reading / British practice / Vol D - 1990 (ocr) ELECTRICAL SYSTEM & EQUIPMENT

Power system performance analysis

TABLE

2.14

Overlapping forced outages — third order (TYPE 5), with third order CM?

Residence time

(TYPE 5)

Event

Failure rate

Restoration by

Restoration

component

from LES

repair

X

rik

rn

_

1, 2,3

XiX2riX3rir2/(r1

4- r2)

1/A

r1r2t e /(rir2 4- rItc

+ r2tc)

(a)

(b)

1, 3, 2

XiX3riX2rir3/(ri + r3)

Up.

rir3t e /(rir3 + rit e + r3te)

(c)

2, 3,

1

X2X3r2X1r2r3/0. 2 + r3)

1/kt

r2r3t c /(r2r3 + r2te + r3te)

(d)

3, 2,

1

X3X2r3X1r3r2/(r3 + r2)

1/ p,

r3r2t c /(r3r2 4- r3tc + r2te)

(e)

2, 1,3

X2Xir2X3r2rii(r2 + rI)

1/p,

r2rit c /(r2ri + r2tc + rite)

(I)

3, 1,2

X3Xir3X2r3r1/(r3 + rI)

Pp

r3rit c /(r3ri + r3t c

+ rite)

(g)

1, 2 + 3

X 123

1/11

tc

For each event: average repair time r

rA P + rg (1 — P)

Where:

l

= Al + 12 2 14 3

P

=

= failure rate of component 'n'

X123

CIMF rate for components 1, 2 and 3

r n

= Repair time of component 'n'

tc

Switching and start-up time of LES

t L

Time limit of LES

Indices for TYPE 5 events are:

8

8

X5 = EX, U5 = EXr, r5 = U5/X5

a

a

tanee matrices derived from power station electrical systems are well conditioned, also one or more voltages on the system are specified, usually the super- grid and/or grid voltage. These factors remove the problems associated with ill-conditioned admittance matrices.

The specified voltage can now be eliminated from the nodal equations. Assuming for convenience that the first node voltage is known, and considering the equation I = YV, the new equations become:

lk Ykl VI =E YkiVi for k = 2 to k = n

The set of equations can then be solved using one of the

available techniques, some of which will be briefly described later,

Busbar type definitions

There are three basic types of busbar defined in load flow analysis:

•PQ busbar, where net active (real and reactive power) are specified; i.e., the watts and VARs supplied from generation sources, minus those consumed by loads at that busbar. This is normally a busbar where only load is connected.

•PV busbar, where net active power and voltage magnitude are specified. Net reactive power is not specified and its value for the busbar emerges as part of the load flow solution. Typically, this is a busbar where generating plant or synchronous compensation is connected and the voltage magnitude is controlled by regulating the reactive power output of the generator.

123

•▪

Electrical system analysis

Chapter 2

TABLE 2.15

Forced outages overlapping

maintenance — third order (TYPE 5)

Residence time

(TYPE 5)

Event

Failure rate

Restoration by

Restoration

component

from LES

repair

X"

rA"

113

(a)

l",

2,

3

Xi'X2r 1'X3ri'r2/(ri'

+

r2)

1/y"

rjr2t c /(r; r2 +- CI t c + r2 tc)

(b)

/",

3, 2

Xi'X3r 1'X2il'r3 /(r1'

4-

r3)

l/A"

rcr3tc /(r i r3 + rct c + r3(e)

(c)

2, 3,

1

XY X 3 T5X 1 q r3 /( .5 + r3)

1/A"

rir3t c /(111- 3 + rp c + r3t e )

(d)

3",

2,

I

X3X2riX1rir2/(ri + T.))

