalachian Electric Power Co. and interconnected systems, approx. 1,100 Mva. 60

60

Off

Lancaster, future interconnection to west, approx, 170 Mva.

10

10

Off

Cabin Creek generators 1 to 8, 232 M va,

t

t

Logan generators, 102 Mva. Plants south of Logan and interconnected systems, 375 Mva,

t

t

Total, Logan and south *Estimated future increase. [Combined with Rutland.

15 4.4

22.6 27

294


Wayne. Furthermore, double-circuit transmission lines were represented on the board by single impedance units set at the impedance of two lines in parallel; when one circuit was supposed to be switched out, the setting of the impedance unit was doubled. The short-circuit studies showed that the reactance between bus sections A and B at Philo should be 7.5% on a 100-Mva. 132-kv. base in order to limit the interrupting duty of any circuit breaker to a value slightly less than 2,500 Mva. The 7.5% reactance would be obtained from two sets of reactors in parallel, as shown in Fig. 24, each having 15% reactance.j Load studies were made to determine the proper normal-load rating of the reactors and to find the arrangement of transmission lines on the two bus sections that would give the best conditions of transmission. In these studies estimated 1942 peak loads were used. In the course of the load studies, three transient-stability runs were made to test the stability of the system (modified by the addition of new generating units and bus reactors at Philo) with the interconnecting lines to other companies (Chicago, Indiana and Cincinnati, and western Pennsylvania) first open, and then closed. The three transient-stability runs are herein designated cases 1, 2, and 3. In cases 1 and 2 proposed generating stations or interconnections at Lancaster and Rutland were assumed to be in operation, and the existing generating stations at Logan and Cabin Creek were combined in an equivalent generator at Rutland. The flow of active and reactive power in both cases was as shown in Fig. 26. The flow from Philo west toward Fort Wayne was heavy. There were also heavy flows from Rutland to Philo and from Philo to Canton. In both cases 1 and 2 a three-phase fault was assumed to occur on section Al of the Philo 132-kv. bus and to be cleared in 0.15 sec. (9 cycles) by the action of I-cycle bus diffffi.ential relays opening the following 8-cycle circuit breakers (see Fig. 24): Howard line 1, Crooksville line 1, generator 1, and bus-tie reactor 1. The assumptions which have been stated represent a severe condition regarding stability, because of the heavy power flow over long transmission lines, the severe type of fault, the fault location at the sending end, and the loss of a long, heavily loaded circuit in clearing the fault. A three-phase fault near Philo on one of the lines to Howard might be expected to produce very nearly the same effect as a fault at the location assumed. In case 1 the interconnections to other companies were open; in case 2 they were closed. The swing curves for cases 1 and 2 are given in tThe reactors actually installed were rated 125 Mva., 138 kv., 23% reactance (equivalent to 20% per reactor on 10o-Mva. 132-kv. base).

FIG.

26.

-of-

Rutland andSouth

47.2 + j13.0 106

166 + j36.2

, ,

rh t

C'f ~

+

an

~ .....

30.3 -j3.0

Newcomerstown

trs

Akron Alliance

Initial load conditions of cases 1 and 2, Study 2. Flow of active and reactive power is given in Mw. voltages are given in per cent of 132 kv.

N

a\

N

:;-

o

Toledo t s

+i

Mvar.

Bus

t..:>

~

~

~

~

§


296

Figs. 27 and 28, respectively. In case 1 Fort Wayne and Twin Branch pulled out of step with the rest of the system, but in case 2 these stations, ed by the interconnections to the Chicago and Indiana

,

60

I I

I

40

~

//

VV __lZVI/ -

20

~r-,

"\

I

Phil0I:!~.

---....

I

Rutland~

~I

~

.......

~~

~

~

V

I

I

-

0

en

~cv

:e -0

~Canton

I Windsor-"'\ I I ~ ~~ ~' -~---...

I

-20

0

en

,

" .,

8- -40 «I

~

ioc

l,~

"i'-L ~I

.

Y

111

Lorain

,

I

;1

I

~Fort

-,

I

I

J

I

"

I

Wayne and West

'"

<,

~I

r-,

"'I'--

!llo...

<,

I

o

0.1

-I----

'" ~ ~o

·~·t ~I

-120

~

/ ~ Fostoria condenser ~~

dr

-100

I

~,

-~ / ~~ ~

I~I

.1"

-80

,

~

"

I

\1

~I

I

~~

I

\~ ~~

I

I

<

-60

......

-

l-/ ZAkron - Alliance

'"

""--.

t~ ~ I I ~

~"

I

~I I Lanc,aster

.i~

I...........

c

condenser

I

0.2

0.3

0.4

r-,

0.5

0.6

Time (seconds) afteroccurrence of fault FIG. 27. Swing curves, Study 2, case 1. Future set-up. Generating plants at Lancaster and Rutland. Interconnections open. Three-phase fault on Philo 132-kv. bus section Al cleared in 0.15 sec. by opening Howard line 1, Crooksville line 1, generator 1, and bus-tie reactor 1. Unstable.

systems, stayed in step. The motion of the other machines was very nearly the same in both cases. Philo A, the machine nearest the fault, lost all its load while the fault was on and speeded up more than any other machine. Philo B, separated from Philo A by the bus-tie reactor, also speeded up, although less than Philo A did because it lost

STUDY 2

297

less load. The other generating stations were affected but .little. The synchronous condensers, whose swing curves are shown by broken lines, had very low inertia and swung with a shorter period than did the generating stations. Because of their small inertia they had little effect on the rest of the system, and they were not represented separately in case 3. 60

j

.... Fault

I I

1

I

cleared

40

~

/v

0.15 sec. ~

y I

lancaster, c

I

I

.~

~,

~

'

""""""'"

~Canton

,-

.

I

I I

v\

0.2

<-Lorain

\ ---Y

-""" ~

\

~

I I

1oledo

..

I

i

I

r\ R= r---

0.3

............

c: ~

--a--~.

condenser~~

"

I 0.1

"

,--....

c-

-----

r-.

I' -...'

-60

I

condenser

'" C>< ~ ~/~ ....... / ~~~', ,~ . ~--

Akro~ s, ~~~ II ,~ i Fostoria

"'I o

-:

~

__V C>< ~ ~

I'R.utland

rr-

~

I

~ K Alliance

-40

I

/,-Windsor

-.1-, " -""""--- --~~I ..

:t:i

r\. ___

~Philo B

II

o

"- " ~

. ./

--~ [.,.7' I

T

~ ~PhiloA

//

V~ ~

~

I

~

\.

~J

,

'J-/

~

t--- ............

---.

FortWayne and West I

I

0.4

I

I

0.5

0.6

0.7

Time (seconds) after occurrence of fault FIG. 28. Swing curves, Study 2, case 2. Future set-up. Generating plants at Lancaster and Rutland. Interconnections closed. Three-phase fault on Philo 132-kv. bus section Al cleared in 0.15 sec. by opening Howard line 1, Crooksville line 1, generator 1, and bus-tie reactor 1. . Stable.

Case 3, representing immediate-future conditions, was set up without the proposed generating stations at Lancaster and Rutland, but with the existing stations south of Philo at Logan and Cabin Creek in operation, and with interconnections to other companies closed. Loads were as shown in Fig. 29, substantially the same as for cases 1 and 2. A three-phase fault was assumed to occur on section A2 of the 132-kv. bus at Philo and to be cleared in 0.15 sec. by the opening of Howard line 2, Crooksville line 2, Rutland line 2, generator 2, and bustie reactor 2. Note that the two groups of lines at Rutland are not bussed. The swing curves (Fig. 30) show that Cabin Creek and


298

lorain 11.4 - j3.6

Fort Wayne and West

-........- - 100.5

40-jl0

Crooksville

South Point

.............. 106

t

139 + j6.0

Logan and South FIG.

29. Initial load conditions of case 3, Study 2. Flow of active and

STUDY 2

299

Akron, Alliance, and Canton condensers

s 40.0 + nos

Po: 40.0 +jl1.0 ~ I

~

-e--

o ~

North east 98.0

....---.....-98.0

Massilon

t$

~

24.0 + j19.6

:s It'

28.0 - ,56

B Philo

Newcomerstown

Windsor andWest

Penn

~...... -

....

102

('t)

e",

I

....... ('f)

Cabin Creek 1040

G

reactive power in Mw.

+j

t

190 - ;14.0

Mvar.

Bus voltages are given in per cent of 132 kv.


300

Logan, which were sending power to Philo, speeded up and pulled out of step with Philo. The pull-out might be attributed partly to the loss of a Philo-Rutland-South Point-Turner circuit and partly to the initial angular displacement between Logan and Cabin Creek, on the one 80

I <.$1 ~I

~I

60

01

Cabin

~I

../

gJl

~I

40

"31

~I

~

"......,..-

~

&

.......-LOga~-r

l.a-- ~

-40

~

o

I

I

J J

I

I

"'lIiI

I

-,

-

~

~~

~

~

/Lorain ../V

~

""".--~

~

~ .......

I -~ ...............

~

-,

.-'!

Toled0-:L ~ ~

-

~

~ .......... ----. r---. Fort Wayne and West ~

----r---

l 0.1

~

•

,

~

I I

PhiloB~ ~

Windsor",

!

-60

<,

I

I

r----..

/

..-Logan

/

~~ r-, Akron - Alliance, Canton condenser -.......-..

I

....---...

~

..--........

?~

I

I

r----.. ~

~ III""""""

/

fI'

/

I

I

-80

~

~

/

v::

",

PhiloAJ

I

A

------

~

71

~

Creek~

~

0.2

0.3

0.4

0.5

0.6

~

--~

~

0.7

Time (seconds) afteroccurrence of fault

30. Swing curves, Study 2, case 3. Immediate-future set-up. No generation at Lancaster or Rutland. Three-phase fault on Philo 132-kv. bus section A2 cleared in 0.15 sec. by opening Howard line 2, Crooksville line 2, Rutland line 2, generator 2, and bus-tie reactor 2. Unstable. FIG.

hand, and Philo, on the other hand, being larger than that between Rutland and Philo in case 2. Philo A and B swung more nearly together than they did in cases 1 and 2, probably because Philo A was more lightly loaded than before (175' instead of 215 Mw.). Conclusions. The following conclusions were reached: 1. The proposed bus reactors at Philo were found to have little effect on system stability, as evidenced by the fact that the generators

STUDY 3

301

in sections A and B of the plant swung approximately together, even for a fault on one bus section. 2. It was also found that, without the benefit of the added inertia of the Chicago and Cincinnati interconnections (although these connections are normally closed), the system under the assumed load conditions would lose synchronism between Twin Branch and Philo after a three-phase bus fault at Philo cleared in 9 cycles with the loss of one Philo-Howard circuit. With the interconnections closed the results indicated that synchronism would probably be maintained under this fault condition. To increase the margin of stability under such conditions a program of modernizing the Philo line breakers was worked out (and has subsequently been completed), calling for a reduction from 8 to 5 cycles in the breaker-operating time and from 9 to 6 cycles in over-all clearing time of faults. Also, the system tie lines between Ohio and Indiana have been strengthened by the construction of a new 132-kv. line from Portsmouth to Muncie, Indiana, providing a direct line from Turner to the Indiana area. 3. Likewise, with a three-phase bus fault at Philo followed by loss of a Philo-Rutland-Turner circuit, the system, under the heavy load conditions assumed, showed a probable loss in synchronism between Philo and the south. Here again, the subsequent improvements which have been mentioned were designed to eliminate this difficulty, and they have undoubtedly provided the necessary margin of stability to prevent loss of synchronism under these conditions. 4. The system engineers have also pointed out that the future installation of 3-cycle breakers to replace 5-cycle breakers at the most important locations, together with more extensive application of highspeed reclosing, will provide still greater gains in system stability. STUDY 3

Figure 31 represents a 132-kv. 60-cycle transmission network connecting a large steam generating 'station A (on the extreme right-hand side of the diagram) to a large metropolitan receiving system, which may be considered an infinite bus and which is represented by system J on the extreme left. Connected to the same 132-kv. network are two other steam generating stations, E and I, and a number of substations with loads. The circuit breakers and relays on the network were too slow to maintain synchronism between A and J through severe faults. Therefore, it was believed best to operate the system sectionalized by open breakers marked with crosses in Fig. 31. Under this plan each of the

FIG.

31.

SystemJ

Circuit breaker openfor sectionalized operation.

18.2+j4J.lO

Substa.D 9.9+J38.J.O "-45-j6 Mva

One-line diagram of the power system of Study 3, showing impedances in ohms at 132 kv., initial loads in megavolt-amperes, initial voltages in kilovolts, and assumed fault locations.

6. FaultlocatiOn assumed in the study.

Circuit breakernormallyclosed.

