computerized protection of a three phase transmission line
TRANSCRIPT
Retrospective Theses and Dissertations Iowa State University Capstones, Theses andDissertations
1973
Computerized protection of a three phasetransmission lineToshio KuriharaIowa State University
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KURIHARA, Toshio, 1946-COMPUTERIZED PROTECTION OF A THREE PHASE
TRANSMISSION LINE.
Iowa State University, Ph.D., 1973
Engineering, electrical
University Microfilms, A XEROX Company, Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.
Computerized protection of a three phase
transmission line
by
Toshio Kurihara
A Dissertation Submitted to the
Graduate Faculty in Partial Fulfillment of
The Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major: Electrical Engineering
Approved;
In Charge of Major Work
For the Major Department
For the Graduate College
Iowa State University Ames, Iowa
1973
Signature was redacted for privacy.
Signature was redacted for privacy.
Signature was redacted for privacy.
ii
TABLE OF CONTENTS
Page
I. INTRODUCTION 1
A. Preamble 1 B. Advantages of the Computerized Relay Scheme
over the Conventional Relay 2
II. LITERATURE REVIEW . 5
III. OBJECTIVES 8
IV. SIMULATION OF THE POWER SYSTEM 9
A. Lagging Current Circuit 9 B. Leading Current Circuit 17 C. Results of the Simulation 23
V. ALGORITHM 70
A. Development of the Algorithm 70 B. Tests of the Algorithm 79 C. Comparison of the Algorithms 93
VI. IMPLEMENTATION 96
A, Constraints for the Implementation 96 B. Example of the Implementation 99
VII. RESULTS AND CONCLUSIONS 105
VIII. LITERATURE CITED 107
IX. ACKNOWLEDGEMENTS 109
X. APPENDIX A: PROGRAMS OF THE SIMULATION 110
A. Program for the Case 1, the Three Phase Fault 111 B. Program for the Cases 24, 25, the Single-Line-to-
Ground Fault 115
XI. APPENDIX B: TABLES OF Za, Zb, AND Zc OF THE CASES IN TABLES 3 AND 4 120
A. Tables of Za, Zb and Zc of the Cases in Table 3a 120 B. Tables of Za, Zb and Zc of the Cases in Table 3b 145 C. Tables of Za, Zb and Zc of the Cases in Table 4a 152 D. Tables of Za, Zb and Zc of the Cases in Table 4b 159
iii
TABLE OF CONTENTS (Continued)
Page
XII. APPENDIX C: THE CSMP PROGRAM FOR THE TESTS OF THE ALGORITHM 166
XIII. APPENDIX D: PDP-11/45 AND THE ASSEMBLY PROGRAM OF THE REVISED ALGORITHM 171
J
A. PDP-11/45 171 B. The Assembly Program of the Revised Algorithm 172
1
I. INTRODUCTION
A. Preamble
Ever since the invention of the computer, applications of the
computer have reached many areas. Power systems are not an exception.
Many power companies have installed general-purpose computers in their
central dispatch offices in order to assist system operators. If the
general-purpose computer controls all substations that are a part of
the controlled system, costly and complex communications equipment must
be installed. In addition, even the fastest computer on the market may
not be fast enough to perform all the functions (1). An effective
alternative is a small special-purpose computer installed at the large
substations. The purpose of this investigation is to develop an
algorithm for a computerized relay scheme, which is one of the most
important functions at substations, using a substation computer for
the protective system logic and control.
The approach employed herein to develop the algorithm is the
simulation of a transmission line under faulted situations. The
System / 360 Continuous System Modeling Program (360A-CX-16X) is used
for the simulation (2).
The implementation of a computerized relay scheme is presented
which consists of two analog input subsystems, an interface to a sub
station computer, a substation computer as well as a local man-machine
interface.
The flowchart of the program for the relay functions is also
2
presented. This program is written in assembly language, and the exe
cution time is determined. An interruption scheme is employed in order
to share computer time with some other future possible programs which
would perform routine substation functions such as event recording and
data logging.
B. Advantages of the Computerized Relay Scheme
over the Conventional Relay
The implementation of relay functions on a substation computer is
attractive for the following reasons (3). First, the computer can be
progranmed to implement any desired relay. Second, once a characteris
tic is implemented it can be monitored, and even modified, to adjust
for any change in operating conditions if an appropriate communication
is provided between the central office and the substation computer.
Third, the relay speed can probably be increased using a computer.
Figure 1 shows how effective a fast relay system is in increasing the
allowable power transmitted in a typical power system (4). The opera
tion time of the fastest electromechanical relay is limited to about
one cycle time (16 ms). Solid state relays are much faster but are
often delayed to a cycle or so to assure accurate trip logic and proper
coordination. Nevertheless, it is possible to operate relay functions
much faster by using a substation computer if an appropriate reliable
tripping logic can be developed. Fourth, conventional relays have
been found to have a typical availability of about 97.57. when evaluated
on situations where relay action should have taken place, or which took
place inadvertently (5). By comparison, digital computer availability
3
SLGF
LLP
DLGF
Relay plus breaker time
Figure 1, The effect of the speed of isolation of a fault on the transmitted power
4
well in excess of 99% can be reasonably attained. Fifth, a 1958
CIGRE survey of failures in fault-clearing equipment showed that the
largest single cause of failure was attributed to relay malfunction.
The attributed causes are shown in Table 1 (6). Relay failure was
most commonly attributed to dirty contacts, open coil circuits, in
correct setting or adjustment. A computerized relay scheme could con
ceivably eliminate problems related to incorrect setting or adjustment,
as well as costly maintenance and testing. This would be offset, at
least partially, by required computer maintenance. However, since
the computer can perform a number of functions in addition to relaying,
including self checking, the unit maintenance cost per failure can be
quite low.
Table 1. Attributed causes of failure in fault-clearance equipment
Cause Percentage
Relay failure 437.
Circuit breaker interrupter 13.5%
A.C. wiring 127.
Breaker trip mechanism 77.
Current transformer 7%
D.C. wiring 57.
Potential transformer 37.
Breaker-auxiliary switches 37.
Breaker trip coils 2.5%
D.C. supply 1%
5
II. LITERATURE REVIEW
The demand for power is increasing so rapidly that most power
companies must plan expansion of the power system at a high rate,
usually doubling approximately every 10 years. This is one reason why
computerized substation functions could be attractive, due to the
flexibility offered by the software and the reliability of the digital
hardware. A few papers of this area are presented below.
The Consumers Power Company and Westinghouse Electric Corporation
formed a Digital Computer Protective Device Co-ordination Program Task
Force to investigate to what extent a digital computer could replace
the routine aspects of the relaying art. R. E. Albrecht, M. J. Nisija,
G. D. Rockefeller, W. E. Feero and C. L. Wagner (7) describe the general
aspects of the resulting program and its status, as well as an example
of its application.
Grimes (4) wrote his master's thesis in this area in 1968. The
relay functions chosen to be implemented in his work are over-current
amplitude comparison and calculation of the positive-, negative- and
zero-sequence symmetrical components of the phase currents.
Rockefeller (8) proposed a complete group of programs for the pro
tection both internal (transformers and busbars) and external (trans
mission lines) to the substation. The principle of the computerized
relay scheme in his paper is: upon detection of a likely disturbance,
calculate impedances as if the fault were a phase fault and then as
if it were a ground fault, then make a decision of a trip.
6
Mann and Morrison presented two papers in 1971. One of them (9)
proposed a method of distance-type protection suitable for on-line
digital computer protection of transmission lines. The basic principle
is the predictive calculation of peak fault current and voltage from
a small number of sample values. In a second paper (10), they also
concentrated upon the protection of transmission line by digital com
puter as part of a large project embracing a range of substation func
tions, The principle developed in the paper is: once the disturbance
is detected, the type of the fault (phase or ground) is determined and
a suitable single phase set of relaying voltages and currents can then
be chosen for subsequent impedance calculation. They tried to avoid
working with healthy data.
Rao (11) also devoted himself in this area in 1971. His work was
based on the idea that upon the detection of a suspicious fault, calcu
late impedance by the predictive calculation of peak potential and
peak current.
Rockefeller (1) described the prospects for substation-computer
relaying in 1972. He presents Prodar 70 which is a time-shared process-
control digital computer system designed to provide high-speed phase-
and ground-distance fault protection for a single transmission line.
The Prodar 70 relaying system was placed in experimental service in
February, 1971, on the Tesla terminal of a 38-mile 230-KV line on the
Pacific Gas and Electric System.
PG and E (12) reported that their eight 500 KV substation com
puters log analog data and report those within 10 minutes before and
7
after the fault to the central computer along with the sequence of
events. This information provides a basis for fault analysis.
Lewis Walker (12) reported that Tri-State Utilities presented
plans to install a substation computer which will perform voltage con
trol by switching shunt capacitors and reactors on the same bus. It
will also perform supervisory control and data acquisition, and, protec
tion functions will be attempted in the future.
8
III. OBJECTIVES
The major objective of this dissertation is to develop an algorithm
of computerized relay scheme which would detect the four major faults
(13) such as the Single Line-to-Ground (SLG) Fault, the Line-to-
Line (LL) Fault, the Double Line-to-Ground (2LG) Fault and the Three-Phase
(3 0) Fault. The algorithm has as its goal, the operation of the relay
functions in one half cycle of time (8 m sec). The interruption scheme
is to be employed on the substation computer in order to share com
puter time with other prospective substation functions programs.
9
IV. SIMULATION OF THE POWER SYSTEM
The power system is simulated by the CSMP program (2), and con
sists of a generator, a three phase transmission line 40 miles long, a
series load bus, a single ground wire, and supply transformers as
shown in Figure 2.
Two possible circuits are to be simulated: the former is a lagging
current load, and the latter is a leading current load. That is, the
series load bus consists of a resistance and an inductive reactance
for the lagging power factor load. Figure 3, and it consists of a
resistance and a capacitive reactance for the leading current circuit,
Figure 4.
A. Lagging Current Circuit
Figure 3 shows the detail of the lagging current circuit. The
notations of the circuit are explained in Table 2. The generator and
transformers are replaced by Thevenin equivalent (13) wherein the im
pedance is a quantity to represent the worst condition immediately after
fault occurs. The capacitance from line to neutral in Figure 2 is ne
glected in Figure 3 because it is very small in p.u. value, and its
effect on the phase current was shown in this investigation to affect
only the fifth significant digit. The effect of the capacitance from
line to neutral has to be considered in cases of higher voltages and longer
transmission lines.
The following equations are established in order to simplify the
circuit.
10
Tl / Z/2 ^ — Z/2 )(T2
Ir
U -LN Load
Generator: Z=0.01+j0.05 p.u. (Thevenin equivalent)
Tl, T2: Z=0.01+j0.05 p.u. (Thevenin equivalent)
^3-3^ = 25 MVA (base volt amperes)
Vg l-L = ^9 KV (base voltage)
IB = 209.19 A (base current)
Zq = 190.44A(base impedance)
Figure 2. A simplified power system for the simulation
re 20 miles
'W\/-Laal Relay a Ral2/2
A
-xMir Laa2/2 yi
20 miles
AAA \MSb-A' Ral2/2 Laa2/2
I Lag
4^ AM—\JtM
Relay b Rbl2/2 Lbb2/2 ! B' f Rbl2/2 Lbb2/2 /A
be
—j I ••VtA-^-njOV %
RC12/2 '|Lcc2/2 ' I ^cl2/2 ^ccl/l
Lbg
Leg
?Rc2
sLcc3
AM-
Rgl
-dmv-
Lgg
±.
Figure 3, Lagging current circuit
Ra2
Laa3
Rg2
12
Table 2. The notations of the lagging current circuit
Rail, Rbll, Rcll: Total resistance of the generator and transformers
Ral2, Rbl2, Rcl2: Resistance of transmission line
Ra2, Rb2, Rc2: Resistance of the series load bus
Rgl: Resistance of the ground wire
Rd; Resistance in the off-diagonalpart of matrix R in the Equation [8], Reference (13),
Rg2; Resistance which would be used to provide 30F and LLF.
Laal, Lbbl, Lccl: Total inductance of the generator and two transformers
^aa2' ^cc2- Self inductance of transmission line
Laa3, Lbb3» Lcc3: Inductance of the series load bus
Lgg: Self inductance of transmission line ground return
Lab, Lac, Lag, Lbg, Leg, Lbc: Mutual inductance of lines
Relay a, Relay b, Relay c: Locations of the relays
A, B, C: Locations of faults
A', B', C; Alternate locations of faults
R^ - Rall+ &al2 + a2
Ry = Rbll + Rbl2 + Rb2
Rc = Rcll + Rc12 + Rc2
[1]
[ 2 ]
[3]
Rg - Rgl + Rg2 [4]
Laa~ Laal + Laa2 + Laa3 [5]
Lbb= Lbbl + Lbb2 + Lbb3 [ 6 ]
L^o" Lccl + Lcc2 + LCC3 [7]
13
The following are differential equations describing the circuit.
[ 8 ]
Va' Rg Rd Rd Rd' 'la' Laa Lab Lac Lag
1 MO
(0
Vb =
^d ^b d ^d lb + ^ba bb ^bc bg
u
:b
Vc Rd R-d Rc R-d Ic Lea Lcb Lcc Leg Ic
yg. Rd Rd Rd Rg J-ga Lgb Lgc Lgg
o
where
Vc = V. m
sin wt sin (wt - 120)
sin (wt + 120)
0
Rj = 0.001588 f A/mile
Rewrite the equation. [8] as follows.
o Vabcg = Rabcg'^abcg + Labcg . abcg [9]
-1 1 9 L^abcg . Vabcg = -^abcg • Rabcg • labcg + ïabcg [10]
o
jLabcg ~ &abcg "Zabcg " ilabcg ^ * Êabcg * ^abcg [11]
I^abcg = (Labcg ^ • Vabcg - Labcg~^ '^abcg '^abcg) dt [12]
The solution of equation [12] provides instantaneous values of the
three phase currents and the ground wire current.
The following are the equations which provide instantaneous voltages
of each relay location, relay a, relay b and relay c.
[13]
• ^
^^a
• r > V a
o ' R«Ia + L«Ia
RVb =
Vb - R'ly + L'îb
RVc ^ J
Vc « ,
R'lc + L'Kc,
= Rail - 0. 01, L = ^aal - 0.05
Za = | / | '4 [14]
14
Zb = iRVbj/ llb[ [15]
Zc = |RVc|/ [icl [16]
The equations [14-16] provide important information to establish
the algorithm of a computerized relay scheme. All the calculations
are carried out based on the per unit values (13), for simplicity.
Typical values of matrixes Wbcg and Rabcg are provided from Anderson
(13).
As mentioned before, the four major faults are to be simulated in
the circuit of Figure 3. The following is a scheme to simulate all
faults by changing values of the loads and Rg2.
(a) The Three Phase Fault
When the fault occurs; = Rb2 ~ c2 ~ (p.u.)
^aa3 " bb3 " cc3 " 0 (p.u.)
Rg2 = Large values (p.u.)
(b) The Single Line-to-Ground Fault
When the fault occurs; R^2 = 0 (p.u.)
Laa3 = ° (P.".)
Rg2 = 0 (p.u.) (same as before the fault)
(c) The Line-to-Line Fault
When the fault occurs; Ra2 = Rb2 = 0 (p.u.)
Laa3 = l-bbS = 0 (p.u.)
Rg2 = Large values (p.u.)
(d) The Double Line-to-Ground Fault
When the fault occurs; Ra2 = Rb2 = ^ (p.u.)
Laa3 = Lbb3 = 0 (P-"-)
Rg2 = 0 (p.u.) (same as before the fault)
15
The various simulated cases are listed in Tables 3a and 3b. The
program of the simulation for the Three Phase Fault, case 1 in Table 3a,
is provided in Appendix A,
Table 3a. The various cases of the lagging current circuit^
Case TSTEP P.F. P.L. L.F. T.F.
1 0.022915 0.7 1 ABC 30F
0.026040 0.7 1 ABC 30F
2 0.022915 0.8 1 ABC 30F
0.026040 0.8 1 ABC 30F
0.022915 0.9 1 ABC 30F 3
30F
0.026040 0.9 ABC 30F
0.022915 0.7 0.5 ABC 30F 4
0.026040 0.7 0.5 ABC 30F
5 0.022915 0.8 0.5 ABC 30F
0.026040 0.8 0.5 ABC 30F
0.022915 0.9 0.5 ABC 30F 6
0.026040 0.9 0.5 ABC 30F
20 0.022915 0.8 1 A SLGF
0.026040 0.8 1 A SLGF
0.022915 0.9 1 A SLGF 21
0.026040 0.9 1 A SLGF
22 0.022915 0.8 0.5 A SLGF
22
0.026040 0.8 0.5 A SLGF
0.022915 0.9 0.5 A SLGF 23
0.026040 0.9 0.5 A SLGF
&TSTEP = when fault occurs; P.F. = power factor; P.L. = power ab
sorbed by load; L.F. = locations of fault; T.F. = type of fault.
16
Table 3a. (Continued)
Case TSTEP P.P. P.L. L.F. T.F.
0.022915 0.8 1 AB LLF 40
0.026040 0.8 1 AB LLF
0.022915 0.9 1 AB LLF 41
0.026040 0.9 1 AB LLF
0.022915 0.8 0.5 AB LLF 42
0.026040 0.8 0.5 AB LLF
0.022915 0.9 0.5 AB LLF 43
0.026040 0.9 0.5 AB LLF
0.022915 0.8 1 AB DLGF 60
0.026040 0.8 1 AB DLGF
0.022916 0.9 1 AB DLGF 61
0.026040 0.9 1 AB DLGF
0.022915 0.8 0.5 AB DLGF 62
0.026040 0.8 0.5 AB DLGF
0.022915 0.9 0.5 AB DLGF 63
0.026040 0.9 0.5 AB DLGF
17
Table 3b. The various (Locations
cases of the lagging current circuit of fault: A', B', C»)
CASE TSTEP P.F. P.L. L.F. T.F.
9 0.022915 0.8 1 A'B' c ' 30F
10 0.026040 0.8 0.5 A'B'C' 30F
26 0.022915 0.8 1 A« SLGF
27 0.026040 0.8 0.5 A« SLGF
46 0.022915 0.8 1 A'B' LLF
47 0.026040 0.8 0.5 A'B« LLP
66 0.022915 0.8 1 A'B' DLGF
67 0.026040 0.8 0.5 A'B' DLGF
B. Leading Current Circuit
Figure 4 shows the detail of the leading current circuit. The
notations of Figure 4 are almost the same as Table 2 except the
capacitances of the load, Ca, Cb and Cc. The generator and transformers
are the same as Figure 3. Again, the capacitance from the line to
neutral is neglected.
In order to simplify the circuit, the following are established.
Ra - Rail + Ral2 + Ra2 [17]
Rb = Rbll + Rbl2 + Rb2 [18]
Rc = Rcll + Rc12 Rc2 [19]
Rg = Rgl + Rg2 [20]
Laa - l-aal + a.a.2 [21]
20 miles 20 miles
Laal iRelay a Ral2/2 J Laa2/2 A' R
-w—imv-
al2/2 Laa2/2
Lab
/ /
Lag
Lac
1, -xJUir—
Relay b Rblt/2 Lbb2/2 — u e f i j e / —
Rbl2/2 Lbb2/2
C \
RC12/2 LCC2/2
nV c Relay C Rcl2/2 Lcc2/2
"cg \
?Rc2
:Cc
Lbg
^gg
l
Figure 4. Leading current circuit
:Rb2
~r^b
;Rg2
Ra2
-T-Ca
00
19
Lbb = Lbbl + Lbb2 [22]
= Lccl ^cc2 [23]
The following are differential equations describing the leading
current circuit.
Va ' Ra&d&d&d ' la"
Vb Rd&bRd&d lb =
+ Vc Rd^d^c^d :c
Rd'^d^d^g I L g.
'îa" 'l/Ca
îb 1/Cb
0 le
+ 1/Cc
0 0
J" ladt
where
Lba^bbLbcLbg
LcaLcbLccLcg
LgaLgbLgcLgg
sin (wt)
sin (wt - 120)
sin (wt + 120)
0
Taking the derivative of both sides of equation [24], we have
[24]
'm
O 00
Xabcg = abcg'iabcg + Labcg'T^bcg + la/Ca Ib/Cb Ic/Cc 0
[25]
The equation [25] is the second degree differential equation and has a
form:
CD o 2 1 + 2 P 1 P 2 I + P 2 I = X
The CSMP has a MACRO for this function. So, the equation [25] is
reformed as follows,
o o o o ^ 00 00. . (V^-Rd(Ib + Ic Ig) - Lab*lb ~ ^ac Ic ~ Lag Ig)/l'aa =
00 I_ + (Ra/Laa)?a + (l/C_"Laa) 1, [26]
20
Q O O O Oo oo oo (Vy - RjCla + Ic+ Ig) - ba'^a - Lbc'^c - bg'Ig) / =
\ + (Rb/Lbb) lb + (1/Cb'Lbb) lb [27]
o o O (D OO OO OO
(V{, - Rd (la + lb + Ig) ~ Lea la ~ Lcb' b " Lcg*Ig)/I'cc =
% + (Rc/Lcc) Ic + (1/Cc'Lcc) Ic [28]
la, lb and can be solved by the MACRO as follows.
la = CMPXPL (ICA, IICA, PAl, PA2, XA) [29]
= CMPXPL (ICB, IICB, PBl, PB2, XB) [30]
!(. = CMPXPL (ICC, IICC, PCI, PC2, XC) [31]
where ICA=I^(0), IICA = la(0), PAl = 0.5 RgCCa/Laa)^, 2A.2=(C^'L^^y^
o O o O oo oo oo XA = (Va - Rd(Ib+lc+^g) ~ Lab Ib'^ac ^c'^ag Lg)/Laa
For Ig, the first degree differential equation can be established.
0 = Rd(Ia+lb'''^c)/Rg+(Lga la +Lg^I^ +Lgj,Ic)/Rg+Ig+(Lgg/Rg) Ig [32]
The MACRO for the integration can be used in this case.
Ig = INTGRL (ICG, XG) [33]
where ICG = lg(0),
o o o XG = (-Rd(la+Ib+lc)/Rg-(LgaIa+LgbIb+LgcIc)/Rg-lg)(Rg/Lgg)
RVa, RVb and RVc are from the equation [13].
Za = |RVa| /(Ia( [34]
Zb = IRVbI /fib) [35]
Zc = |RVc [ /jlcf [36]
The equations [34-36] are also the same as the equations [14-16], and
21
provide the information to establish the algorithm.
The following describes the method of simulating the four major
faults in Figure 4.
(a) The Three Phase Fault
When the fault occurs; Ra2 = Rb2 ~ c2 ~ ® (p.u.)
Ca = Cb = Cc = Large value (p.u.)
Rg2 = Large value (p.u.)
(b) The Single Line-to-Ground Fault
When the fault occurs; Ra2 = 0 (p.u.)
Ca = Large value (p.u.)
Rg2 = 0 (p.u.) (same as before the fault)
(c) The Line-to-Line Fault
When the fault occurs; Ra2 = Rb2 ~ ® (p.u.)
Ca = Cb = Large value (p.u.)
Rg2 = Large value (p.u.)
(d) The Double Line-to-Ground Fault
When the fault occurs; Ra2 = Rb2 = 0 (p.u.)
Ca=Cb = Large value (p.u.)
Rg2 = 0 (p.u.)(same as before the fault)
The various possible cases in Tables 4a and 4b are also simulated
in the leading current circuit in order to generalize the algorithm.
The program of the Single Line-to-Ground Fault, the cases 24, 25 in
Table 4a, is presented to demonstrate the programming method of the
simulation in Appendix A.
22
Table 4a. The various (Notations
cases are the
of the leading current circuit same as Table 3a)
Case TSTEP P.P. P.L. L.F. T.F.
7 0.018749 0.022915
0.7 0.7
1 1
ABC ABC
30F 30F
8 0.018749 0.022915
0.8 0.8
0.5 0.5
ABC ABC
30F 30F
24 0.018749 0.022915
0.8 0.8
1 1
A A
SLGF SLGF
25 0.018749
0.022915 0.8 0.8
0.5 0.5
A A
SLGF SLGF
44 0.018749 0.022915
0.8 0.8
1 1
AB AB
LLF LLF
45 0.018749 0.022915
0.8 0.8
0.5 0.5
AB AB
LLF LLF
64 0.018749 0.022915
0.8 0.8
1 1
AB AB
DLGF DLGF
65 0.018749
0.022915
0.8 0.8
0.5 0.5
AB AB
DLGF DLGF
23
Table 4b The various (Notations
cases of the leading current circuit are the same as Table 3a)
CASE TSTEP P.F. P.L. L.F. T.F.
11 0.018749 0.8 1 A'B'C* 30F
12 0.022915 0.8 0.5 A'B'C' 30F
28 0.018749 0.8 1 A' SLGF
29 0.022915 0.8 0.5 A' SLGF
48 0.018749 0.8 1 A'B« LLF
49 0.022915 0.8 0.5 A«B« LLF
68 0.018749 0.8 1 A«B« DLGF
69 0.022915 0.8 0.5 A'B' DLGF
c . Results of the Simulation
The various possible cases shown in Tables 3 and 4 have been
simulated by the CSMP programs. The programs provide la, lb, Ic' ^a»
RV]j, RVç, Zg, Z]3 and Zg as outputs. The macro "PRTPLT" was used to
plot the outputs, and typical graphs for the visibility of the charac
teristics of the faults are prcvided in the following cases;
(a) Lagging current circuit;
Figure 5a-i shows complete outputs of the case 2 (TSTEP =
0.022915), 30F, in Table 3a. The d-c offset can be seen in Figures 5a,
b and c which would decay to zero in a short time. This phenomenon can
be explained by the principle of constant flux linkages (14). The
24
amplitudes of the currents increase suddenly to about 4.5 times com
pared with the corresponding sample in the prefaulted cycle in about
four sampling intervals (4 ms) after the fault is applied. Figures
5d, e and f show that the relay voltages suddenly decrease about 50
percent within four sampling intervals after the fault. Figures 5g, h
and i show Za, Zb and Zc. These have a cyclic characteristic where
the period of the cycle is eight sampling intervals, which is a half
cycle of the system frequency (60 Hz). Za(Tl = 0.0062496) is almost
equal to Za(T2 = 0.014582) in Figure 5g where time between T1 and T2
is one cycle, and the fault did not occur during the time interval. But
Za(Tl = 0.018749) is 0.88338, and Za(T2 = 0.027082) is 0.11359 where
the fault occurred during the interval. This characteristic can be
used to establish the algorithm to detect the faults.
Figure 6 shows la, lb and Ic in the case 2 (TSTEP = 0.026040),
30F. The comparison of Figure 5a and Figure 6a shows that d-c offset
would be different even in the same case if time of the fault is dif
ferent, That is why each case in Tables 3a and 4a are simulated twice:
one is the simulation where the fault occurs when the current is at its
peak, and another is the simulation where the fault occurs when the
current is at a zero crossing point.