1/it"

ri. r2t c /(r3r2 + qtc + r2lc)

(e)

2",

1,3

.X . ''XA riX3 *i/(ri' + ri)

1/a"

Tirit c /(rri + r5t c + rite)

(f)

3",

1,2

X3X1r3X2ri'ri/(ri' + r1)

1/g"

ti r 1 t c /(r3r i i- tit c + r t tc)

(g)

2",

2 +3

X I"X23 II'

1/p,"

rjt c /(ri + t e )

For each event: average repair time r" = rA" P + r (1—P)

Where:	t"	f4 2 + A3 or it 1 + 1.4 '2 + A3 or AI + Az
	P	e — A tL
	X n = Failure rate of component 'n'
	X23	= CMF rate for components 2 and 3
	r n	Repair time of component 'n'
	1c	= Switching and start-up time of LES
	I L	= Time li mit of LES
	Component out on maintenance indicated thus ('')
Indices for TYPE 5 events are:
X; = EX", U =	X"r",	r = 1.4/XE
a	a

• Slack busbar. This is a busbar nominated by the analyst from the PV busbars for analysis purposes only. There is only one slack busbar on each system. Voltage magnitude is specified, but the net active power is designated as unknown. This is because, prior to solving the network equations, the system losses are unknown and it is not possible to specify the total generated power exactly. During analysis, these losses are assumed to be taken from the system at this busbar. The busbar with the greatest amount of generating capacity connected is usually chosen to be the slack busbar.

Complex variables — definitions

The complex variables in load flow analysis are the voltage and current at each busbar or node. These are

defined by the linear nodal equations I = YV and the busbar constraints, as follows:

(a) A t a PQ busbar

V I* = (net active power) + j (net reactive power) (net active generated power x net active power supplied to loads) + j (net reactive generated power — net reactive power supplied to loads) at that busbar.

(h) At a PV busbar

Re VI* = net active power (Re = real part of)

=(net active generated power — net active power supplied to loads) at the

busbar.

and IV! voltage specified at that busbar VI = modulus of V).

124

III e –ii

I)

Power system performance analysis

FIG. 2.20 Reliability analysis of failure rate (FR) — presentation of results

(c) At the slack busbar	• Voltage magnitude at slack and PV busbars.
V = voltage specified at that busbar	•	Net active power input at PV and PQ busbars.
	•	Net reactive power input at PQ busbars.

Footnote:

Complex power S = P -s-jQ, and V =		and I = Ifleil
l* is defined as II: e	and called the complex conjugate of I

The product VI * = S because VI * = I VI

= IVI III

=

= IVIII1cos 4, +jVfl sin 01 = P + jQ

si milarly V * I can be shown to equal P – jQ .

Simplified system representation data requirements and outputs

A simplified system representation has the following data input requirements for analysis purposes:

•The impedances of network branches. (Circuits or transmission lines, transformers, series and shunt reactors, static capacitors. Transmission circuits with significant charging currents are represented by a pi network, i.e., a series impedance and two shunt capacitances, one at each end of the circuit.)

The data outputs are:

•Voltage magnitude and angle at PQ busbars.

•Voltage angle at PV busbars.

•Active power generation at the slack busbar.

•Reactive power generation at the slack and PV busbars.

•Power flows at both ends of each network branch.

•Losses in each branch and total system losses.

125

Electrical system analysis	Chapter 2

RELIABILITY ANALYSIS-

AVERAGE OUTAGE TIME(H)

FIG. 2.21 Reliability analysis of average outage duration (A0D) — presentation of results

Solution of network equations

As mentioned earlier, the solution of the network equations in matrix form developed from the nodal representation of the power system is done by computer programs, using one of several methods. These include the Gauss-Seidel method, the Newton-Raphson method and the fast-decoupled method. These are now briefly described.

Gauss-Seidel method

This is one of the easiest methods to program. It is adapted from the Gauss-Seidel iterative method of solving simultaneous linear differential equations.

Nominating the slack busbar as number I, purely for presentation purposes, the algorithm is:

(i - 1)	E Yik Vr 1 )/Yii
— Yil VI — E Yu, V
	k=0+1)

where p is the iteration number.

Assuming for the moment that all busbars other than the slack busbar are the PQ type, for a PQ busbar:

1 i (PS! jQS, P)/V7 (Superscript SP ---

specified value)

and substituting for I; in the above:

V;? =

	(Pr iQr)					(i -1)	P	- I 7
	—						n	/1 i;
						- _ YjkV k - E Yik \q
	v7				YilVi
						E	k =6+ I)
[						k=2
					for k = 2 to k = n			(2.1)

This algorithm can be applied iteratively until convergence is reached.