12-j101Mva 129kv

o

118 - j 52 Mva.

~

~~

~/220.

~

12.0+jS1.50 j35.Hl --14 + j 10 Mva.

Substa. F

5.7+j24.3n

1m

3.3+j12.S0 .... 23+j24 Mva

124 kv

Substa. G

4O-j2S Mva.

Sta.A

CIo:)

00

t:rj

t1 t-4

c:

~

to<

~

t-4 t-4

t:d t-4

:>

~

00

t-4

:>

o

t-4

"'d

t-3 ~

~

STUDY 3

303

three generating units at station A fed toward the metropolitan system J over a separate circuit. A fault on anyone circuit and its subsequent clearing disconnected one of the units at A from the rest of the system. The disadvantage of this mode of operation was that, after clearing a fault in or near stations A, B, or C, the loads at A, B, C, or D (depending on the fault location) were fed from stations E, I, and J over such long lines that the voltages at the loads became too low. To take the most extreme case, the load at station A could not be supplied over the long line from system J if a fault occurred in generator At. The study here described was made to determine which breakers would have to be speeded up and possibly also equipped with highspeed relaying to make station A stay in synchronism with the rest of the system for faults on the I32-kv. network if the network were to be operated unsectionalized, that is, with the breakers marked by crosses closed. The breaker-opening times were about 20 cycles, except for some 8-cycle breakers at stations A and E; and the relay times were from 1 to 60 cycles, depending on the fault location. It was assumed in the study that any breakers requiring faster opening would be rebuilt to open in 8 cycles. The line relaying would be changed where necessary to high-speed distance relays having I-cycle operating time; and, if necessary for stability, carrier-current pilot equipment would be added to give simultaneous tripping of the breakers at both ends of the line for faults at any point of the line. It was further assumed that high-speed bus differential protection would be installed where necessary for stability. It is common to use two-line-to-ground faults in stability studies, inasmuch as three-phase faults are very infrequent on high-voltage steel-tower transmission lines. In this study, however, three-phase faults were assumed, because they are not much more severe than twoline-to-ground faults in their effect on stability and because they are much simpler to represent. Only the positive-sequence network need be considered for three-phase faults, whereas the zero- and negativesequence networks must also be considered for two-line-to-ground faults. Three-phase faults were assumed at various locations, the choice of which will be discussed, and for each location swing curves were calculated to determine whether the system would be stable with the existing breakers and relays and with faster breakers and relays. To obtain the swing curves, the network of Fig. 31 (with all breakers closed) was set up on an a-c. calculating board. Each line was represented by a nominal e, although the shunt capacitance at each end of

304


the line is not shown in Fig. 31. The three generators of station A were represented by a single equivalent machine on the assumption that, being connected to the same bus-although sectionalized by reactorsthey would swing nearly alike. The generators of station E were not represented, except in one case in which the fault location was near this station. For other fault locations it appeared that these generators were electricaliy close enough to the metropolitan system J and far enough from the fault so that they would not have much effect. In such cases the load of station E was set to represent the true load there less generation there. At station I two groups of generators on separate busses were represented. All generators were represented by transient reactance and voltage behind transient reactance. At station A the reactances of the three generators were represented by separate impedance units connected at one end to different bus sections (separated by reactors) and connected at the other end to one power source, the voltage of which represented voltage behind transient reactance of all three generators. Generator A1 had two windings, the self and mutual reactances of which were represented by a Y circuit. In accordance with usual practice voltages behind transient reactance were kept constant during the study. Data on the inertia, speed, and kinetic energy of each actual and equivalent generator are given in Table 8. An interval of 0.1 sec. was used in the point-bypoint calculations. The outputs of the various machines, the values of the loads, and the bus voltages were adjusted to typical values (shown on Fig. 31) as nearly as possible to values observed on the system with sectionalized operation at time of maximum system load. Loads were represented by constant impedances, which were not changed after the fault was on. The most severe fault location is ordinarily close to the sending end of the system. Therefore it is logical to try first a fault so located. The worst fault location on the system of Fig. 31 would be on the low-voltage bus of station A. A three-phase fault here would entirely interrupt any exchange of power between the generators of this station and the rest of the system. This bus, however, was of isolated-phase construction; therefore the only type of fault which could occur here was one-line-to-ground, which is not a severe fault from the standpoint of stability. The closest point to the bus at which a three-phase fault could occur was on the high-voltage side of one of the transformers (there was no high-voltage bus at A) or on the far side of a set of reactors feeding a local load (fault location 2). As a matter of fact, a more severe fault location than either of these was on the bus of sub-

STUDY

305

3

TABLE 8 INERTIA OF GENERATORS (STUDY

Rating (Mva.)

WR2 (lb-ft.2 )

AS

100 50 50

859,000 475,000 475,000

A, total

200

1,809,000

Generator

At AS

Et E2

ES E4

12.5 12.5 35 30

51,000 51,000 357,000 29,100

E, total

90

Ila Ilb

50 60

390,000 391,000

11, total

110

781,000

ISa ISb

35 100

243,000 750,000

IS, total

135

993,000

3)

Speed (r.p.m.)

Stored energy (Mj.)

1,800 1,800 1,800

640 365 365

6.40 7.29 7.29

1,370

6.85

38 38 266 87

3.04 3.04 7.60 2.90

429

4.76

291 292

5.81 4.86

583

5.30

181 560

5.17 5.60

741

5.49

1,800 1,800 1,800 3,600

1,800 1,800

1,800 1,800

H

11M

7.89

25.2

18.5

14.6

station B, where a three-phase fault would entirely interrupt the flowof power between generators A and the other generators. It wasimmaterial, while the fault was on, whether it was on one or another bus section of substation B or whether it was just outside the substation on anyone of the seven lines. The worst location as regards conditions after the fault was cleared, however, was on the upper bus section (fault location 1), where clearing a fault would require the opening of a bus-tie breaker and of three line breakers, whereas a fault in any of the other locations mentioned would require the opening of fewer lines. Consequently, the effect of a three-phase fault at location 1 on the bus of substation B was examined first. A three-phase fault at this location was known to put station A out of step if slow breakers were used. Therefore 8-cycle breakers and 1cycle bus differential relays were assumed, giving a clearing time of 9 cycles. The swing curves are shown in Fig. 32. Clearly the system was stable, although without much margin. Twelve-cycle clearing would probably be too slow.


306

100

V

e

~

~

60

E

u Q) "ii)

40

~

V

Q)

\ 20

~V

V

~

f--

Faultcleared completely 9 f"'v

'-

"l

<,

.....

I--. t---.

",.

~

~

----

I

~~-

/ ~ ~ "".,..--

It- Faulton I

o

" r-, r-,

1/

I

1i".o e

<

A-- r-;

~

~ Q) Q)

r--- ~

-~

./

80

-0 jij

~

~'

-

............... ~2 ~,

"

~

¥ 'r-J

I

o

--

8

32

24

16

48

40

'"

.........

K-~ ~ r-I""""""

56

64

72

Time (cycles) FIG.

32. Swing curves, Study 3, fault at location 1, cleared completely in 9 cycles. System stable.

100

.

80

-:

~V

~

~

~

1/

~

I I

..................

Fault cleared 15~

12

~~

-

~

~1,,7

~~

k-- Fault on- ~ I

-~

-11

J~

I

8

~

"'

I

o o

r-,

VII

~~

20

-- -

A

~

1 I

16

24

32

40

48

Time (cycles) FIG.

33. Swing curves, Study 3, fault at location 2, cleared in 15 cycles. System stable.

STUDY 3

307

It could be inferred, without taking more swing curves, that the system would likewise be stable for a fault on either of the other bus sections at substation B cleared in 9 cycles, but would be unstable with the slow breakers. Therefore all the breakers at B should be speeded up, and each bus section should be equipped with high-speed protection. It could be inferred further that the system would be stable for a fault close to substation B on anyone of the seven lines cleared in 9 cycles at both ends. Whether sequential clearing would give stability 60 _ _-...--r--~~-.... is not yet clear, but this point will be considered later. Attention was next turned to fault ~ location 2, separated by a reactor ~ 40 I---+---+~ from the low-voltage bus of station ~ o A. Fifteen-cycle clearing time was 12 assumed, corresponding closely to 20 ~~~..........-t-e~~ the existing relay and breaker times ~ 11 of 6 and 8 cycles, respectively. The ~ swing curves are given in Fig. 33. J o~...._ ........._~-- ...... Again, the system was stable. 24 8 16 o Time (cycles) Fault location 3, just outside station A on one of the lines to sub- FIG. 34. Swing curves, Study 3, fault at location 3, cleared at both station B, was considered next. ends in 9 cycles. System stable. Nine-cycle clearing was assumed at both ends. The swing curves, shown in Fig. 34, indicate that the system was stable by a big margin. Sequential clearing will be considered later. The next fault was taken at location 4 near station E on one of the lines to substation C. The machine at E was represented by a power source on the supposition that this machine would be a significant factor in the stability of the system. Sequential clearing was assumed, 9 cycles (1 cycle relay time plus 8 cycles breaker time) at E, where the breaker already was an 8-cycle one, and 80 cycles (60 cycles relay time plus 20 cycles breaker time) at C. The swing curves are given in Fig. 35. The system was stable and would be so even if the line were not opened at C. There seems not to be much margin, however, in the clearing time at E. For this fault location the existing breakers were satisfactory. When the fault was taken at the other end of the same line (location 5), however, 21-cycle clearing at C (1 cycle relay time plus 20 cycles breaker time), followed by 68-cycle clearing at E (60 cycles relay time plus 8 cycles breaker time) was not fast enough for stability. The

-

:a

!

120 r-----r----,...--~-__r__-_,_--.,...__-~--

100

&/)

Q)

80

~

aD

Fault cleared at C

Q)

"tJ

co

.g 60 1J Q)

'ii Q)

bo

e

-e 40

20

I.-I---.l----. ... 12 36

O'-......

o

J

-~-----

24

48

60

....- . . . . . . 84

-~

96

Time (cycles)

FIG. 35. Swing curves, Study 3, fault at location 4, cleared at station E in 9 cycles and at substation C in 80 cycles. System stable. 140

r

/

/

120

en

bO

"'0

80

~

~~

60

Q)

./

co ~

40

----

o

..........

~

I

I

I

Fault cleared at station C

21'"'-'

68'"'-'

--J I

~~

~

I I

~~

I

_1/

.......... J

16

I

I

I

......... ........... 8

I

I I

~

./

;

~

I

I I

~

;

~ll

I

J ____

I I

o

Fault cleared at station E--.

I I

Fault on-

-

V-

I-

I

~~

~-

20

I

I

J II

tV u "ii

I

I

V

~

Q)

:

V

I I

Q)

I I

j~

I

100

!

~

I

-

I

!

I

I

24

32

40

48

56

64

72

Time (cycles) FIG.

36. Swing curves, Study 3, fault at location 5, cleared at substation C in 21 cycles and at station E in 68 cycles. System unstable. 308

STUDY 3

309

swing curves of Fig. 36 show that station A went out of step with the rest of the system. Therefore the breaker at C must be speeded up. Nine-cycle clearing at both ends would certainly give stable operation, because 9-cycle clearing sufficed in the more severe fault locations 1, 2, and 3. Sequential clearing (not tried) probably would be satisfactory. Attention was now turned to the circuit from B to D, and a threephase fault was taken at location 6 near D. Generator 12 was not 140 - - - _ -_ _-...-----.~...,.___,...._-~__r____r__=__.,....__r____r_r_...__ ......

I

120

-e

1I~~>--FaU" cleared in 27'"

100

.1/

( /) Q,)

1/

bO

Q,)

"0

80

(ij

I

u t)

'':;:

-

;

I I I I I Fault cleared at B'-.. --L.-68"" ~

T

I

Q,)

Q)

;

60

Q,)

~

c

I

<

20 '--

/~- --~~ ~

......

_ ~ ~

o

r

I

I

Il-r---'----~~..........~- _ ...... I'--Fault cleared in 68/V

O'--...I.....

FIG.

!