Figure 7a shows la in the case 9 (TSTEP = 0,022915), 30F. The
magnitudes of the currents rapidly increase ten times compared with
the corresponding sample in the prefaulted cycle within four sampling
intervals. This fault has a magnitude of twice that of the case 2 in
25
Figure 5a, 30F, in the comparison of the currents because the locations
of the fault are A', B' and C rather than A, B and C. RVa in Figure
7b decreases rapidly to 1/15 its former value compared with the correspond
ing sample in the prefaulted cycle, and this occurs within four sampling
intervals after the fault. This fact proves that the location of the
fault is very important in the analysis of the faults.
Figures 8a-i show 1^, lb, le» R^a, RVb, RVc, and in the
case 42 (TSTEP = 0.026040), LLF. The currents and increase about
ten times compared with the corresponding sample in the prefaulted
cycle within four sampling intervals after the fault, while Ic stays al
most the same as before the fault. The voltages RVa and RVb decrease
a little less than 50%, while RVc stays almost the same as before the
fault. Za, Zb and Zc again show the characteristics of cycle.
Figures 9a-i show la, lb, Ic, RVa, RVb, RVc, Za, Zb and Zc in the
case 61 (TSTEP = 0.022915), DLGF. Again, the currents increase rapidly,
and the voltages decrease after the fault in the faulted phases, while
the healthy phase stays almost the same as before the fault.
(b) Leading current circuit:
Figure 10 shows la, lb, Ic, RVa, RVb, RVq, Zy and Zc in the
case of 24 (TSTEP = 0.018749), which is the single Line-to-Ground Fault,
la increases three times compared with the corresponding sample in the
prefaulted cycle within five sampling intervals (5 ms) after the fault.
D-c offset is greater than in the lagging current circuit. Figures 10a
and d show that I^ is leading with regard to Va before the fault, but Va
is leading with regard to la after the fault because the circuit consists
26
of R(resistance), L(inductance) and C(capacitance) before the fault, but
the circuit consists of R and L only after the fault. This phenomenon
provides difficulty in establishing an algorithm of the computerized
relay functions in the leading current circuit. In another word, more
time is necessary in the operation to compute relay functions. Figures
lOh and lOi again show the characteristic of cycle where the period of
the cycle is eight sampling intervals.
Figures 11a and b show and RV^ in the case 28 (TSTEP =
0.018749), SLGF, where the location of the fault is A' rather than A.
la increases about five times compared with the corresponding sample in
the prefaulted cycle within five sampling intervals after the fault.
decreases about five times compared with the corresponding sample
in the prefaulted cycle within five sampling intervals after the fault.
The comparison of Ig in Figure 10a and la in Figure 11a proves that
the location of the fault is important factor for the analysis of the
faults.
Appendix B presents Tables of Za, Z,b and Zc of all the cases in
Tables 3 and 4. The tables contain sixteen samples of two cycle
(prefaulted and faulted) for Za, Zb and Zc. In addition, first
eight samples (A, B, ..., H) are compared with the last eight samples
(a, b, ..., h) correspondingly, and the ratios such as a/A, b/B,...,h/H
are presented.
27
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RVC -4.28036--1.62296 1.28276 3.99356 6.09656 7.27146 7.33956 6.29046
MINIMUM -7.33996-01
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RVC VERSUS TIME m a x i ^ u " 7.33956-01
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4.28366-01 1.62496-01
•1.28136-01 -3.9924E-01 -6.09586-01 -7.2712E-01 -7.33996-01 -6.29126-01 -4.28496-01 -1.62646-01 1.27986-01 3.99116-01
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T IKE 0. 0 l .0«.l6E-03 2.09J2E-03 ). l24@E-03 4.16S4E-03 5. 2080E-03 6.2496E-03 T. 2912E-03 6. 3328E-03 9. 3744E-03 1.041&E-02 1.1458E-02 1.2499E-02 1.3541E-02 1.4582E-02 l.5624E-02 1. 66A6E-02 1.7707E-02 1.8749E-02 1.9790E-02 2.0832E-02 2. 1874E-02 2. 29155-32 2. 3957E-C2 2. 4998E-02 2.6040Ç-02 2.7082E-02 2. 8123E-02 2.9165E-02 3. 0206E-02 3.1248E-02 3. 2290E-02 3. 3331E-02 3.4373E-02 3. 5414E-02 3. 6456E-02 3. 7498E-02 î . 8539E-02_ 3. 9531E-02 4.0622E-02 4.16&4E-02 4. 2706E-02 4.3747E-02 4.4789E-02_ 4.5830E-02 4.6872E-02 4. 79l*E-0Z 4.8955E-02 4.9997E-02 5.1038E-02 5. 2080E-02
28 3.SblSE 3.1160E 1.7243E 1.2977E 1.0333E 7.9449E 4.9397E-9.3106E-3.8541E 3.1240E 1.7261È 1.2985E 1.03 37É 7.9480E-4.9426E-9.2659E-3.8490E 3.1250E 1.7263E 1.2986e 1.0Î38E 7.9491E-3 .3245É-2.0476E-1.6070E 3.93466^ 2-6527E-2.0075E-1.5409E-l.U73E_-6.9432E-
, l..5?4le-7.1316E-
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_«G 9.6018Ê-01
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00 oi_ 01 q2_ 00 ol 01 PL •01 •01_
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B.6018E-01 8 .6018E-01
3.6013^-01 3.601bg-31
1.0840E-01 1.0B40E-Q1 1.0840E-01 1.0840E-01
8.6018E-01 8.60186-01 8.6018E-01 8.6018E-01
1.0840E-01 1.0840E-01
8. 6018E-01 8. 6018E-01
1.0840E-01 1.0840E-01
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8.601SE-01 8.6013E-01 8. &ai8E-01 8.60186-01
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8. 6018E-01 a.60ia':-oi
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Figure 9h. Zb, Case 61 (TSTEP = 0.022915), DLGF
MINIMUM ZC VERSUS TIME MAXIMUM 4.8531E-02 1.69215 03
TIME IC 1 1 RC RG 0.0 7.1491E-31 1 .0084E 00 8.6018E-01 l.a4l6E-03 3.5544E-01 1.0084E 00 8.60iep-0l 2.0832E-03 5.2407E-01 1.0084E 00 8.6018E-01 3. 1248E-03 9.1408E 01 • 1 .0084E on 8.6018F-01 4. 1664E-03 2.4116E 00 I.0084E 00 8.6018E-01 5.2080E-03 1.5715E 00 1.0084E 00 8.6018Ç-01 6.2496E-03 1.21B9E 00 1.0084E 00 8.60186-01 7. 2912E-03 9.6790E-01 1.0084E 00 8.6018E-01 8.3328E-03 7.1548E-01 1.0084E 00 8.6D18E-01 9. 3744E-03 3.5&05E-01 1.0084E 00 8.6018E-01 1.0416E-02 5.2396E-01 1.0084E 00 8.6018E-01 1. 1458E-02 8.8421E 01 — • 1.0084E 00 8.6018E-01 1.2499E-02 2.4105E 00 1.0084E 00 8.6018E-01 U3541E-02 1.571ÎE 00 1.0084E 00 8.6013E-01 1.45B2E-02 Î.2188Ê 00 1.0084E 00 8.6018E-01 1.5624E-02 9.6791E-01 1.0084E 00 8.6018E-01 I •&6&6E-02 7.1557E-01 1.0084E 00 8.60186-01 1.77076-02 3.5627E-01 1.0084E 00 8.6018E-01 I.8749E-02 5.2301E-01 1.0084E 00 8.6018E-01 I. 9790E-02 9.1494E 01 — • 1.0084E 00 8.6018E-01 2.0832E-02 2.4114E 00 1.00S4E 00 8.6019E-01 2. 1874E-02 1.5715E 00 1.0084E 00 8.6018E-01 2. 2915E-02 1.1844E 00 1.00845 00 8.60186-01 2.3957E-02 8.9017E-01 1.0084E 00 8.6018E-01 2, 4q98E-02 6.4511E-01 1.0084E 00 8. 60186-01 2.6040E-02 3.2700E-01 1.0084E 00 8.6018F-01 2. 7082E-02 4.2598E-01 1.0084E 00 8.60186-01 2.8123E-02 6.5164E 01 1.0084E 00 8.b6l8E-01 2. 9165E-02 2.0504E 00 1.0084E 00 8.6018E-01 3.0206E-02 1.3433E 00 1.0084? 00 8.6018E-01 3.1240E-O2 1.0464E 00 1.0084E oo 8.&018E-01 3. 2290E-02 8.3563E-01 1.00846 00 8. 6018E-01 3.3331E-02 6.2457Ç-01 1.0084E 00 8.6018E-01 3.4373E-02 3.2518E-01 1.0084E QO 8.6018E-01 3.5414E-02 4.0579E-01 1.00S4É 00 8.60186-01 3. 6456E-02 7.3762E 01 - • 1.0084E 00 0.6O18E-O1 3. 7498E-02 2.0276E 00 1.0084E 00 8.6018E-01 3.8539E-02 1.3306E 00 1.0084E 00 8.60186-01 3.9581E-02 1.0377E 00 1.0084E 00 8.60186-01 4.Q622E-02 8_.2900E-01 1.0084E 00 8.6C(18E-01 4.16&4E-02 6.1935E-01 1.0084E 00 8.6018E-01 4. 2706E-02 3.2191E-01 1.00S4E 00 B.6018E-01 4.3747E-02 3.9701E-O1 1.0084E 00 8.6018E-01 4. 4789E-02_ 1.6921E 03 — — • -—- — 4- I .0084E QO 8.6018E-01 4.5830E-02 2.0594E 00 1.0084E 00 8.6018E-01 4.68726-02 1.3409E 00 1.0084E 00 8.6018E-01 4.7914E-02 1.0433E 00 • 1.0064E 00 8.6018E-01 4. 8955E-02 _ 8.32 766-01 . . 1.0084E oo 8.6018E-01 4.9997E-02 6.2215E-01 1.0084E 00 8. 60186-01 5.1038E-02 3.2374E-01 , 1.0084E 00 8.6018E-01 5. 2080E-02 4.0062E-01 1.0084E 00 8.6018E-01
Figure 91. Z^, Case 61 (TSTEP = 0.022915), DLGF
MINIMUM lA VERSUS TIME HàXlMUH -7 . 3625E 00 8 .9000Ç-01
TI-E I& I I RA RG 0 .0 9 .72406- 01 9 .0840E-01 8 .60186-01 l . 7 .23226- 31 9 .08406-01 8 . 60iaF - 0 l 2. 0932Ç-CÎ 4 .7337 '=- 0 1 — — — — — — — 9.0840F-01 8 .60186-01 3. iz' .ar-oj 1.5410F- 31 9 .0840E-01 9 .60l8r-0 l 4 .1664E-31 - l .d950P- 01 ' • 9.0840E-01 8 . 60186-01 5 .2090E-03 -5 .0O42C- 01 9 .0840E-01 3 .60186-01 6 .24956-03 -7 .49536- 01 9 .0B40E-01 8 .60186-01 7 . ?9 l2f -03 -8 . 78656- 01 ' ; 9.0840E-01 8 .60186-01 a.33236-03 -8 .7655c- Ôl —-
9.0840E-01 8 .60186-01 9 . 3744C-0Î -7 .4216E- 01 9 .08406-01 8 . 60186-01 1 . 0416E-02 -4 .9558E- 01 9 .08406-01 8 .60186-31 1 .145Se-02 -1 .7409c-•01 9 .0840F-01 8 .6016F-01 1.24)9^-02 1.73546- 31 9 .0840C-01 a .ôOlQc-Ol 1 . 3541E-02 4 .94546-01 9 .0840E-01 8 .60186-01 1 . 45326-02 7 .40116-•01 9 .08406-01 8 . 60186-01 l . 56246-02 9 .72956- 01 9 .08406-01 B.601BF-01 1 . 6666 6-02 8 .72966-01 9 .08406-01 0 . 6018E - ô r l .77076-02 7 .39386- 01 9 .0840E-01 8 .60186-01 U 57496-02 4 .94276- 01 1 .0840E-01 8 . 60186-01 1 .97906-02 9 .9915F-•02 — • 1.0840E-01 8 .60186-01 2 .0332E-02 -5 .14006-01 1 .0840E-01 8 .60186-01 2 . 18746-02 -1 .32906 00 I .D840E-01 8 . 60186-01 2 . 291 5E-0? -2 . 23746 00 1 .08406-01 8 . 60196-31 2 . 39576-02 -3 .30246 00 1.08406-01 8 .60186-01 2.49986-02 -4 .2729c 00 1 .0840 '=-01 8 .60186-01 2 . £ .04 JE-02 -5 .3995= 30 1 .08406-01 9 .60136-01 2 . 70326-02 -5 .70C9E 00 1.0840E-01 8 .60186-01 2 .3123E-02 -6 .C261C 00 1 .08406-01 8 .60186-01 2 . 91656-02 -6 .0627c JO 1 .0840C—01 3 .60186-01 3.02066-02 -5 .93996 00 1 .0840E-01 8 .60186-01 3 . 1248E-02 -5 .4192E 00 I .OB40E-01 8 .60186-01 3 . 22906-02 -4 . 99486 30 1 .0840E-01 8 . 60186-01 3 . 3331E-02 -4 .36986 00 1.08406-01 8 .60186-31 3 . 4373E-02 -3 .9453e 30 1.08406-01 8 . 60186 -01 3 .5414E-02 -3 .70456 00 1.08406-01 8 .6018r-01 3 . 54566-02 -3 .69996 00 1.0840E-01 8 .60186-01 3 . 7^93^ -02 -3 .94S36 00 1 .08 406-01 8 .60186-01 3 .35396-02 -4 .4142= 00 1.08406-01 8 .6018^-31 3 . '»! 316-02 -5 .04 396 00 1 .0840E-01 8 . 60186-01 4 .C622E-02 -5 .74 516 30 1 .08406-01 8 .60186-01 4 .16646-02 -6 .41 566 30 1 1.08406-01 8 . 60186-01 4 . 27 :66 -02 -6 -?5 i26 . 30 — — * 1.08406-01 8 .60186-01 4 . 37476-02 -7 .2B57C 00 t 1 .0340E-01 8 .60186-01 4 .4799Ç-02 _-7 . 3535:_ 00 • f 1.08406-01 8 . 601 86-01 4 .5830^-02 - 7. 14756 30 - • • 1.0840F-01 8 .60186 -01 4 .59726-02 -6 .69503 00 — 4 1.0840E-01 8 .6018 ' : -01 4 . 79146-02 -6 .36:26 30 • 1.0840E-01 8 .60186-31 4 .59556-02 -5 . 34 03C 30 1 .0B40i : -0 l 8 .6C18F-01 4 . 99976-02 -4 .63 :76 30 1.08406-01 8 .60186-01 5. 10386-02 -4 .3 :53 - 30 1.08406-01 8 .60186-31 5 . 2 0906-02 -3 .6155C 30 1.08406-01 8 . fcO l a r -o i
Figure 10a. la, Case 24 (TSTEP = 0.018747), SLGF
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09
MINIMUM IC VERSUS TIME MAXIMUM
J 0. Ô 1.0416e-03 2.Ô8Î2E-03 J. 1?^BE-0Î_ 4.16S4E-Ô3 5. 2090E-03_ 6. 24969-03 7.2912E-03 8. 3328E-03 9.3744E-03 i.3416Ê-62" J. l4595-02_ 1. 2499E-02
J.3541E-02 r. 4582E-02 l.i624e-p2 1 «66b6É~b2
_l.J707F-02_ 1. @749E-02 J . 9 790E-02
2.083"2É-Ô2 ^.1874e-02_ 2.29Ï5É-02 2.3957?-02_ 2.49985-02
_2.6040E-02 2. 7082^-02"
_2. 9123F-02_ 2. 9ï6sp-62 1.0206ç-02 3. 124 8F-02 j.-2290f-p.2_ 3.33315-02 3. 4 3_73F-02 3. 5414E-Ô2
JL. 6456E-02 3. 74995-02 Ji . 3539P-02_ 3.959lf-02 4.062 2F-02_ 4.1664E-02 4.27CtE-02_ 4. 3747F-02 4. 4739Et02 4.5830F-02 4.6872E-02
'4. 7914E-02 4.8955F-02 4.9997E-02 5.1038E-02 5. 20aOE-02
1 - 2 . b. 3. &.
' 8 . 8. 7. 5. 27
-5. -3. " 6 #
-8." - 8 . -1 -5. • 2 . 5. ' 3 ". Ai. 8. 9.
-9.4642E-I
01
7307E-01 545ÎÇ-02 —-9i4'0r=0t :— S747E-31 2435E-31 6692E-01 7864E-01 —-72736-01 8031E-01 43026-02 80315-01 48205-01 — t725f=^5t •
— 8$ 6180E —01 • 75095-01 ——f 70345-01 ———
9.925ÔE-01 l ic
9'.0840E-9.C840E-9.0840E-9.0840'=-
— *
78735-01 5339E-02 BÎôiE-Ol —-6230E-01 —-73905-01 83225-01
9. _?• 5. 2
- I .
-7.
-9. - 8
6973E-01 296t>e-31 73135-01 4775F-31 15945-01 61275-01 3836E-01 06825-01 42165-01 3961E-01
- 6 #
-3^ "5.
6 .
8. 7. 5. 2
- 1 . -4 A -7. -3. -9. -a. -5. - 2 -6.
1499E-0244E-
01 01
06395-îpul-66465 36045-73965-74 71=: 53635-4462f: Û509Ê-42135: 15035-92 095: 17805-16615-9332c 33226-90325-
02 31 01 01 01 01 01 01 01 01 01 01_ 01 01 31 01_ 02
9.0840E-9.09405-9.0840É-9.0840E-9.08405-9.0840E-
^ 01 8.6018f-31 01 8.6018E-01 01 8.60i8e-0f 01 8 .60ier- 0 i 01 e. 'fc0l8F-0l 01 9.6018F-01 01 8.6018ê-oi 01 3,60l8E-0l 0 1 01
9.0840E-9.0840E ~970b40e-
01 01 01
8.60186-01 8.6018E-OI 8.6018F-31 8.60185-01 8. 60t8î-01
9.08406-01 8.6018E-01 9.0840E-9.08405-9.08406-9.08405-9.O840E-9.0840E-
01 8.6018E-01 01 8.6018E-01
5.6078 f=0t 01 8.6018E-01 01 01
B.6018E-01 8. 6018E-01
9.0840E-9.0840E-
01 8 .6018F-01 01 B.6018E-01
9.0840E-9.0B40Ë-9.084ÔE-9.08405-
01 8 .6018F-01 01 8.6018E-01 0 1 01
8.60186-01 8.6018F-01
9.08405-9.08405-
0 1 01
8. 6018E-01 8.60185-01
9.08406-9.08405-
0 1 0 1
8. 601 8E-01 8.60185-01
9. 9 .
9.0840E-9.0840E-
084ÔF-£^01-0840E-p8406_: 08 40E-0840E-C94ÔE-08405-
01 01
8. 6018E-01 8. 6018E-01
01 01
8.60185-01 8. 6018e-01
9. 9 . 9 . 9 .
31 31
8.60185-01 e . 6 0 1 8 6 - 0 1
01 31
9. 9j, 9. 9.
0840E' 08405
01 0 1 01 0 1
6018P-01 60185-31 60185-31 60186-01 6018F-01 60186-01
9. 9 . 9 . 9 . 9 . ji. 9. 9 . 9 .
0840E-M40E-0840?-m40^ 084ÔE-08406-08405-0840EJ 084of-
01 01
6018E-31 60185-01
01 01 01 0 1
8 .
6 0 1 8 6 - 0 1 6018E-01 6Ô18P-31 60185-31
01 01
6018E-01 60185-31
01 8. 60186-31
Table 10c. Ic, Case 24 (TSTEP = 0.018747), SLGF
TIME
MINIMUM RVA VERSUS TIME
RVA -8.306ie-0l
I
MAXIMUM 8 . 3362e -0l
I RA PG 0. 0 1.04I6E-03 2.0832E-03 3. 1248E-03 4. 'l664E-03 5.2080E-03 6.2496E-03 7.2912E-03 t7y3?8ë-ôî~ 9. 3744E-03 I.Q4I6E-02 I. 1458E-02 tr2<?9e-'ô2~ 1.3541E-02 t.45«e-0r 1.5624E-02 la 565'6î-ô2~~ X.7707E-02
01 01
8.I770E-8.2001E 6.9626E—01 ' — 4*6865^-01 —
9,0840E-01 9.0840E-01
8.6018E-01 8.6013E-01
1.7046E-•1.5349E
oi 01
9.0840E-01 9.0640E-01 9.0840E-01 9.0840E-01
8.6018E-01 8. 6018ç-01 8.6018E-01 e.601bç-01
•4.5409E--6.8562E-•8.1287E--8.15456-
01 01
-6.9582E -4.Ô931E-
i5r 01
9.0840E-01 9.0840E-01
"T.0840E-0i 9.0840E-01
8.6018E-01 8.6018E-01
•1.7139E 1.5259E-
"trstssr-6.8506E-t;i2'5ûr=
•ôï • 01 ïït 01 or 01
"9^840E-3l 9.0840E-01 "^rô84ôe=l^r"
9.0840F-01
9.0840E-01
8.60181^01 8.6018E-01
1787496-02 1.9790E-02
m —
1625E-01 b.9iTW-5.1209E-
?r 01
"TTOffÇOE-Ol 9.0840E-01 1.0840E-01 1.0840E-01
8.60I8E-01 3.6018E-01 "f. 60t8f=ôi 8.6018E-01
8.6018F-01 "Ï;6618E-01 8.6018E-01
2.083ZE-02 2.1874E-02
2.7455E-1.9323E-
01 02
2. 2915E-02 2. 3957Ç-02
•2.1598E--3.9605E-
01 31
2.4998E-02 2.6Q40E-02
-4.9376E--4.94.38E-
01 01
l .OB40F-01 1.0840E-01 1.08406-01 1.08406-01 1.0840E-01 1.0840E-01
"87S51BE-01 8. 6018E-0I 8.6018E-01 8. 6018E-01
2.7082E-02 2. 8123E-02
-3.9791E--2.1915E-
01 01
2.9165E-02 3. 0206E-02 3. 124BE-02 3.22906-02
1.4568E 2.67466 -01 5.0086E-6.78986-
01 01
3. 3331E-02 3.4373E-02 3. 5414E-0Z 3.64566-02 3.7498É-0r 3.8539E-02
7.7444E-7.7237E-6.7277E-4.90456-
01 01
1.0840E-01 1.084OT-01 1.08406-01
8.60186-01 8.6018E-01 8.6018E-01 8.6018E-01 8. 60185-01 8.6018E-01 8.60186-01
1.0840E-01 8.6018F-01 1.0840E-01 1.08406-01
8.6018E-01 8.6018E-01
01 01
1.08406-01 1.0840E-01 1.08406-01 1.0840E-01
8.60186-01 8.6018E-01 8.60186-01 8.6018E-01
2.5279E--4.3807E-
01 03
1.0840E-01 1.08406-01
8.60186-01 3.60186-01
3.9581E-02 .J^.0622Ej-02_ 4.1664E-02 .4.2706ezo^
4.3747E-02 4.4789E-02
-2.4232E -4.2517E
•01
•01 -5.2550E--5.28406-
0 1 01
-4.Î384E--2.5657E-
01 01
4.5830E-02 4.6872E-02 4.79146-02 4.89556-02 4.9997Ê-02 S.1038E-02 5. 20adË-Ô2
-2.3967E-2.2816E
02 01
4.6109E-6.3898E-
01 01
7.3437E-7.3240E-
01 01
1.08406-01 1.08406-01
8.6018E-01 8.60186-01
1.0840E-01 1.0840E-01
8.6018F-01 8. 60186-01
1.08406-01 1.08406-01
8.6018E-01 8.6018F-01
1.0840E-01 1.0840E-01
8.6018E-01 8.6018E-01
1.08406-01 1.0840E-01
8. 60186-31 8.60186-01
1.08406-01 1.0840E-01
6.3298E-01
8.6018E-01 8.6018E-01
1.08406-01 8.60186-01
Figure lOd. RVg, Case 24 (TSTEP = 0.018747), SLGF
o> n)
MINIMUM «VB VERSUS TIME MAXIMUM
T IUÇ
3. : v.ombe-oi 2. 0f l32E-03 Î . 124 BE-03
i 6%4f-03 5 . 2030 t -03
6. 2<.9b?-03 7 . ;<112E-03 8 . 3326E-03 9 . 3744F-03 I . ? ' . i6r -o2 i. i-tssf-oz I .2499F-02 I . 25<, lS-02 i .4532F-02 I. ;624r-02 i .6666E-02 I .77JTF-02 1 . :749F-02 I .O79JÇ-02 2 . DS3 2E-02 2. 1874t-02 2. 20! 5Ç-Ô2 • . ? ' !57r-02 2 . ' .9 î3Ç-02 2 . t04 0F-02_ 2 .7032E-32 2 . . s l2 3E-02_. 2 .9165S-02 3 . Q2?fcE-02_ 3 . I248E-02 3 . 229_0t-g2_. 3 . 3331E-02 3 . -373t -32_ 3 . Ï41 4E-02 3 . &4â6E-02_ ) .7493E-Ô2 3 . f 539f-02_ 3 . Î53 1<^-Û2 4 . :522f-02 _ 4* I 66 4E-02 4 .27:&r-02_ 4 . 3747F-02 4.4739s-02 4 . 5330f-32 4 .1872E-02 4 , 79l4«^-02 4 . ;9 i5e-c? 4 . n iTS-OZ
5. 103f-C2 5 . 2030F-02
8 . 6018E-01 8.60i8e-01 8.60186-01 a.fcoiap-oi 8#6018E—01 8 . 60I8E-01 B.60I8E-01 8 . 6018E-01 S .6018E-01 8 .6018E-01 B.6Ô18E-01 8 .60I9E-01 3 .60Î8E-31 8 . fc0l8E-0l
b75612e-01 I
-8 .5403f-0 l
2 .6377E 9 .0840E-3.45095 9.0840E-7.4230= 0940E-
9 .0940E-804E 7 .3329E 9 .0840E
9 .0840E-9.0840E-9.0840E-
2871E
3.7345E 6 .1340E
9 .0840E-9.0840E-9.0846E-9.0840E 9 .0840É 9 .0840»:-9 .0840E 9 .0840E 97Ô840E 9 .0940E 9 .OèMJÊ 9 .0840E
6014= 5 .4204E 7 .4145= B.2800F
7.8853E 2903E
3 .7379= 8 .6018E-31 8 . 6018E-01 6 .1649E
5938E 8 .6018E-01 8. 6018F-01 5.4134E
7 .4132 = B.6018E-01 8 .6018E-01 8 .3741E
8 .0612F 9 .0840E 9 .0840E
8 .6018E-01 8.6018F-01 6 .5220E
3.9d02E 9 .0B40E 9 .0840=
8 . 6018E-01 3 .6018E-OV 8 . 6018E-01 3 .6018E-01
8 .1957F
4797E 9.0840= 9 .0840E Ï .4136E
5340 = 9 .08408 9 .0840E
3 .60186-31 8 . 601BE-01 8.601be-01 8.6018E-01
8 .5156E 8 .2074E 9.0840E
9 .0840E 6 .6544E 9 .0840E B.6018E-D1
e .6O10f-Ol 4 .0917E 9 .0840E 9 .0840E
4118ë 9 .0840= 9 .0B40E
B. 6018E-01 8 .6018E-01 3640E
7 .4992E 8 .4924=
9 .0840E 9 .0840E
8 . 6018E-31 8 .6018E-01
9 .0840E 9 .0840E 9 .0840E 9._08^0E 9 .084ÔE 9 .0840E
8 . '9266 8 .6018E-01 8 .601BF-31 f i .6456E
4 .0866Ç B.6018E-01 8 . 601 8E-01 B.6Ô18E-Q1 8 .60I8E-01
9 .Û537E 41 38=
5 .3658E 7 .5012E 0840 = 8. 6018F-01
8.6019E-0JI 01
8.4951: 0B40E 19613 9.0840E 8. 6018=
8.6018E-01 9 .0840E 6 .64 98c 9.0840= 4 .0915c
9.0840E 9 .1008= 0940E 2.h335 = 084CE 5 .3638 =
4970 = 0840E
8.6018F-01 8.6018E-01 8 .6Ô18E-01 8. 6018F-01 8.6018E-01
Figure LOe. RVb, Case 24 (TSTEP = 0.018747), SLGF
TIME 0 .0 1.041&P-03 irswzi-inr 1. 1248F-03 ~r.t6t4e-iïr
•>. 2080E-03 6.2vç6e-03 T. 2912E-03 tttîîlf^c 9.3744F-03 "t:ff4ue-0z-l . I458E-.02 -n24me-?2-l . 3541E-02 -174582^-02-l . 5624E-02 l « 6666E—^ 1.7707E-02
MINIMUM -8.2897E-0r
RVC VERSUS TIME MAXIMUM
RVC ^îvîî43é-0l" -2.7376E-01 "^76t8e-02
3.6122E-31
-8.2925^=01 I RC RG
6» 1960E-O1 T. 83 77E—Q1 — 8.28*57E—01 7.4729E-01 - - - - - - - - -
2.73265-01
•3.6072F-01 01 ———• 01
-6.1915^ -7.8338E-=t;787zf -7.4718E -5.5229P-01
2.7334E—01 — —
OT" 01
1.0749E-O2 1.9790E-02 2.0832E-02 ?. 1874E-02
"272915Ë-W 2.3957E-02 2.4998É-Ô2 2 . 6040E-02 t7t0t2f^-2.8123E-02 2;9l65E-02 3.0206E-02 3. 1248E-02 3.2290E-02
4.7231E-3.5292E 6 . 0 3 7 7 e -7.6256F
ôf 01
8.0607E 7.2804E 5.4û44E-5r 2.7172E-01
•ôï •01 >-f" •01 —4--•01
9 . 08406-01 8.60IW-01 9.0840E-01 8.6018F-31
'trôsw-isrr 9.084OE-0J 9.084àe=ôrr 9.0840E-01
8.6018K-01 8. 6018e-01 ~57ïôl8e-0l 8. 6018E-01
9. 9
9.