If there is a PV busbar present, it is necessary to calculate QT) before the above algorithm can be used. This is done by calculating:

I r i = E Yik	(Superscript P = the last
k =1	calculated value(s))

126

Power system performance analysis

—RELIABILITY ANALYSISANNUAL OUTAGE TIME(H/YR)

FiG. 2.22 Reliability analysis of annual outage time (AOT) — presentation of results

and Qr 1 = Im [Vr 1 x IF -1 1

(Im = Imaginary part of and superscript P- I = the last but one calculated value) and Qi can be used in Equation (2.1).

Finally, for PV busbars, the calculated value of V? is reduced to its original specified value without changing its angle.

A logic flow diagram for the Gauss-Seidel method is given in Fig 2.48. The diagram illustrates the successive stages in the computation. Once the initial data are read into the computer and the iteration counter set to zero, the program calculates the voltage at each node in turn. It then compares the set of voltages just calculated with the preceding set of calculated voltages. If the two sets of values are within a predefined tolerance, then the solution is said to have converged. If convergence does not occur, a further iteration takes place and comparison between the two most recent sets of voltages is made. At some stage either convergence is achieved, and the results of the analysis are displayed, or the preset maximum number

of iterations is reached. In that case, a message is displayed to the user informing him that the maximum number of iterations has been reached without the solution converging. (The user then seeks the reasons for this.)

To reduce computation time, an acceleration method can be inserted into the program in conjunction with the process described. One method is to project each voltage variable linearly in the direction the solution is seen to be moving. This may require additional storage and care must be taken not to 'over project' the variables, otherwise no solution will emerge.

Newton-Raphson method

This method is based on the Newton-Raphson general algorithm for the solution of a set of simultaneous nonlinear equations. F(X) = 0, where F is a vector of functions f1 to f n in variables x1, to xn

At each iteration of this method, the non-linear problem is approximated by a linear matrix and solved for X.

127

Electrical system		analysis										Chapter 2
			•
			••		INPUT OATA FOR SUBSYSTEM NUMBER
			000
					DATA	TO	CALCULATE PATHS AND CUTS
					NUMBER	OF	BRANCHES • 7
BRANCN	EOuIvftLENT
NUMBER	SYSTEm NO.		0-ENO		5-END		COmPONENTS
	t	i t	12/1		SI/1
2	(	2)	SIll		53/1
3	(	3)	41/1		S1/1
•	(	4)	142/1		02/I
5	(	5)	$2II		$4/1
6	(	6)	04/1		$3/1				510/1
	(	7)	03/1		$4/1
7
	TNE FIRST		2 BRANCKES ARE ASSUMED UNIDIRECTIONAL
	SOURCES CONNECTED TO NODES :
							hi 2
56111041110 EFFECTS OF COmPONENT ACTIVE FAILURES
	COMPONENT ACTIv LY FAILED						BREAKERS		THAT TRIP.
				1					2
				2			85		9
				3					4
				4
								2
				2			3
							2
				2			1		7
				3			a
				4			9	5	9
				5			6	4
							•	7	2
				7				6	2
				9			1 c,	10	4
								5
							7
			1	0			9
				2			3
				1			8	I D	9	5	4
				2			5
							9	10
							5	4	9
				2			7
	NORm4LLY OPEN COMPONENTS
	P	I
:ONT/NuE'IYIN
											06459-2	FR/, 03 JUN 1988 I

FtG. 2.23 Input data to calculate paths and cuts, and the switching effects of component active failures

F(XP -1 ) = - J(XP - I) x AXP

i.e., AXP = -1.1(XP -1 )] - I F(X 1 ' -1 )

X is then updated by XP = XP - I + XP

The square matrix J is the Jacobian matrix of F(X). This contains partial derivatives and has general element.

afi

for ith row, k' column

ax k

There are several ways to write the load flow equations; one popular way is to substitute for 1,, obtained from I = YV, into the PQ and PV busbar equations given earlier.

The Newton-Raphson method is widely used and is efficient in solving large networks. Convergence is rapid when approaching correct values. Efficient solution is very dependent on accurate calculation of the elements in the Jacobian matrix. This matrix is sparse, and this property is exploited by use of sparsity-programmed ordered elimination. This means the way in which rows

and columns are written is changed, and only non-zero elements of the Jacobian matrix are stored and operated on. This is difficult to program but improvement in solution efficiency makes it worthwhile.