I

40

8

16

:

__..._"___'_....__....._........

I

~-..~

24

I

32 40 Time (cycles)

48

56

64

72

37. Swing curves, Study 3, fault at location 6, cleared at substation B in 27 cycles, system stable; or in 68 cycles, system unstable.

represented for this case, because, with a fault at 6 either on or cleared,

12 could exchange no power with station A and hence would have no effect on the swinging of A with respect to the infinite bus J. For similar reasons the clearing time at substation D was immaterial to the stability of A. The clearing time at substation B was first assumed as 68 cycles (60 cycles existing relay time plus 8 cycles new breaker time) ; generator A pulled out of step as shown by the swing curves in broken lines in Fig. 37. The clearing time was then reduced to 27 cycles, resulting in stability without much margin, as shown by the swing curves in solid lines in Fig. 37. The required clearing time of 27 cycles or less at B was too short to permit selective operation between the

310


breaker at B and the 20-cycle breaker to the left of D for a fault just to the left of D. Either the breaker to the left of D must be speeded up, or else carrier-current relaying must be installed on the line from B to D. With an 8-cycle breaker and J-cycle relay at D, and a margin of 20 cycles (required by good practice), the clearing time at B would be 1 + 8 + 20 + 8 = 37 cycles, which was still too slow. Hence carriercurrent relaying was needed, and by its use the clearing time at substation B could be reduced to 9 cycles. As already stated, the clearing time at D was not important; therefore the breaker at D need not be changed. Consider next fault location 7 near substation C on the line from B to C. Before the fault was cleared, conditions were the same as they B

E

A

c

• 8 - cycle breakers needed. - - - Carrier - current relaying needed. FIG. 38. Location of 8-cycle breakers and carrier-current relaying necessary to maintain stability with interconnected operation under the conditions of Study 3.

were for the fault at location 5, for which 21-cycle clearing was too slow. Hence the breaker at C must be speeded up to 8 cycles, giving a clearing time at C of 9 cycles. After the breaker at C was opened, conditions were similar to what they were for the fault at location 6, except that the shock during the first 9 cycles was greater for location 7 than for 6, as the reactance between the fault and the bus of substation B was slightly less for location 7 than for location 6. Therefore the critical clearing time at B for a fault at 7 must be considerably less than the 27 cycles found for location 6. A clearing time of 68 cycles (existing relay time of 60 cycles plus new breaker time of 8 cycles) was far too slow, and even 37 cycles (second-zone impedance relay time of 29 cycles plus breaker time of 8 cycles) was too slow, Carrier-current relaying was therefore needed on each of the two lines between Band C. For similar reasons carrier-current relaying should be used on each

STUDY 4-DESCRIPTION OF SYSTEMS

311

of the four lines between A and B. Sequential clearing of end-zone faults appeared to be permissible, however, on the two lines between C and E. Eight-cycle bus-tie breakers and high-speed bus relaying should be used at stations C and E, as well as at B. Faults were not taken any farther from station A than stations D and E, because it was known that station A would stay in step, even with the existing breakers and relays. The conclusions of the study are set forth in Fig. 38, which shows'the locations where 8-cycle breakers were required and the lines to which carrier-current relay channels should be added. STUDY 4§

Three power companies (herein called companies A, B, and C), operating in neighboring states but having no physical connections with one another, were considering the construction of interconnecting transmission lines to permit the interchange of power between them in either direction. Study 4 was made to determine (a) the transmission capacity or power limit of the proposed interconnecting lines, (b) the required capacity of line terminal equipment, such as transformers and shunt reactors or synchronous condensers, and (c) the modifications necessary within the three systems to permit utilization of the full power capacity of the interconnecting lines. Description of systems. The systems involved in the study are portrayed by the impedance diagrams (Figs. 39 and 40), and they are also described in the following paragraphs. Stations of these systems are designated by two-letter combinations, the first letter of which denotes the company owning the station. Company A operated a metropolitan power system having a steamelectric generating station of 112-Mva. capacity (station AA) which fed a number of substations through a l3.8-kv. transmission network (shown in upper right-hand corner of Fig. 40). Station AA was connected through two 66-kv. lines to substation AB, from which connections were made to companies D and E and-by one of the proposed lines-to company B. Company B had a system consisting of two distinct parts. (See Fig. 39.) One part, shown on the left-hand side of the diagram, had two steam-electric plants (station BF, 25 Mva., and station BG, 32.5 Mva.) a few miles apart. These plants served the area in which they §Information concerning this study was obtained through the courtesy of Ebasco Services, Inc., New York City, with the concurrence of the operating com... panies concerned.


312

lOMv. . .

DC

- - Usbnllines. - - Proposed lines.

0-

Synchronous condenser,

0Ste.m·· electric I.ner.ti", ~st.tlon. rji\.-

v:..;-

Hydroelectnc I.ner.tin, 'tatlOn.

0- [qulv.le~t lenerator ~

representln,.power system.

LOid.

:I: :'~r~~~=.:n=...

+

Tr.ns'ormer.

39. Study 4, part 1. Impedance diagram showing impedances of transmission lines and transformers in per cent on 4o-Mva. base. Values of capacitive susceptances of lines are in per cent on the same base. Proposed interconnections are shown in broken lines.

FIG.

STUDY 4-DESCRIPTION OF SYSTEMS

313

AIC

H DA

~

(I:U~:~)

DA 110 kv

'i.

~

N

3.28

DB HOb -

DC 4.15 ltv. N

In

::

3.28

~

~

.;-

it

"!

~ C?

~

fIi

cd

;-

,.,;

+'

III

....,.,;

N

2 C

7~

",

~

aMva.

1.!9

:;-

DC 110 kv.

AG

138 ltv.

I

iii

+ ~ ltl

;-

AC

0;

~ 0

~

~ It' N N

2

.;-

~

s

;~

cd

AA

::

~

~

~

AJ

:;-

II' IQ IQ

~+

IE

~

AD

13.8

N

~ .,

20 Mva.

+ In

... 'It

C?

~

40 Mva.

H

DE 110 kv

R

D F69 kv.

/'Ii

:;-

,...

~ 25

Mvt. G

9.92

DF 110kv

SA 601tv

1210

8j:t

System C equlVlltnt

+

all ~

FIG.

CA 1l0kv

~

N

46. Study 4, part 2. Impedance diagram with values in per cent on 25-Mva. base. Line capacitance values are in microfarads on calculating board.

314


were situated and fed a number of smaller towns through a 60-kv. transmission system. The other part, shown on the right-hand side, had one steam-electric plant (station BD, 50 Mva.) feeding a number of substations through 66-kv. radial transmission lines. These two parts were ed through a two-circuit 132-kv. line about 100 miles long from substation BC to station BD. The system of company B was: interconnected at substation BA to the system of company F and at station BD to the system of company G. Company C, having an extensive system with both steam and hydroelectric plants, was interconnected with several other large systems in adjacent states. It is represented in Figs. 39 and 40 as an equivalent generator on the bus of substation CA. Company D operated system (shown in Fig. 40) which included several hydroelectric stations (DA, DE, DF, DG, and DH) of capacities ranging from 8 to 40 Mva. and a I IO-kv. transmission system connecting these generating stations to one another, to substations, and to companies A and E. This system extended several hundred miles from the city served by company A. Company E had a 28-Mva. steam-electric plant located in a city about 60 miles distant from company A. Company F had a large system; with both steam and hydroelectric plants. Of these only station FA, the plant nearest system C, was represented individually in the study, the remainder of the system being represented by a single equivalent generator (shown in lower lefthand corner of Fig. 39). Company G had steam plants at GB and GD and' a hydroelectric plant at GC. It was interconnected with company B at station BD (lower right-hand corner of Fig. 39). Company H operated a metropolitan system, situated about half way between companies A and B but not connected to them. Its generating capacity was approximately 275 Mva. (See Fig. 40.) A list of synchronous machines of companies A, B, D, E, and G will be found in Table 9. As the systems of companies C, F, and H were not represented in detail, their machines are not listed. Proposed interconnecting lines. Two lines were proposed, the first connecting company A to company B, the second connecting company B to company C. Both are shown by dashed lines in Fig. 39. For the first line two alternative routes were studied, one from substation AB to station BD, the other from substation AB to substation Be. The lengths of these routes are about 280 and 244 miles, respectively. The second line was to have terminals at BD and CA, and a length of about 256 miles.

a

STUDY 4-PROPOSED INTERCONNECTING LINES

315

TABLE 9 LIST OF SYNCHRONOUS MACHINES (STUDY

Number Station of Like Kind of Machine Units

AA " "u

1

Steam

1

H

1 1

u

u

1

u

" H

,

Rating kv.

13.2

" "

13.8 u

Total

4)

Total

Kinetic Energy

Xd

WR2

(%)

(kilo-lb-

18,750 1,800 H 25,000 H 25,000 31,250 " 12,500 3,600 112,500

28.0 24.0 21.0 11.0

"

140 207 237 237 15.5

104.8 154.0 176.4 176.4 46.4 658.0

6,250 3,600

14.0

10.2

29.9

kva.

r.p.m.

ft. 2 )

(Mj.)

AJ

1

Steam

BA

1

Condo

4.16

3,500

720

29.0

38.9

4.7

1

Condo

12.47

H

"

3,750 3,750 7,500

720

1

29.1 32.0

38.9 106.3

4.7 12.7 17.4

BC u Ie

BD

"

Total

1 1

Ie

12.0

"

"

18,750 1,800 31,250 " 50,000

18.0 22.0

102 235

12.5

25,000 3,600

11.0

62*

186

7,500 3,600 12,500 1,800 32,500

15.8 19.0

19* 85.3*

56.8 64 184.8

14,000 42,000

150

36.0

Total

1

Steam

BO

1 2

Steam

DA

"

Steam

BF

" "

13.8

Ie

4.16

"

Total

13.8

76.4 176 -252.4

38.8 116.4

3

Hydro Total

DC

1

Condo

4.15

8,000

900

34.9

DE

2

Hydro Total

6.9

10,000 20,000

180

35.0 4,650

34.8 69.6

DFl DF2 DF

2 1

Hydro

6.9

10,000

"

u

20,000

180 164

35.0 4,650 42.0 9,870

34.8 60.8 130.4

DG

3

H

a

" *Estimated.

40.5

40,000

Total Hydro Total

7,500

6.9

"

2,750 8,250

112.5 39.0

1,250

7.6

3.6

-10.8


316

TABLE 9 (Continued) Number Station of Like Kind of Units Machine

Rating

kv.

kva.

r.p.m.

,

Total

(%)

Kinetic Energy (kilo-Ib- (Mj.) ft. 2 )

Zd

WR2

2

Hydro Total

13.8

14,500 29,000

257

32.0

1,950

29.6 -59.2

DI

1

Condo

13.8

15,000

900

35.7

93

17.4

EA

2

Steam

4.15

II

1 1

"

"u

8,575 3,600 II 6,250 5,000 " 28,400

12.0 14.0 16.0

19.6 12.4 10.0

58.4 37.2

900

35.1

38.0

DH II

II

" EA

Total

1

FA

II

7,500

162.4 31,250 1,800 33,000 300 64,250

19.0 36.0

414 6,200

164

32.0

1,010

Steam Freq.chr. Total

4

Hydro Total

5,000 20,000

1

Steam

15,625

II

GD

4.15

1 1

II

GO

Condo Steam Total

II

GB

"

29.9 -183.9

7.1 121

23.0

Both interconnecting lines were to be of 154-kv. single-circuit construction, with hollow 250,OOO-circular-mil copper conductors of O.766-inch diameter, ed by wooden H frames. The equivalent spacing between phase conductors would be 17 ft. Ground wires would be installed to protect the line against lightning. The necessary terminal equipment would include transformers, shunt reactors or synchronous condensers, circuit breakers, and protective relays. The transformers at substation AB would have three windings: high voltage, 154 kv. Y; low voltage, 66 kv. Y; and tertiary, 13.8 kv. ~. Those at the other end of the A-to-B line (at either BC or BD, depending upon the route selected) would be 154-to-132 kv. Y...connected autotransformers with 12.5-kv. d-connected tertiary windings. For the B-to-C line the transformers at station BD would be autotransformers similar to those on the A-to-B line, and, even if the terminals

STUDY 4-SCOPE OF THE STUDY

317

of both lines were to be at station BD, it was assumed that separate transformers would be used for each line. At substation CA 154-to110-kv. autotransformers with tertiary windings would be required. The tertiary windings of these transformer banks would be used for connection of shunt reactors or synchronous condensers to supply the reactive power needed to hold the voltages of the terminal busses at suitable values. They would also provide a path for zero-sequence current. To attain reliable transmission of power over the proposed singlecircuit interconnections, without the aid of any parallel ties between the systems, it would be necessary to use high-speed clearing of faults on these lines and high-speed reclosure of terminal breakers. Obtainable circuit breakers had an opening time of 5 cycles and a reclosing time of 20 cycles with three-pole operation. The assumption of I-cycle carrier relaying gave an assumed fault-clearing time of 6 cycles and a reclosing time of 21 cycles, measured from the instant of occurrence of the fault. Scope of the study. The study included power-flow and transientstability studies. The power-flow studies served to determine: (1) approximate steady-state power limits of the interconnections, (2) the reactive power which had to be supplied by reactors or synchronous condensers at the terminals of the proposed lines and elsewhere, (3) the required capacity of the terminal transformers, (4) the need for additional generating, transformer, or transmission-line capacity within the several systems, and (5) initial conditions for the transient-stability studies. Transient-stability studies served to determine: (1) the transient power limits of the interconnections for faults on the interconnecting lines, and (2) the required clearing times of faults on other lines to prevent impairing the power capacity so determined. The study was divided into two parts, made at different times and with different a-c. calculating boards. Part 1 dealt with the power capacity of the A-to-B and B-to-C interconnections for power flow in each direction, also with the choice between BC and BD as southern terminal of the A..to-B line; it dealt also with the effect on stability of interconnecting systems F and G to system B, and with various internal problems of systems Band C arising from the proposed interconnections. For brevity those portions of the study pertaining primarily to system C and to the B-to-C line are omitted from the present report. Part 2 was concerned with internal problems of system A, including the required speed of clearing of faults on lines within that system, and also with the effect on stability of connecting system H to a tap on the A-to-B line.