0840G=ÔI 876018E-01 0840E-01 S.6018E-01 tîirot^
9.
9. •tjt 9.
0840E-01 bffçoe-ol
8.6018E-01 8.6018E-31 8.601 aç-oi
08406-01 8.6018E-01 lOBiVO^iyr 87Ï018Ç-11 0840E-01 8.6018E-01 bffçoe^oi 8.6018?=^îr 0840E-01 8.6018E-01
9.0840E-01 4.0840E-01 9.0840E-01 9.0840€-01 9.0840E-01
-ff.6018E-01 8.6018E-01 8.6018E-01 8.6018E-01
- — —— f 9.0840E-OF 9.0840F-01 •t73ô4ôe-01
9.0840E-01
8. 6018E-01 8. 6018E-01
3.7363T: -3.3V97E-5.9023t:
-7.5020E-
TiT 01
-7.V566E-•7.1979E-
? r 01
9.0840E-01 9.0840E-01 9.0840e-or 9.0840Ç-01
9.60iaE-0l 8.6018F-31 "stsotsf^ôt 8. 6018E-01
7.6018E-31 8.6018E-01 8.6018E-01 8.6018E-01
01 01
9.0840E-01 9.0840E-01
8.6aiaE-01 8.6018E-01
3. 3331E-02 3.4373E-02
-5.3423Ê--2.67266-
01 01
9.0840F-01 9.0840E-01
8. 6Q18E-01 8.60181-31
3.5414E-02 ^a4a6e-02_ 3.7498E-02 1. 8539E-02 3.958 lE-^02 4.0622E-02 4. 1664E-02 4.2706E-02 4. 3 747E-02 4 .4789E-02 4.5836E-02 4 .68726-02
4.0437E-3.4201E-
02 01
9.0840E-01 9.0840e-01
8.6018E-31 8.6018E-01
5.9155E-_7.J . l 06E-7 .9627F-7 .2029E-
0 1 — — 01
9.08406-01 9.0B40E-01
01 —!
01 — — 9.0840E-01 9.08409-01
8.60l8E -ai 8.6018E-01 8.6018E-01 8.6018e-01
5.3468E-2.6771E-
01 01
9.0840E-31 9.0840E-01
e .60l8E-3l 8.60186-01
-4.0000E--3.4159E-
02 01
9.08405-01 9.0840E-01
8.60186-01 8.6018E-31
4.7914E-02 4.89556-02 4.9997E-02 x-ioi8j-p2_ 5.20806-02
•5 .91156--7 .5070E-
01 —-—-01 !->-•
9.0840E-01 9.0840E-01
-7.9595E--7.2000E-
•-5.3443E--2.67496-
01 01 01 01
"Ï.0840E-01 9.0B40E-01 9.0840E-01 9.08406-01
8.60186-01 8. 60186-01 8.6018Ç-01 8.6018E-01 8.6018E-01 8.60186-01
4.01816-02 9.08406-01 8 .6018E-01
Figure lOf. RVc, Case 24 (TSTEP = 0.018747), SLGF
o\ 4>
MINIMUM ZA VERSUS TIME MAXIMUM ttâ^osë'or
TIME ZA I I HA KG 0.0 9.17296-01 + 9.0940E-01 8.60186-01 1. 0<il6f-03 1.13186 00 4 9,08406-01 8. 60186-01 2. :83?f-03 1.47096 00 4" 9,0840E-Ô1 8.60186-01 3. IZ-Sf-03 3.3411F 00 9.0840E-01 8.60186-01 *»• 1 664^-03 8.99496-01 4 9.0840E-01 8.60186-01 5. :0'CÇ-J3 3.33096-01 4 9.08406-01 8. 60186-01 b. Z'-'itf-ôi 6* 0664 E — 01 9.0840E-01 8. 60186-01 ?.:9:2E-o3 7.80316-01 4 9.0840E-01 8.60186-31 8. ?3?3'-03 9.27346-01 4 9.08406-01 8.60186-01 9. 3 7-.<.f-03 l.lOOlE 00 • f 9.0840E-01 8. 60166-01 1. C4;6E-02 1.40416 00 4 9.08406-01 8.60186-31 1. I458C-02 2.69586 00 4 9.08406-01 8.60186-01 1. S.Û7<iOE-Ôl 4 • 9.0840Ë-01 8.60166-01 I . 3541f-0? 3.08666-01 4 9.08406-01 8. 60186-01 l.<.592r-:2 6. 12536-01 4 9.08406-01 8.60186-31 1. 7.64776-31 4 9.08406-01 8. 60186-31 1* 'jbbbi' J2 9. 30356-01 4 9.0840E-01 8.60186-01 l.77:76-32 1.10326 00 4 9.08406-01 8. 6018^-01 1. 3749F-02 1.4C76P 03 4 1.08406-01 8. 60186-01 1.97406-02 5.1252E 30 4 1.0840E-01 8.60186-01 2. Û8326-02 5.34156-01 4 i .0840E-0l 8. 60186-01 2. ! 97<iF-02 1.4540E-02 4 1.08406-01 8. 60186-01 2.29156-02 9.4423E-02 4 1.08406-01 8.60186-01 2. 39576-02 1.19936-31 4 1.08406-01 8. 60186-01 2. '•9?9f-02 1.1556P-31 1.08406-01 8.60186-01 >. t.04Dt-02 9.6946E-02 4 1.0840E-01 8. 60186-01 2. 'a=2t-G2 6.97976-02 4 1.08406-01 8. 60186-01 2. 31236-02 3.6jo6C-0? 4 1.08406-01 8.60186-01 2. Qlb56-02 2.40296-03 4 1.0840E-01 8. 60166-01 3.0236^-02 4.5 9076-02 4 1.08406-01 8.60186-31 3. Î243F-02 9.2424E-02 4 1.06406-01 6. 60186-01 3. 23906-02 1.38726-01 4 1.0840E-01 8.60186-01 3.33216-02 1.77236-01 4 l .08406-01 8.60186-01 3.4373^-02 1.95776-01 4 1.0840E-01 8. 60166-01 3. ;4I4F-02 1.81616-01 4 1.08406-01 8.60186-01 3. 04566-02 1.32566-01 4 1.08406-01 8. 60186-01 3. 74936-02 6.40746-02 4 U0840E-01 8.60186-01 3. P!3«;6-Û2 9.92416-04 4 1.08406-01 8. 60186-01 3. -în6-02 4.80426-02 " t 1.0840H-01 8. 60166-01 4.:t:26-ù2 7.4036<^-32 1.0840E-01 fl.60186-31 4. 166'.«-02 6.19106-02 4 1.08406-01 8. 60186-01 4. 2736<=-02 „ 7.59616-02 4 I .0840E-01 8. 60186-31 4. 37476-:,2 5.95476-02 1.08406-01 8.60186-01 4. «.7696-02 3.40906-3? 1.0840E-01 8.60186-01
58306-32 3.35326-33 4 1.08406-01 8 .60186-01 4. 68726-02 3.40746-02 + 1.0840E-01 8. 60186-01 4."91,«-02 7.60476-02 4 1.0840E-01 8. 60186-01 4. ^9;56-02 1.19656-01 4 1.08406-01 8. 60186-01 4. ( .9976-02 1.58596-31 4 1.0840E-01 8.60186-31 5- 1 0326-32 1.01506-31 •f 1.0940E-01 8.60186-01 5. i0?06-C2 1.74076-31 1.08406-01 8. 60186-01
Figure 10g. Za, Case 24 (TSTEP = 0.018747), SLGF
on Ln
MINIMUM ' Z8 VERSUS TIME MAXI>WH 1.84056-03 e . i rSSE 02
T_[Hi I I_ 58 «G 0.0 4.4948F-01 9 .0840E-01 8.6018Ç-31 1.0416E-03 6.9864E-J1 9.0840E-01 8.6Q18E-01 2.0832E-03 8.61UE-01 *• • ; 9.0840E-01 8.6018E-01 3. 1248E-03 1.0154E 00 9.0840E-01 8.60186-01 4.1664E-03 I.2212E 00 *• 9,0840E-01 8.6018E-01 5.2080E-03 1.66341: 30 *• ' 9.0840E-01 8.6018^-01 t.2496E-03 7.4443E 00 + 9.0840€-0l 8.6018E-31 r . 2912E-03 2.1539E-01 9.0840E-01 8.60186-01 6.3328E-03 4.5092Ç-01 • . ; 9.0840E-01 9.6018E-01 9.3T44E-03 6.9343E-01 4- ' i ' • ; 9.0840Ç-01 8.6018F-01 1.0416E-02 8.S455E-0I + 9.08406-01 8.6018E-01 I. 1458E-02 1.0075E 00 • 9.08405-01 8.6018E-01 1.2499E-02 1.2112E 00 4- 9.0840E-01 8.6018E-01 1.3541E-02 1.6499E 00 + 9.0840E-01 8.60186-01 1.4582E-02 6.9839E 00 • ! 9.0840E-01 8.6018E-01 1.5624E-02 2.1839E-01 *• 9.0840E-01 8.6018E-01 l« 66&6E-02 4.5189E-01 + 9.08406-01 8.60I8E-01 1. 7707E-02 6.9439E-01 • : 9.Q840E-01 8 .60186-01 1.8749E-02 8.5532E-01 + • i 9.0840E-01 8.60186-31 1. 9790E-Û2 1.0409E 00 + • 1
1 • : 9.0840E-01 8.60186-01
2.0832E-02 1.3791E 00 * . i 9.08406-01 8.60186-01 2. 1874E-02 2.6929E 00 t 1 ! ! ; ' I 9.08406-01 8.6018E-01 2.2915E-02 2.4087E 00 9.0840E-01 8.6018E-01 2.3957E-02 1.4401E-31 * 9.0840E-01 8.6018E-01 2.4998E-02 2.7514E-01 9.0840E-01 8. 6018E-01 2.6040E-02 4.89928-01 9.08406-01 8.60186-31 2.7082E-02 6.5834Ç-0X * ^ 1 9.0840E-01 8.60186-01 2.B123E-02 8.4341E-01 I : 9.0840E-01 8.6018E-01 2.9165E-02 1.1422E 00 * 1 ; 9.08406-01 8.6018E-01 3. 0206E-02 2.1Z19E 00 . ! ! • 9.0840E-01 8.6018E-01 3. 1Z48E-02 2.8267E 00 • • ! 1 1 i 9.0840E-01 8.6018E-01 3. 2290E-02 1.5467E-01 1 i
• 1 ! • : 9 . 08 406-01 8. 60186-01 3.3331E-02 2.5470E-01 . • . ! ' 9.0840E-01 8.6Qil8E-0l 3.4373F-02 4.5895E-Û1 + 1 1 : 9.08406-01 8. 6018E-01 3. 5414E-02 6.1$28E-01 , ; ! • • • 9.0840E-01 8.60186-01 3.6456E-02 7.79688-01 + 1 I ; . 9.0840E-01 9.60186-01 3. 7498E-02 1.0249E 00 + ; 9.0840E-01 8. 6018E- 01 3.89396-02 1.69Q8IE 00 +: ' ! ' ; 9.0840E-01 8.60186-01 3.9581E-02 5.9911E 00 + , ; ! : 9.08406-01 8.6018E-01 4.0622E-02 1.7599E-31 • : : 9.0840E-01 8.6019E-01 4.1664E-02 2.74936-01 + , ; • , . 9.08406-01 8.60186-31 4. 2706E-02 4.8611E-01 +• : 1 i : : : 9.08406-01 8. 60186-01 4. 3747E-02 6.4791E-01 *• ; 9.08406-01 8.6018E-01 4.4789Ê-02 8.2334E-01 + ' : 9.0840E-01 8.6018E-01 4.5830E-02 1.0998E 00 *: 9.08406-01 8.60186-01 4.6872E-02 1.9448E 00 1- 9.0840E-01 8.6018E-01 4. 7914E-02 3.5082E 00 9.08406-01 8.60186-01 4.8955E-02 1.6247Ç-01 ; 1 : 9.08406-01 8.60186-01 4.9997E-02 2.6121E-01 9.0840E-01 8. 6018E-01 5.1038E-02 4.6778E-01 *; I : ! 1 ; 9.0840E-01 8.6018E-31 S. 2080E-02 6.2969E-01 ^ : 1; 9.08406-01 8.60186-01
Figure lOh. Zb, Case 24 (TSTEP = 0.018747), SLGF
MINIMUM IC VERSUS TIME MAXIMUM 1. 08346-01 1.74496 01
T IMf zc 1 I RC • KG 0.0 2.0267E 00 9.0840Î-01 8.6O1BE-0I 1.04I6E-03 4.1826E 00 9.0840E-01 8.6I018E-01 2.0832F-03 1.2166Ç-01 »• 9.ÛS40C-01 8.6018E-01 3. 1248Ç-03 5.4941E-01 9.0B40E-01 8. 6{018E-01 4.1664E-03 7.5172E-0X 1 : 9.0840E-01 8.601BE-01 5. 2080C-03 9.0408E-01 9.0840E-01 B.6018E-01 6.2496F-03 1.0A41E 00 9.0840E-01 e.6018E-01 7. 2912F-03 1.3048E 00 9.0840f-0l 8.6018E-01 8, Î328E-03 l.iToit 00 - • - — + , 9.0840E-01 8. 6\01 BF—01 9.3744E-03 S.0323E 00 ' 9.0840E-01 B.6D18E-01 1.0*l6f-0i 1.2449E-31 * 9.0B40f-Ol 8. 60l BE—01 I.I4S8E-02 5.5650E-01 9.0840E-01 8. 6^18E-01 l.2*99t-02 7« S764E-01 9.0840E-01 fl.4;01«F-ôl I . 35'.lf-02 9.0900E-0i 9.0B40E-01 8.&018E-01 1.4592E-Ô2 1.0637E 00 — + ^ 1 • 9.0840F-01 8. 6l018E-01 1. 5624E-02 t.3101E 00 9.0840E-01 8. 6018F-01 1.66b6E-02 1.9814E 00 9.08406-01 8.6aiSE-01 I. 7737E-02 4.9393E 00 9.0B40E-01 8.6018E-01 I. 8749E-02 1.23966-01 9.0840E-01 8.6018F-01 1.9790E-02 5.3287E-01 : 9.0840E-01 8.6018E-01 2. 0832E-02 6.9090E-01 ^ ; ' • 1 9.084k)E-01 8.6,0186-01 2. 1874*-02 7.7558E-01 9.0B40E-01 8.6018E-01 2.2915E-02 8.3124F-01 —' • 9.0840E-01 8.6'018E-01 2. 3957E-02 8.7752E-01 9.0840F-01 8.6018E-01 2. ' .5?SE-02 9.34S0E-01 — + 9.0840E-01 8. 6018E-01 2. 6040E-02 1.0967E 00 — • 9.0840E-01 8.60186-01 2.70B2E-02 3.2227E-J1 9.0840E-01 8.6018E-01 2. 8123E-02 7.3704F-01 9.0840F-01 8. 6018E-01 2.9I65Î-02 7.9937E-01 9.0840E-01 8.6018E-01 3. 02066-02 8.2728E-01 9.0840E-01 8. 60I8E-01 3. 1248E-02 8.4451E-31 9.0840E-01 a. 6018F-01 3.2270F-02 8.5729E-01 9.084'0E-01 8.60186-01 3. 3331E-02 9.6868F-01 9.0840E-01 8.60186-01 3.4373E-02 8.9367E-01 9.0840E-01 8.6018E-01 3.5414E-02 7.9854E-UI ! . 9.0840E-01 8. 60186-01 3. 6456E-C2 8.7524E-0I ' : 9.0840«:-0l 8.6018E-01 3.749BF-02 8.3760E-01 9.0840E-01 8.6018E-01 3. 6539E-02 9.9836E-01 9.0840Ê-01 8. 6018E-01 3.9591E-02 9.1110E-01 ! 9.0840E-01 8.6018E-01 4.06225-02 9.2976E-01 9.0840Ç-01 8. 601BF-01 4.1664E-02 9.6577F-01 9.0840E-01 8. 6(018E-01 4.2706E-02 1.0944E 00 9.0840E-01 8.60186-01 4, 3747E-02 3.8061E-01 •f , . 9.0840E-01 8. 6018E-01 4. 47395-02 7.7260F-31 9.0B40E-01 8.6018E-01 4.5830F-02 8.2676E-01 9.0840E-01 8. 60186-01 4.6872Ç-02 9.5105t-0l 9.084OT-01 8.60186-01 4.7914E-02 9.6724E-01 9.0840E-01 8.60186-01 4, 8955E,-02 . 8.81709-01 9.0840E-01 8. 60186-01 4.9997F-02 8.99989-01 9.0840E-01 8. 60186-01 5. 1038E-02 9.44435-01 9.0840E-01 8.60186-01 5.2080E-02 5.9062E-31 «-4 ! 9.0840E-01 8.60186-01
Figure 101. Zq, Case 24 (TSTEP = 0.018747), SLGF
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r I RA PO 9.:^?0F-C! 8."?00' =-01
9.0840;-0l 9.0840:^-01
8.63!?F-"l 8.601 "5-01 1 .0-Vl
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8.63!?F-"l 8.601 "5-01
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9.09405-01 9.0840':-0l
8.60MF-OI 8.6018F-01
4.\66 4E-0? fi.208CF-03
1.70465-01 -1.9349F-01
9.0940E-01 9.0940C.-01
8.601"5-Ul 8.60195-01
6.3490F-Ô3 7.2O12C-03
-4.54C9F-01 -6.8S62F-01
9.0940E-01 9.39409-01
8.60195-0! 8.6018F-01
8.î3l9=-6i q.?744F-03
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1 .04160-0' ' . .14585-02
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9.08405-01 9.0840E-01
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T . 2 4 q q E - 0 2 l .?541«--02
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9.0840=-01 9.0840F-01
8.60185-01 8.60185-01
T,45S»e-()d~ ' .5t>4E-02
4.5334F-01 fc.8506F-0l
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8.60185-01 8.60185-01
T.<6ttr^2 1 .7TCTF-02
8.1250E-01 8.1625F-01
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6.9575F-CI 5.6322E-C1
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2.0832E-02 2.1874F-C2
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7.2915E-02 2.3957F-02
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8.60185-01 0.6O18F-O1
4.16f4E-02 4.?7C6«r-02
-3.lfc02'=-0l -3.3519Ç-CI
H 11 • : 1 11 '1 5.42015-02 5.420ie-02
8.6018E-0! 8.60185-01
4.3747F-02 4.4789E-02
-7.fl5Z9f-0l -1.75C6E-01
5.42015-02 5.42015-02
8.6013F-01 8.60195-01
4.5830E-02 4. fcf l72F-02
-2.23t4E-02 1.4952F-G1
5.42015-02 5.4201Ç-0Z
8.60185-01 8.60185-01
4.7914F-0? 4.0955F-O2
• 3.1057E-01 4.3812F-01
5.42015-02 5.420XF-02
P.60185-01 8.6019F-Û1
4.9997F-02 5.10386-02
5.1086E-01 5.1677F-CI
5.4201E-02 5.4201E-02
8.60185-01 8.60185-01
5.2080F-C2 4.5410F-01 5 .4201F-02 8.60185-01
Figure lib. RVa, Case 28 (TSTEP = 0,018749) , SLGF
/
70
V. ALGORITHM
A. Development of the Algorithm
An algorithm is developed to establish computerized relay functions
which would detect four major faults, viz., the Three Phase Fault (30F),
the Line-to-Line Fault (LLF), the Double Line-to-Ground Fault (DLGF)
as well as the Single Line-to-Ground Fault (SLGF). In addition, the
algorithm contains a distance relay function, which would detect
whether the fault occurred in the primary zone (within 20 miles from the
relay position) or in the secondary zone (20-40 miles from the relay).
The algorithm is based upon intuitive, empirical tests on measured
simple phenomena rather than a theoretically oriented solution. The
empirical pattern technique utilizes the fault induced differences in
cycle to cycle current, voltage and instantaneous values where an
instantaneous voltage is divided by an instantaneous current. The
local intelligence used for the algorithm is the following.
Voltages: 16 samples per cycle (60 Hz)
Currents: 16 samples per cycle (60 Hz)
Important factors in the decision making for the algorithm were
the following: a) The accuracy of detection of faults should be op
timized rather than the selectivity, i.e., if an error is to be made it
is preferable to trip falsely than to fail to clear a fault, b) The
algorithm should work for every possible situation of the power system,
namely, the maximum and minimum loads throughout a year as well as for
different power factors.
71
The algorithm was developed based upon the results of the simula
tion such as that in Appendix B which covered a reasonable range of
conditions. The power system status in real time is broken down into
four possible situations in this research as follows;
(a) The three phases fault situation: The Three Phase Fault
(b) The two phases fault situation; The Line-to-Line Fault and
the Double Line-to-Ground Fault
(c) The one phase fault situation; The Single Line-to-Ground Fault
(d) No fault situation; Normal Condition.
The classifications are necessary because the severity of the fault de
pends upon what kind of fault it is, and how far it is from the relay.
Different threshold values have to be used for different situations to
get better accuracy in relay functions.
The results of the simulation provided two important facts for
the algorithm as follows;
(a) A type of pattern recognition can be used to detect a fault.
For example, Za(Tl) is almost equal to Za(T2) where time between T1 and
T2 is one cycle, and a fault did not occur during the interval, as in
Figure 5g. Nevertheless, Za(T2) is substantially smaller compared
with Za(Tl) when the fault occurred during the interval.
(b) Severity of a fault depends on the location of the fault.
This characteristic can be used to improve the selectivity of the
algorithm.
The algorithm is presented below, and was developed based upon
the above facts.
72
Step 1. Detect the first Rx by the comparison with the threshold
value 1 (THl).
Rjf = Z^(T2)/Zx(Tl) where x = a, b and c.
Time between T1 and T2 is one cycle. If Rx ^THl, the condi
tion could conceivably be a fault. If Rx — THl, then Rx
also is compared with Threshold values TH2 and TH3.
Step 2. After detection of the first suspicious Rx, eleven Rx*s are
compared with THl, TH2, and TH3. A total twelve Rx's are
compared with THl, TH2 and TH3 consecutively. Four counters,
CTl, CT2, CT3 and CT4 are provided. CTl counts how many times
Eix's are compared with THl. CT2 counts how many times Rx's
are less than or equal to THl. CT3 counts how many times Rx's
are less than or equal to TH2. CT4 counts how many times Rx's
are less than or equal to TH3. Values of THl, TH2, and TH3 are
selected based on the data in Appendix B.
The reason why twelve consecutive Rx's are compared with
THl, TH2 and TH3 is as follows; These comparisons provide
enough information to classify the power system into four pos
sible situations and to detect whether a fault is in the pri
mary zone or in the secondary zone.
Step 3. Classify the power system into four situations when CTl is
equal to twelve,
CT2 is used for this classification as follows:
73
(1) 10 CT2 612
The three phases fault situation.
(2) 6 f: CT2 £:9
The two phases fault situation.
(3) 3 CT2 i 5
The one phase fault situation.
(4) 1^ CT2 6 2
The normal situation (No fault).
How the above logic is determined is the following:
When a fault is SLGF, the results of the simulation indicate
that CT2 is more than two and less than six.
When a fault is DLGF or LLF, the results of the simulation
indicate that CT2 is more than five and less than ten.
When a fault is 30F, the results of the simulation indicate
that CT2 is more than nine and less than or equal to twelve.