Fast decoupled method

Based on the Newton-Raphson method, this solution takes advantage of two practical characteristics of power systems:

•That active (watt) power flow between nodes is strongly dependent on the difference in phase angle between nodes.

•That reactive (VAr) power flow between nodes is strongly dependent on the difference in voltage between nodes.

If Newton-Raphson is formulated as:

[ A QP	ef)
— =	A V
A	V

128

Power system performance analysis

COAP0NEAT	IFS	AFR	MT	ST	SP	AT	AR
-------	-	---	---	---		---	- -
1	0.0090	0.0030	48.0000	. 0000		8.0000	0.5000
2	0.0030	0.0090	48.0000	. 0000		8.0000	0.5000
3	0.0030	0.0030	48.0000	. 0000		8.0000	0.5000
	0.0030	0.0030	48.0000	. 0000		8.0000	0.5000
	0.0700	0.0140	120.0000	. 0000		72.0000	L.0000
	0.0200	3 • 01 •0	1 20.0000	. 0000		72.0000	1.0000
	0.0120	0.0050	48.0000	. 0000	0.0010	24.0000	L.0000
2	0.0120	0.0050	36.0000	. 0000	0.0010	24.0000	1.0000
3	0.0120	0.0050	46.0000	. 0000	0.0010	24.0000	1.0000
	0.0120	0.0050	36.0000	. 0000	0.0010	24.0000	1.0000
5	0.0120	0.0050	36.0000	. 0000	9.oaLo	24.0000	1.0000
6	0.0120	0.0050	36.0000	. 0000	0.0010	24.0000	1.0000
7	0.0120	0.0050	36.0000	. 0000	0.0010	24.0000	1.0000
8	0.0120	0.0050	36.0000	. 0000	0.0010	24.0000	1.0000
	0.0120	0.0050	36.0000	. 0000	0.0010	24.0000	1.0000
1 0	0.0120	0.0050	36.0000	. 0000	0.0010	24.0000	1.0000
1	0.0750	0.0100	2.0000	. 0000		30.0000	1.0000
	0.0750	0.0200	2.0000	. 0000		30.0000	1.0000
1	0.0120	0.0050	36.0000	. 0000	0.0010	24.0000	1.0000
1	0.0200	0.0140	1 20.0000	. 0000		72.0000	1.0000
2	0.0200	0.0140	1 20.0000	. 0000		72.0000	1.6000
	0.0040	0.0040	24.0000	. 0000		0.0000	0.0000
2	0.0040	0.0040	24.0000	. 0000		0.0000	0.0000
			CONAON mODE FAILURE DATA
		THERE ARE NOT G.A.F. DATA
	INCONPATIBLE COAPONENTS
	NONE
	CONTROL PARANETERS
	DO N/0 PATHS FAIL 62E8 REQUIRED TO DPERATE , .					A
	5923009 NumBER of OvERLAPP/NG OuTACES					2
	DO YOu HAAT TO CONSIDER STUCK BREAKERS ,
						Y
	APE THE 0811,060 EvENTS OF Ni0 PATHS OF FIRST ORDER ,					Y
	THE mAxImum PERMITTED NuNBER OF 14/0 BREAKERS IN A 300E4 PATH					2
	DOES THE $YSTEM CONTAIN INCOAPATIBLE COMPONENTS ,					N
	00 YOu HAAT THE PROGRAA 70 DEDUCE THE BREAKERS WHICH TRIP DURING AeF ,					I

0000P-2 FR:. 83 JUN 1 986

FIG, 2.24 Reliability and CMF input data

2000 CONNECTED TO NOOE HISSER 0 3 /62130001E5 ND. 1

LIST IF SYSTEN PATHS

N096E0 OF FATHs . 4

CO ,, PONEHT NumBERS
00/1 S1/1 52/1 02/1	2	1
S0/1 04/1 22/i 42,1	0	5	4
53/1 01 , 1 111/i 2	3
PATHS NOFAALLY oFEN
53/i 64/1 52/1 02/1	2	5	4