318


Loads. Estimated July and December peak loads three years after date of study were used. The July loads have practically the same active-power component as the December loads but have higher reactive-power components. Method of determining power limits. Approximate steady-state momentary power limits of the interconnections were found by increasing the electrical angle between bus voltages at major generating stations of the different power systems to approximately 45°. The practical steady-state power capacity was taken as 5 Mw. below the value so found, to allow for momentary fluctuations of about 5 Mw. above the average power carried by the line. Such fluctuations could be caused by sudden changes in loads or by faults at distant points. Transient-stability power limits were found principally by obtaining swing curves by the point-by-point method in order to determine whether the system would be stable or unstable with various values of initial power flow if a two-wire-to-ground fault should occur on the interconnecting line, followed by 6-cycle clearing and 21-cycle reclosure. The effects of faults on other lines were investigated in similar fashion, except that three-phase faults were assumed on overhead lines without ground "vires, on double-circuit lines, and on cables, and that various clearing times were assumed. The number of machines represented in these transient-stability runs varied from four to ten. In two instances the number of machines was reduced to two, power-angle curves were obtained by use of a calculating board, and curves of power limit versus switching time were then computed. The rated power capacity of the interconnecting lines was taken as 5 Mw. lower than the transient power limit, to allow for fluctuations. Simplification of systems. In representing the systems only major stations, substations, and transmission tie lines were included. Radial lines were represented as loads on the busses from which they radiated. For example, the entire 66-kv. system supplied from station BD was represented as a load on the BD 66-kv. bus. Lines were represented by nominal sr circuits, the capacitances of all lines on a given bus being represented by one capacitor on that bus. Proposed interconnecting lines were represented by several sections, depending upon location of proposed taps to other systems. Three-circuit transformers were represented by their equivalent Y circuits. If one branch of such a circuit had negative reactance (as indicated, for example, in Fig. 39 on the 66-kv. side of the transformer at substation AB), the following change was made in setting up the circuit on the calculating board: The branch with negative reactance was set up with zero reactance, and its negative reactance was com-

STUDY 4-SIMPLIFICATION OF SYSTEMS

319

bined with the greater positive reactance of the opposite branch, thus preserving the correct value of through reactance from primary to secondary terminals. TABLE 10 GROUPING OF SYNCHRONOUS MACHINES FOR REPRESENTATION BY POWER SOURCES ON THE CALCULATING BOARD (STUDY

4)

Power Source Numbers Station

DE, DF, DH

Run

Run

Run

Run

Run

Run

Run

Runs

1

2

3

4

5

6

7

8 to 11

1

1

1

I

1

1

- - --

1

DA,DG

2

EA

3

AJ

- -- - - -- - - - - - - - -l '- -- -- 2

2

2

3

3

3

4

AA (south bus)

BD BF

5

BG

7

System C

6

00*

GB GC,GD FA

Off

Rest of system F System H

1

1

2

2

3

3

4

4

5

5

6

6 --

00

00

Off

Off

- - - - - - - - - - - - - - --

- - --

- -- - - 2

AA (north bus)

Run

4

4

2

2

4

- - --

- - --

- -- -- -- - - -- -- 3 5 5 3 3 7 5 7 - - - - - - - - - - - - - - -Off 6 Off -- 6 ---- 4 8 4 8 7 4 6 - - - - - - - - - - - - - - -- - - - - - - - - - - -- - I 00

00

8

7

00

---9 8 ---9 Off 10

------00

Off

00

00

5

5

-6

--

00

Off

-6 -7 --

00

00

00

Off

-Off

-Off

-9

* 00 = Infinite bus. Synchronous condensers were not represented dynamically during stability studies, but merely by capacitors or reactors. System C was represented by an equivalent generator on the substation CA bus. System F, exclusive of station FA, which was shown individually, was represented by an equivalent generator. In stability studies the equivalent generators at CA and FA were regarded as infinite. Station DG was combined with station DA. Stations GC and GD were represented by an equivalent generator on the sub-

320


station GA bus. In part 1 of the study the portion of system D beyond substation DD was represented as an equivalent generator on the DD bus, and station AJ was combined with station AA. In part 2 the portion of system B near stations BF and BG (including transformers, lines, and loads, but not generators) was represented by an equivalent circuit (shown in Fig. 40), the terminals of which were: station BF 12.5-kv. bus, station BG 4.16-kv. bus, substation Be 132-kv. bus, and substation BA 60-kv. bus. Since generators were not included in this network, it was possible to change the generator reactances according to the number of machines assumed in operation at each station and according to whether adjusted synchronous reactances were wanted for steady-state runs or transient reactances for transient-stability runs. In many of the transient-stability runs further combinations of generators were made, as shown in Table 10; for example, stations BG and BF were combined in some runs. Swing curves, part 1. Twelve transient-stability runs were made in the part of the study considered. The swing curves are reproduced in Figs. 41 to 52, inclusive, and the conditions under which they were taken and the resulting conclusions concerning the stability or instability of each case are summarized in Table 11. The grouping of generators for each stability run is shown in Table 10. Each run will now be discussed. Run 1. The company B terminal of the A-to-B line was assumed to be at substation BC. As a step in finding the power limit of this line for northward power flow, the power sources on the board were adjusted to give 39.6 Mw. of power received at substation AB from Be. To supply this power, 13 Mw. was transmitted to system B from system C, and a new generating unit was assumed installed and operating at station BF, resulting in a need also of 20 Mva. of additional transformer capacity at BF. A two-line-to-ground fault was assumed to occur on the 154-kv. A-to-B line near the sending end. The assumed clearing and reclosing times were 6 and 21 cycles, respectively. The sending end of any line is ordinarily the worst fault location on the line from the standpoint of stability; but" since acceleration of the generators during the 6 cyclesin which the fault is on is small compared with that during the following 15 cycles In which the line is open and the systems are completely separated, a fault of given type anywhere on the line would have substantially the same effect on stability as a fault at the sending end. Inspection of the swing curves (Fig. 41) shows that the machines of system B swing approximately together and that they speed up. The speeding up is due to their dropping load during the disturbance. The machines of systems A, D, and E also swing

BC CA

AB BD AB

2LG

"

AB

"

"

35.4

"

" "

45.0

"

u

"u

+

2LG

" " "

3q,

AB

CA

"

BD

BD

154 kv.

AA(S) AA(N) AB

"

AA(N)

13.8 kv.

" " "

"

BD

u

AB

BD

"

"

" u

154 kv.

132 kv.

"

BC

Near Station

d/lo"

Location of Fault

Line Voltage 154 kv.

,

AF BD

" t

AF

"

AB

"

Be

Be

u

AB

"

18 18 6

12 18

" " "

6

9

"

6

A

"

21

.. ·. ·.

·.

" "

II

21

30

"

Reclosing 21

Switching Time (cycles)

4)

Clearing

r

(STUDY

On Line to

TABLE 11 STABILITY RUNS

*Systems F and G connected to system B with 0 + }Opower interchange. t Fault 1,000 ft. from station AA on 700,000 eir .-mil cable. tsystem H connected with 0 jO power interchange.

10 11 12

" " u

II

8

25.5

BD BD

12.4 26.0

9

AB AB

30.0

40.6

6 7

II

CA

3q,-3q,

5

40.4

58.0

AB

4

3

BD

"

2LG

Type of Fault

BC BD Be

Be

From

Rec'd at

-a

CA BC

13.0 45.6 21.4 40.0

39.6

Mw.

A.

Power Transfer

BD

2

1

Run

r

SUMMARY OF

Stable Unstable Stable Unstable Unstable

Stable but critical Unstable Unstable Stable Stable

Unstable

Stable

Stable or Unstable

~

AA bus split

•

•

*

*

Remarks ~

tr:

~ to-

.....

~

>-

t-d

..til

;3

:;:0

~

a

Z 0

~ ~

in

~

,

Jo<

tj

d

322


together, but they slow down because they take on additional load during the disturbance to make up in their systems for the power which was received from system B. System C was represented as an infinite bus and, therefore, did not swing. Station BD, which was closer to system C than were stations BF and BG, swung less than did stations BF and BG, as though it were attracted by system C. The curves clearly show that the system was stable. 80

r-------,----.-----r---r-~-...._-_r_____,r____,..-__T'"-.....,.

I

"Of ,.... en

t...

~I~

~l~ ~I\O ;1.5 L.l. t I

40

~ Q)

"0 Q)

~ e ta

I

0

Q) ~

ta %! 0 :>

ti c

...

.!

..

-40

.5 cQ)

·Cii

c

l!'

to-

-80

- 120

1 (DE, DF. DH)

'-----a_.....&._~-....._'__

o

0.2

0.4

0.6

........

~

0.8

....

1.0

Time (seconds) FIG. 41. Swing curves, Study 4, part 1, stability run 1. 39.6 Mw. received at substation AB from substation Be, and 13.0 Mw. received at station BD from substation CA. Two-line-to-ground fault-near substation BC on proposed 154-kv. line to substation AB, cleared in 6 cycles, line reclosed in 21 cycles. Stable.

Run S. The power received at substation AB was increased to 45.6 Mw. To supply this increased power, the power sent to system B from system C was increased to 21.4 Mw., while the outputs of stations BD, BF, and BG were left unchanged. Systems F and G were connected to system B with zero interchange of active and reactive power. With the same fault location and with switching times the same as those in the preceding run, the system proved unstable. Therefore the power limit oj the A-to-B line for northward r'power flow is between J,.O and J,.5 Mw. received. The rated power capacity of the line may be taken as 35 Mw, Study of the swing curves (Fig. 42) shows that the machines

STUDY 4-SWING CURVES, PART 1

323

of systems Band G and of station FA swung ahead together. The rest of system F was represented as one infinite bus, and system C was represented as another infinite bus. The generators of system B did not swing as far ahead of system C in this run as they did in the preceding run, in spite of the increase in power transfer. This fact shows 80 ..--.....----r---,...__--r--_,._-_-

- __- _ -__

40

-... en

G) G)

r

"0

0

G)

00 e co

Wo .19

'0 >

System F as infinite bus

-40

n;

...e

.!

.5

't: -80 G) 'Cij

c

e

~

-120

- 160 '"'---..- --'--...L-....-"'---..L_--L.._-'-_..L-_I-....--J._...L.........J o 0.2 0.4 0.6 0.8 1.0 1.2 Time (seconds)

42. Swing curves, Study 4, part 1, stability run 2. 45.6 Mw. received at substation AB from substation BO, and 21.4 Mw. received at station BD from substation CA. Systems F and G connected to system C with 0 + jO power interchange. Two-line-to-ground fault near substation Be on proposed 154-kv. line to substation AB, cleared in 6 cycles, line reclosed in 21 cycles. Unstable. FIG.

that the connection of systems F and G to system B has a beneficial effect on stability. Nevertheless, the generators of systems A, D, and E, again swinging together, swung farther behind than previously and pulled out of step with the other generators. Hence the power limit, either with or without the connection of systems F and G to system B, is between 40 and 45 Mw., and it may be concluded that the connection of those systems has only a small effect. Run 3. One consequenceof selecting substation Be as the company


324

B terminal of the A-to-B line, while station BD is the terminal of the B-to-C line, is that, when it is desired to transfer power from company C to company A, this power must flow over the double-circuit 132-kv. 120__......__---t"---.--......--_-__-....--__-

__

-~-_

~I=

j.~

80

~I~

;1. 5

'-

-i f -i

I

40

-a

...