When a suspicion of a fault is unfounded, the results of the
simulation indicate that CT2 is less than three. This threshold
method could avoid a detection of a small disturbance which does
not require tripping of the circuit breaker.
4. Detect whether the fault is in the primary zone or in the
secondary zone. Two counters CT3 and CT4 are used for this pur
pose. That is, CT3 is used for the one phase fault situation, and
CT4 is used for the two phases fault situation and the three phases
fault situation as follows:
1. The one phase fault situation:
(a) 2 £:CT3 £ .12
The fault is in the primary zone.
(b) CT3 -2
The fault is in the secondary zone.
2. The two phases fault situation:
(a) 2 CT4 é:12
The fault is in the primary zone.
(b) CT4-^ 2
The fault is in the secondary zone.
3. The three phases fault situation;
(a) 3 dCT4 6 12
The fault is in the primary zone.
(b) CT4.Z 3
The fault is in the secondary zone.
The basis for the above logic is the following: Even though two
faults are in the same classification the fault severity is quite dif
ferent if the locations of the faults are different. That is why CT3
counts more than or equal to two when a fault is in the primary zone,
and CT3 is less than two if a fault is in the secondary zone. This de
cision also is based upon the results of the simulation. In other words,
CT3 and CT4 indicate the severity of the fault.
Step 5. Initiate a trip for the circuit breakers.
Step 6. Reset counters CTl, CT2, CT3 and CT4.
Step 7. Clear interruption.
75
Figures 12a and b present the flowchart of the subprogram for the
algorithm mentioned above. The supervisory program and the subprogram
of the relay functions always reside in the main memory of the substation
computer. The supervisory program keeps the track of the highest pri
ority of the interruption and decides which interruption would be
served first. In addition, it provides housekeeping such as setting
initial values which is necessary for the subprograms of the algorithm.
The subprogram of the relay functions should always have the highest
priority because the availability of the relay functions must be almost
100 percent in order to avoid damage to the power system. Other sub
programs, such as data logging, should be given a lower priority and
must be interrupted when a fault occurs.
As soon as samples of the voltage and the current are stored in
the main memory of the substation computer through an interface, the
subprogram of the relay functions is initiated. Explanation and nota
tions of the flowchart are presented below.
HOUSEKEEPING A: ZA9=ZA8 ZA8=ZA7 ZA7=ZA6 ZA6=ZA5 ZA5=ZA4 ZA4=ZA3 ZA3=ZA2 ZA2=ZA1
(Update values
of ZA2, ,ZA9 for the cycle to cycle comparison)
HOUSEKEEPING B; ZB9=ZB8 ZB8=ZB7 ZB7=ZB6 ZB6=ZB5 ZB5=ZB4 ZB4=ZB3
ZB3=ZB2
ZB2=ZB1
(Update values of ZB2, ZB9 > • • • • • >
for the cycle to cycle comparison)
76
HOUSEKEEPING: ZC9=ZC8 ZC8=ZC7 ZC7=ZC6
ZC6=ZC5
ZC5=ZC4 ZC4=ZC3 ZC3=ZC2
ZC2=ZC1
CTl, CT2, CT3 and CT4: Counters
THl, TH2, and TH3: Threshold values
THl = 0.4, TH2 = 0.1, TH3 = 0.05
These values have been chosen arbitrarily, based on the
results of the simulation.
I; Counter
This counter I keeps track of phase identification. That
is, when I is equal to one, samples of voltage and current
come from phase a. When I is equal to two, the samples
are from phase b. When I is equal to three, the samples
are from phase C.
lA, IB and IC; Phase currents
RVA, RVB and RVC: Relay voltages of each phase
10FS; The one phase fault situation
20FS: The two phases fault situation
30FS: The three phases fault situation
RESULT; When Result = 1, the fault is in the primary zone, and
it is in the secondary zone when Result = 2
P.Z.; Primary Zone (0-20 miles from the relay)
S.Z.; Secondary Zone (20-40 miles from the relay)
(Update values of ZC2,...,ZC9 for the cycle to cycle comparison)
77
Interruption
I = 14-1
HOUSEKEEPING
HOUSEKEEPING 1=0
lA
HOUSEKEEPING IA=0.0001
IB:0
15=0.0001
IC;0 ZA1=RVA/IA
IC=0.0001
ZB1=RVB/IB
ZA9=0.0001 ZCI=RVC/IC
ZB9 :0.00l
R=ZA1/ZA9
;C9:0. OOi ZB9=0.0001
R=ZB1/ZB9 2C9=0.0001
R=ZC1/ZC9
Figure 12a. Flowchart of the algorithm
78
R:TH1
ct2=ct2+1
ctlio
R:TH2
:T3=CT3+1
R;TH3
CT1=CT1+1
CT1;12 Clear inter s nifii
(Fault) (No fault) CT2:3
CTl, CT2 CT3 CT4<r-0
CT2:6 (10FS)
CT3;2 %2:10
(30FS) Result=2 Result=l CT4:2
:T4:3
Figure I2b. Flowchart of the algorithm
79
B. Tests of the Algorithm
The algorithm for the computerized relay scheme was developed
based on the two important facts, stated before, which have been found
in the results of the simulation. The algorithm must be tested to
evaluate an accuracy in the operation of relay functions. The various
tests have been done on different locations of faults, viz., 5 miles,
15 miles, 25 miles, 35 miles and 50 miles away from the relay location,
which are shown in Tables 5a, b and c. The program for the tests of
the algorithm consists of two parts as follows.
Part I. Simulation of the faulted power system by the CSMP program
Part II. The program of the algorithm written in the CSMP program.
This program also uses a macro "PRTPLT" for the output and plots a graph
which indicates Ra, Rg, and the time of the detection of fault as well
as locations of fault. One exangle of the outputs is presented in
Figure 13, which is TEST 19 in Table 5a. The program for TEST 19 of
the algorithm is presented in Appendix C.
The results of all tests are presented in Tables 6a, b and c.
Tables 6a and b show that the algorithm has an accuracy of 100 percent
in the detection of faults. However, there are two errors in Table 6a
and three errors in Table 6b in the detection of locations. These five
errors are selectivity errors, i.e., even though the faults are in the
secondary zone, the algorithm has detected as faults in the primary
zone. The reasons why more errors have been found in the leading cur
rent circuit than in the lagging current circuit are: d-c offset and
80
Table 5a. Tests of the algorithm
P.L. = power absorbed by load a = 5 miles P.P. = power factor R.F. = location of fault T.F. = type of fault K.C. = kind of circuit TSTEP = time of fault
b = 15 miles c = 25 miles d = 35 miles T1 = 0.012499 sec T2 = 0.016666 sec
TEST TSTEP P.L. P.P. l.p. T.F. k.c.
1 T1 1 0.8 a SLGF Lagging
2 T1 1 0.8 c SLGF If
3 T2 0.5 0.8 b SLGF ft
4 T2 0.5 0.8 d SLGF tf
5 T1 1 0.8 a DLGF ff
6 T1 1 0.8 c DLGF ft
7 T2 0.5 0.8 b DLGF ft
8 T2 0.5 0.8 d DLGF If
9 T1 1 0.8 a LLP If
10 T1 1 0.8 c LLP ff
11 T2 0.5 0.8 b LLP ft
12 T2 0.5 0.8 d LLP ft
13 T1 1 0.8 a 30F ff
14 T1 1 0.8 c 30P ft
15 T2 0.5 0.8 b 30P ff
16 T2 0.5 0.8 d 30P ff
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
81
Tests of the algorithm (Notations are same as Table 5a)
TSTEP P.L. P.P. L.F. T.F. K.C.
T1 1 0.8 a SLGF Leading
T1 1 0.8 c SLGF If
T2 0.5 0.8 b SLGF ft
T2 0.5 0.8 d SLGF If
T1 1 0.8 a DLGF II
T1 1 0.8 c DLGF II
T2 0.5 0.8 b DLGF If
T2 0.5 0.8 d DLGF It
T1 1 0.8 a LLF II
T1 1 0.8 c LLF n
T2 0.5 0.8 b LLF II
T2 0.5 0.8 d LLF It
T1 1 0.8 a 30F II
T1 1 0.8 c 30F It
T2 0.5 0.8 b 30F It
T2 0.5 0.8 d 30F If
82
Table 5c. Tests of the algorithm (Notations same as Table 5a) (e = 50 miles)
TEST TSTEP P.L. P.F. . L.F. T.F. K.C.
33 T2 1 0.8 e SLGF Lagging current
34 T2 0.5 0.8 e SLGF "
35 T2 1 0.8 e DLGF "
36 T2 0.5 0.8 e DLGF "
37 T2 1 0.8 e LLF "
38 T2 0.5 0.8 e LLF "
39 T2 1 0.8 e 30F "
40 T2 0.5 0.8 e 30F "
83
Table 6a. Results of the tests of the algorithm
C: Correct in the detection of location P.Z,: Primary zone
NC: Not correct in the detection of location S.Z.: Secondary zone
Result Delay to be detected Test Detection of fault Detection of location (ms)
1 Detected P.Z. (C) 4
2 Detected P.Z. (NC) 6
3 Detected P.Z. (C) 4
4 Detected s.z. (C) 5
5 Detected P.Z. (C) 5
6 Detected P.Z. (NC) 5
7 Detected P.Z. (C) 5
8 Detected S.Z. (C) 5
9 Detected P.Z. (C) 5
10 Detected S.Z. (C) 5
11 Detected P.Z. (C) 5
12 Detected S.Z. (C) 5
13 Detected P.Z. (C) 6
14 Detected S.Z. (C) 4
15 Detected P.Z. (C) 4
16 Detected S.Z. (C) 5
84
Table 6b. Results of the tests of the algorithm (Notations are same as Table 6a)
Result Delay to be de-Test Detection of fault Detection of location tected (ms)
17 Detected P.Z. (C) 9
18 Detected S.Z. (C) 6
19 Detected P.Z. (C) 5
20 Detected P.Z. (NC) 7
21 Detected P.Z. (C) 6
22 Detected S.Z. (C) 5
23 Detected P.Z. (C) 5
24 Detected P.Z. (NC) 7
25 Detected P.Z. (C) 6
26 Detected S.Z. (C) 5
27 Detected P.Z. (C) 5
28 Detected P.Z. (NC) 7
29 Detected P.Z. (C) 5
30 Detected S.Z. (C) 5
31 Detected P.Z. (C) 6
32 Detected S.Z. (C) 6
85
Table 6c. Results of the tests of the algorithm (Notations are same as Table 6a)
Result Delay to be detected TEST Detection of fault Detection of location (ms)
33 Detected S.Z. (NC) 6
34 Detected S.Z. (NC) 5
35 Detected S.Z. (NC) 6
36 Detected S.Z. (NC) 5
37 Detected S.Z. (NC) 5
38 Detected S.Z. (NC) 5
39 Detected S.Z. (NC) 7
40 Detected S.Z. (NC) 5
86
SLCP TEST - PAGE 1
MINIMUM RESULT VERSUS TIME MAXIMUM 0.0 l.OOOOE 00
TIME RESULT I R* RC 0.0 0.0 1.7084E 00 8.6018E-01 1.0416E-03 0.0 1.7084E 00 8.6018E-01 2.0832E-03 0.0 1.7084E 00 8.6018E-01 3. 1248E-03 0.0 1-7084E 00 8.6013E-01 *.1664E-03 O.O 1.7084E 00 8.6018E-01 5.2 080E-03 0.0 1.7084E 00 8. 6018E-01 6.2496E-03 0.0 1-7084E 00 8.6018E-01 7.2912E-03 0.0 1.7084E 00 S.6018E-01 8.3328E-01 0.0 1.7084E 00 8.6018E-01 9.3744E-03 0.0 1.7084E 00 8.6018E-01 1.0416E-02 0.0 1.7084E 00 8.6018E-01 1.1458E-02 0.0 1.7084E 00 8.6018E-01 1.2499E-02 0.0 1-7Û84E 00 8.6018E-01 1.3541E-02 0.0 "1.7084E 00 8.6018E-0: 1.4582E-02 0.0 1.7084E 00 8.6018E-01 I. 5624E-Û2 0.0 1.7084E 00 8.6018E-01 1•6666E—02 0.0 4.0651E-02 "8.6O18E-0I I.T7Û7E-02 0.0 4-0651E-02 8.6018E-01 1.8749E-02 0.0 4.0651E-02 8.6018E-01 1.9790E-02 "" 0.0 •4:065ÏE- oz •8.6018E-01 2.0832E-02 0.0 4.0651E-02 8.6018E-01 2.1874E-02 0.0 4.0651E-02 8.6018E-01 2-291SE-02 l.OOOOE 00 — —: —-— + 4.0651E-02 ff.60l8E-01 2.39S7E-02 l.OOOOE 00 4.0651E-02 a.6018E-01 2.4998E-02 l.OOOOE 4.0651E-02 8.6018E-01
Figure 13. Result of Test 19
87
change of a phase angle occur more in the leading current circuit than
in the lagging current circuit when the fault is provided in the simu
lation. These errors would provide unnecessary trips of the breakers,
which are not desirable for the stability. The algorithm was developed
to detect every fault in the primary zone, even the far end of the zone,
rather than optimizing selectivity. This approach is meaningful be
cause failure to trip for a fault in the primary zone would provide
damage to the power system, and this must be avoided.
Tables 6a and b also show delays for the detection of faults as
follows;
Average delay in the lagging current circuit 4.7 ms
Average delay in the leading current circuit 5.9 ms
Table 6c shows the results of the tests where the faults have been
provided at 50 miles from the relay, which is outside of the secondary
zone. The algorithm should not detect faults outside the secondary zone.
All the tests clarified that the algorithm detected as faults in the sec
ondary zone. That is, the relay often provides unnecessary trips for the
breakers in the secondary zone. This must be avoided to get better
stability in the power system. There are two methods of determining
whether a fault is located in the secondary zone or outside (15). A
power network for protective relaying is considered to explain the
methods, which is shown in Figure 14. It is assumed that the location
A is at the far end of the secondary zone, and the location B is just
outside of the secondary zone in Figure 14.
s.z.
p.z
Computerized relay
Gl, G2: generators
1, 2 , 3, 4, 5, 6; circuit breakers
P.Z,; Primary Zone
S.Z.: Secondary Zone
Figure 14. A simplified power network for the protective relaying
89
(a) Method 1:
Although from conditions at the computerized relay a fault at A
cannot be distinguished from one at B, at breaker 5 the distinction is
easily made. The currents in breaker 5 vary, especially in phase ac
cording to which side of the breaker the fault is on. A fault at A
causes power to flow into the secondary zone; a fault at B causes
power to flow out of the secondary zone. Therefore, a signal transmitted
from breaker 5 to the computerized relay at breaker 2 over a carrier-
current channel, or other communication channel, can be used either to
permit or to block tripping of breakers 4 and 5. This method is called
pilot relaying.
(b) Method 2;
The method 2 of distinguishing between internal and external faults
near the far end is to delay the tripping of the breaker long enough to
allow some other breaker to open first. When a fault is at B, breaker 6
is immediately tripped. As soon as the breaker is opened, conditions
at the computerized relay change so as to indicate clearly that there is
no fault in the secondary zone. When a fault is at A, breaker o is
not tripped,, and after a certain delay the computerized relay still
sees a fault in the secondary zone. Then the computerized relay ini
tiates a trip for breakers 4 and 5.
The method 2 is employed in this research because of the following;
The method 1 requires a communication channel and another computer in
order to detect the direction of power flow at breaker 5. Nevertheless,
the method 2 does not require additional hardware at all, and time delay
90
can be programmed easily.
The step 5 of the algorithm must be revised as follows;
Step 5. As far as the algorithm detects as a fault in the
primary zone, a trip is initiated immediately for the
breakers, which is the same as before. Neverthe
less, once a fault is detected in the secondary zone,
a trip should not be immediately initiated. As soon
as a fault is detected in the secondary zone, a timer
is initiated to count a delay time. A trip is initiated
for the breakers if the algorithm still sees the fault in
the secondary zone between the minimum delay time and
the maximum delay time.
Delay time = Breaker time + Relay time + Margin
Assume: Breaker time = 2 cycles
Relay time = 0.25/^0.6 cycles
Margin = 0.5 1 cycle
Maximum delay time = 3.6 cycles
Minimum delay time = 2.75 cycles
The flowchart of the revised algorithm is presented in Figure 15a
and b. A part of the flowchart from the interruption to the connection A
in Figure 12a is exactly the same, and it is abbreviated. Figure 15a
shows the flowchart from the connection A. Notations of Figures 15a and
b are same as Figures 12a and b except the following.
CT5, CT6: Counters
CT5 is used to indicate whether a timer is set to count a delay
91
CT6=CT6+1
CT6=300 CT6;600
R:TH1
CT1;0
CT3=CT3+1 R:TH2
CT1=CT1+1
R:TH3
;T1;12
Clear interrxJpt No faul Fault
CT2:3
10FS CT2:6
CTl, CT2, CT3, CT4 =0 CT3;2
Figure 15a. Flowchart of the revised algorithm
92
30FS 20FS CT2:10
CT4:3
CT4:2
(Suspicion of a fault) in S.Z.
CT5:1
CT6;176
CT6; 228
CT5=0
CT6=0
CT5=CT5+1 RESULT = 1 F. in P.Z
RESULT = 2
Figure 15b. Flowchart of the revised algorithm
93
time or not. That is, when CT5 is equal to one, the timer is not
yet set, and it will be set to count the delay time; CT6 is set
to be zero in Figure 15b. When CT5 is greater than one, it indi
cates the timer was already set to count the delay time.
CT6 is a timer for this purpose. When CT6 is greater than or
equal to three hundred and less than or equal to six hundred,
CT6 is the dumny timer which does not work for counting the delay
time.
C. Comparison of the Algorithms
The algorithm developed herein is compared with the algorithm de
veloped by Mann and Morrison (9). The theory of Mann and Morrison is
the following: They try to calculate the peak voltage and the peak cur,
rent before their occurrence by small numbers of samples. They also cal
culate phase angle 0, and a line impedance Z is calculated for the
distance relay. The equations for this purpose are;
V = Vpk sin wt [37]
Taking derivative of equation [37], we have
V * =w Vpk cos wt [38]
From equations [37, 38]
«here v' = ^ (v^+l - v^) t = 0.5 ms.
Similarly
i = Ipk sin (wt + 0) [40]
94
i' = w Ipk cos (wt + 0) [41]
ai 1 where i' = — = (i k+i - ik)
0 = arc tan (-^) - arc tan [43] i' v'
where v, i = instantaneous phase voltage and current
vk = value of v sampled at time tk
Vpk, Ipk = peak values of phase voltage and current
v', i' = instantaneous values of first derivative of phase
voltage and current.
From the above, the line impedance can be calculated as follows.
y ^ Vpk.Cos 0 + j Vpk.sin 0 Ipk Ipk L J
This theory works well under the assumption d-c offset is substantially
small after the fault occurs. So they try to use mimic impedance in
the current transformer secondary to avoid d-c offset in the subtransient
condition. This mimic impedance may work to avoid d-c offset only when
the primary X/R is matched to the secondary X/R(16). They also used a
sample rating of 40 samples per cycle (SOH^) in order to get accurate
values of v' and i'.
On the other hand, the algorithm of this research is developed
based upon the two facts, stated before, as follows;
(a) Cycle to cycle comparison which is a kind of pattern recog
nition is used for the detection of faults.
95
(b) Severity of the faults is used for the detection of distance
from relay to the fault rather than calculating line impedance from the
relay to the fault in order to establish the algorithm which would work
under the d-c offset in the subtransient condition after the fault.
96
VI. IMPLEMENTATION
A, Constraints for the Implementation
The Implementation of the computerized relay scheme must utilize
certain processes such as multiplexing and A/D conversion as well as
programming of the relay functions. A system which performs multi
plexing and A/D conversion is called an analog input subsystem. The
diagram of the implementation is shown in Figure 16. The constraints of
the system are discussed below.
(a) Analog input subsystem
Two analog input subsystems are necessary in this implementation
because the algorithm needs to sample voltage and current of the same
phase at the same time. The capability should be 2880 samples per
second because the algorithm needs 16 pairs of voltages and currents of
each phase per one cycle of the system frequency (60 Hz) for the pro
tection of one transmission line. If the substation computer protects
more than one line, the capability of the analog input subsystem must
be increased by a factor of number of lines. The output should be 12
bits in parallel, which would provide an accuracy of 0.025 percent of
full scale (17),
(b) Substation computer
The substation computer should be designed in detail considering
the entire investigation of the computerized substation functions rather
than only the computerized relay scheme presented in this dissertation.
Nevertheless, the minimum requirements are;
Voltages ^
(Analog) _) Outputs
12 bits parallel (Digital)
Current
(Analog) 12 bits parallel
(Digital)
Local man-machine interfaces
Substation computer
Circuit breakers
Analog input subsystem
Analog input subsystem
DMA Channel
DMA Channel: Direct Memory Access Channel
Figure 16, Implementation of the computerized relay scheme
98
The computer should be capable of high-speed real time appli
cations and multi-task applications
Internal priority interruption
Main memory with parity
8K 16 bit words core memory is required for the assembler
(1 yj,s cycle time), A IK 16 bit bipolar memory is necessary to
store the subprogram of the computerized relay functions in order
to optimize the execution time (300 ns cycle time).
Floating point processor
If the floating point software package is employed, the execu
tion is very slow. So, the hardware floating point is es
sential. Hardware Multiply/Divide could be used instead of
the hardware floating point processor.
Direct Memory Access (DMA)
DMA allows memory data storage or retrieval at memory cycle
speeds.
Programmable real time clock
The clock provides programmed real time interval interrupts and
interval counting in several modes of operation.
Power Fail and Restart
Power fail and restart, not only protects the memory when power
fails, but also allows the subprogram to save the existing
program location and status, thus preventing harm to devices,
and eliminating the need for reloading programs. Automatic
99
restart is accomplished when power returns to safe operating
levels, enabling unattended operations.
8. 0-55°C operatable
9. Basic assembly language with a sufficient instruction set for
this purpose
(c) Local man-machine interface
1. Teletype
A teletype where the speed is 30 characters per second is used
to print important events such as time of a fault and a loca
tion of the fault.
2. Paper Tape Punch
A high-speed paper tape punch is used to punch data such as
voltage and current for later off-line analysis.
B. Example of the Implementation
It might be desirable to design a special purpose processor for the
cumputerized relay scheme to satisfy all constraints stated before. But
it would take time and would be very expensive to do so. That is why
one example of the implementation of the computerized relay scheme is
presented herein using a 16-bit computer already on the market as a
substation computer. Upon a brief survey of 16-bit computers on the
market, PDP-11 family of Digital Equipment Corporation is considered for
this application. Table 7 shows the summary of PDP-11 family computers
(18). PDP-11/45 is selected as a substation computer because of the
following:
100
Table 7. PDP-11 Family Computers
PDP-11/05 PDP-11/15 PDP-11/20 PDP-11/45
Central processor KD 11 KG 11 KA 11 KB 11
General purpose registers 8 8 8 16
Instructions Basic set Basic set Basic set Basic set and: MUL, Div, xoa. ASH, ASHC, MARK, SXT, SOB, SPL, RTT, MTPX, MFPX
Segmentation No No No Optional
Hardware stacks Yes Yes Yes Yes
Stack overflow detection Fixed Fixed Fixed Variable
Interrupts Single-line Single-line Multi-line Multi-line multi-level multi-level multi-level multi-level
plus 7 software levels
Overlapped instructions
Extended arithmetic
Floating point
Basic memory
No
Option
No
Option
No
Option
Yes
Standard
Software Software Software Internal to CPU
Core Core Core Core, MOS or Bipolar
The floating software package is too slow for this real time appli
cation, and the hardware floating point is essential. The bipolar memory
is required to optimize the execution time of the subprogram of the relay
functions.
LOI
The subprogram of the relay functions in Figures 15a and b is
written in PAL-llS (Program Assembly Language for the PDP-11) (19), and
the program is presented in Appendix D. The execution is calculated
based upon the assumption that the subprogram of relay functions are
already stored in the bipolar memory of 300 ns cycle time, and it is
available as follows;
When the power system is not faulted: 67.8yus (worst case)
When the power system is faulted: 74.1 yMs (worst case).
This result indicates that the computerized relay scheme using
PDP-11/45 can protect up to four transmission lines because the analog
input subsystem in Figure 16 provides 11,520 samples per second, and the
sample interval is 86,7 s when four lines are protected.
The implementation of the computerized relay scheme requires the
following DEC equipment (20).
a) 11/45 Central Processor with $21,250
8K core memory with parity. Includes
power supply, cabinet, power fail/
restart, and teletype (30 cps).
b) IK bipolar memory (MSll-CM) with $ 3,900
the bipolar memory control (MSll-CC).
c) Floating point processor (FPll-B) $ 4,900
Performs hardware operations on 32-bit
and 64-bit floating point as well as
integer to floating conversions.
/
102
d) Programmable Real Time Clock (KWll-P)
$ 600
e) 300 cps Paper Tape Reader plus 50 $ 3,900
cps Paper Tape Pimch combination
with the controller
f) Two analog input subsystems of $ 9,400
Modular computer systems (21).
+ 102.4 Volts Full Scale Inputs
12 bit output including sign up to
50,000 conversion per second
Total cost of the above equipment is $43,950, which does not in
clude software and installation charge. The cost of the conventional
relays installed at Sycamore substation of Iowa Power and Light Company
is presented below in order to compare the costs of the computerized
relay scheme and the conventional relays.
Two relays to protect 345 KV line $90,000
One relay to protect 161 KV line $12,000
One relay to protect 69 KV line $ 7,500
Total cost of the conventional relays is $109,500, which protects
three transmission lines. The algorithm is developed for 69 KV line,
but it can be similar for 161 KV line or 345KV line. That is why the
computerized relay scheme using PDP-11/45 would protect these lines as do
the conventional relays. As a result, the computerized relay scheme can
save more than 50 percent over the conventional relays in this case,
under the assumption the maintenance cost of the computer is cancelled
103
out by the maintenance cost of the conventional relays.
Figure 17 shows the interesting fact that while the conventional
hardware costs increase year after year because the continuing pressure
for improved power system reliability increases relaying sophistication
and complexity, the cost of the computer system is dropping year after
year (1).
104
50
To^al computer system
40
Software
30
10 • Conventional relay hardware
1970 1975 1980
Figure 17. Cost comparison between the computer (software and hardware) and the conventional relay hardware (1)
105
VII. RESULTS AND CONCLUSIONS
The simulation of the power system has proved to be an effective
way to get characteristics of the substransient condition of the faulted
power system. The characteristics presented in the results of the simu
lation and the data in Appendix B have provided enough information to
establish the algorithm of the computerized relay scheme which would work
for the power transmission line shown in Figure 2.