66439-2	142 . 02 JUN 533 1

FIG. 2.25 List of system minimal paths

129

Electrical system analysis	Chapter 2

LOPS CONNECTED TO NODE NUMBER S 3 /SUBSYSTEM NO. 1

LIST OF CUTS

11)48E8 OF CUTS . 61

CUT	COmPONENT NumEERS

1$31L

2S1/1 52/1

381/1 W2/I

45171 837i

551/1 T2/I

S1/I 84/1

75111 $4/1 SI/I 89/1

9	S1/I 02/1
L O	si/1 elo/L
1 1	81/1 01/1
1 2	52/i 87/1
1 3	52/1	02/1
1 4	5211 8811
1 5	52/1	WI/I
1 6	52/1 81/1
1 7	52/I 11/1.
1 8	52/1 82/1
I N	W2/I 67/1

20U2/1 02/I

211) 2/I 88/1

228211 81/1

23w211 8111

2442/I T1/I

2542/1 8271

2687/1 83/1

2787/1 T2/1

2887/1 84/1

2902/1 83/1

3002/I 12/1

3102/1 84/1

328811 83/1

3388/1 T2/I

348811 84/1

15

11 7/1

04/1

360271 $4/I

3788/1 $4/1

39

87/1 89/1

3987/1 02/I

4087/1 B1011

410211 89/1

420211 82/1

4302/1 81011

:90777,u2 ,, ,6

G805P-2

THJ, 02 JUN 1060 1

		44		88/1	89/1
		45		88/1	0211
		46		88/I	810/1
		47		87/1	01/1
		48		02/1	01/1
		49		88/1	01/1
		50		83/1	81/I
		51		T2/I	W1/1
		52		8411.	8111
		53		83/1	81/1
		54		83/1	TI/1
		55		83/I	02,1
		56		T2/I	81/1
		57		12/1	Ti/I
		58		12/1	82/i
		59		84/1	81/1
		60		04/1	T111
C u TS 1m41 ARE ELIA/mATE0		El		84/1	82/1
		01 CLOSING A		NORMALLY OPEN PATH
CUT			PATHS	THAT 495	60 CLOSED
	8		4
	9		4
1	0
38			4
39			•
4 I)
•1			4
42			4
43
44		4
45		4
46		4

06900-2 THU, 02 JUN 1988 I

FIG. 2.26 List of minimal cuts

130

Y i k,

Power system performance analysis

•

LOAD CONNECTED TO MODE MURDER 6 3

ISUOSYSTEM $0. I

•

•

•

................................. R 6LIAOILITY RESULTS

EvENT

FORCED F. RATE

AY. DURATION

TOTAL 0,1 T . TINE

14/1)0. F,

RATE

As.

OuRnTION

TOTAL OUT, TINE

S 3/1

'PILED

0.30008E-02

0.90000E

00

0.008000

GB

0.400000

02

0.11400E

0

0.00000E

00

8.240000

02

S1/1

S2/1

0.96630E-07

0.27397E-85

8.688710

el

8.23671E-0

0.18707E-01

C UT

8.192000

81

11/1

02/1

0,12842E-05

0.24E56E-0

CUT

0.44521E-04

0.54911E

91

0.240B8E

82

0.24447E-83

S1/1

03/ 1

0.39452E-06

0.13699E-04

0.123480

02

0.94685E-0

OUT

0.042860

42

6.109080-03

S1/1

T2/1

0,11507E-G5

0.39452E-0

0.93790E-04

0.230430

02

0.77882E-83

OUT

0.34521E-06

0.203710

02

S1/1

04/1

0.13699E-84

8.12210E

82

0.71014E-6

OUT

0.240000

02

0.16737E-03

51/1

14/1

0.96630E-07

6.2)9971-05

0.80871E

02

0.21671E-0

CUT

8.976760

00

0.16787E-04

69/1

OUT

0.34521E-06

0.13699E-04

0.931560

00

0.33716E-0

m/0 PATH CLOSED

0.11507E-05

8.908270

00

0.12761E-04

Si/1

11 2/2

OUT

0.337900-04

8.939970

00

0.11372E-0

1/0

O.I.Ti4

CLOSED

0.970700

00

8.32437E-04

Si/1

610/1 OUT

0.34521E-06

0.13699E-04

0.931560

00

0.31710E-8

..6 /0

AATW

CLOSED

0.1819000

92

0.12761E-04

Si./1

01/1

0.96630E-07

0.18265E-85

8.666080

01

0.15761E-0

CUT

8.14521E-06

0.295711

02

0.10959E-04

S2/i

67/1

0.13699E-04

0.