0

f

System F

....co c 11 -40 ~5

.. c

.I! en

c

I!

to-

-80

- 120

t---+-..--+---+---f---of---+----f~lE_FI~~-

""""""-_......._ ....... -...l"----""_-'"'-_....._.-..-_.a..-..... 0.4 0.6 0.8 1.0 1.2 Time (seconds) FIG. 43. Swing curves, Study 4, part 1, stability run 3. 40.0 Mw. received at substation AB from substation BC, 58.0 Mw. receivedat substation BC from station BD, and 40.4 Mw. received at station BD from substation CA. Systems F and G connected to system C with 0 jO power interchange. Double-circuit threephase fault near station BD on the 132-kv. lines to substation Be, cleared in 9 cycles, lines reclosedin 30 cycles. Stable but critical. - 160 ..........-"'"

o

0.2

+

line from BD to BC, which at the same time may be required to carry power from station BD to loads in the left-hand portion of system B. To test stability under such conditions, run 3 was made. Generation was adjusted to send 40.0 Mw. (slightly below the power limit found above) to AB from BC and 40.4 Mw. to BD from CA. At the same


325

time station BD was generating more than was required by the load in the adjacent part of System B, and the surplus was transmitted to the distant part, resulting in a total power of 58.0 Mw. received at substation BC over the 132-kv. lines from BD. No new generators were assumed to be running. Systems F and G were connected to system B with zero interchange. A three-phase fault involving both circuits] of the 132-kv. line to Be was assumed to occur near BD and to be cleared in 9 cyclesby simultaneous opening of the breakers at both ends of both circuits, followed by their reclosure in 30 cycles. The swing curves (Fig. 43) show that the system was stable ·but critical. The machines of systems A, D, and E, again swinging together, sloweddown and almost, but not quite, pulled out of step with the other machines. Stations BD, GB, GC, and GD (which are on the sending side of the opened line) swung ahead and then oscillated; of these stations, BD, which is nearest the fault, swung ahead most rapidly. Stations BF and BG slowed down at first but speeded up after reclosure of the faulted lines. The power limit was slightly over 58 Mw. on the BC-BD lines or slightly over 40 Mw, on the AB-BC line. The latter figure is close to the limit already found for a fault on the AB-BC line and is therefore satisfactory. If the fault should involve only one circuit, the power limit would be higher. Run 4. It was next decided to investigate the power limit of the A-to-B line when carrying power southward (from A to B). The power sources were adjusted to give 35.4 Mw. received at Be from AB. To supply this amount of power, the output of station AA generators was increased to 22 Mw. above the rated capacity of the station, requiring that a new generating unit be assumed installed and operating there; and, in addition, 30.5 Mw. was received at substation AB from system D. To absorb this amount of power in system B, station BF was shut down. The B-to-C line was closed with zero active and reactive power interchange as measured at station BD 132-kv. bus. Systems F and G were not connected. Additional transformer capacity was required as follows: 13 Mva. at station AA, 5 Mva, at substation AB, and 12 Mva. at substation Be. A two-line-to-ground fault was assumed to occur near AB on the proposed 154-kv. A-to-B line, with the usual switching times. The swing curves (Fig. 44) show the system to be unstable. Systems A, D, E and system B, although initially swinging apart, came back together and remained in synchronism, but both these groups pulled out of step with system C. [Such a fault might consist either of a three-phase fault on each circuit or of 8, one-line-to-ground fault on one circuit and a two-line-to-ground fault on the remaining phases of the other circuit.

326


Run 5. Conditions in this run were identical to those in the last ron except that systems F and G were connected to system B with zero interchange. The combined system was again unstable (see Fig. 45), but this time systems Band C stayed in step with each other, whereas 320 ---------....--r------r----r---y---.--.---r--,

en

~I

~.f-----+----I \0 .sl -;; 240

e ';1

-

...., ...---+---1

e r

u

"'0

~I

CD

J

ftS

200 1 - - - 1 - - - 4 - - - 1

f

Ci

E 160 1----I---+-~~-!....-4--~-+--+_-+_-..---+__#_-+---+-----t

~

C CD

'iii

c

~ 120 1--,..,~~---+---JC.-+---+--4----+--+_-+_-+-i__l_-+---+---t---i System C 80

as infinite bus

I----J---+-~~--+--+---+-----:~-J#_---+---r--T----t---t

40 '--......_.-L._-A--'-...&.-_...-_.l..--.lIo--..Io_--'-_~_.....- - - '

o

0.2

0.4

0.6

0.8

1.0

1.2

Time (seconds) FIG. 44. Swing curves, Study 4, part 1, stability run 4. 35.4 M w. received at substation BC from substation AB. Two-line-to-ground fault near substation AB on proposed 154-kv. line to substation BC, cleared in 6 cycles, line reclosed in 21 cycles. Unstable,

systems A, D, E pulled ahead and out of step with Band C. The motion of the A, D, E machines was almost exactly the same in both runs, but the motion of the system B machines was greatly reduced by the connection of systems F and G. The power limit of the A-to-B line for southward power flow was less than 35 Mw. whether systems F and G were connected or not, and was estimated as being between

327


30 and 35 Mw.

The practical power capacity was taken as 25 Mw.

Note that the transient-stability limit for southward flow was less than that

for northward flow on the same line. Run 6. The company B terminal of the A-to-B line was now shifted from BC to BD, and northward power transfer was investigated. 300

I

280

I I

i

I

"2' Cl)1

I

~I~

~,

-

ar....

240 ~~I ~I

~

I

I-

~I

I

~ 200 f - "til

fco

8L

I

~ca 160

j

1: QI

.~ 120

...e

80

I

LV ;

V V

1--'

----

I

I

I

o

~I

I V

I

YI

V

/

System C as infinitebus

yo ~

. L~

-.L

V

~~

I~ ,,-I 4 (BG).

I

(AA. AJ, EA)

.~

(3(B~ ~~

I

0.2

14"

V 5 (~~stem G~

I

-

~

~

V i/~

I

I

~

I

40

Y

I

rr

V

V

I

~,

~

./

I

ml

"31

,,-

~'I

....

V

V /

/

~t~

~I

"0

V

1 (systemDl')

-8r~ cu_

I

V

/

""",-

r---""'"

_~

V

~ ~ ~ ""7

~V

""

6(FA)

.

V

System F as Infinite bus

!

0.4

0.6 Titne (seconds)

0.8

1.0

1.2

FIG. 45. Swing curves, Study 4,part 1, stability run 5. Same as run 4 except that systems F and G were connected to system C with 0 + jO power interchange. Again unstable.

First the value of 30 Mw. received at AB from BD was tried. Station BF was generating 54 Mw. with the new generator operating, and 23 M va, of additional transformer capacity was required there. No power was received from system C, and systems F and G were disconnected. Generation at station AA was reduced to such an extent that to supply its local load necessitated the receipt of power from system D as well as from system B and required 11 Mva. of additional trans-


328

former capacity at station AA (slightly less than the amount required in run 4). A two-line-to-ground fault was applied to the A-to-B line at the sending end (station BD), cleared in 6 cycles, and reclosed in 21 cycles. The system was stable; all the swings were of small amplitude. (See Fig. 46.) Run 7. The power transfer on the A-to-B line was increased to 40.6 Mw. received at substation AB from station BD. To provide the increased power, 12.4 Mw. was received at BD from system C. To absorb the increased power, the generation at station AA was further 200 , . - - . . , . - _ , _ - - . . - - - y - - - r - - - . , . - - - - - , - - . . , . . - - - . - - - - - r -_ _

Q)

"60 e

«S

~

B g

co

E .!

.5

~

'in

120 ~~-""'-

L....

1 (System D) 80 t--_--+----+-~_+__-+_~---+--~--+-__t

c

cu

.=

t----t---I--~IIIIIIIl:-_+__-+_~~ 2 (AA, AJ, EA)

40 a-----'_--.L.._.....-'--......_ " " - - - - " _ . - . . _......._ ~_ _ o 0.2 0.4 0.6 0.8 1.0 Time (seconds}

46. Swing curves, Study 4, part 1, stability run 6. 30.0 Mw. received at substation AB from station BD. Two-line-to-ground fault near station BD on proposed 154-kv.line to substation AB, cleared in 6 cycles, line reclosed in 21 cycles. Stable. FIG.

reduced. Power flow conditions were much like those of run 1 except that the generation at AA was smaller, the difference being made up by power received from system D. Also, systems F and G were connected to B. The interconnected systems were stable (Fig. 47), with considerably more margin than there was in run 1 (Fig. 41). The improvement might be due in part to the change in company B terminal and in part to the connection of systems F and G. The power limit of the A-to-B line with terminal-at BD was not exactly determined, but it appeared to be somewhat higher than the corresponding limit with line terminal at Be, at least for northward power flow. Swing curves, part 2. In part 2 of the study the required clearing time of faults on company A's 13.8-kv. system and the effect on stabil-


329

ity of an interconnection with company H were investigated. The network of Fig. 40 was set up on the calculating board. As compared with the set-up of part 1 (Fig. 39), the principal changes were: (a) The base was changed from 40 Mva. to 25 Mva. (b) The l3.8-kv. network of company A was represented in detail, including the separation of station AA bus into two sections by a bus reactor, with the generators of each section represented separately, and the separation 200

r----r---,.---r---r---.....---...r----r-_r_-.......--.....---,

- 160 I----I---+---+--.a..--t-+---+----t-- 6 (Ge, GD)

~

I

~ '0 -

5 (GBl,..,...--+----t 7·(FA~)

I ~ 1 ~ - - t ' - - - t ' - -.....- - + - - - - - t - - i - - - f - - - t - - - - f ml"U ul~

3 t ~ -+---+--+--.--+----+----t---..+---+--t ~

1'-

O--......-~- ..........I.-..&.-_"--.......-

o

0.2

0.4

0.6

......_ - - - - - -.....

0.8

1.0

Time (seconds) FIG. 47. Swing curves, Study 4, part 1, stability run 7. 40.6 Mw. received at substation AB from station BD and 12.4 Mw. received at station BD from substation CA. Systems F and G connected to system C with 0 + jO power interchange. Two-line-to-ground fault near station BD on proposed 154-kv. line to substation AR, cleared in 6 cycles, line reclosed in 21 cycles. Stable.

of station AJ from station AA. The 13.8/66-kv. transformers at station AA were assumed to be of 25-Mva. capacity each, or 50 Mva, total, as found necessary in part 1. (c) The remote part of system D was shown in more detail. (d) The part of system B containing stations BF and BG was represented by an equivalent circuit. The company B terminal of the A-to-B line was placed at -station BD. Systems F and G were not connected. Zero active power and about 17 Mva, reactive power were supplied to system B from system C. Run 8. First the required clearing time of faults on the 13.8-kv.


330

system of company A was examined. The A-to-B interconnection was delivering 26 Mw. to station BD (approximately the practical power capacity of the interconnection as found in part 1). The output of station AA generators was 101.5 + j40.7 Mva. Since this output exceeded the capacity of the station, a new generator was assumed on the north bus, and the aggregate capacity was divided equally 160 ...----r---.--r---r-.,...--..,r-----r---r----,r----,--.,.--.,---, 140

120

en Q)

e t)I) Q)

~ 100 Q)

1io c: co

~

80 t--+----I-t--++--~~~-+--f-~f---+---+-----f--4

.B

g co

E

60

t---+~_t_-hr--.L-.-f"'"___I__r_~r---+--t-~t:::::;a.....Iiiii:::+---t~......