The algorithm has attained one of the goals, a half cycle of the
operation time of the relay functions as far as a fault is located in
the primary zone. The algorithm takes a few cycles to detect a fault
in the secondary zone because of the Method 2 stated before. The de
tection of the fault in the secondary zone is necessary for the back-up
protection, and this time delay is desirable because a computerized
relay located at the secondary zone detects the fault quickly and has
a chance to trip for the breakers. This fact indicates that the com
puterized relay scheme might provide better stability in the power sys
tem over the conventional where the operation time is limited around one
cycle. The algorithm also has attained the other goal, the detection
of the four major faults, judging from the results of the tests. The
algorithm has an accuracy of 84.5 percent in the detection of locations
which is called selectivity.
It is found out that the cost of the computer system is less than
50 percent of the conventional hardware from the example of the imple
mentation. Furthermore, the substation computer could perform supervisory
control, voltage control and data logging at the substation, and it
106
could conceivably save money.
The constraints show that a fairly sophisticated computer system
is necessary for the relay functions rather than a mini computer with
only 4K memory.
The further investigations recommended are; 1) The simulation
should be carried out for the other specifications of the power system
such as longer transmission line as well as higher voltages. This
has not been done in this research because of the limited computer
time. Then a prototype system should be constructed so that field
tests could be made to confirm the simulated results.
2) Theoretical analysis of the pattern recognition technique em
ployed herein could be possible.
3) Time delay is employed in this research to distinguish whether
a fault is located inside or outside of the secondary zone. Nevertheless,
further investigation of pattern recognition technique might provide a
solution for this purpose.
4) The investigation of a digital directional element should be
attempted so that the digital relay system is not dependent on an
external directional relay.
107
VIII. LITERATURE CITED
1. Rockefeller, G. D. "What are the prospects for substation-computer relaying?" Westinghouse Engineer, 32 (1972), 152-156.
2. System/360 Continuous System Modeling Program (360-CX-16X), Application Description. White Plains, N.Y.: IBM, Technical Publication Department, 1970.
3. I.B.M. Research Division. Power system computer feasibility study. Vol. I. San Jose, Calif.: I.B.M. Research Division, 1960.
4. Grimes, J. D. "A computer-centered relaying scheme for the protection of bulk power transmission lines." Unpublished M.S. thesis. Library, Iowa State University of Science and Technology, Ames, Iowa, 1968.
5. Harder, E. L. and Cunningham, J. C. Relay and circuit breaker operation. Electrical Transmission and Distribution Reference Book. East Pittsburgh, Philadelphia: Westinghouse Electric Corporation, 1950.
6. Van C. Warrington, A. R. Protected relays. New York: John Wiley and Sons, 1962.
7. Albrecht, R. E., Nisija, M. J., Rockefeller, G. D., Feero, W. E. and Wagner, C. L. "Digital computer protective device coordination program. I - general program description." IEEE Trans, on Power Apparatus and Systems, 83 (1964), 402.
8. Rockefeller, G. D. "Fault protection with a digital computer." IEEE Trans. Power Apparatus and Systems, 88 (1969), 438-451.
9. Mann, B. J. and Morrison, I. F. "Digital calculation of impedance for transmission line protection." IEEE Trans, on Power Apparatus and Systems, 90 (January/February, 1971), 270-279.
10. Mann, B. J. and Morrison, I. F. "Relaying a three phase transmission line with a digital computer." IEEE Trans, on Power Apparatus and Systems, 90 (March/April, 1971), 742-750.
11. Rao, A. M. "Computerized substation protection." Unpublished M.S. thesis. Library, Iowa State University of Science and Technology, Ames, Iowa, 1972.
12. Computer Relaying Subcommittee. Minutes submitted by Ruben Zimering. St. Petersburg, Florida. January, 1973.
13. Anderson, P. M. Analysis of Faulted Power Systems. Ames, Iowa; Iowa State University Press, 1973.
108
14. Kimbark, E. W. Power system stability. Vol. III. Synchronous machines. New York: John Wiley and Sons, Inc., 1956.
15. Kimbark, E. W. Power system stability. Vol. II. Power Circuit Breakers and Protective Relays. New York; John Wiley and Sons, Inc., 1950.
16. Wright, A. Current transformers: Their transient and steady-state performance. London: Chapman and Hall, 1968.
17. Hanna, W, A. "Analog-to-digital conversion system design considerations." Unpublished M.S. thesis. Library, Iowa State University of Science and Technology, Ames, Iowa, 1971.
18. Digital Equipment Corporation. PDP-11/45 processor handbook. Maynard, Mass.: Author, 1972.
19. Digital Equipment Corporation. PAL-llS Assembler and Link-llS Linker programer's manual. Maynard, Mass.: Author, 1972.
20. Digital Equipment Corporation. PDP-11/45 Systems, Services and Options. Maynard, Mass.: Author, 1972.
21. Modular Computer Systems. Modcomp price list. Fort Lauderdale, Florida: Author, 1972.
109
IX. ACKNOWLEDGEMENTS
The author would like to thank Dr. A. V, Pohm for his excellent
advice on the computer area throughout the course work and the re
search as a major professor as well as for his suggestion to do this
topic. He also wishes to thank Dr. P. M. Anderson for his expert
advice on the power system and the simulation method of the transient
condition of the fault as a co-major professor. He wishes to acknowledge
Dr. W. B. Boast for his generous support on the computer time. He
wishes to thank the power affiliate group for the financial support.
Finally, he thanks his wife, Mitsuko, for her help in Ames life.
110
X. APPENDIX A: PROGRAMS OF THE SIMULATION
Two programs of the simulation are presented; one is for the case 1,
30F, and another is for the cases 24, 25, SLGF. The programs consist
of three parts such as Initial, Dynamic as well as Terminal, The ad
vantages of the CSMP program (2 ) are the following.
(1) With few exceptions FORTRAN statements can be intermixed with the
S/360 CSMP simulation statements.
(2) The method of integration can be chosen from several standard
options provided in the program.
(3) The output is provided automatically in a fixed format for all
output options.
Notations of the programs are the following.
W = 2/rf(f=60c/s)
ICA, ICB, ICC and ICG: Initial conditions of the currents
TSTEP; Time of the fault
L11-L44; Inverse of Matrix abcg in Equation 8 before the fault
LF11-LF44: Inverse of Matrix Labcg in Equation 8 after the fault
RKSFX: Runge-kutta Fixed Length Integration Method
CAl, CBl, CCI,and CGI: Initial conditions of DIA, DIB, DIC and DIG
where DIA=dIa/dt
CA2, CB2, CC2 and CG2: Initial conditions of DDIA, DDIB, DDIC and
DDIG where DDIA=dDIA/dt
Others can be found in the reference (2) and Figures 3, 4.
Ill
A. Program for the Case 1, the Three Phase Fault
I F$**CONTIN^OUS SYST
riTLE TRANSIENT COND
! _ INITIAL Ir PARAM W=37,6I.M^ iPARAM" RD«Q.:0200168
*>,** VERSIQ
EM MODELIN^
Ml.3
ILTICN OF THREE PHASE
RG1=0.8601 T64,VM"0.8|L 649666 I IPARAM RAlab;.1084026, jpARAM RA2^d.7 iPARAM I CAW©;.377346 , 1^8=0.240616', ICC«-0.6117496 :PARAM R62.i:,L;o;is7^0.03 II i PARAM TSTEP=0.022919 | ; [ .PARAM LLL«Oi.8?8002,L|1:2—O;07336'24,. LI 3—0. C 5 909 72,L14.-#. 259139, [L21"—0.073:3621 ,L22-Ô(.:902687;. .l.r !L23—0. 07 3l3621, L24--j6. 2679 86 , _ L31—0.0590972, L32"NC . 0733624?.:.." L 33«0. 898002, L34—0 .1259139,...!• ,L41 —0 .2 591 39 , L 42I-AV2 679 8477.1.+ L43—0.254139,L44"2J^1409 I : _
JPARAM LFLLP2.62829;LFI2=-0.535K^... LF13—0.3<5i764,LF14^0. 56979, LF22-2.69818,LF23=-0.F35598,.^^ LF24—0.506129,LF33»B:.62829,..,'.' LF34*-0. 569:790, LF44«
JPARAM P11=&,P12=376. 3.23715 e91,P13=l.
P21*0,P22=&76.991,PZ3=3.665r9% P32«376.93l,P33*5.7995865 l21"Li2 j . i :L3 L^L 13 ! 1 J L41*L14 Ti ' n L32.L23 J : M
570796,. ,iP31-0,
PROGRAM****
AT FAULT
112
|L42«L24 L43-L34 RB1«RA1 RC1*RA1__ RB2-RÀ2 :RC2«RA2 :LF21»LF12 LF31-LF13 LF41*LF14 I : :LF32«LF23J_! LF42-LF24 ! LF43-LF34 I ! ic6—( iCAtrta+icc) NOSORT I : CALL CESUGMtO.O)
.. --iJ OYNAfllC i : Y-STEP(TSTiEPJ p:A"BAl+ ( 1-;T)*RA2 RB"RB1*RB2«(1"Y) RC«RCl»RC2r(l-Yl RG-RG1*Y*RG2 LAl.W*Lll*(l-Yl+W*LFill*Y LA2-W*L12*(1-Y)*W*LF12#Y iLA3=W*L13*( IA4.W*L14*I(
iB2«W*L22*'(
1-Y)+W*LF13*Y 1-Y)+W»IF lk*Y l-Y)+W*LFZl*Y
:1-Y)+W*LF%2*Y LB3=W*L23$(l-Y)+W*LFa3*Y LB4=W*L24*(1-Y1+W*LF2A*Y LCl=W*L31*k&-Y)+W*LFKl*Y LC2=W*L32#iM-Y)+W*LFbb#Y |L,C3=W*L 33*t*!l-Y)+W$lFa3*Y 1. C4îW«U34»,l l-Y)+W*LFa4*Y LGl*W*L41#kl-Y)+W*LF#l$Y
113
LG2-k*L42*(l-Y)+W*LF^2*Y LG3"W*L43*!(&-Y) + W*LF&3$Y lG4=W*L444(i-Y)+W*LM44*Y |VA«Vf*SINE!{:Pll,P12,Plli3) VB«VM*SINE(P2l,P22',P}2;3) VC«VM*SINE( P311 P32f P|33 ) DÂ*IA*(LA1*RA+RD*(LA2^LA3+LA4
*IB*(LA2*kB+RD*(LAl I +IC*TLA3*RC+RD*(LA1
+LA3+LA4) +LA2+LA4) +IA2+LA3) '• IG*( LA4*RG+R0*(LAI
DB»IA*(LBltRÂ+RD*(LB2fL83+L84) • IB» (LB2*RB+RD* ( LB1;+:LB3+LB4 )
; +IC»(LB3*RC+RD*(L81ftB2+LB4)) , '•IG»(LB4»R;G*RD»(LB1i+;LB2*LB3) > DC*ÏA*(LCÏfRÂ+RD*ILq&+LC3+LC4) I !• IB* ( LC2*!R;'8-»RD* ( LC 1&LC3+LC41 )
.
: •IC»{LC3»RC+R0»{LC1KLC2»LC4I) ! +IG»(LC4*RG+R0*(LClkLC2*LC3)1 DG«IA*(LG1*RA+RC*(LG2+LG3+LG4) i !• IB* ( LG2»RlB+RD» ( LG1^LG3+LG4) >
+IC»(LG3*RC+R0*(LG%+LG2*LG4)) !*IG»(LG4»R;G*RD»(LGl!tLG2+LG3))
•LCl»VA;+LC2»VB+LlQ3»VC-0C 'LG1*V*+LG2*VB+LG3*VC-DG
1G[A«LA1*VA+LA2*V8+L#3*VC-DA TGCB«LBl*VA+Le2»VB+LB3*VC-CB lIGO'C-1i;gog« lIAsINTGRLCICAtlCCA) irB«INTGRL(i:CB.IGCB) IC«iNtGRL{irCC,IGOC) !I:G«INTGRL ( LICG.lGDG) ;A-IGCA :B' ,C:
•IGCB^ -I GDC
114
%VA.VA-(R*]IA+(L/W^^A RVB»V8-(R*|IB*(L/W)#a); RVC-VC-(R*!lC+<L/W)*C NOSORt •RIA.IA RIB-IB RlIC-IC IF (ABS<RIjA;).GT.0.00 IIF IRIAi lO,20,20 10 RiA«-d.ppooooi GO TO 30 _| 1
30001) GO
20 RIA-O.C&OOOOl !30 IF(ABS(blB).GT.O. jlF (RIB) 401,50,50 40 RIB—O.-OjOOOOOl CP TO 60 (i ?!Q BIB-0.0000001 60 IF(ABS(R;IC).GT.O. 1I:F(RIC) 7OI,18Q,,80_ GO TO 90 j i ISO Ric'o.caooool
):
0 30
OjOOOOOl) GC
(^00001) GlQ
TO 60
i90 ÎA-ÀôyrR;VÀT/A8SC R lA ) jZ;8-ABS(RVB|);/AeS(RIB) j^C»AB5el^C)i7A6SiftICi ITERMINAL ]tlMtft 'OÎUT.p.0002604,|P!NT!H«0. .PRTPLT IA(,RA,RG),I8{R8,RG),IC( PR TPL T" Tvïn;RA,RG},RyBi(RB,R6),Rj
(NA, RG),ZB( PHASE FA
PRTPLT ZA LÏSÏL'Tïîïf METHOD RKSIF^X Ê.NC IPARAH TST^'0.026044 RESET "LÏBÏL; ILABEL 3PF CASE :eko"" —rt— 'STOP
g!8,RG),ZC( iuT T«0.02
TO 90
060,PROEL' R;C,RG) yC(RC,RGi RC,RG) 2pl5 P-0.7
.0010416,
P OF L»0.
auT0EL«0.ac
> CASE 1
10416
1 T»0 026040
115
B. Program for the Cases 24, 25, the Single-Line-to-Ground Fault
TJItleTRAN *1 INITIAL
SilENT COND
PARAM P AR AM PARAM RD=0
ip'ARAM" 'PARAM
;»**»CONTI'WJOUS SYST
E EM MODELING PROGRAM*!»»*
«=3^6!. 991 RATE =1100
.0200168 1094026, 8724,ICB
PAR AM CA1^85.12,CB1 iPARAM RA2=PI.8,CCA=0-!p!aRAM RG24liO , 1=0 . 1, R =0.02 jPARAM TSTE|P'j=0.018749 pÀRAM' Pll«|oi,Pl2^5T6. p21«0,P22=|3i76.991,P2 {Pi32 i3 76 ;9%;TP 3^376 6 [PARAM Lll=m.456527,L iirSïO 71?5SK5",LT4ir(57T
»?*» VERSION 1.3 »*•
t f ION OF S
RI61=0.8601 =!-0.586819
INGLE LINE to GROUND
7&4,VM=0. qT ,llCC«-0.2 730
t-231.75,ciCH=314.04, 6, CCB^O.6,
9I91,P13=1. 3=3.567024
t;22 =0.45652?, L23=0.1 e4=6": IÏÎ I®r,r33^07^i56 5177 4=0.1284W2,L44«0.3f7;S26l
"52^ f2 'JC2«RA2
ISTôT 12=0.14351#,... 218492,... 43515,...
IRBl'W {RpI'RAl
CkC=0.6 CiGl =-0.0051
4?2626,.. ,P3I=0».«.
649666 73,ICG=0
FAULT
00411746
CA2=-H»W«IC!A cb2=-w*w*icb
"|CC"2'— CG2*-W$W*ICG
4-
I
O (h h» h" O O r* h* iTs lA
m M O h» ^ M
• + "A m -4 cg a CL « « m fo
I I
• u m O M w a H ^ m h m M
M 00 Ol oui 3 3 3 3 S, ^ w cg fO ^
H » ;
m U vu
3 3 3 3 V S V V , N m m rHi(Nj fsjlcg m W H • H N M H I H II < (D U O 00 U O O
3 3 S V ^ m L> _ _ m flûK < • u H li O O u.c>
u o o o CD o H M I II H ti M < CDi<t < CÛ O
>-* Ui H 2 +
. , (M NH (/> wl«0 u H —la: oc o w * + + * «lû, ^1
Z,Ui <,IO U|< <|H- ociot oc'o 2: 1/1 H, N N, » ..,.>T'.y..<;B).u!.<.
117
C8=CB7 CC«CC7 RG=RG1 PA I =0 . 5*R4jk;SQRT ( CA/L W ) |P'A2 = l/S0RTjCCA»CAAl^^r P81=0. 5*RBi»iS0RT( CB/LBiBJ >:82«i/SÔRfl^'B«L38>^ jpCl =0. 5»RC*S0RT ( CC/C; PC 2 = 17S QRti( EC*LCCT ViA= VH^S INElllPl 1, P12, P|i;3 ) y:B=VM*SrNEi(P 217^2 2 , P Ë3T VC«VM*SINE!(iP3l,P32f P3:3) bVA=VM*P12fSINE(Pll,P12,PP13) bVB=VM$f22ksiNE(P2l,P22,PP23) 0VC=VM*P32f$INE(P31, bÏB=DERIV(C81,IB) bplS^CERTvjlkSTTnTBT pic=DEaiv(Rci,ic; DDicr.DERi9%cc2,Tncr biA=DERIVlCAl,IA) bûi À-Û ERI v,«A znnrr blG=DERIV(CGl,IG) j&iG^oÉRivïfcGFtiîrsr <A' ( 1/L AA) !•'( DVA-RO* I
P^PP33T
DIB+DIC+DI: l-lAB»DÔiB-VAC»ÔOIC-^G*DOrG) I A=CMPXPLl ICA.Cai .PA!I:,PA2 ,XA) ka.TÎ/LBB ) *ïo Vb-r Dïïïiî A ;ô fc^Ô ïi klBC*DOIC-LBG*DOIG) |IB=CMPXPLMCB,C81fPB lxic« {1 / L CC ) » ( OVC-R 0» {
1,P92,X8) blA+DIB+DIk
-LBA*ÔBTÎ
-LCA»DOIA
118
^iCB*ODIB-LCG*ODIGJ IC=CMPXPL(ICC,CC1,PC
lP,G= LGG/ RG"~n ^fe=^RD*j IAVlIB»rC)/RG ZZ5=(XG-IGp/PG ,lG=INTGRL(;liÇG,ZZ5l R.V A=V A- ( R*.I !A+ ( L /H > *D I A) RVB=VB-(R*iB+(L/W)*DIB] PVC=VC-(R*IC+(L/W)*DIP) NOSORT R:IA=IA RIB-IB
&).GT.0.00 RfC=IC ilF (ABS(RIk :I:F (RIA) 10,20,20 lib RiA=-o.bbooooi IGO TO 30
boool
OOOOl) GO
GO TO 30 I I .feo RI A=o,obj m iFiABsïpi: ajO IF(ABS(MB] .GT.O. |lf (RIB) 4b:.50,50
I 9 RIB=-0.pp00001 b TO 60 u 0 RIB=O.Opp0001 b IF(ABS(R!IC).GT.0.
gF(RIC) 7bi;î80,80 \fo RiC'-o.bbooool 6p TD 90 I I
lO Ric=o.obboooi 0"ZA=ÂBS(p^Â)/ABS(R
jz!B»ABS(RVB3yABS(RIB) }Zp»ABSCRVCj),/ABS(RIC7I
L',PC2,XCJ
-:(LGA*DIA+ L |SB*OIB+LG
TO 30
0000001) G
0ÙOOOO1) G3
IjA)
TO 60
TO 90
C,*OIC)/RG
119
ITERMINAL jTilMEROÇLT PRTPLT lÀTl PAT PL T RVA P'RTPLf ZAl tiABEL SLGF METHOD RkS END jpARAM TSfEp plARAH ICA= IPARAM CÀT= IPARAH R«= RESÈf'LABÊ LABEL SLGF :END jSiTOP
RIAtRG) i IB( (kA,CG),RV otr RiA,RG) ,ZB{
iCASE 24 T rw" '
2.604E-4,
,«0.022915 01.40073It I
16 8". 2 3", eg-; li.6,CCA = l.
CASE 25 T
RINTIM=0.0 RB,RG), IC( Bj tRStRGi tP R|BTRG),ZC( =0.018749
.C&=-0.3373 #42.61,
2,CCB=1.2, CCI
>0.022915
53,PR0EL»C R|C,RG) VC(RCFRG) RC»"RST
96,ICC.-0 . l59.91,CGl=-0.048 CCC-l.Z
.0010416,0UTDEL=0.00
01615972,1C ;X0.002164
10416
3.7
120
XI. APPENDIX B; TABLES OF Za, Zb, AND Zc OF THE CASES IN
TABLES 3 AND 4
The tables contain sixteen samples each for Za, Zb and Zc in the
period of two cycles. The first eight samples (A, B,....,H) are the
prefaulted cycle, and the last eight samples (a, b,...., h) are the
faulted cycle. Each sample in the prefaulted cycle are compared with
the corresponding sample in the faulted cycle, and the ratios (a/A,
b/B,...,h/H) are also presented in the tables.