22100

02

0.71014E-0

OUT

0.98638E-07

6.160000

92

0,16737E-03

S2/1

02/1

0.60009E

01

0.15711E-0

OUT

0.18265E-05

0.20571E

02

0.10959E-04

S2/1

00/1

0.34521E-06

0.13699E-04

8. I22100

02

0.71014E-0

DuT

6.18737E-83

S2/i

01/1

0.12942E-05

0.549110

01

0.192000

01

0.24839E-0

CUT

0. 39452001

0.44521E-04

0.240000

82

8.24447E-03

S2/1

61/1

8.123430

82

0,94605E-0

CUT

0.11507E-85

6.13699E-84

6.342860

92

0-1690A0-03

52/1

T1/1

0.13790E-B4

0.290430

02

0.39452E-0

CUT

0.34021E-06

0.29571E

02

0.77801E-83

S2/I

82/1

0.13699E-64

8.122180

02

0.71014E-0

OUT

0.3904 10-00

0,109470

0.16737E-03

N2/i

67/1

0.240566-83

0.426570

01

01

0.71971E-0

OUT

0.69041E-06

0.184820

0.10516E-02

N2/1

02/1

0.13699E-B4

0.113120

02

81

0.16408E-0

OUT

0.189470

0.10265E-03

14 2/1

80/1

8.39041E-05

0.24658E-03

0.42657E

04

01

0.71971E-0

OUT

0.10516E-02

CONTINUElYIN

58450-2

PRI. 03 JUN 1988

I

Fro. 2.27 List of nodal failure events

and Jacobian matrix J is partitioned

A B aik = -.3AP1/a9k b1k = -V1,-(3APi/aVk

D eik = -a AQi/a0k dik = - vk•OAQi/i3Vk

then it is sections A and D which strongly influence the solution. If sections B and C are approximated to zero, there are now two separate equations:

LP = ALTO and AQ = DAV/V.

This is known as decoupling.

The fast-decoupled method has some advantages over Newton-Raphson:

•No recalculation of the Jacobian matrix is necessary.

•Speed is some 5 times greater.

•Higher initial convergence in most cases.

One disadvantage is worse convergence close to the solution.

Representation of transformer with off-load taps

A transformer having nominal tap setting is, for nodal analysis purposes, represented as a simple lumped impedance.

However, in power systems, transformer taps are often set to an off-nominal value. This requires some modification to the nodal analysis techniques previously described.

The vast majority of tappings used are in-phase, but occasionally phase-shifting tappings are used. In-phase tappings can be accommodated by simple changes in equivalent circuits. Phase-shifts are more difficult. The admittance matrix Y is no longer symmetrical and major changes must be made to the analytical processes.

One method of accommodating off-nominal tappings is to incorporate the off-nominal tap representation directly into the admittance matrix. This is done as follows.

Consider the general case of a transformer with an arbitary complex tap ratio a' + jb' between two busbars (or nodes) i and k (Fig 2.49 (a)).

Replace this by an ideal transformer with nominal admittance inverting the turns ratio for analytical

131

0, the nodal matrix

Electrical system analysis	Chapter 2

••

• REL1ASILITY INDICES FOR NODE S 3 igull002TEN NO. 1

•

FORCED FAILURE RATE

FORCE() OUTAGE AVERAGE DURATION

FORCED OUTAGE TIME PER YEAR

FAILURE RATE DUE TO MAINTENANCE

OUTAGE AvERACE OuRnTION DUE TO mnINT.

TOTAL OUTAGE TInE PER YEAR DUE TO HA/NT.