40

t--:3IIlI'f'"""'---::I~~t---#l--_t_-~--t-__:l~-,t----+---+-----f~r-f

20

~~~_f_--#t----t-~Lt._-~lL-+--+--..--+--f----I---1

.s

.5 C Q)

'Vi c

~

- 20 "'-

o

System as infinite bus ..-..-_""----'-_........_"--...... _ ........C _Il.o.1.0 0,4 0.6 0.8 0.2

.....-.--'_~

1.2

Time (seconds) FIG. 48. Swing curves, Study 4, part 2, stability run 8. 26.0 Mw. received at station BD from substation AB. Three-phase fault on 13.8-kv. feeder near station AA north bus, cleared in 12 cycles. Stable.

between the two busses. The voltages of both bus sections were held at the same magnitude, but a phase difference between them caused 5 Mw. to be transferred from the south to the north bus. From system D 25 Mw. was delivered to system A. One machine was assumed to be operating in each system D plant except station DA, where all machines were assumed to be in service. At station EA 15 Mva. of additional 110/66-kv. transformer capacity was required. A three-phase fault was assumed to occur near station AA on a 13.8-kv. underground cable feeder supplied from the north bus. With 12-cycle clearing the system was stable. (See swing curves, Fig. 48.)


331

260

240 220

200

180

,...

Ir

160

'0 GJ

co c 140 to

InI

g

120

Ci c

"G) .., lOa

.5

~

c: Q)

'iii

c

e

80

to-

60 40

20 0 - 20 .........._-'----a_--'-_...A..---"_........ ..... o 0.2 0.4 0.6 0.8 Time (seconds)

.....

_~

FIG.

1.0

....

1.2

49. Swing curves, Study 4, part 2, stability run 9. Same as run 8, except that clearing time was increased to 18 cycles. Unstable.

Run 9. With I8-cycle clearing of the same fault the systems were unstable (Fig. 49). Run 10. The purpose of this run was to show the effect of threephase faults farther away from station AA. The location selected was a point about 1,000 ft. from station AA on a 700,OOO-circular-mil cable supplied from the south bus. With the additional impedance of this cable between station AA and the fault, the voltage at AA re-

332


mained high enough to prevent the generator output from decreasing as much as before, so that even with I8-cycle clearing the system was stable (see Fig. 50). The extreme oscillations of the station AJ machine, as shown by its swing curve, should be discounted somewhat, as they did not represent the true motion of the machine but were the 200 r----.---r-.-......-......-r--,---r---r--.----.~__y_o-~__..

180 Xli

i

160

~I

f"04

! ..... 4D

"0

~

140

!

.;1 ~I

120

~I

.!! co

I

c

tV

I» 100

~

g

.

80

c

60

to c

~ .., .~

en C

~

40 20 0

- 20

.......

'--.-.._-L---a._~_"--__'__-'--.._

o

0.2

0.4

0.6

0.8

- . r . . _.....

1.0

1.2

Time (seconds) FIG.

50. Swing curves, Study 4, part 2, stability run 10. Same as run 9 except

that fault was 1,000 ft. from station AA south bus on 700,OOO-cir.-mil cable.

Stable.

result of inaccuracies caused by the use in the step-by-step calculations of a time interval too long in comparison to the period of oscillation of the machine. Run 11. The purpose of this run was to show the effect of removing the bus-tie reactor at station AA, leaving the north and south busses tied together through only the I3.8/66~kv. transformers and the 13.8kv. network. In every other respect it was identical to run 9. Com-


333

parison of the swing curves of run 11 (Fig. 51) with those of ron 9 (Fig. 49) shows that splitting the bus at AA decreased the rate at which systems A and B swung apart, but not sufficiently to allow stability to be regained. Although this run was considered unstable, 220 ------.....------,r-----r---w----r-.....-----,---t--r----, 200 180

160 ~

en

Q,)

~

1)0

-

140

Q,)

"'C

Q,)

ao e 120 to

~

S

g

100

-;; E .!

80

.5

....c Q)

.iii c:

...e

60 40 20

0 - 20 ~---------"'-...._I..---L.-...L-.--J._~_a..._.......,L.._.l....---'

o

FIG.

0.2

0.4

0.6 0.8 Time ( seconds)

1.0

1.2

51. Swing curves, Study 4, part 2, stability run 11. Same as run 9 except that station AA bus was split. Still unstable.

it represented a borderline condition, indicatingthat the systems would be stable under slightly more favorable conditions, such as appreciable arc impedance in the fault or decreased power transfer on the interconnection. Run 12. The purpose of this run was to show the effect on stability of connecting system H to the A-to-B line. System H was a metro-


334

politansystem located near the center of the proposed interconnecting line from AR to RD. It was represented as a 275-Mva. generator tied to the 154-kv. line through a 30-Mva. transformer, with zero interchange of active and reactive power. Power of 45 Mw. was trans180 160 140

,.... 120

~

if»

"".....

I., c &

~ ~

Ci

E

f

l S

100

80 60 40

·Iii c:

20 0 ~20

-40 0

0.2

0.4

0.6 0.8 Time (seconds)

1.0

1.2

FIG. 52. Swing curves, Study 4, part 2, stability run 12. 45.0 Mw. received at station BD from substation AB. System H connected to the A-to-B line with 0+;0 power interchange. Two-line-to-ground fault near substation AB on the proposed 154-kv. line to station BD, cleared in 6 cycles, line reclosed in 21 cycles. Unstable.

mitted to BD from AB, necessitating a new generator on AA north bus and requiring only one machine to be operated at BD. The power delivered to system A from system D was raised to 37.4 Mw. by increasing the output of the system D generators already in operation. With a two-line-to-ground fault at the sending end of the 154-kv. inter-

STUDY 4-POWER-ANGLE CURVES, PART 2

335

connecting line and with the usual clearing and reclosing times" the systems were unstable. The swing curves (Fig. 52) show that systems A, D, and E swung ahead together, whereas systems B, C, and H swung very little. The power limit of the A-to-B line for southward power transfer with system H connected to the line was thus shown to be less than 45 Mw. This limit was found more exactly as described below. 240

-V I

220

/

200

I

/"

I I

Q)

00

I;

I~

/1

l:c

I

I~

I

A,D,E, no fault

'"

-, '\

I~

I I

I<.J II

I

I

, I I

.YV I

80 -100

~Systems

I~

I

100·

~

I~

I

i20

""'"'<

1

/ v

~

-80

~

-60

I ~

~Systems

/ ""- A,D,E, --...... fault on

-40 -20 0 20 Rotor electrical angle (desrees)

40

60

80

FIG. 53. Power-anglecurves, Study 4, part 2, run 13. 5 Mw. receivedat station BD from substation AB. Three-phase fault on 13.8-kv. feeder near station AA north bus. Critical clearing angle, determined by equal-area method, is -34°; critical clearing time, 24 cycles.

Power-angle curves, part 2. Power-angle curves and the equal-area criterion were used for (1) obtaining a curve showing the critical clearing time of a three-phase fault close to the north bus of station AA required to maintain stability of systems A, D, E with systems Band C for any amount of power transferred from A to B; (2) determining the power limit of the A-to-B line for southward transfer with system H connected. Critical·clearing time of a three-phase fault near 8tation AA 19.8-kv.


336

north bus. To use the method mentioned it was necessary to reduce

the system to an equivalent two-machine system. Inspection of the swing curves for runs 8 to 11 showed that the machines of systems A, D, and E swung together and, therefore, could be combined with very little error to one equivalent machine. The machines of system B likewise swung together and could be combined, and system C was regarded as an infinite bus. It was decided to combine the machines 240 220 200 cJ)

t:: eu ~

iE

,

180

~ 160

...o

::J

...

Q)

~ 140

/

v:

V

/

v

/

I

t'O

00

,-5 I'~

I~

I-

I.~

I

I=E

IU

I

I

I I I

I I

80 -100

I

~V

-80

"-

1.=

I

I

"" ."

~

,e:

I

100

"<

leu 1"6"0

I I

120

...-Systems A.D,E. no fault

I

1

I I

Q.,0

~

I I

I

V

",-

~

-60

~

------

.....

...........

J

-40 -20 a 20 Rotor electrical angle (degrees)

I-Systems A.D,E, fault on.

""'"

~

40

60

80

FIG. 54. Power-angle curves, Study 4, part 2, run 14. 15 Mw. received at station BD from substation AB. Three-phase fault on 13.8-kv. feeder near station AA north bus. Critical clearing angle is -37°; critical clearing time, 18 cycles.

of system B with those of system C and to represent them all as an infinite bus. This procedure could be expected to give somewhat pessimistic results because the speeding up of the machines of system B to follow those of systems A, D, E was beneficial to stability. Power-angle curves were obtained on the calculating board by the following procedure: when possible, the initial conditions of each run were made identical to those of a previous load-flow study. The power received at station BD from substation AB was set to 5 Mw., 15 Mw., 25 Mw., and 35 Mw. in runs 13, 14, 15, and 16, respectively;


337

and the voltages of the 66-kv. bus at AB and of the 132-kv. bus at BD were adjusted to normal values. The phase angle of the power source representing the voltage back of transient reactance of the finite A-D-E machine was then advanced in steps of about 15°, both this voltage and the voltage representing the infinite bus being held constant, and at each step the power output of the finite machine was recorded. These results were plotted as the curves labelled "no fault" 240 220

/

200

V' I

l:i

I.~

I' "

I.e: ~

I~

1«1 I~

I

18

I

I

I I

~Systems

I

~

~~

L-.-

li

-40

-20

A, D, E, fault on

---...

~

!~ -60

A,D,E, no fault

-,

I~

IJ

-80

~

I

I I

-100

"<

~Systems

I

VI!

80

.........

~

I

/~

100

-

I

1/

120

-

0

20

~--

40

60

80

Rotor electrical angle (degrees) FIG. 55. Power-angle curves, Study 4, part 2, run 15. 25 Mw. received at station BD from substation AB. Three-phase fault on 13.8-kv. feeder near station AA north bus. Critical clearin.g angle is -33°; critical clearing time, 16 cycles:

in Figs. 53 to 56, inclusive. The angle scale in these figures is arbitrary and is not the angle between the internal voltage of the finite machine and the voltage of the infinite bus. For each value of power transfer a second curve was obtained in similar manner, except with the fault applied, and with the internal voltages held at the same values as before. The critical clearing angle for each value of power transfer was then obtained by the equal.. area method, and the corresponding critical clearing time was found from a swing curve calculated by use of the values of power read from the power-angle curve.

338


The kinetic energy of the A-D-E machine was taken as 1,361 Mi. The swing curves are given in Fig. 57. The desired curve of power transfer versus critical clearing time was then plotted and is shown in Fig. 58. The results of stability runs 8 and 9 are consistent with this curve: for 25-Mw. transfer the system was stable with 12-cycle clearing and unstable with 18-cycle clearing. 240--------.....------.---.-----,---,...----., I

~ -_~~systems

~

I ~ A,D,E• ./ -, no fault 220 1----+----4--~+--__+_----t'----_f__:::lIillE:_--t---t---t

Vi

~

'"

120 I - - - - f - - - - + - - ! - " f - - - - - f - - - i - - - + - - - - i - - - f - - - - - t 100 t----t--~........- + - - _ + _ - _ _ _ t - - _ + _ _ - _ _ P - - - - - t _ _ - _ _ t

80'---......- .....- -.....--....------......- .....--~-~ 60 80 -100 -80 .. 60 -40 -20 0 20 40

Rotor electrical angle (degree~) FIG. 56. Power-angle curves, Study 4, part 2, run 16. 35 Mw. received at station BD from substation AB. Three-phase fault on lS.8-kv. feeder near station AA north bus. Critical clearing angle is -45°; critical clearing time, 11 cycles,

Power limit oj A-to-B line with system H connected. The swing curves of run 12 indicate that the machines of systems B, C, and H may be regarded as an infinite bus and those of systems A, D,and E as a single finite machine. Power-angle curves with no fault and again with a two-line-to-ground fault near AB were obtained. It seemed permissible to use the same power-angle curves for all values of initial power transfer, because in runs 13 to 16 the voltage behind transient reactance varied only 3% over the range of power transfer that was considered. With the fault cleared, the power transfer over the inter-


339

connection became zero, but the output of the finite machine did not become zero because it was supplying local load. Its output with the line open, however, was independent of angle and was found from previous load studies, as follows: The total load on systems A, D, and E, when operating with zero interchange with system B, was determined from one load run, and the total load plus losses when delivering -20

0

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en Q)

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FIG.

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i J

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Time (seconds)


(c)

(cl)

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57. Swing curves used in connection with the power-angle curves of Figs. 53 to 56 to determine critical clearing times.

45 Mw. to system B was obtained from another load run. The difference between these figures gave the decrease in losses which would occur on opening the 154-kv. interconnecting line, normal voltage conditions being assumed. This decrease in losses was expressed as a percentage of the power sent into the 154-kv. line. To determine the output with the fault cleared, the percentage found as just described was added to the input to the 154-kv. line, and the augmented input was subtracted from the output of the initial operating conditions.


340