A. Tables of Za, Zb and Zc of the Cases in Table 3a
121
Table Bl. 30F, Case 1 and TSTEP = 0.022915
ZA ZB ZC
A 5.3488 0.39549 1.4463 B 4.2467 0.11810 1.0057 C 1.9339 1.3606 0.65191 D 1.2728 85.288 0.25597 E 0.87340 2.8668 0.39186 F 0.51950 1.6330 2.5864 G 0.082661 1.1209 7.3065 H 0.74450 0.75843 2.2931 a 3.4913 0,20012 0.87237 b 0.61121 0.081086 0.29254 c 0.29659 1.0180 0.14484 d 0.18385 0.73530 0.049462 e 0.11529 0.32867 0.055354 f 0.060616 0.20959 0.26341 S 0.0068821 0.14236 2.8774 h 0.058569 0.091903 0.57987 a/A 0.652 0.512 0.601 b/B 0.143 0.686 0.290 c/C 0.153 0.748 0.222 d/D 0.144 0.008 0.192 e/E 0.132 0.201 0.140 f/F 0.116 0.127 0.101 g/G 0.083 0.127 0.393 h/H 0.078 0.121 0.252
Table B2. 30F, Case 1 and TSTEP = 0.026040
ZA ZB ZC
A 1.2728 85.288 0.25597 B 0.87340 2.8668 0.39186 C 0.51950 1.6330 2.58684 D 0.082661 1.1209 7.3065 E 0.74450 0.75843 2.2931 F 5.3407 0.39569 1.4466 G 4.2495 0.11774 1.0059 H 1.9343 1.3594 0.65210 a 0.74526 64.857 0.14455 b 0.24412 0.47748 0.24893 c 0.10245 0.25090 2.6776 d 0.0041768 0.15949 0.47547 e Oill517 0.099768 0.28097 f 0.40446 0.048550 0.19240
122
Table B2. (Continued)
ZA ZB ZC
g 4.4960 0.0065489 0.13297 h 0.44173 0.083075 0.083309 a/A 0.585 0.760 0.564 b/B 0.279 0.166 0.635 c/C 0.196 0.153 1.035 d/D 0.050 0.142 0.065 e/E 0.154 0.131 0.122 f/F 0.075 0.122 0.133 g/G 1.058 0.055 0.132 h/H 0.228 0.061 0.127
Table B3. 30F, Case 2 and TSTEP = 0.022915
ZA ZB ZC
A 26. 207 0.43385 1.3291 B 2.8662 0.10745 0.98236 C 1.6585. 1.8640 0.67442 D 1.1871 7.1128 0.29347 E 0.86338 2.2091 0.43045 F 0.54720 1.4576 4.5153 G 0.11253 1.0742 3.8896 H 0.88051 0.77024 1.9019 a 17.232 0.20838 0.79506 b 0.55867 0.084782 0.28301 c 0.28602 1.1284 0.14178 d 0.17979 0.70872 0.048680 e 0.11359 0.32484 0.054024 f 0.059945 0.20828 0.25521 g 0.0069307 0.14178 2.3911 h 0.057647 0.091628 0.59815 a/A 0.657 0.480 0.594 b/B 0.194 0.789 0.288 c/C 0.172 0.605 0.210 d/D 0.151 0.098 0.165 e/E 0.131 0.147 0.184 f/F 0.109 0.142 0.056 g/G 0.061 0.132 0.614 h/H 0.065 0.118 0.310
123
Table B4. 30F, Case 2 and TSTEP = 0.026040
ZA ZB ZC
A 1.1871 7.1128 0.29347 B 0.86338 2.2091 0.43045 C 0.54720 1.4576 4.5153 D 0.11253 1.0742 3.8896 E 0.88051 0.77024 1.9019 F 26.022 0.43406 1.3293 G 2.8674 0.10705 0.98253 H 1.6588 1.8618 0.67458 a 0.68856 4.3373 0.15272 b 0.23704 0.44373 0.26980 c 0.10059 0.24287 2.1261 d 0.0043436 0.15633 0.46302 e 0.11257 0.098351 0.27772 f 0.39070 0.048048 0.19108 g 6.0350 0.0062826 0.13234 h 0.45070 0.081568 0.083012 a/A 0.580 0.609 0.520 b/B 0.274 0.200 0.625 c/C 0.183 0.166 0.470 d/D 0.038 0.145 0.119 e/E 0.127 0.127 0.145 f/F 0.015 0.110 0.143 g/G 2.103 0.058 0.134 h/H 0.281 0.043 0.123
Table B5. 30F, Case 2 and TSTEP = 0.022915
ZA ZB ZC
A 5.8731 0.49425 1.2188 B 2.0189 0.092652 0.96790 C 1.4193 3.8487 0.071556 D 1.1125 3.1251 0.35627 E 0.86883 1.7262 0.52297 F 0.59885 1.2986 91.598 G 0.16328 1.0338 2.4114 H 1.2580 0.79490 1.5715 a 3.9467 0.22274 0.72009 b 0.49994 0.091210 0.27257 c 0.27272 1.3567 0.13835 d 0.17493 0.67114 0.047794 e 0.11135 0.31910 0.052528 f 0.059052 0.20631 0.24616
Table B5. (Continued)
ZA ZB ZC
S 0.0069946 0.14090 1.9934 h 0.056421 0.091212 0.62126 a/A 0.671 0.450 0.590 b/B 0.247 0.984 0.281 c/C 0.192 0.352 0.193 d/D 0.157 0.214 0.134 e/E 0.128 0.184 0.100 f/F 0.100 0.158 0.002 g/G 0.042 0.136 0.726 h/H 0.044 0.114 0.395
Table B6. 30F, Case 3 and TSTEP = 0.026040
ZA ZB ZC
A 1.1125 3.1251 0.35627 B 0.86883 1.7262 0.52297 C 0.59885 1.2986 91.598 D 0.16328 1.0338 2.4114 E 1.2580 0.79490 1.5715 F 5.8809 0.49444 1.2190 G 2.0194 0.092149 0.96803 H 1.4195 3.8405 0.71571 a 0.63573 1.9210 0.16809 b 0.22978 0.40724 0.31035 c 0.098648 0.23339 1.6041 d 0.0045278 0.15250 0.44453 e 0.10985 0.096613 0.27269 f 0.37668 0.047429 0.18900 S 9.6256 0.0059524 0.13133 h 0.46087 0.079715 0.082534 a/A 0.571 0.614 0.471 b/B 0.263 0.235 0.593 c/C 0.164 0.180 0.017 d/D 0.027 0.147 0.184 e/E 0.087 0.122 0.173 f/F 0.064 0.096 0.155 g/G 4.767 0.064 0.135 h/H 0.324 0.020 0.115
125
Table B7. 30F, Case 4 and TSTEP = 0.022915
ZA ZB ZC
A 17.092 0.69951 2.5668 B 6.5028 0.30898 1.8088 C 3.3478 3.0396 1.1723 D 2.2751 31.477 0.42853 E 1.5822 4.7884 0.866615 F 0.93717 2.8895 6.1336 G 0.097637 2.0214 10.006 H 1.6381 1.3751 3.9251 a 10.361 0.35189 1.4721 b 0.56823 0.14405 0.33968 c 0.28831 45.170 0.15867 d 0.18094 0.52761 0.052692 e 0.11409 0.29247 0.061649 f 0.060193 0.19668 0.30243 g 0.006995 0.13652 14.601 h 0.057822 0.089107 0.51828 a/A 0.606 0.500 0.572 b/B 0.087 0.466 0.187 c/C 0.086 14.863 0.135 d/D 0.079 0.016 0.122 e/E 0.072 0.060 0.072 f/F 0.064 0.068 0.049 g/G 0.072 0.067 1.459 h/H 0.035 0.064 0.131
Table B8. 30F, Case 4 and TSTEP = 0.026040
ZA ZB ZC
A 2.2751 31.477 0.42853 B 1.5822 4.7884 0.86615 C 0.93717 2.8895 6.1336 D 0.097637 2.0214 10.006 E 1.6389 1.3751 3.9251 F 17.049 0.69990 2.5673 G 6.5062 0.30825 1.8092 H 3.3485 3.0366 1.1727 a 1.2689 17.774 0.25382 b 0.28814 0.46888 0.58479 c 0.11362 0.24892 0.89004 d 0.003575 0.15871 0.39205 e 0.12997 0.099419 0.25673 f 0.49323 0.048426 0.18204 g 1.9311 0.0064858 0.12783 h 0.39963 0.082708 0.080779
126
Table B8. (Continued)
ZA ZB ZC
a/A 0.557 0.564 0.592 b/B 0.182 0.097 0.674 c/C 0.121 0.085 0.145 d/D 0.036 0.078 0.0392 e/E 0.079 0.072 0.065 f/F 0.028 0.068 0.070 g/G 0.296 0.020 0.070 h/H 0.119 0.027 0.068
Table B9. 30F, Case 5 and TSTEP = 0.022915
ZA ZB ZC
A 30.986 0.76969 2.3654 B 4.5644 0.33338 1.7758 C 2.8941 4.8300 1.2218 D 2.1387 8.9039 0.49327 E 1.5796 3.7448 1.0410 F 0.99845 2.5902 16.495 G 0.13639 1.9452 5.8488 H 2.1633 1.4034 3.2682 a 19.021 0.37442 1.3485 b 0.53751 0.15260 0.33241 c 0.28169 10.164 0.15658 d 0.17847 0.51647 0.052175 e 0.11299 0.29003 0.060762 f 0.059760 0.19576 0.29643 g 0.0070264 0.13609 9.3045 h 0.057227 0.08890 0.52623 a/A 0.613 0.486 0.570 b/B 0.117 0.450 0.187 c/C 0.097 2.103 0.128 d/D 0.083 0.057 0.107 e/E 0.071 0.077 0.058 f/F 0.059 0.075 0.017 g/G 0.051 0.069 1.593 h/H 0.026 0.063 0.161
127
Table BIO. 30F, Case 5 and TSTEP = 0.026040
ZA AB ZC
A 2.1387 8.9039 0.49327 B 1.5796 3.7448 1.0410 C 0.99845 2.5902 16.495 D 0.13639 1.9452 5.8488 E 2.1633 1.4034 3.2682 F 31.121 0.77014 2.3659 G 4.5660 0.33255 1.7761 H 2.8946 4.8243 1.2221 a 1.1845 5.0317 0.27629 b 0.28307 0.44807 0.67773 c 0.11244 0.24394 0.83352 d 0.0036818 0.15675 0.38526 e 0.12833 0.09854 0.25455 f 0.48300 0.048114 0.18109 g 2.0412 0.0063704 0.12736 h 0.40319 0.081772 0.080551 a/A 0.554 0.564 0.563 b/B 0.179 0.119 0.651 c/C 0.112 0.093 0.050 d/D 0.026 0.080 0.065 e/E 0.059 0.070 0.077 f/F 0.015 0.062 0.076 g/G 0.446 0.019 0.071 h/H 0.139 0.016 0.065
Table Bll. 30F, Case 6 and TSTEP = 0.022915
ZA ZB ZC
A 6.5752 0.89843 2.1937 B 3.3385 0.39956 1.7742 C 2.5106 26.474 1.3205 D 2.0288 4.6375 0.61233 E 1.6130 2.9715 1.5138 F 1.1135 2.3325 12.962 G 0.20706 1.8972 3.8451 H 4.0475 1.4749 2.7364 a 4.0423 0.41918 1.2393 b 0.50204 0.16899 0.32508 c 0.27349 4.4581 0.15444 d 0.17534 0.49905 0.051633 e 0.11160 0.28608 0.059857
128
Table Bll. (Continued)
ZA ZB ZC
f 0.059208 0.19425 0.29034 g 0.0017069 0.13539 6.7143 h 0.0564662 0.088560 0.53499 a/A 0.614 0.466 0.565 b/B 0.150 0.423 0.183 c/C 0.108 0.168 0.116 d/D 0.086 0.107 0.084 e/E 0.069 0.096 0.039 f/F 0.052 0.083 0.022 g/G 0.034 0.071 1.745 h/H 0.013 0.060 0.195
Table B12. 30F, Case 6 and TSTEP = 0.026040
ZA ZB ZC
A 2.0288 4.6375 0.61233 B 1.6130 2.9715 1.5138 C 1.1135 2.3325 12.962 D 0.20706 1.8972 3.8451 E 4.0475 1.4749 2.7364 F 6.5797 0.89882 2.1940 G 3.3392 0.39834 1.7744 H 2.5107 26.316 1.3208 a 1.1121 2.6261 0.32097 b 0.27831 0.42402 0.89759 c 0.11134 0.23785 0.75865 d 0.00379 0.15432 0.37512 e 0.12677 0.097437 0.25121 f 0.47351 0.047721 0.17962 g 2.1594 0.0061123 0.12662 h 0.40665 0.080598 0.080194 a/A 0.548 0.566 0.524 b/B 0.172 0.142 0.592 c/C 0.099 0.101 0.058 d/D 0.018 0.081 0.097 e/E 0.031 0.065 0.091 f/F 0.071 0.053 0.081 g/G 0.646 0.015 0.071 h/H 0.161 0.003 0.060
129
Table B13. SLGF, Case 20 and TSTEP = 0.022915
ZA Z B ZC
A 20.207 0.43385 1.3291 B 2.8062 0.10745 0.98236 C 1.6585 1.8640 0.67442 D 1.1871 7.1128 0.29347 E 0.86338 2.2091 0.43045 F 0.54720 1.4570 4.5153 G 0.11253 1.0742 3.8890 H 0.88051 0.77024 1.9019 a 18.475 0.45601 1.3108 b 0.7111b 0.07246 0.91404 c 0.38098 3.1708 0.58862 d 0.24869 2.8947 0.22829 e 0.16282 1.5608 0.33633 f 0.090539 1.1661 2.0118 g 0.015758 0.92545 10.145 h 0.080569 0.71153 2.2804 a/A 0.704 1.051 0.992 b/B 0.248 0.676 0.931 c/C 0.229 1.704 0.872 d/D 0.208 0.406 0.786 e/E 0.188 0.706 0.781 f/F 0.165 0.800 0.445 g/G 0.139 0.864 2.608 h/H 0.091 0.923 1.198
Table B14. SLGF, Case 20 and TSTEP = 0.026040
ZA ZB ZC
A 1.1871 7.1128 0.29347 B 0.86338 2.2091 0.43045 C 0.54720 1.4576 4.5153 D 0.11253 1.0742 3.8896 E 0.88051 0.77024 1.9019 F 26.022 0.43406 1.3293 G 2.8674 0.10705 0.98253 H 1.6588 1.8618 0.67458 a 0.78595 6.8489 0.25999 b 0.31761 1.8802 0.41985 c 0.14990 1.2654 2.7324 d 0.020044 0.96657 6.4098 e 0.15235 0.72940 2.1300 f 0.63779 0.45198 1.3385 g 2.5577 0.065877 0.92181 h 0.55548 3.1056 0.59002
Table B14. (Continued)
130
ZA ZB ZC
a/A 0.662 0.962 0.885 b/B 0.367 0.851 0.973 c/C 0.274 0.868 0.605 d/D 0.178 0.899 1.648 e/E 0.173 0.947 1.119 f/F 0.024 1.041 1.006 g/G 0.891 0.614 0.938 h/H 0.334 1.669 0.880
Table B15. SLGF, Case 21 and TSTEP = 0.022915
A 5.8731 0.49425 1.2188 B 2.0189 0.092652 0.96790 C 1.4193 3.8487 0.71556 D 1.1125 3.1251 0.35627 E 0.86883 1.7262 0.52299 F 0.59885 1.2986 91.598 G 0.16328 1.0338 2.4114 H 1.2580 0.79490 1.5715 a 4.1997 0.52232 1.1999 b 0.62082 0.041707 0.89660 c 0.35920 90.059 0.61713 d 0.24126 1.8283 0.27106 e 0.16080 1.2877 0.37194 f 0.091131 1.0627 3.6155 S 0.017743 0.90386 3.8385 h 0.077939 0.74548 1.7834 a/A 0.715 1.052 0.985 b/B 0.307 0.450 0.926 c/C 0.252 23.399 0.862 d/D 0.216 0.585 0.761 e/E 0.185 0.746 0.711 f/F 0.152 0.816 0.039 g/G 0.108 0.873 1.591 h/H 0.061 0.938 1.133
131
Table B16. SLGF, Case 21 and TSTEP = 0.026040
ZA ZB ZC
A 1.1125 3.1251 0.35627 B 0.86883 1.7262 0.52299 C 0.59885 1.2986 91.598 D 0.16328 1.0338 2.4114 E 1.2580 0.7949 1.5715 F 5.8809 0.49444 1.2190 G 2.0194 0.092149 0.96803 H 1.4195 3.8405 0.71571 a 0.72554 2.9767 0.31087 b 0.30713 1.4806 0.49084 c 0.14987 1.1282 6.2722 d 0.023162 0.93027 3.2805 e 0.14815 0.75532 1.7221 f 0.64082 0.52184 1.2160 g 2.3948 0.035193 0.89918 h 0.54985 51.532 0.61613 a/A 0.652 0.952 0.873 b/B 0.355 0.857 0.938 c/C 0.250 0.869 0.068 d/D 0.142 0.899 1.360 e/E 0.117 0.950 1.095 f/F 0.108 1.055 0.997 g/G 1.183 0.382 0.928 h/H 0.386 13.411 0.861
Table B17. SLGF, Case 22 and TSTEP = 0.022915
ZA ZB ZC
A 30.986 0.76969 2.3654 N 4.5644 0.33338 1.7758 C 2.8941 4.8300 1.2218 D 2.1387 8.9039 0.49327 E 1.5796 3.7448 1.0410 F 0.99845 2.5902 16.495 G 0.13639 1.9452 5.8488 H 2.1633 1.4034 3.2682 a 20.943 0.80309 2.3406 b 0.72987 0.31842 1.6392 c 0.40120 18.251 1.0488 d 0.26303 4.1432 0.37769 e 0.1717 2.6236 0.73000 f 0.094307 2.0551 4.5618
132
Table B17. (Continued)
ZA ZB ZC
S 0.014309 1.6733 11.734 h 0.087869 1.3066 3.7779 a/A 0.675 1.043 0.989 b/B 0.159 0.954 0.922 c/C 0.138 3.778 0.858 d/D 0.123 0.465 0.764 e/E 0.108 0.700 0.701 f/F 0.094 0.793 0.276 g/G 0.104 0.860 2.006 h/H 0.040 0.931 1.155
Table B18. SLGF, Case 22 and TSTEP = 0.026040
ZA ZB ZC
A 2.1387 8.9038 0.49327 B 1.5796 3.7448 1.0410 C 0.99845 2.5902 16.495 D 0.13639 1.9452 5.8488 E 2.1633 1.4034 3.2682 F 31.121 0.77014 2.3659 G 4.5660 0.33255 1.7761 H 2.8946 4.8243 1.2221 a 1.3828 8.6864 0.44890 b 0.40849 3.1846 0.93767 c 0.17972 2.2382 6.7611 d 0.019707 1.7482 8.6998 e 0.19279 1.3373 3.5798 f 0.88910 0.81629 2.3320 S 1.7854 0.32423 1.6265 h 0.54794 24.198 1.0404 a/A 0.646 0.975 0.910 b/B 0.258 0.850 0.900 c/C 0.180 0.864 0.409 d/D 0.144 0.898 1.487 e/E 0.089 0.952 1.095 f/F 0.028 1.059 0.985 g/G 0.390 0.974 0.915 h/H 0.189 5.015 0.851
133
Table B19. SLGF, Case 23 and TSTEP = 0.022915
ZA ZB ZC
A 6.5752 0.89843 2.1937 B 3.3385 0.39956 1.7742 C 2.5106 26.474 1.3205 D 2.0288 4.6375 0.61233 E 1.6130 2.9715 1.5138 F 1.1135 2.3325 12.962 G 0.20706 1.8972 3.8451 H 4.0475 1.4749 2.7364 a 4.4282 0.94467 2.1653 b 0.66952 0.39771 1.6206 c 0.38681 7.1663 1.1098 d 0.25849 2.8239 0.45053 e 0.17077 2.2121 0.90142 f 0.094985 1.9077 12.692 g 0.015675 1.6703 5.4451 h 0.086466 1.4101 2.9700 a/A 0.673 1.051 0.986 b/B 0.200 0.995 0.913 c/C 0.154 0.270 0.839 d/D 0.127 0.608 0.735 e/E 0.105 0.744 0.595 f/F 0.085 0.817 0.979 g/G 0.075 0.880 1.416 h/H 0.019 0.959 1.085
Table B20. SLGF, Case 23 and TSTEP = 0.022915
ZA ZB ZC
A 2.0288 4.6375 0.61233 B 1.6130 2.9715 1.5138 C 1.1135 2.3325 12.962 D 0.20706 1.8972 3.8451 E 4.0475 1.4749 2.7364 F 6.5797 0.89882 2.1940 G 3.3392 0.39834 1.7744 H 2.5109 26.316 1.3208 a 1.2978 4.4896 0.54615 b 0.40292 2.5447 1.2342 c 0.18075 2.0191 45.519 d 0.022313 1.7105 4.9340 e 0.19131 1.4196 2.9126 f 0.91083 0.98244 2.1257 g 1.7074 0.43116 1.5956
134
Table B20. (Continued)
ZA ZB ZC
h 0.54393 6.2741 1.0958 a/A 0.639 0.968 0.891 b/B 0.249 0.856 0.815 c/C 0.162 0.865 3.512 d/D 0.107 0.901 1.283 e/E 0.047 0.962 1.064 f/F 0.138 1.093 0.968 g/G 0.511 1.083 0.898 h/H 0.216 0.238 0.829
B21, LLF, Case 40 and TSTEP = 0.022915
ZA ZB ZC
A 26.211 0.43386 1.3291 B 2.8662 0.10745 0.98237 C 1.6585 1.8640 0.67442 D 1.1872 7.1126 0.29347 E 0.86339 2.2091 0.43046 F 0.54721 1.4576 4.5154 G 0.11253 1.0742 3.8896 H 0.88052 0.77025 1.9019 a 19.508 0.30995 1.2984 b 0.74305 0.023131 0.90462 e 0.35202 2.0973 0.61223 d 0.20635 0.41743 0.27377 e 0.11965 0.27706 0.36598 f 0.053479 0.20820 4.4527 g 0.0075488 0.15894 3.0795 h 0.075570 0.11557 1.5844 a/A 0.744 0.714 0.976 b/B 0.259 0.215 0.923 c/C 0.212 1.125 0.907 d/D 0.173 0.058 0.932 e/E 0.138 0.125 0.850 f/F 0.097 0.142 0.988 g/G 0.066 0.147 0.791 h/H 0.085 0.149 0.833
135
Table B22. LLF, Case 40 and TSTEP = 0.026040
ZA ZB ZC
A 1.1872 7.1126 0.29347 B 0.86339 2.2091 0.43046 C 0.54721 1.4576 4.5154 D 0.11253 1.0742 3.8896 E 0.88052 0.77025 1.9019 F 26.026 0.43407 1.3294 G 2.8674 0.10705 0.98253 H 1.6588 1.8619 0.67459 a 0.69727 4.2260 0.29650 b 0.23298 0.44896 0.41986 c 0.083599 0.26831 6.2205 d 0.017737 0.18799 2.8437 e 0.12684 0.13112 1.5486 f 0.31763 0.077715 1.1147 S 1.2486 0.013275 0.83855 h 0.92975 0.092864 0.58705 a/A 0.587 0.594 1.010 b/B 0.269 0.203 0.975 c/C 0.152 0.184 1.377 d/D 0.157 0.175 0.730 e/E 0.144 0.170 0.810 f/F 0.012 0.179 0.838 g/G 0.435 0.124 0.853 h/H 0.560 0.049 0.870
Table B23. LLF, Case 41 and TSTEP = 0.022915
ZA ZB ZC
A 5.8728 0.49426 1.2188 B 2.0189 0.092659 0.96791 C 1.4193 3.8490 0.71557 D 1.1125 3.1250 0.35627 E 0.86884 1.7263 0.52301 F 0.59886 1.2986 91.494 G 0.16328 1.0338 2.4114 G 0.16328 1.0338 2.4114 H 1.2581 0.79491 1.5715 a 4.4618 0.33169 1.1847 b 0.65769 0.032238 0.88154 c 0.33751 1.3393 0.64146 d 0.20276 0.38927 0.32836 e 0.11887 0.26675 0.42377
136
Table B23. (Continued)
ZA
f 0.053545 g 0.0072228 h 0.075086 a/A 0.759 b/B 0.325 c/C 0.237 d/D 0.182 e/E 0.136 f/F 0.089 g/G 0.044 h/H 0.059
ZB ZC
0.20308 36.493 0.15633 2.0029 0.11459 1.3248 0.671 0.972 0.347 0.910 0.347 0.896 0.124 0.922 0.154 0.810 0.156 0.398 0.151 1.274 0.144 0.843
Table B24. LLF, Case 41 and TSTEP = 0.026040
ZA ZB ZC
A 1.1125 3.1250 0.35627 B 0.86884 1.7263 0.52301 C 0.59886 1.2986 91.494 D 0.16328 1.0338 2.4114 E 1.2581 0.79491 1.5715 F 5.8805 0.49445 1.2190 G 2.0194 0.092157 0.96804 H 1.4195 3.8408 0.71572 a 0.64582 1.8604 0.35884 b 0.22855 0.40649 0.50741 c 0.083382 0.25470 12.421 d 0.017102 0.18180 1.9307 e 0.12552 0.12834 1.3105 f 0.31285 0.077162 1.0326 g 1.1714 0.014769 0.82993 h 0.99499 0.088344 0.62394 a/A 0.580 0.595 1.007 b/B 0.262 0.235 0.970 c/C 0.139 0.196 0.135 d/D 0.104 0.175 0.800 e/E 0.099 0.161 0.034 f/F 0.053 0.156 0.847 g/G 0.580 0.160 0.857 h/H 0.700 0.023 0.871
137
Table B25. LLF, Case 42 and TSTEP = 0.026040
ZA ZB ZC
A 30.982 0.76970 2.3654 B 4.5644 0.33338 1.7758 C 2.8941 4.8302 1.2218 D 2.1387 8.9037 0.49328 E 1.5796 3.7448 1.0410 F 0.99846 2.5903 16.496 G 0.13640 1.9452 5.8488 H 2.1634 1.4034 3.2682 a 21.813 0.58524 2.3307 b 0.73457 0.081558 1.6330 c 0.35210 0.54927 1.1042 d 0.20491 0.33703 0.45828 e 0.11704 0.25577 0.92208 f 0.050456 0.20218 24.028 S 0.010189 0.15852 4.5074 h 0.076678 0.11739 2.6653 a/A 0.704 0.760 0.985 b/B 0.160 0.244 0.919 c/C 0.121 0.113 0.903 d/D 0.095 0.037 0.928 e/E 0.074 0.068 0.885 f/F 0.050 0.078 1.457 g/G 0.074 0.081 0.770 h/H 0.035 0.083 0.815
Table B26. LLF, Case 42 and TSTEP = 0.026040
ZA ZB ZC
A 2.1387 8.9037 0.49328 B 1.5796 3.7448 1.0410 C 0.99846 2.5903 16.496 D 0.13640 1.9452 5.8488 E 2.1634 1.4034 3.2682 F 31.118 0.77016 2.3660 G 4.5660 0.33256 1.7762 H 2.8946 4.8245 1.2222 a 1.1991 4.9132 0.49856 b 0.27115 0.46675 1.0517 c 0.086725 0.28016 76.290 d 0.02567 0.19763 4.2762 e 0.14125 0.13896 2.6210 f 0.34644 0.083401 1.9565 g 1.4673 0.015357 1.4980 h 0.86395 0.099671 1.0532
138
Table B26, (Continued)
ZA ZB ZC
a/A 0.560 0.551 1.009 b/B 0.171 0.124 1.010 c/C 0.086 0.108 4.626 d/D 0.183 0.101 0.731 e/E 0.065 0.098 0.801 f/F 0.011 0.108 0.826 g/G 0.321 0.046 0.843 h/H 0.298 0.020 0.861
Table B27. LLF, Case 43 and TSTEP = 0.022915
ZA ZB ZC
A 6.5750 0.8946 2.1938 B 3.3386 0.39959 1.7742 C 2.5106 26.485 1.3206 D 2.0289 4.6375 0.61235 E 1.6130 2.9715 1.5139 F 1.1135 2.3326 12.960 G 0.20707 1.8973 3.8451 H 4.0479 1.4750 2.7364 a 4.6341 0.65654 2.1525 b 0.68107 0.11853 1.6101 c 0.34318 0.48712 1.1770 d 0.20272 0.32276 0.56408 e 0.11656 0.24945 1.3169 f 0.050431 0.19893 8.8410 S 0.010128 0.15694 3.0986 h 0.076464 0.11685 2.2607 a/A 0.704 0.737 0.981 b/B 0.204 0.296 0.907 c/C 0.136 0.018 0.891 d/D 0.099 0.069 0.915 e/E 0.072 0.083 0.869 f/F 0.045 0.084 0.682 g/G 0.048 0.083 0.805 h/H 0.018 0.079 0.826
139
Table B28. LLF, Case 43 and TSTEP = 0.026040
ZA ZB ZC
A 2.0289 4.6375 0.61235 B 1.6130 2.9715 1.5137 C 1.1135 2.3326 12.960 D 0.20707 1.8973 3.8451 E 4.0479 1.4750 2.7364 F 6.5795 0.89885 2.1940 G 3.3392 0.39838 1.7744 H 2.5109 26.327 1.3209 a 1.1287 2.5538 0.61877 b 0.26872 0.43772 1.5609 c 0.086702 0.27116 7.5353 d 0.024857 0.19365 3.0367 e 0.14085 0.13728 2.2472 f 0.34404 0.083187 1.8352 g 1.4074 0.016442 1.5070 h 0.89665 0.096743 1.1451 a/A 0.556 0.550 1.010 b/B 0.166 0.147 1.031 c/C 0.077 0.116 0.581 d/D 0.120 0.102 0.789 e/E 0.034 0.093 0.821 f/F 0.052 0.092 0.837 g/G 0.421 0.041 0.851 h/H 0.356 0.003 0.866
Table B29. DLGF, Case 60 and TSTEP = 0.022915
ZA ZB ZC
A 26.211 0.43386 1.3291 B 2.8662 0.10745 0.98237 C 1.6585 1.8640 0.67442 D 1.1872 7.1126 0.29347 E 0.86339 2.2091 0.43046 F 0.54721 1.4576 4.5154 G 0.11253 1.0742 3.8896 H 0.88052 0.77025 1.9019 a 19.468 0.30854 1.2988 b 0.71809 0.013617 0.91272 c 0.34761 2.7641 0.61554 d 0.20632 0.42129 0.27274 e 0.12086 0.27529 0.36815 f 0.054963 0.20574 4.2747
140
Table B29. (Continued)
ZA ZB ZC
S 0.0063315 0.15666 3.1796 h 0.075172 0.11377 1.6094 a/A 0.742 0.711 0.777 b/B 0.250 0.126 0.929 c/C 0.209 1.482 0.912 d/D 0.173 0.059 0.929 e/E 0.139 0.124 0.856 f/F 0.100 0.141 0.946 g/G 0.056 0.145 0.817 h/H 0.085 0.147 0.845
Table B30. DLGF, Case 60 and TSTEP = 0.026040
A 1.1872 7.1126 0.29347 B 0.06339 2.2091 0.43046 C 0.54721 1.4576 4.5154 D 0.11253 1.0742 3.8896 E 0.88052 0.77025 1.9019 F 26.026 0.43407 1.3294 G 2.8674 0.10705 0.98253 H 1.6588 1.8619 0.67459 a 0.70706 4.1473 0.29247 b 0.23432 0.44704 0.42167 c 0.085836 0.26522 6.0227 d 0.016046 0.18523 2.9077 e 0.12691 0.12900 1.5684 f 0.32378 0.076448 1.1243 g 1.3423 0.013258 0.84294 h 0.89753 0.090497 0.58780 a/A 0.595 0.583 0.996 b/B 0.271 0.202 0.979 c/C 0.156 0.181 1.333 d/D 0.142 0.172 0.747 e/E 0.144 0.167 0.824 f/F 0.012 0.176 0.845 g/G 0.468 0.123 0.857 h/H 0.541 0.048 0.871
141
Table B31. DLGF, Case 61 and TSTEP = 0.022915
ZA ZB ZC