TOTAL FAILURE RATE

AyERAGE OUTAGE Tint

TOTAL OUTAGE TimE PER YEAR

0.10189E-01 (OUTAGES PER YEAR)

0.80546E 01

(HOURS)

0.161.06E 00 (POURS PER TEAR)

0.67427E-02 fOuTAGES PER YEAR)

B.94170E 01

(HOURS)

0.03502E-B1 (HOURS PER VERDI

0.24932E-81 (OUTAGES PER YEAR)

0.90069E Al

(HOURS)

8.22156E BO (HOURS PER YEAR)

CONTRIBUTION TO THE ABOVE IMOICES(uS/NC THE SHAD DINE/ASTONS)

OLE TD CUTS ELI.Im.rE0 sr

OuE TO CUTS THAT HAY BE

DUE TO ACTIVE FAILURES

DUE TO ACTIVE FAILURES

A REPAIR ACTION

EI,InxNATE0 CLOSING A N/0 PATH

I A STUCK SHEARER CONO/T/ON

FFR

•

0.31000E-02

FED

•

0.1.7098E-04

FFR

•

0.15014E-01

FFR

0.29019E-04

(CAD

•

0.46645E

02

FOOD

•

0.9822I€ DO

F000 •

0.99997E 00

FOAO

• 0.9999110 00

FO

Y•

0.14600E

00

FOT/re

0.16792E-04

EDT/V.

B.150 14E-01

POT/Y. 0.20019E-04

'FR

•

0.55059E-02

'FR

•

0.60501E-03

mFR

•

0.55023E-03

rIFT1

•

0.76712E-06

9000

0.11310E

02

ROAD

• 0.96769E 00

m040

•

0.97245E 00

ROAD

• 8.97103E 00

0.62303E-01

MOT/r•

0.66360E-03

nOTIF.

0.53507E-03

1400/Y.

0.74490E-06

TFR

•

0.60360E-02

TED

•

0.70294E-03

TFR

•

0.15564E-01

TFR

-

0.20787E-04

NOT

0.241200

02

AOT

•

0.96804E 00

OUT

•

0.99900E 00

AOT

•

0.99921E OD

T01/0.

0.20830E

02

10T/T•

0.68047E-03

TOT/V.

0.15519E-01

TOTer•

0.28764E-04

IONTTALET+,4

GRASP-2

FRI, 03 JUN 1 988 1

17n3. 2.28 List of calculated busbar indices

convenience, Fig 2.49 (b). Suppose the transformer is on nominal tap. The ideal transformer is eliminated and the nodal equations for the network branch are:

Ilk = Yik (V1			Vk)
	V V	i	Yik Vk		(2.2)
	ik
Iki =	Yik (Vi		Vk)
=	v		+ Lik	v	(2.3)
	.ik

Equations (2.4) and (2.5) showing the general case, can be compared with Equations (2.2) and (2.3) showing a transformer with nominal tap setting.

Thus, in the case of a transformer with nominal tap setting, the nodal matrix is constructed by adding Yik to each diagonal term Yu and Ykk, and inserting - Yik in the off-diagonal terms Yik and Y. Similarly, for the general case with off-nominal taps,

and 1,1, = - Iki

In the general case, let the voltage on the k side of the ideal transformer be V„

(a 2 + b 2 )

- (a - jb) and -(a + jb)

Yik is added to diagonal term Yii Yik is added to diagonal term Ykk

Yik is inserted in off-diagonal term Yik Yik is inserted in off-diagonal term Yki

then	V, = (a + jb) Vi
and Vil k =		-
	Ilk =	- (a - jb) Iki
and	Iki = Yik (Vk		Vt)
Eliminating V,
lik	(a 2	+ b 2 ) Yik V, - (a - jb) Yk Vk		(2.4)
Iki = Yik		Vk - (a +	jb) Yik Vi	(2.5)

It can be seen here that, when b is no longer symmetrical.

In most cases, the off-nominal taps are in phase (i.e., b = 0) and the simple ir circuit shown in Fig 2.50 can be deduced from Equations (2.4) and (2.5).

A second approach to representing off-nominal transformer taps is to ignore the off-nominal taps in constructing the nodal matrix and to introduce their effects by modifying the nodal injected currents I,

and Ik by amounts li and AIk, as in Fig 2.51.

132