The critical reclosing time was found by use of the equal-area method and swing curves (Figs. 59 to 61) for transfers of 25, 35, and 45 Mw., using a clearing time of 6 cycles in every case. The clearing angle corresponding to the clearing time of 6 cycles was found by computing the swing curve up to this point. The critical reclosing angle was then 40 found by the equal-area method, and the corresponding reclosing time was obtained by extending the swing curve to this angle. ~ 30 ; A curve of critical reclosing time \ Unstable ~ versus power transfer was plotted E Stable and may be seen in Fig. 62. From "0 20 ~ this curve the power limit for 21 .~ cycles reclosing time is read as 42 Q) "0 Mw, This is consistent with the re~ sult of stability run 12, in which the system was found to be unstable with , 45 Mw. transfer and 21-cycle recloso 30 ing. The practical power capacity 10 20 o was taken as 35 Mw. Clearing time (cycles) Conclusions. The following cOI\FIG. 58. Curve of power delivered to station BD from substation AB elusions were among those reached over proposed 154-kv. interconnect- as a result of Study 4: ing line versus critical clearing time Capacity of interconnection between of a three-phase fault on a 13.8-kv. systems A and B. The power cafeeder near station AA north bus. pacityof the proposed 154-kv. interBased on the results of runs 13 to 16, connection between systems A and Study 4, part 2. B, based on the maintenance of synchronism through a two-line-to-ground fault on the interconnecting line, with 6-cycle clearing and 21-cycle reclosing, and allowing a margin of at least 5 Mw. for momentary fluctuations in power on the line, would be 25 Mw. received in system B or 35 Mw. received in system A. These values would hold for either of the two routes considered. If system H were connected to the line, the power received in system B might be increased to 35 Mw, In arriving at these limits it was assumed that automatic equipment for controlling the interchange power flow would be installed and used, and that high-speed relaying and switching would be provided on critical circuits within systems A and B as well as on the interconnecting line itself. The connection of systems F and G to system B would be beneficial to stability but would not have enough effect to warrant the assignment of higher power ratings to the interconnection. The capacity of the intercon-

\

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Q,)

,

1\

\,

\

STUDY 4-CONCLUSIONS

341

nection, if a loss of synchronism were allowable during a fault on the interconnecting line 'or on other critical circuits, would be approximately 35 to 40 Mw. in either direction. Speed of fault clearing. To avoid instability when the interconnection was delivering its rated load of 25 Mw. to system B, it would be 0.8

I

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140

120 60

80

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180

200

FIG. 59. Power-angle curves and swing curve, Study 4, part 2, run 17. 25 Mw. received at station BD from substation AB. Two-Jine-to-groundfault near substation AB on proposed 154-kv. interconnecting line to station RD. Clearing time, 6 cycles. Critical reclosing time, 42 cycles.

necessary to use high-speed relaying and switching on some of the circuits of company A. Three-phase faults near station AA 13.8-kv. or 66-kv. busses and near substation AB 66-kv. bus must be cleared in 12 cycles or less. The time for clearing three-phase faults on the 13.8-kv. network might be increased to 18 cycles if the fault was separated from the bus by 1,000 feet or more of 700,OOO-circular-mil cable or by an equivalent impedance. If the station AA bus was sectionalized, three-phase faults on or near the bus must be cleared in 16 cycles or less.

342


High-speed relaying and switching might be necessary also on several of the transmission lines of company B, but further studies would be required, after the terminal of the line had been selected, to determine the extent of such changes. Terminal transformer capacity. The transformer at substation AB should have a capacity of 45 Mva. and that at either station BD or 0.6

1

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80

100

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140

160

180

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Rotor electrical angle (degrees) FIG. 60. Power-angle curves and swing curve, Study 4, part 2, run 18. 35 Mw. received at station BD from substation AB. Two-line-to-ground fault near substation AB on proposed 154-kv. interconnecting line to station RD. Clearing time, 6 cycles. Critical reclosing time, 31 cycles.

substation BCshould have a capacity of 50 Mva. Both these transformers should have tertiary windings rated 15 Mva. for connection to synchronous condensers or static reactors. The capacity of the 13.8/66-kv. transformers at station AA should be increased from 20 Mva. to 45 Mva, The line as built. The interconnection between companies A and B was built and put into service in 1942. The line was 268.5 miles long

STUDY 4-THE LINE AS BUILT

343

with terminals at substations AB and Be and no intermediate switching stations nor taps. (The route finally selected was longer than the one assumed in the preliminary studies.) It was a single-circuit 154kv. line. The phase conductors were 250,OOO-circular-mil, hollow, hard-drawn copper cables of O.u83-inch diameter, ed by strings 0.4

,

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Rotor electrical angle (degrees)

61. Power-angle curves and swing curve, Study 4, part 2, run 19. 45 Mw. received at station BD from substation AB. Two-line-to-ground fault near substation AB on proposed 154-kv. interconnecting line to station BD. Clearing time, 6 cycles. Critical reelosing time, 16 cycles. FIG.

of ten 10-inch suspension insulators. They were spaced 14 ft. 6 in. apart in a horizontal plane (18 ft. 3 in. equivalent spacing). There were two i-inch high-strength galvanized-steel ground wires 12 ft. above the phase conductors .at structures and 20 ft. above them at midspan. The structures were wooden H frames. There were three complete transposition barrels in the line. At substation AB there was a 40-Mva. 161/69/13.8-kv. three-winding transformer bank, and at substation Be, a 45-Mva. 161/138/12.5-


344

kv. autotransformer bank. At each terminal there were three 5-Mvar. static reactors connected to the tertiary windings. Because of the length of the line it was necessary to develop and use a new type of high-speed relay system which could distinguish between faults and power swings even though the magnitude of indicated impedance might be nearly the same. 2 ,3 Internal faults in the terminal transformers were cleared by differential relays which tripped 50

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50

Reclosing time ( cycles) Constant clearing time, 6 cycles FIG. 62. Curve of power delivered to station BD from substation AB over proposed 154-kv. interconnecting line versus critical reclosing time for a two-line-toground fault on this line near substation AB, cleared in 6 cycles. Based on the results of runs 17 to 19, Study 4, part 2.

open the low-voltage oil circuit breakers (66 kv. at AB, 132 kv. at BC)and tripped closed a three-pole grounding switch on the 154-kv. side, causing the breakers at the distant end of the line to open. Staged-fault tests." After the line had been constructed and was ready for service, a number of staged faults were placed on the line to check the operation of the new long-line relays and to check the transient-stability power limits found in the calculating-board studies which have been described. Some faults were placed on the 154-kv. interconnecting line itself, others on 60-kv., 66-kv., and 132-kv. lines near its terminals. The first group of fourteen tests was made with systerns A and B synchronized but with almost no power flow on the interconnection. In these tests a few incorrect relay operations were

31 34

AB

BD

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5

*Synchronism lost between systems Band F. tRelay trouble at station AB caused slow tripping.

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26

154 154

4

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(kv.)

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, A

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Rec'd at

Power Flow on Interconnection

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,

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Fault Location

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TABLE 12

,

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6.8 6.5 10.5

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Reclosing Time (cycles)

STAGED Two-LINE-To-GROUND FAULT TESTS ON 154-Kv. INTERCONNECTION OF SYSTEMS A AND B WITH POWER INTERCHANGE (STUDY 4)

Stable Stable Stable Stable Unstable* Unstablet Stable Stable

Stable or Unstable

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00 ~

d


346

discovered, which led to some changes in the connections and adjustments of the relays and circuit-breaker control. The second group of eight tests were made with various amounts and directions of power interchange. Oscillographic records were made of eighteen currents or voltages in the main and control circuits, making possible the determination of the sequence and time of various operations. The tests of the second group are summarized in Table 12. 50

M,

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20

A

• 30

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Reclosing time of terminal breakp.rs (cycles)

FIG. 63. Approximate curve of transient-stability power limit as a function of reclosing time for faults on the 154-kv. interconnection of companies A and B, Study 4. Points from staged-fault tests. and from calculating-board results are plotted.

With the exception of tests 4 and 4A, in which synchronism was lost, power swings observed throughout the interconnected systems were small. After the application of faults on the 154-kv. line the terminal breakers successfully cleared the fault and reclosed in every case where they were intended to do so. ·The same breakers remained closed in every test in which a fault was applied to other lines, indicating proper functioning of the relays during external faults. Study of the test results revealed that the expected 21- or 22-cycle over-all time from fault to final reclosure of the second breaker was not being obtained on of a delay of from 2 to 4 cycles in the relay action at the end of the line remote from the fault. This trouble was corrected subsequently.

REFERENCES

347

In order to correlate the tests with the results of stability runs made on the calculating board, Fig. 63 was prepared. Here points representing the staged-fault test results and other points representing the calculating-board results with line terminal at substation Be are plotted with delivered power as ordinate and reclosing time as abscissa. The sloping line MN separates the region of stability from the region of instability. The test results agree reasonably well with the results of the board studies. Exact agreement cannot be expected because the conditions existing at the time of the staged-fault tests differred from the conditions assumed in the calculating-board studies. In particular, the interconnection between systems Band C, assumed in the calculations, had not yet been built. Even if it' had been, it would not have been feasible to schedule power flow and generation in five or more power systems in several states in such manner as to duplicate the conditions assumed in calculation.

REFERENCES 1. R. C. BERGVALL, "Series Resistance Method of Increasing Transient-Stability Limit," A.I.E.E. Trans., vol. 50, pp. 490-4, June, 1931; disc., pp, 494-7. 2. F. C. POAGE, C. A. STREIFUS, D. M. MACGREGOR, and E. E. GEORGE, "Performance Requirements for Relays on Unusually Long Transmission Lines," A.l.E.E. Trans., vol. 62, pp, 275-83, June, 1943. 3. A. R. VAN C. WARRINGTON, "Protective Relaying for Long Transmission Lines," A.I.E.E. Trans., vol. 62, pp. 261-8, June, 1943. 4. C. W. MINARD, R. B. Gow, W. A. WOLFE, and E. A. SWANSON, "StagedFault Tests of Relaying and Stability on Kansas-Nebraska 27Q-Mile 154-kv. Interconnection," A.I.E.E. Trans., vol. 62, pp. 358-67, 1943; disc., pp.425-6.

INDEX Braking effect of grounding resistor, 241 Breakers, data required on, 63 eight-cycle, need for, Study 3, 310 Bus, infinite, 124 Bus reactors at Philo, 290, 294, 300 Busses, effect of, on stability, 190 Byrd, H. L., 168

Acceleration, 15, 16 angular, 17, 19 Adjustment of initial operating conditions, 77 ittances, driving-point, 80-6 mutual, 80-6 self-,80-6 terminal, 80-6, 95 transfer, 80-6 Algebraic solution of networks, 78-101 Alpha, beta, zero components, 245 Alternating-current calculating board, 64 American Gas and Electric Company, 288,290 Amplifiers for instruments on a-c. board, 70, 73 Analogies between translation and rotation, 20 Analogue, mechanical, 7 Angle, 17, 19 Angular acceleration, 17, 19 Angular momentum, 19 Angular velocity, 17, 19 Apparatus base, 54 Assumptions made in stability studies, 43 Autotransformers, 58 on a-c. board, 68, 69, 72, 73, 74 zero-sequence equivalent circuit of, 239

Cables, data required on, 63 equivalent circuits, 59 Calculating board, a-c., 64 procedure for using, 75 Capacitance of transmission lines, representation of, 60 Capacitors, on a-c. board, 67, 68, 73, 74 series, 189 Carrier-current relaying needed, Study 3,310 Check list of data required for transientstability study, 62 Checks on readings, 78 Circuit breakers, data required on, 63 eight-cycle, need for, Study 3, 310 Cleared fault, application of equal-area criterion to, 129 Clearing angle, 130 determination of, by pre-calculated . swing curves, 157, 158 Clearing time, 139 allowable, Study 1, 286 critical, curves for determining, 16885 determination of, by pre-calculated swing curves, 157, 1,P8 effect of, on transient stability, 159 Combining machines, 111, 264, 319 Common voltage, referring quantities to, 56