A 5.8728 0.49426 1.2188 B 2.0187 0.092659 0.96791 C 1.4193 3.8490 0.71557 D 1.1125 3.1250 0.35627 E 0.86884 1.7263 0.52301 F 0.59886 1.2986 91.494 G 0.16328 1.0378 2.4114 H 1.2581 0.79491 1.5715 a 4.4664 0.33245 1.1844 b 0.63688 0.020476 0.89017 c 0.33308 1.6070 0.64511 d 0.20253 0.39346 0.32700 e 0.12000 0.26527 0.42598 f 0.055028 0.20075 65.164 S 0.0059688 0.15409 2.0504 h 0.074583 0.11273 1.3433 a/A 0.760 0.672 0.971 b/B 0.315 0.220 0.919 c/C 0.234 0.417 0.901 d/D 0.182 0.125 0.917 e/E 0.138 0.153 0.814 f/F 0.098 0.154 0.712 g/G 0.036 0.149 0.850 h/H 0.059 0.141 0.854
Table B32. DLGF, Case 61 and TSTEP = 0.026040
ZA ZB ZC
A 1.1125 3.1250 0.35627 B 0.86884 1.7263 0.523011 C 0.59886 1.2986 91.494 D 0.16328 1.0338 2.4114 E 1.2581 0.79491 1.5715 F 5.8805 0.49445 1.2190 G 2.0194 0.092157 0.96804 H 1.4195 3.8408 0.71572 a 0.65602 1.8213 0.35270 b 0.22956 0.40530 0.51060 c 0.085524 0.25197 13.460 d 0.015332 0.17917 1.9633 e 0.12542 0.12621 1.3256 f 0.31867 0.075813 1.0409
142
Table B32. (Continued)
ZA ZB ZC
g 1.2562 0.014621 0.83394 h 0.95478 0.086109 0.62440 a/A 0.589 0.582 0.674 b/B 0.264 0.234 0.963 c/C 0.142 0.194 0.147 d/D 0.093 0.173 0.814 e/E 0.099 0.158 0.843 f/F 0.054 0.153 0.853 g/G 0.622 0.158 0.861 h/H 0.672 0.022 0.872
Table B33. DLGF, Case 62 and TSTEP = 0.022915
ZA ZB ZC
A 30.982 0.76970 2.3654 B 4.5644 0.33338 1.7758 C 2.8941 4.8302 1.2218 D 2.1387 8.9037 0.49328 E 1.5796 3.7448 1.0410 F 0.99846 2.5903 16.496 G 0.13640 1.9452 5.8488 H 2.1634 1.4034 3.2682 a 21.830 0.58620 2.3305 b 0.70705 0.048422 1.6491 c 0.34747 0.58765 1.1112 d 0.20516 0.33835 0.45691 e 0.11861 0.25337 0.91996 f 0.052207 0.19926 20.612 g 0.0088875 0.15590 4.6446 h 0.076372 0.11534 2.7083 a/A 0.704 0.761 0.985 b/B 0.154 0.145 0.928 c/C 0.120 0.121 0.909 d/D 0.095 0.037 0.926 e/E 0.075 0.067 0.883 f/F 0.052 0.076 1.249 g/G 0.065 0.080 0.794 h/H 0.035 0.082 0.828
143
Table B34. DLGF, Case 62 and TSTEP = 0.026040
ZA ZB ZC
A 2.1387 8.9037 0.49328 B 1.5796 3.7448 1.0410 C 0.99846 2.5903 16.496 D 0.13640 1.9452 5.8488 E 2.1634 1.4034 3.2682 F 31.118 0.77016 2.3660 G 4.5660 0.33256 1.7762 H 2.8946 4.8245 1.2222 a 1.1991 4.9132 0.49856 b 0.27115 0.46675 1.0517 c 0.086725 0.28016 76.290 d 0.025067 0.19763 4.2762 e 0.14125 0.13896 2.6210 f 0.34644 0.083401 1.9565 S 1.4673 0.015357 1.4980 h 0.86395 0.099671 1.0532 a/A 0.560 0.551 1.010 b/B 0.171 0.124 1.010 c/C 0.086 0.108 4.626 d/D 0.183 0.101 0.731 e/E 0.065 0.098 0.826 f/F 0.011 0.108 0.826 g/G 0.321 0.046 0.843 h/H 0.298 0.020 0.861
Table B35. DLGF, Case 63 and TSTEP = 0.022915
ZA ZB ZC
A 6.5750 0.89846 2.1938 B 3.3386 0.39959 1.7742 C 2.5106 26.485 1.3206 D 2.0289 4.6375 0.61235 E 1.6130 2.9715 1.5139 F 1.1135 2.3326 12.960 G 0.20707 1.8973 3.8451 H 4.0479 1.4750 2.7364 a 4.6341 0.65654 2.1525 b 0.68107 0.11853 1.6101 c 0.34318 0.48712 1.1770 d 0.20272 0.32276 0.564081 e 0.11656 0.24945 1.3169 f 0.050431 0.19893 8.8410 g 0.010128 0.15694 3.0986
144
Table B35. (Continued)
ZA ZB ZC
h 0.076464 0.11685 2.2607 a/A 0.704 0.730 0.982 b/B 0.203 0.296 0.907 c/C 0.136 0.018 0.891 d/D 0.099 0.069 0.921 e/E 0.072 0. 083 0.869 f/F 0.045 0.084 0.685 g/G 0.048 0.082 0.805 h/H 0.018 0.079 0.826
Table B36. DLGF, Case 63 and TSTEP = 0.026040
ZA ZB ZC
A 2.0289 4.6375 0.61235 B 1.6130 2.9715 0.15139 C 1.1135 2.3326 12.960 D 0.20707 1.8973 3.8451 E 4.0479 1.4750 2.7364 F 6.5795 0.89885 2.1940 G 3.3392 0.39838 1.7744 H 2.5109 26.327 1.3209 a 1.1287 2.5538 0.61877 b 0.26872 0.43772 1.5609 c 0.086702 0.27116 7.5354 d 0.024857 0.19365 3.0367 e 0.14085 0.13728 2.2472 f 0.34404 0.083187 1.8352 g 1.4074 0.016442 1.5070 h 0.89665 0.096743 1.1451 a/A 0.556 0.550 1.010 b/B 0.166 0.147 10.310 c/C 0.077 0.116 0.581 d/D 0.120 0.102 0.789 e/E 0.034 0.093 0.821 f/F 0.052 0.092 0.836 g/G 0.421 0.041 0.849 h/H 0.357 0.003 0.866
145
B. Tables of Za, Zb and Zc of the Cases in Table 3b
146
Table B37. 30F, Case 9 and TSTEP = 0.022915
Zk ZB ZC
A 26.207 0.43385 1.3291 B 2.8662 0.10745 0.98236 C 1.6585 1.8640 0.67442 D 1.1871 7.1128 0.29347 E 0.86338 2.2091 0.43045 F 0.54720 1.4576 4.5153 G 0.11253 1.0742 3.8896 H 0.88051 0.77024 1.9019 a 3.3361 0.014643 0.12042 b 0.043452 0.0019173 0.031322 c 0.019017 0.23068 0.018362 d 0.010150 0.017774 0.011244 e 0.0050374 0.011146 0.0041845 f 0.0012576 0.0083685 0.0089393 g 0.0021281 0.0066170 0.16137 h 0.0058073 0.0052483 0.043057 a/A 0.127 0.033 0.090 b/B 0.015 0.017 0.031 c/C 0.011 0.123 0.027 d/D 0.008 0.002 0.038 e/E 0.005 0.005 0.009 f/F 0.002 0.005 0.001 g/G 0.018 0.006 0.041 h/H 0.006 0.006 0.022
Table B38. 3^F, Case 10 and TSTEP = 0.026040
ZA ZB ZC
A 2.1387 8.9039 0.49327 B 1.5796 3.7448 1.0410 C 0.99845 2.5902 16.495 D 0.13639 1.9452 5.8488 E 2.1633 1.4034 3.2682 F 31.1121 0.77014 2.3659 G 4.5660 0.33255 1.7761 H 2.8946 4.8243 1.2221 a 0.12625 0.30093 0.13907 b 0.012802 0.016334 0.22521 c 0.0010042 0.00993818 0.020993 d 0.0062335 0.0072862 0.020285 e 0.015191 0.0055596 0.017390 f 0.039446 0.0041042 0.014676 g 0.15653 0.0025946 0.012154 h 0.024121 0.0006351 0.0096397
147
Table B38. (Continued)
ZA ZB ZC
a/A 0.059 0.033 0.281 b/B 0.008 0.004 0.216 c/C 0.001 0.003 0.001 d/D 0.045 0.003 0.003 e/E 0.007 0.003 0.005 f/F 0.001 0.005 0.006 g/G 0.034 0.007 0.006 h/H 0.008 0.0001 0.007
Table B39. SLGF, Case 26 and TSTEP = 0.022913
ZA ZB ZC
A 26.207 0.43385 1.3291 B 2.8662 0.10745 0.98236 C 1.6585 1.8640 0.67442 D 1.1871 7.1128 0.29347 E 0.86338 2.2091 0.43045 F 0.54720 1.4576 4.5153 G 0.11253 1.0742 3.8896 H 0.88051 0.77024 1.9019 a 9.9801 0.57419 1.1919 b 0.23557 0.24031 0.84672 c 0.12905 1.0733 0.64485 d 0.085295 0.82602 0.37307 e 0.056531 0.76417 0.47061 f 0.032326 0.72870 3.9283 g 0.0076702. 0.69797 1.4941 h 0.022920 0.66044 1.1291 a/A 0.380 1.323 0.896 b/B 0.082 2.236 0.861 c/C 0.077 0.575 0.956 d/D 0.071 0.116 1.271 e/E 0.065 0.345 1.093 f/F 0.059 0.500 0.870 g/G 0.068 0.649 0.384 h/H 0.026 0.857 0.593
148
Table B40. SLGF, Case 27 and TSTEP = 0.026040
ZA ZB ZC
A 2.1387 8.9039 0.49327 B 1.5796 3.7448 1.0410 C 0.99845 2.5902 16.495 D 0.13639 1.9452 5.8488 E 2.1633 1.4034 3.2682 F 31.121 0.77014 2.3659 G 4.5660 0.33255 1.7761 H 2.8946 4.8243 1.2221 a 0.77470 5.3329 0.90746 b 0.16636 1.5308 4.7408 c 0.072358 1.4397 2.9812 d 0.011805 1.4297 2.1374 e 0.065215 1.4437 1.8174 f 0.31394 1.5049 1.6034 g 0.62268 0.64601 1.4041 h 0.19324 1.3066 1.1515 a/A 0.362 0.598 1.839 b/B 0.105 0.408 4.554 c/C 0.072 0.555 0.180 d/D 0.086 0.734 0.365 e/E 0.030 1.028 0.556 f/F 0.010 1.954 0.677 g/G 0.136 1.942 0.790 h/H 0.066 0.270 0.942
Table B41. LLF, Case 46 and TSTEP = 0.022915
ZA ZB ZC
A 26.207 0.43385 1.3291 B 2.8662 0.10745 0.98236 C 1.6585 1.8640 0.67442 D 1.1871 7.1128 0.29347 E 0.86338 2.2091 0.43045 F 0.54720 1.4576 4.5153 G 0.11253 1.0742 3.8896 H 0.88051 0.77024 1.9019 a 12.094 0.19906 1.1244 b 0.27031 0.10084 0.77609 c 0.12436 0.15956 0.62956 d 0.067209 0.089459 0.44069 e 0.032518 0.069046 0.55766 f 0.0061758 0.055195 1.4185 g 0.017420 0.043670 0.96912 h 0.042128 0.032770 0.83708
149
Table B41. (Continued)
ZA ZB ZC
a/A 0.461 0.459 0.846 b/B 0.094 0.934 0.790 c/C 0.075 0.085 0.933 d/D 0.056 0.012 1.501 e/E 0.037 0.031 1.282 f/F 0.011 0.037 0.314 g/G 0.154 0.040 0.249 h/H 0.047 0.042 0.440
Table B42. LLF, Case 47 and TSTEP = 0.026040
ZA ZB ZC
A 2.1387 8.9039 0.49327 B 1.5796 3.7448 1.0410 C 0.99845 2.5902 16.495 D 0.13639 1.9452 5.8488 E 2.1633 1.4034 3.2682 F 31.121 0.77014 2.3659 G 4.5660 0.33255 1.7761 H 2.8946 4.8243 1.2221 a 0.69504 2.1085 0.88944 b 0.093688 0.14535 8.3210 c 0.011319 0.089338 1.7342 d 0.035651 0.064324 1.5369 e 0.083175 0.046787 1.4491 f 0.16301 0.030809 1.3824 g 0.50065 0.012623 1.3112 h 0.46568 0.014142 1.2037 a/A 0.325 0.236 1.804 b/B 0.059 0.039 7.993 c/C 0.011 0.034 0.105 d/D 0.261 0.033 0.264 e/E 0.038 0.033 0.443 f/F 0.005 0.040 0.584 g/G 0.109 0.038 0.738 h/H 0.160 0.002 0.984
150
Table B43. DLGF, Case 66 and TSTEP = 0.022915
ZA ZB ZC
A 26.207 0.43385 1.3291 B 2.8662 0.10745 0.98236 G 1.6585 1.8640 0.67442 D 1.1871 7.1128 0.29347 E 0.86338 2.2091 0.43045 F 0.54720 1.4576 4.5153 G 0.11253 1.0742 3.8896 H 0.88051 0.77024 1.9019 a 12.089 0.19888 1.1247 b 0.26475 0.090619 0.78546 c 0.12254 0.17979 0.63578 d 0.066740 0.091017 0.44137 e 0.032675 0.068993 0.56975 f 0.0066562 0.054686 1.4539 g 0.016779 0.043013 0.98453 h 0.041452 0.032091 0.84739 a/A 0.461 0.458 0.846 b/B ' 0.092 0.843 0.816 c/C 0.073 0.096 0.942 d/D 0.055 0.012 1.517 e/E 0.037 0.031 1.323 f/F 0.012 0.037 0.323 g/G 0.149 0.040 0.253 h/H 0.047 0.041 0.445
Table B44. DLGF, Case 67, and TSTEP = 0.026040
ZA ZB ZC
A 2.1387 8.9039 0.49327 B 1.5796 3.7448 1.0410 C 0.99845 2.5902 16.495 D 0.13639 1.9452 5.8488 E 2.1633 1.4034 3.2682 F 31. 121 0.77014 2.3659 G 4.5660 0.33255 1.7761 H 2.8946 4.0243 1.2221 a 0.69654 2.1007 0.88699 b 0.093674 0.14554 7.6308 c 0:012520 0.088267 1.7620 d 0.034384 0.063177 1.5579 e 0.082288 0.045764 1.4644 f 0.16369 0.029992 1.3929 g 0.52184 0.012118 1.3167 h 0.44552 0.014068 1.2025
151
Table B44. (Continued)
ZA ZB ZC
a/A 0.325 0.235 1.810 b/B 0.059 0.038 7.329 c/C 0.012 0.034 0.106 d/D 0.252 0.032 0.266 e/E 0.038 0.032 0.448 f/F 0.005 0.038 0.588 g/G 0.114 0.036 0.741 h/H 0.154 0.002 0.984
152
Tables of Za, Zb and Zc of the Cases in Table 4a
153
Table B45. 30F, Case 7 and TSTEP = 0.018749
ZA ZB ZC
A 1.4041 0.85455 0.12449 B 2.6958 1.0075 0.55650 G 0.98760 1.2112 0.75764 D 0.30856 1.6499 0.90900 E 0.61253 6.9839 1.0687 F 0.78477 0.21839 1.3101 G 0.93085 0.45189 1.9814 H 1.1032 0.69439 4.9393 a 1.4076 0.85532 0.12396 b 5.8070 0.83019 0.29792 c 0.51244 0.74702 0.27177 d 0.048444 0.70638 0.21988 e 0.046367 0.88798 0.17015 f 0.067476 0.86041 0.12399 g 0.064141 0.022257 0.07906 h 0.050577 0.089070 0.032127 a/A 1.002 1.001 0.995 b/B 2.152 0.924 0.535 c/C 0.518 0.616 0.358 d/D 0.157 0.428 0.241 e/E 0.075 0.127 0.159 f/F 0.085 3.939 0.094 g/G 0.068 0.049 0.039 h/H 0.045 0.128 0.006
Table B46. 30F, Case 8 and TSTEP = 0.022915
ZA ZB ZC
A 1.2404 6.7885 2.4053 B 1.6437 0.073474 3.3016 C 2.0424 0.90548 8.1362 D 2.5988 1.4000 2.2381 E 3.9061 1.7898 0.36863 F 71.901 2.2188 1.1008 G 0.77488 2.9040 1.5404 H 0.66146 5.0848 1.9282 a 1.2399 6.7945 2.4047 b 1.0508 0.29974 3.9984 c 0.73104 0.095627 3.8392 d 0.51861 0.14452 0.51460 e 0.37948 0.13589 0.097828 f 0.27577 0.11317 0.021384 g 0.17518 0.086034 0.058512 h 0.069129 0.056709 0.064703
154
Table B46. (Continued)
ZA ZB ZC
a/A 0.999 1.000 1.000 b/B 0.639 4.083 1.211 c/C 0.357 0.105 0.469 d/D 0.199 0.103 0.229 e/E 0.097 0.075 0.265 f/F 0.003 0.051 0.019 g/G 0.226 0.029 0.037 h/H 0.104 0.011 0.033
Table B47. SLGF, Case 24 and TSTEP = 0.018749
ZA ZB ZC
A 1.4041 0.85455 0.12449 B 2.6958 1.0075 0.55650 C 0.9876 1.2112 0.75764 D 0.30856 1.6499 0.90900 E 0.61253 6.9839 1.0687 F 0.78477 0.21839 1.3101 G 0.93085 0.45189 1.9814 H 1.1032 0.69439 4.9393 a 1.4076 0.85532 0.12396 b 5.1252 1.0409 0.53287 c 0.53415 1.3791 0.69090 d 0.014540 2.6929 0.77558 e 0.094423 2.4087 0.83124 f 0.11993 0.14401 0.87752 S 0.11556 0.27514 0.93480 h 0.096946 0.48992 1.0967 a/A 1.002 1.000 0.995 b/B 1.901 1.033 0.957 c/C 0.540 1.138 0.911 d/D 0.047 1.631 0.852 e/E 0.154 0.344 0.778 f/F 0.132 0.659 0.667 g/G 0.124 0.597 0.471 h/H 0.087 0.710 0.221
155
Table B48. SLGF, Case 25 and TSTEP = 0.022915
ZA ZB ZC
A 1.2404 6.7885 2.4053 B 1.6437 0.073474 3.3016 C 2.0424 0.90548 8.1362 D 2.5988 1.4000 2.2381 E 3.9061 1.7898 0.36863 F 71.901 2.2188 1.1008 G 0.77488 2.9041 1.5404 H 0.66146 5.0848 1.9282 a 1.2399 6.7945 2.4047 b 1.1793 0.10956 2.9905 c 0.94302 0.76439 4.5960 d 0.74852 1.2198 4.9691 e 0.60131 1.6426 0.41109 f 0.47843 2.2380 1.0864 g 0.33845 3.6439 1.3704 h 37.591 41.269 1.5524 a/A 0.999 1.000 1.001 b/B 0.717 1.500 0.902 c/C 0.461 0.844 0.564 d/D 0.288 0.871 2.220 e/E 0.153 0.917 1.115 f/F 0.006 1.008 0.986 g/G 0.436 1.256 0.889 h/H 56.868 8.122 0.805
Table B49. LLF, Case 44 and TSTEP = 0.018749
ZA ZB ZC
A 1.4041 0.85455 0.12449 B 2.6958 1.0075 0.55650 C 0.98760 1.2112 0.75764 D 0.30856 1.6499 0.90900 E 0.61253 6.9839 1.0687 F 0.78477 0.21839 1.3101 G 0.93085 0.45189 1.9814 H 1.1032 0.69439 4.9393 a 1.4076 0.85532 0.12396 b 3.7933 0.90974 0.50286 c 0.60487 0.98269 0.66700 s 0.022490 1.3701 0.78736 e 0.077563 4.6906 0.91836 f 0.092496 0.25665 1.1229
156
Table B49. (Continued)
ZA ZB ZC
g 0.08156 0.0072122 1.6978 h 0.060892 0.057409 4.7518 a/A 1.002 1.005 0.995 b/B 1.407 0.903 0.903 c/C 0.612 0.811 0.881 d/D 0.072 0.830 0.865 e/E 0.126 0.371 0.859 f/F 0.117 1.175 0.857 g/G 0.087 0.015 0.856 h/H 0.055 0.082 0.961
Table B50. LLF, Case 45 and TSTEP = 0.022915
ZA ZB ZC
A 1.2404 6.7885 2.4053 B 1.6437 0.073474 3.3016 C 2.0424 0.90548 8.1362 D 2.5988 1.4000 2.2381 E 3.9061 1.7898 0.36863 F 71.901 2.2188 1.1008 G 0.77488 2.9041 1.5404 H 0.66146 5.0848 1.9282 a 1.2399 6.7945 2.4047 b 1.0278 0.29000 3.4929 c 0.67816 0.12155 16.157 d 0.44754 0.17864 1.5069 e 0.29499 0.17356 0.26731 f 0.17885 0.15150 0.88316 g 0.069525 0.12271 1.2760 h 0.078371 0.089266 1.6342 a/A 0.999 1.000 0.999 b/B 0.625 3.947 1.057 c/C 0.322 0.134 7.219 d/D 0.172 0.127 0.673 e/E 0.075 0.097 0.726 f/F 0.002 0.068 0.802 g/G 0.089 0.042 0.828 h/H 0.117 0.017 0.847
157
Table B51. DLGF, Case 64 and TSTEP = 0.018749
ZA ZB ZC
A 1.4041 0.85455 0.12449 B 2.6958 1.0075 0.55650 G 0.98760 1.2112 0.75764 D 0.30856 1.6499 0.90900 E 0.61253 6.9839 1.0687 F 0.78477 0.21839 1.3101 G 0.93085 0.45189 1.9814 H 1.1032 0.69439 4.9393 a 1.4076 0.85532 0.12396 b 4.0028 0.90052 0.50785 c 0.59133 0.97547 0.06563 d 0.017165 1.4135 0.77425 e 0.084093 3.3132 0.88789 f 0.09806 0.24791 1.0621 S 0.086711 0.013154 1.5305 h 0.064356 0.051467 8.5070 a/A 1.002 1.001 0.995 b/B 1.484 0.893 0.911 c/C 0.598 0.591 0.877 d/D 0.055 0.856 0.851 e/E 0.137 0.378 0.831 f/F 0.126 1.135 0.810 g/G 0.093 0.029 0.772 h/H 0.058 0.074 1.722
Table B52. DLGF, Case 65 and TSTEP = 0.022915
ZA ZB ZC
A 1.2404 6.7885 2.4053 B 1.6437 0.073474 3.3016 C 2.0424 0.90548 8.1362 D 2.5988 1.4000 2.2381 E 3.9061 1.7898 0.36863 F 71.901 2.2188 1.1008 G 0.77488 2.9041 1.5404 H 0.66146 5.0848 1.9282 a 1.2399 6.7945 2.4047 b 1.0393 0.29194 3.4465 c 0.68708 0.11891 14.830 d 0.44902 0.17888 1.5021 e 0.28947 0.17698 0.25179 f 0.16776 0.15765 0.84115 g 0.055415 0.13093 1.2070 h 0.084780 0.098702 1.5327
158
Table B52. (Continued)
ZA ZB ZC
a/A 0.999 1.000 0.999 b/B 0.633 3.976 1.043 c/C 0.336 0.130 1.822 d/D 0.172 0.128 0.671 e/E 0.073 0.098 0.683 f/F 0.002 0.071 0.764 g/G 0.071 0.045 0.783 h/H 0.128 0.019 0.794
159
Tables of Za, Zb and Zc of the Cases in Table 4b
160
Table B53. 30F, Case 11 and TSTEP = 0.018749
ZA ZB ZC
A 1.4041 0.85455 0.12449 B 2.6958 1.0075 0.55650 C 0.98760 1.2112 0.75764 D 0.30856 1.6499 0.90900 E 0.61253 6.9839 1.0687 F 0.78477 0.21839 1.3101 G 0.93085 0.45189 1.9814 H 1.1032 0.69439 4.9393 a 1.4076 0.85532 0.12396 b 12.337 0.77259 0.15145 c 0.40007 0.70401 0.11707 d 0.11776 0.85818 0.087814 e 0.044179 31.138 0.066545 f 0.018317 0.31016 0.049728 g 0.0090930 0.093522 0.034978 h 0.0070647 0.036240 0.020458 a/A 1.002 1.000 0.995 b/B 4.586 0.766 0.472 c/C 0.405 0.581 0.154 d/D 0.381 0.520 0.095 e/E 0.072 4.459 0.062 f/F 0.023 1.420 0.037 g/G 0.009 0.206 0.017 h/H 0.006 0.052 0.004
Table B54. 30F, Case 12 and TSTEP = 0.022915
ZA ZB ZC
A 1.2404 6.7885 2.4053 B 1.6437 0.073474 3.3016 C 2.0424 0.90548 8.1362 D 2.5988 1.4000 2.2381 E 3.9061 1.7898 0.36863 F 71.901 2.2188 1.1008 G 0.77488 2.9041 1.5404 H 0.66146 5.0848 1.9202 a 1.2399 6.7945 2.4047 b 0.76027 0.38097 6.3831 c 0.42790 0.053659 1.4417 d 0.28375 0.0029557 0.35765 e 0.21627 0.016179 0.13374 f 0.18872 0.017624 0.056110 g 0.20925 0.014823 0.023837 h 1.5442 0.010035 0.0097696
161
Table B54. (Continued)
ZA ZB ZC
a/A 0.999 1.000 0.999 b/B 0.462 5.185 1.933 c/C 0.209 0.059 0.177 d/D 0.109 0.002 0.159 e/E 0.055 0.009 0.362 f/F 0.002 0.007 0.050 g/G 0.270 0.005 0.015 h/H 2.334 0.001 0.005
Table B55. SLGF, Case 28 and TSTEP = 0.018749
ZA ZB ZC
A 1.4041 1.2112 0.12449 B 2.6958 1.6499 0.55650 C 0.98760 6.9839 0.75764 D 0.30856 0.21839 0.90900 E 0.61253 0.45189 1.0687 F 0.78477 0.69439 1.3101 G 0.93085 0.85532 1.9814 H 1.1032 1.0562 4.9393 a 1.4076 1.4162 0.12396 b 37.721 2.9356 0.52353 c 0.43056 2.0738 0.67896 d 0.091756 0.14171 0.75949 e 0.0088464 0.25351 0.80572 f 0.017563 0.45624 0.83297 g 0.024111 0.61175 0.84738 h 0.021741 0.77666 0.84592 a/A 1.002 1.169 0.995 b/B 13.992 1.779 0.940 c/C 0.436 0.296 0.896 d/D 0.297 0.649 0.836 e/E 0.014 0.561 0.753 f/F 0.022 0.657 0.635 g/G 0.025 0.715 0.427 h/H 0.019 0.735 0.171
162
Table B56. SLGF, Case 29 the TSTEP = 0.022915
ZA ZB ZC
A 1.2404 6.7885 2.4053 B 1.6437 0.073474 3.3016 C 2.0424 0.90548 8.1362 D 2.5988 1.4000 2.2381 E 3.9061 1.7898 0.36863 F 71.901 2.2188 1.1008 G 0.77488 2.9041 1.5404 H 0.66146 5.0848 1.9282 a 1.2399 6.7945 2.4047 b 0.95006 0.10716 2.9685 c 0.62638 0.75684 4.4809 d 0.45600 1.2016 5.6560 e 0.36729 1.6101 0.43130 f 0.33145 2.1836 1.1141 S 0.40014 3.5379 1.388,3 h 0.28528 38.307 1.5533 a/A 0.999 1.001 0.999 b/B 0.578 1.459 0.899 c/C 0.306 0.836 0.551 d/D 0.175 0.858 2.527 e/E 0.094 0.904 1.171 f/F 0.004 0.984 1.012 g/G 0.516 1.218 0.901 h/H 0.431 7.540 0.805
Table B57. LLF, Case 48 and TSTEP = 0.018749
ZA ZB ZC
A 1.4041 0.85455 0.12449 B 2.6958 1.2112 0.75764 C 0.98760 1.2112 0.75764 D 0.30856 1.6499 0.90900 E 0.61253 6.9839 1.0687 F 0.78477 0.21839 1.3101 G 0.93085 0.45189 1.9814 H 1.1032 0.69439 4.9393 a 1.4076 0.85532 0.12396 b 15.937 0.92747 0.50010 c 0.45030 1.2238 0.65220 d 0.10611 20.639 0.75418 e 0.02883 0.59437 0.85809 f 0.0067227 0.18973 1.0126 g 0.0022012 0.077787 1.4046
163
Table B57. (Continued)
ZA ZB ZC
h 0.0045248 0.032524 48.293 a/A 1.002 1.001 0.995 b/B 5.911 0.920 0.898 c/C 0.456 1.010 0.860 d/D 0.343 12.516 0.829 e/E 0.047 0.085 0.802 f/F 0.008 0.870 0.772 g/G 0.002 0.172 0.708 h/H 0.004 0.046 9.855
Table B58. LLP, Case 49 and TSTEP = 0.022915
ZA ZB ZC
A 1.2404 6.7885 2.4053 B 1.6437 0.073474 3.3016 C 2.0424 0.90548 8.1362 D 2.5988 1.4000 2.2381 E 3.9061 1.7898 0.36863 F 71.901 2.2188 1.1008 G 0.77488 2.9041 1.5404 H 0.66146 5.0848 1.9282 a 1.2399 6.7945 2.4047 b 0.72227 0.36749 3.4702 c 0.36967 0.029631 15.956 d 0.21382 0.031888 1.4668 e 0.13319 0.047352 0.24458 f 0.083101 0.049717 0.82314 S 0.045319 0.046897 1.1800 h 0.0062819 0.041152 1.4939 a/A 0.999 1.000 0.999 b/B 0.439 5.001 1.051 c/C 0.181 0.032 1.961 d/D 0.082 0.022 0.655 e/E 0.034 0.026 0.663 f/F 0.001 0.022 0.747 g/G 0.058 0.016 0.766 h/H 0.009 0.008 0.774
164
Table B59. DLGF, Case 68 and TSTEP = 0.018749
ZA ZB ZC
A 1.4041 0.85455 0.12449 B 2.6958 1.0075 0.55650 C 0.98760 1.2112 0.75764 D 0.30856 1.6499 0.90900 E 0.61253 6.9839 1.0687 F 0.78477 0.21839 1.3101 G 0.93085 0.45189 1.9814 H 1.1032 0.69439 4.9393 a 1.4076 0.85532 0.12396 b 13.554 0.93393 0.50125 c 0.45406 1.2405 0.65311 d 0.10632 27.375 0.75506 e 0.028698 0.58951 0.85650 f 0.0064291 0.18938 1.0043 g 0.0018764 0.078018 1.3703 h 0.0042169 0.032880 47.232 a/A 1.002 1.000 0.995 b/B 5.038 0.926 0.900 c/C 0.459 1.024 0.862 d/D 0.344 16.195 0.830 e/E 0.046 0.084 0.801 f/F 0.008 0.867 0.766 g/G 0.002 0.172 0.691 h/H 0.003 0.047 9.553
Table B60. DLGF, Case 69 and TSTEP = 0.022915
ZA ZB ZC
A 1.2404 6.7885 2.4053 B 1.6437 0.073474 3.3016 C 2.0424 0.90548 8.1362 D 2.5988 1.4000 2.2381 E 3.9061 1.7898 0.36863 F 71.901 2.2188 1.1008 G 0.77488 2.9041 1.5404 H 0.66146 5.0848 1.9282 a 1.2399 6.7945 2.4047 b 0.72227 0.36749 3.4702 c 0.36967 0.029631 15.956 d 0.21382 0.031888 1.4668 e 0.13319 0.047352 0.24458 f 0.083101 0.049717 0.82314
165
Table B60. (Continued)
ZA ZB ZC
g 0.045319 0.046897 1.1800 h 0.0062819 0.041152 1.4939 a/A 0.999 1.000 0.999 b/B 0.439 5.001 1.051 c/C 0.181 . 0.032 1.961 d/D 0.082 0.022 0.655 e/E 0.034 0.026 0.663 f/F 0.001 0.022 0.747 g/G 0.058 0.016 0.766 h/H 0.009 0.008 0.774
166
XII. APPENDIX C; THE CSMP PROGRAM FOR THE TESTS OF
THE ALGORITHM
The program consists of two parts; the first part is the simula
tion of the fault same as in Appendix A, and the second part is the
program of the algorithm. The notations are the same as in Figures 12a
and 12b.