Bad effects of instability, 9 Base impedance, 55 Base quantities, for a-c. board, 66, 72 choice of, 75 for per-unit system, 54 Board, a-c. calculating, 64 349

350

INDEX

Components, alpha, beta, zero, 245 symmetrical, 193 Composite line, 60 Condenser, synchronous, data required on, 63 mechanical analogue of, 8 Study 1, 296-7 treatment of, in transient-stability study, 119 Connections on a-c. board, 69, 77 changes of, 78 Conversion, of impedances, 54 star-mesh, 86-7 Coordinates, symmetrical, 193 two-phase, 245 Correction factors for equivalent 1r circuit of transmission line, 59 Coupled lines, equivalent circuits of, 234 Critical clearing angle, 130, 136 determined by pre-calculated swing curves, 157, 158 Critical clearing time, curves for determining, 168-85 determined by pre-calculated swing curves, 151, 157 Damper winding, 11 D8IDping, 5,43,122,123 Data required for transient-stability study, 62Dead sequence networks, 210 Difference equation (problems), 51 Dimensions of quantities of mechanics, 15, 17-9 Discontinuities in point-by-point solution of swing equation, 38 Double unbalances, 232 Duration of fault, effect of, on stability,

159, 185, 224

Ebasco Services, Inc., 254, 311 Energy, kinetic or stored, 16, 20, 23-6 of synchronous machines, 25, 256-8, 292, 305, 315-6; seealso Inertia constant rotational, 20, 23-6 Equal-area criterion, 122

Equal-area criterion, Study 4, 335-43 Equipment, miscellaneous, 61 Equivalent circuits, of coupled lines, 234 of remote portions of system, 61 of transformer, 56 negative-sequence, 235 zero-sequence, 235-40 of transmission lines and cables, 59 Equivalent generator, 6 Equivalent impedance of transformers, 54,56 Equivalent inertia constant of twomachine system, 133 Equivalent input of two-machine system, 133, 136 Equivalent motor, 6 Equivalent output of two-machine system, 133, 136 Equivalent pi (71") circuit of transmission line, 59 Equivalent power-angle curve of two finite machines, 133 Equivalent T circuit of transformer, 56 Factors affecting stability, 187 Fault, close to generator, 78 effect on stability, 5 of duration of, 159, 185, 224 of type of, 223 shown by mechanical analogue, 9 Fault impedance, 228 Fault location, effect of, on stability, 189 Fault locations, Study 1, 265 Study 3,302 Fault shunts, 220, 231 Faulted three-phase networks, solution of, 193 Faults, relative number of various types of, 225 relative severity of various types of, 223 representation of, in symmetrical components, 205, 228-30 in two-phase coordinates, 247 simultaneous, 232

INDEX Finite machines, 132 Flux linkage of field winding, 123 Force, 15, 16 Formal solution of swing equation, 28, 32 Frequency, effect of, on stability, 189 used on a-c. board, 66, 72 Frequency changer, 189 (problem), 51 General Electric A-c. Network Analyzer,66 Generator units of a-c. board, 66, 68, 72,74 Generators, data required on, 62 representation of, 56 Governor, 5, 122 Graphical integration, determination of swing curve by, 139 Grounding, effect of, on stability, 240 Grounding impedance, 237 effect of, on stability, 240 of autotransformer, 240

H (constant), 24-6 High-voltage bussing, effect of, on stability, 190 Historical review, 11 Hunting, 11 Impedance, base, 55 fault, 228 grounding, 238 negative-sequence, 201 per-unit, 55 positive-sequence, 201 series, representation of, 231 transformer, 56 zero-sequence, 201 effect of grounding on, 241 mutual, of lines, 233 Impedance diagram, 53 Impedance units on a-c. board, 67, 68, 73,74 Inertia, moment of, 18, 19 Inertia constant, 22 effect of, on stability, 188

351

Inertia constant, equivalent, of. twomachine system, 133 of combined machines, 111 of remote portion of system, 62 Infinite bus, 124 Initial operating conditions, 77, 78 Input assumed constant, 43 Instability, bad effects of, 9 definition of, 1 Instantaneous clearing, stability limit for, 160, 164 determination of, by curve, 176 Instruments on a-c. board, 70, 71 Integration, graphical, determination of swing curve by, 139 Interconnecting lines, Study 4, 314, 342 Interconnection, 12 Interconnections, effect of, on stability, Study 2, 294-301 Intermediate busses, effect of, on stability, 190 Internal voltages, effect of, on stability, 2,188 Joule (unit), 15 Jumper circuits on a-c. board, 69 Junction, three-phase, 202 I{inetic energy, 16, 20, 23-6 in equal-area criterion, 125 of synchronous machines, 25, 256-8, 292, 305, 315-6; see also Inertia constant rotational, 20, 23-6 Kirchhoff's laws, 202 Laws of mechanics, 15 Length, 15 Limitations of system, 77 Line-impedance units of a-c. board, 67, 68,73,74 Line-to-ground fault, 207, 248 fault shunt, 220, 231 with fault impedance, 229 Line-to-line fault, 207, 212, 248 fault shunt, 220-231 with fault impedance, 229 Lines, transmission, representation of, 59

INDEX

352

Lines, zero-sequence mutual impedance of, 233 Load adjuster, 73 Load-impedance units of a-c. board, 67,

68;73, 74 adjustment of, 73, 77 Loads, data required on, 63 representation of, 61 Location of fault, effect of, on stability, 189

Mass, 15 Master instruments of a-c. board, 70, 73,75

McClure, J. B., 149 Mechanical analogue, 7 Mechanics, laws of, 15 Mesh, three-phase, 203 Mesh circuit, 82, 110 Metropolitan systems, 12, 187 M.k.s. system, 15 Modified time, 149, 169, 178 Moment of inertia, 18, 19 Momentum, 15, 16 angular, 19 Motor-generator set for a-c. board, 71 Motors, large synchronous, data required on, 63 Multicircuit transformers, zerosequence equivalent circuit of, 238

Multimachine system, 6, 122, 191 Mutual impedance base, 55 Mutual impedance of parallel transmission lines, 61 zero-sequence, 233 Mutual reactors on a-c. board, 68, 69, 74 Mutual transformers on a-c. board, 68, 69, 74 Negative sequence, 193, 194 Negative-sequence equivalent circuit of transformer, 235 Negative-sequence impedance, 201 Negative-sequence network, 203, 212 Network, negative-sequence, 203, 212 poffitive-sequence, 53,203,212

Network, zero-sequence, 203, 212 Network analyzer, 64 General Electric, 66 Westinghouse, 72 Network calculator, 64 Network reduction, 83, 213-5 algebraic, 83 calculating-board method of, 109 symbols used in, 90 Network re-expansion, 216-8 Network solution, algebraic, 78-101 Networks, sequence, 203 solution of, 53 solution of faulted three-phase, 193 substitute, 247 Newton (unit), 15 Nodes, 83 Nominal pi (1r) circuit of transmission line, 59 Nominal voltage and current of a-c. board, 66, 72 Ohio Power Company, 290 One-line diagram, 53 Open circuits, representation of, by connections between sequence networks, 231 Parallel transmission lines, 61 effect of, on stability, 189, 190 Per-cent quantities, 56 Per-unit quantities, 54 Phase shift in transformers, 59, 235 Phase shifters on a-c. board, 66, 72, 73 Philo station, 290-2 Pi (1r) circuits of transmission line, 59 Point-by-point solution of the swing equation, 27 discontinuities, 38 errors, 34, 42 Polarity of connections on a-c. board, 69,76 Positive direction of circuit units on a-c. board, 69, 76 Positive sequence, 193, 194 Positive-sequence impedance, ·201 Positive-sequence network, 53, 203 use of, for x and y networks, 248

INDEX Post-fault output curve, 129, 131 Power (mechanics), 15, 16 in rotation, 19 Power-angle curve, 2, 3, 125, 126 equivalent, of two finite machines, 133, 135 Power-angle curves, Study 4, 335-8, 341-3 Power-angle equation, 2, 78 one machine and infinite bus, 126 two finite machines, 133 two-machine reactance system, 135 Power limit, 6 method of determining, Study 4, 318 Power supply of a-c. board, 71 Pre-calculated swing curves, 149-59 Pre-fault output curve, 129-31 Pritchard, S. R., Jr. 168 Radian, 17, 19 Reactance, effect of, on stability, 2, 188 of synchronous machines, 56 of transformers, 57 of transmission lines, 61 zero-sequence, 233 transient, 256-8, 292, 315-6 decrease of, to aid stability, 276 Reactive power, sign of, 79 Reactors, data required on, 63 grounding, effect of, on stability, 241 on a-e. board, 67, 68, 73, 74 correction for resistance of, 76 shunt, use of, to aid stability, 273 special, on Westinghouse a-c. board, 73 Readings on a-c. board, 78 Reclosing, high-speed, Study 4, 317 Reclosing time, critical, Study 4, 340, 344, 346 Reduction, of network, 83 algebraic, 83 calculating-board method of, 109 symbols used in, 90 of sequence networks, 213-5 Re-expansion of networks, 216-8 Referring quantities to a common voltage,56 Regulating transformers, 59

353

Regulator, induction voltage, on a-c. board, 66, 72 voltage, 11 effect of, 123 Relays, long-line, 344 protective, data required on, 63 false operation of, 10 Remote portions of system, 62 Resistors, grounding, effect of, on stability, 241 on a-c. board, 67, 68, 73, 74 series, use of, to aid stability, 273 Rotation, analogies of, to translation, 20 laws of, 17, 19 Scalar instruments on a-c. board, 73 Sectionalised operation, 301 Sequence, negative, 193-4 positive, 193, 194 zero, 194 Sequenceirnpedances, 200 Sequence networks, 203, 212 connections between, for representing faults, 205 for representing, faults with impedance, 228-30 for representing open circuits, 231 Series capacitors, 189 Series impedances, representation of, by connections between sequence networks, 231 Series resistors, use of, to aid stability, 273 Severity of types of faults, 223 Short circuit, effect of, on stability, 5 representation of, by connections between the sequence networks, 205, 228-30 by connections between the substitute networks, 247 types of, 205 Short-circuit impedance of transformers, 54, 56 Shunt reactors used to aid stability, 273 Shunts for representing faults, 220, 231 Simplification of systems, Study 4,318 Simultaneous faults, 232

354

INDEX

Solution, of faulted three-phase networks, 193 of networks, algebraic, 78-101 Stability, certain factors affecting, 187 definition of, 1 transient, summary of methods of calculating, 185 Stability limit, effect of fault-clearing time on, 159, 167 effect of type of fault on, 223 steady-state, 4 transient, 5, 6 Stability studies, assumptions made in, 43 typical, 253 Staged-fault tests, Study 4, 344 Star-mesh conversion, 86-7 Steady-state stability limit, 4 Study 4,318 Steam turbogenerators, stored energy of, 25 Step-by-step solution of the swing equation, 27 discontinuities, 38 errors, 34,42 Stored energy, see Kinetic energy Studies, typical stability, 253-347 Study 1, 254 Study 2,290 Study 3,301 Study 4,311 Substitute networks, 247 Summary of methods of calculating transient stability, 185 Summers, I. H., 149 Sustained fault, application of equalarea criterion to, 127 stability limit for, 161, 164 determination of, by curve, 176 Swing curves, 28, 32, 48, 103, 1.09, 144 determination of, by graphical integration, 139 pre-calculated, 149-59 Study 1, 265-85 Study 2, 296,.297, 300 Study 3, 306-9 Study 4, 322-34, 339 Swing equation, 20

Swing equation, formal solution of, 28, 32 point-by-point solution of, 27 errors, 34, 42 Symmetrical components, 193 Synchronous condenser, data required on, 63 mechanical analogue of, 8 Study 1, 296-7 treatment of, in transient-Btability study, 119 Synchronous machines, kinetic energy of, 25, 256-8, 292, 305, 315-6 reactance of, 256-8, 292, 315-6 representation of, 1, 43-4,56 Synchronous motors, data required on, 63 System base, 55 T circuit of transmission line, 60 Tapped loads, 61 (problem), 120 Three-circuit transformer, 318 zero-sequence equivalent circuit of, 238 Three-phase fault, 209, 248 fault shunt, 220, 231 with fault impedance, 229 Three-phase networks, solution of faulted, 193 Three-phase transformers, zerosequence equivalent circuit of, 239 Time, 15 modified, 149, 169, 178 Torque, 17, 19 Transformer banks, 59 Transformers, data. required on, 63 equivalent circuits of, 56 impedance of, 56 multicircuit, 58 zero-sequence equivalent circuit of, 238 negative-sequence equivalent circuit of, 235 per-unit quantities of, 54 phase shift in, 235 reactances of (table), 57

3S5

INDEX Transformers, regulating, 59 representation of, 56 in the sequence networks, 235 three-circuit, 57, 318 zero-sequenceequivalent circuit of, 238

three-phase, 59 zero-sequence equivalent circuit of, 239 two-circuit,' 56 zero-sequence equivalent circuit of, 235 Transient reactance, decrease of, to aid stability, 276 values of, 256-8, 292, 315-6 voltage behind, 123 Transient stability,certaiIL factors affecting, 187 summary of methods of calculating,

185

Transient stability limit, 5, 6 determination of, by equal-area oriterion,128 Study 4, 318, 340, 346 Translation, analogies of, to rotation, 20 laws of, 15 Transmission, long-distance, 12 Transmission lines, constants of, 61 data required on, 63 equivalent circuits, 59 Turbogenerators, stored energy of, 25, 256-9, 292, 305, 315-6 Two-line-to-ground fault, 209, 248 fault shunt for, 220, 231 with fault impedance, 229 Two-machine system, 1, 7, 122, 149 equivalent inertia constant of, 133 equivalent input of, 133, 136 equivalent output of, 133, 136 Two-phase coordinates, 245

Unbalances, double, 232 Units of quantities of mechanics, 15, 19 Varmeter on a-c. board, 70, 73 Vector instruments, 71, 73 Vector measurements on a-c. board, 71, 73 Vector power, 79 Velocity, 15 angular, 17 Voltage, behind transient reactance, 123 of transmission, effect of, on stability, 189 Voltage regulators, 11 effect of, 123 Water-wheel generators, stored energy of,- 25, 256-9, 315-6 Wattmeter-varmeter on a-c. board, 70, 73 Westinghouse .A-c. Network Calculator, 72 Work, 15 of rotation, 18 WR2 (moment of inertia), 23, 256-9,

305, 315-6

Zero sequence, 194 Zero-sequence equivalent circuit of transformers, 235-40 Zero-sequenceimpedance,201 effect of grounding OD, 241 of transformer, 236 Zero-sequence mutual impedance of lines, 233 Zero-sequence network, 203, 233 representation of lines in, 233 Zero-sequence reactance of lines, 233

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