167
" •«CONTINUOUS SYSTEM MODELING PROGRAM»**»
»»» VERSION 1.3 »»»
TITLE TEST OF THE ALGORITHM » THE PROGRAM CONSISTS OF THE SIMULATION OF FAULTS AND THE ALGORITHM
» THE SIMULATION OF FAULTS INITIAL : PARAM 1*2" PARAM W=376.S9I PARA^^AII'Ef 1000 PARAM RC=0.0200168 PARAM RA1=0.1084026,RG1=0.8601764,VM=0.81649666 PARAM TSTEP=0.016666 PARAM T0C=0.625 PARAM ICA«0.400731, ICB=-0.33739o, ICC=-0.0615972, ICG=0.00216437 PARAM CA1*-68.23,CB1=-?2.61,CC1=159.91,CG1=-0.048 PARAM RA2^1.6,CCA=1.2»CC8=1.2,CCC=1.2 PARAM RG2:=0,L=0.1,R=0.02 PARAM Pl l^O,PI2=376.991,PI3=1.472626, . . . P21=ÔVP22^376.991,P23=3.567020,P31=0, . . . P32^376.991,P33=5.6614165 PARAM L11*0.456527,L12=0.143515, . . .
' L13%0.125845,L14=0.128492, . . . L22«0.456527,L23 = 0.143 51 5 , . . . L24»0.133182,L33=0.456527, . . . L34=d.128492,L44=0.378261 PARAM A1=1,A2=1,A3=1 A4=J. ,A5*r , m6=1, A7=l , A8=1»A9=1 PARAM 61=1,82=1,83=1, . . . 84=1,85=1,86=1,87=1,88=1,69=1
. PARAM Cl_f^C2=l .C3 = l , . . . "C4=r , 'C5*r ,C"6=l ,C7=l ,C8=l ,C9=r
PARAM CTlx0,CT2=0,CT3=0,CT4=0,RESULT=0 R82=RA2 ;
- - - - . - -: R8l=RAl I
RC1=RA1 : " c.A2v=w»w»rcA • • " "
• CB2=-W»W*ICB CC2»-W»W»ICC ___ _ _
• CG2*-W»W«:CG" ! PP13=P13+1.57079 : PP23=P23*1.57079 • PP33=P33+:1TS7079 " . - . _ . . _
CA7«1/(CCA*W) CB7«1/(CC8»W)
I CC7«l/tCCC»WJ ; » i ' DYNAMIC ; ; Y=STEP(TSTEP) ! YY=1-L0C*Y i
LAA«L11»YY/W ; LAB=L12»YY/W i LAC«L13»yY/W I LAG=L14»YY/W i i LB8=L22/W, : " • ; - t ; LBC»L23/W; : : j : ' i LBG=L24/W '
158
LCC=L33/W LCG=L34/W LGG=L44/W L8A=LAB LCB=L3C LCA=LAC LGA=LAG LGB=L8G LGC=LCG RA=RAl*YY+(1-YJ*RA2 RB=RB1*RB2 RC=RC1+PC2 CA=CA7*i l -YJ+RATE*y»CA7 CB=CB7 CC=CC7 RG=RG1+RG2*Y PA1=0.5*RA»S0RT(CA/LAA) PA2=l /S0RTtCA*LAA) P8i=U.S*RB*S0RT{C3/L68) P82=1/S0RT(CB*LB6) PC1 = 0. j *BC*SORTlCC/LCC J PC2=1/S0RT{CC»LCC) VA=VH«SINE(P11.P12,P13) V8=VM*SINE(P21TP22,P23) VC=VM»SINEtP31,P32.P33) 0VA=VM«P12*SINE( PU, P12.PP13J CVB=VM*P22»S!NF(P21,P22,oP23) 0VC=VM*P32*SINe(P31,P3 2.PP33J OI8=OEft IV(CBl , I5J DDIB=0ERIVICB2,0IB) OIC=OERIV<CC1.IC) 0CIC=ùSRIV(CC2.0IC) 0IA=DERIV(CA1,IA) DDI4=0ERIV(CA2.DIA) 0IG=D£RIV<CG1,IG) 0DIG=DERI\ / (CG2.DIG) XA=(1/LAA)»(0VA-RD»(0I8+0IC+0IG). . . -LAB*D0I8-LAC»D0IC-LAG*00IG) IA=CMPXPL(ICA,CA1,PA1,PA2,XA) XB^( l /LB8)* tOV8-RD»(CIA+OIC+0IG)-L8A*D0IA. . . -LBC»DDIC-L6G«DDIG) IB=CMPXPL(ICB,C8l ,P81,PB2,X3) XC = <1/LCC)»(0VC-RL>»J DIA«-OIB<-OIG)-LCA*DDIA-- . -LC8*0DI3-LCG»00IG} IC=CMPXPL<ICCtCCl,PCl ,PC2,XC) PG=LGG/RG XG=-Rû* l îA+IB«- i r . ) /RG-{LGA*OIA+LGB*OI B+L GC*DIC) /RG ZZ5=(XG-IG)/PG 1G=INTGRL( ICG.ZZ5) RVA=VA-tR*I A«-(L/WJ*OIA) RV8 = VB-(R*I3<- tL/W)»0IB) RVC=VC-(R*IC»(». /W)*OIC) NO SORT RIA=IA SIB=IB RIC=IC IF (ABSiRIA).GT.O.ûOûOdOlJ GO TO 30 IF IRIA» 10,20,20 10 R1A=-0.0000001 GO TO 30
169
20 RIA=0.0000001 30 IF(A8StRI3) .GT.0.0000001) GO TO 60 IF (RIB) 40,50,50 40 RIB=-0.0000001 GO TO 60 50 RIB=0.0000001 60 IF(ABS{RIC).GT.0.0000001) GO TO 90 IF! RIO 70,00,80 70 RIC=-0.0000001 GO TO 90 80 RIC=0.0000001 90 ZA=ABSIRVAJ/A3SIRIA) Z8=ABS(RV3)/A8S( i< i8) ZC=AB5lRVC)/A8SfRICJ * END OF THE SIMULATION *
« THE ALGORITHM 1=1 + 1
IF(AMOO(I ,4.0J.NE.O.O) GO TO 700 A1 = 2A B1 = ZP C1 = ZC RRA=A1/A9 PR3=B1/B9 RRC=C1/Cy WRITE(9,U TIME,A1,81,CI ,RRA,RR3,RRC 1 FOPMATJ«O*. 10E12.4) IFIRRA.GT.0.4) GO TO 250 CT2=CT2»1 IF(RPA.GT.O-1) GO TO 300 CT3=CT3+I IF(RRA.GT.0.05J GO TO 300 CT4=CT4+1 GO TO 300 250 IFICTI .EO.O) GO TO 310 300 CT1=CT1+1 310 IF(RR8.GT.0.4) GO TQ 320 CT2=CT2+1 IF(RR3.GT.O.IJ GO TO 330 CT3=CT3+1 IFIRRB.GT. 3.05) GO TO 330 CT4=CT4+1 GO TO 330 320 IFICTl .E3.0J GO TO 340 330 Cr i=CTl+l 340 IF<RRC.GT.0.4J GO TO 350 CT2=CT2*1 IFtRPC.GT.O. lJ GO TO 360 CT3=CT3»1 IF(PBC.GT.0.05) GO TO 360 CT4=r T4+1 GO TO 3bC 350 IF(CTl .EO.O) GO TO 600 Î60 CTl=CTl+l l t-(Cn .LT.12) GO TO 600 IF (CT/ .LT. j ) GO TO 500 IP (CT2-LT.6) GO TO- 400 IF (CT2.LT.1C) GO TO 370 IF (CT4.LT.3) GO TO 460 ." I I TO ' t ' lO
170
no IF [CT4.LT.2 j GO TO 4oO GO TO 430 400 IP 1CT3.LT.2» GO TO 460 430 RESULT=1 GO TO 500 460 RESULT=2 500 CT1=0 CT2=0 CT3 = 0 CT4=0 600 A9=A8 A8=A7 A7=A6 A6=A5 A5=A4 A4=A3 A3=A2 A2 = A1 WRITE(9,2J 0£LT,Al ,A2,A3,A4,A5,Ab,A7,Ad,A9 2 FORMATC • .10E12.4) 89=68 88=87 87=86 86=85 85=84 84=83 83=82 82=81 C9=C8 C8=C7 C7=C6 C6=C5 C5=C4 C4=C3 : - " • - -C3=C2 C2=C1 700 CONTINUE TERMINAL TIMER 0£LT=1.0416E-3,F INTIM=0.O25tPR0tL = 0.0010416,0UTDKL = 0.0010416 PRTPLT RESULTfRA.RGJ LABEL SLGF TEST fJ METHOD RKSFX END • STOP
171
XIII. APPENDIX D: PDP-11/45 AND THE ASSEMBLY PROGRAM OF THE
REVISED ALGORITHM
A. PDP-11/45
The PDP-11/45 is a 16-bit computer representing the large computer
end of the PDP-11 family of computers. It is designed for high-speed
real-time applications and multi-user, multi-task applications requiring
up to 124K words of addressable memory space. It operates with solid
states and core memories. Its major features are a fast central proces
sor with choices of 300 or 450 nanosecond memory, a Floating Point
Processor and a sophisticated memory management. All computer system
components and peripherals connect to communicate with each other on a
single high-speed bus which is called UNIBUS, Since all system-elements,
including the central processor, communicate with each other in identi
cal fashion via the UNIBUS, the processor has the same easy access to
peripherals as it has to memory.
I/O devices transferring directly to or from memory are given
highest priority, and may request bus mastership and steal bus cycles
during instruction operations. The processor resumes operation im
mediately after the memory transfer. Multiple devices can operate simul
taneously at maximum direct memory access (DMA) rates by stealing bus
cycles.
The central processor, connected to the UNIBUS as a subsystem,
controls the allocation of the UNIBUS for peripherals and performs 172
172
arithmetic and logic operations and instruction decoding. It contains
multiple high-speed general-purpose registers which can be used as ac
cumulators, pointers, index registers, or as autoindexing pointers in
autoincrement or autodecrement modes. The processor can perform data
transfers directly between I/O devices and memory without disturbing
the registers. In addition, it allows nested interrupts and automatic
reentrant subroutine calling by using its dynamic stacking technique.
All operations in PDP-11/45 are accomplished with one set of in
structions, unlike conventional 16-bit computers, which usually have
three classes of instructions such as memory reference instructions, AC
control instructions and I/O instructions. Since peripheral device
registers can be manipulated as flexibly as core memory by the central
processor, instructions that are used to manipulate data in core memory
may be used equally well for data in peripheral device registers. For
example, data in an external device register can be tested or modified
directly by the CPU without bringing it into memory or disturbing the
general registers.
Floating point instructions overlap CPU instructions and can continue
without CPU intervention, leaving the CPU free to execute other in
structions.
B. The Assembly Program of the Revised Algorithm
PAL-llS and Link-llS (17) are stand-alone programs which are com
patible subsets of the Disk Operating System (DOS) , PAL-llR and Link-11
system programs. Minimum hardware requirements are an 8K PDP-11 with a
173
teleprinter, A high-speed paper tape punch and/or a line printer may
be used also. PAL-llS enables you to write source programs using
letters, .numbers and symbols which are meaningful to you. The revised
algorithm is written in PAL-llS, and it is presented below.
*s L *B L *L L/2 *T T
ZAl 007000 ZA2 007004 ZA3 007010 ZA4 007014 ZA5 007020 ZA6 007024 ZA7 007030 ZA8 007034 ZA9 007040 ZBl 007044 ZB2 007050 ZB3 007054 ZB4 007060 ZB5 007064 ZB6 007070 ZB7 007074 ZB8 007100 ZB9 007104 ZCl 007110 ZC2 007114 ZC3 007120 ZC4 007124 ZC5 007130 ZC6 007134 ZC7 007140 ZC8 007144 ZC9 007150 I 007154 CTl 007156 CT2 007160 CT3 007162 CT4 007164 CT5 007166 CT6 007170 THl 007172 TH2 00717d4
TH3 007176 BC2 005210 A 005402 STl 005562 TB3 005702 TB7 006020 TBll 00 6120 R1 = 7. 000001 R5 = 7.000005
RESULT 007200 B 005222 ST0 005400 TBI 005624 TB4 005734 TB8 006032
R2 = 7.000002 SP = 7.00^6
FLOAT 007204 BBl 005334 BAl 005516 CI 005646 TB5 005754 TB9 006036 PC=7,000007 R3 = 7.000003 ALG. 005002.
END?
BCl 005154 BB2 005370 BA2 005552 TB2 005650 TB6 006012 TB10 006070 R0=7.000000 R4 = 7.000004 = 005002
; SUBPROGRAM OF THE ALGORITHM FOR THE
; COMPUTERIZED RELAY SCHEME
000000 R0 = 7.0 000001 R1 = 7.1 000002 R2 = 7.2 000003 R3 = 7.3 000004 R4 = 7.4 000005 R5 = 7.5 000006 SP = 7.6 000007 PC = 7.7 005002 = 5002
174
005002 012501 005004 011502 005006 014503 005010 012704
000001 005014 060437
007154 005020 020437
007154 005024 001566 0050 26 013703
000002 005032 020337
007154 005036 001476 005040 170001 005042 172437
007144 005046 174037
007150 005052 172437
007140 005056 174037
007144 005062 172437
007136 005066 174037
007140 005072 172437
007132 005076 174037
007136 005102 172437
007126 005106 174037
007132 005112 172437
007120 005116 174037
007124 005122 172437
007114 005126 174037
007120 005132 172437
007110 005136 174037
007114
ALG: MOV (R5); Rl; Get current MOV (R5), R2; Get voltage MOV -(R5), R3; Rest R5 MOV^l, R4 ; R4 4-1
ADD R4, I ; I = I+l
CMP R4, I ; (I:R4)
BEQ A ; Branch if I = 1 MOV # 2, R3 ; R3 2
CMP R3, I ; (I;R3)
BEQ B ; Branch if I = 2 SETF ; Set Floating Mode LDF ZC8, AC0 AC0 <^ZC8
STF AC0, ZC9 ZC9<r-AC0
LDF ZC7, AC0 AC0 <- ZC7
STF AC0, ZC8 ZC8 <- AC0
LDF ZC6, AC0 AC0 «-ZC6
STF AC0, ZC7 ZC7 <- AC0
LDF ZC5, AC0 AC0 4-ZC5
STF AÇ0, ZC6 ZC6 AC0
LDF ZC4, AC0 AC0 ZC4
STF AC0, ZC5 ZC5 "f- AC0
LDF ZC3, AC0 AC0 4-ZC3
STF AC0, ZC4 ZC4 <- AC0
LDF ZC2, AC0 ACO V ZC2
STF AC0, ZC3 ZC3 4-AC0
LDF ZCl, AC0 AC0 ZCl
STF AC0, ZC2 ZC2 4- AC0
175
005142 022701 CMP #^0, RI ; (IC:0) 000000
005146 001003 BNE BCl ; Branch if IC 0 005150 012703 MOV #1, R1 ; Rl<e 1
000001 005154 177001 BCl: LDCIF Rl, AC0 ; Integer to floating 005156 177102 LDCIF R2, AGI ; Integer to floating 005150 174401 DIVF AC0, AGI ; RVC/IC 005162 174137 STF AGI, ZGl ; ZGl«e-ACl
007110 MOV #0, R3 005166 012703 MOV #0, R3 ; R3<-0
000000 005172 020337 CMP R3, ZC9 ; (ZC9 ; R3)
007150 005176 001005 BNE BC2 ; Branch if ZC9 0
005200 012703 MOV#0.00Qa, R3 ; R3^/).(?(W1 174100
005204 010337 MOV R3, ZG9; ZG9<e-R3 007150
005210 172537 BC2: LDF ZCl, ZCl ; AG1<- ZCl 007110
005214 174537 DIVF ZC9, AGI : ZG1/ZG9 007150
005220 000470 BR ST0 005222 172437 B : LDF ZB8, AC0 ; AC0^ZB8
007100 005226 174037 STF AC0, ZB9 ; ZB9 <-AC0
007104 005232 172437 LDF ZB7, AC0 ; AC0 4-ZB7
007074 005236 174037 STF AC0, ZB8 ; ZBS-^ AC0
007074 005242 172437 LDF ZB6, AC0 ; AC0 — ZB6
007070 005246 174037 STF AC0, ZB7 ; ZB7 «- AC0
007074 005252 172437 LDF 2B5, AC0 ; AC0^ZB5
007064 005256 174037 STF AC0, ZB6 ; ZB6^AC0
007070 005262 172437 LDF ZB4,AG0 ; AC0<r-ZB4
007/860 005266 174037 STF AC0, ZB5 ; ZB5^AC0
007064 005272 172437 LDF ZB3, AC0 ; AC0 <-ZB3
007054 005276 174037 STF AC0, ZB4 ; ZB4 AC0
007060 005302 172437 LDF ZB2, AC0; AC0 <- ZB2
007050
176
005306 174037 007054
STF AC0, ZB3 ; ZB3 AC0
005312 172437 007044
LDF ZBl, AC0 ; AC0 ZBl
005316 174037 007050
STF AC0, ZB2 Î ZB2 <- AC0
005322 022701 000000
CMP #-0, R1 ; (IB ; 0)
005326 001003 BNE BBl ; Branch if IB 0 005330 012703
000001 MOV #1, R1 ; R1 1
005334 177001 BBl: LDCIF Rl, AC0 ; Integer to floating 005336 177102 LDCIF R2, AGI ; Integer to floating 005340 177401 DIVF AC0, ACl; RVB/IB 005342 174137
007044 STF AGI, ZBl ; ZBl^AGI
005346 012703 000000
MOV #0, R3 ;
005352 020337 007104
CMP R3, ZB9 ; (ZB9 : 0)
005356 001005 BNE BB2 ; Branch if ZB9 ÇS 005360 012703
174100 MOV ^0.ÇfÇfÇfl, R3 ; R3<-0.0001
005364 010337 007104
MOV R3, ZB9 ; ZB9 <-R3
005370 172537 007044
BB2: LDF ZBl, AGI ; AGI ZBl
005374 174537 007104
DIVF ZB9, AGI ; ZB1/ZB9
005400 000473 ST0: BR STl ; Branch 005402 170001 A: SETF ; Set Floating Mode 005404 172437
007034 LDF ZA8, AG0 ; AC0 ZA8
005410 174037 007040
STF AC0, ZA9 ; ZA9 AG0
005414 172437 007030
LDF ZA7, AG0; AG0 <- ZA7
005420 174037 007034
STF AG0, ZA8 ; ZA8 AG0
005424 172437 007024
LDF ZA6, AG0 ; AG0 ZA6
005430 174037 007030
STF AC0,. ZA7 ; ZA7 AC0
005434 172437 007020
LDF ZA5, AG0 ; AC0 .t- ZA5
005440 174037 007024
STF AG0, ZA6 ; ZA6w6r-AG0
005444 172437 007014
LDF ZA4, AC0 ; AG0 4r- ZA4
177
005450 174037 007020
STF AC0, ZA5 ; ZA5^AC0
005454 172437 007010
LDF ZA3, AC0 ; AC0<-ZA3
005460 174037 007014
STF AC0, ZA4 ; ZA4 -e-AC0
005464 172437 007004
LDF ZA2, AC0 ; AC0 ZA2
005470 174037 007010
STF AC0, ZA3 ; ZA3 AC0
005474 172437 007000
LDF ZAl, AC0 ; AC0-f-ZAl
005500 174037 007004
STF AC0, ZA2 ; ZA2 -é- AC0
005504
II
CMP #:0, RI ; (lA : 0)
005510 001003 BNE BAI ; Branch if lA ? 005512 012703
000001 MOV # 1, RI ; RI 1
005516 177001 BAI: LDCIF RI, AC0 ; Integer to floating 005520 177102 LDCIF R2, AGI; Integer to floating 005522 174401 DIUF AC0, ACl ; RVA/IA 005524 174127
007000 STF ACl, ZAl ; AClZAl
005530 012703 000000
MOV #0, R3 Î R'3<-0
005534 020337 007040
CMP R3, ZA9 ; (ZC9 : 0)
005540 001005 BNE BA2; Branch if ZA9 0 005542 012703
174100 MOV #0.0001, R3 ; R3^ 7.0001
005546 010337 007040
MOV R3, ZA9 ; ZA9-t-R3
005552 172537 007000
BA2; LDF ZAl, ACl ; ACl-é-ZAl
005556 174537 007040
DIVF ZA9, ACl ; RVA/IA
005562 174137 007204
STl: STF ACl , Float ; Float 4- ACl
005566 170637 007204
ABSF FLOAT ; {FLOAT |
005572 172537 007204
LDF ACl, FLOAT; ACl FLOAT
005576 060337 007170
ADD R4, CT6 ; CT6 = CT6 + 1
005602 012703 000454
MOV # 300, R3 ; R3 <-300
005606 020337 007170
CMP R3, CT6 ; (CT6 : 300 )
178
005612 001003 BNE TBI ; Branch if CT6 = 300 005614 012703 MOV #200, R3 ; R3<-200
000310 005620 010337 MOV R3, CT6 ; CT6^ R3
007170 005624 173537 TBI; CMPF ACl, THl ; (THl : ACl)
007172 005630 170000 CFCC ; Copy.floating.condition code 005632 001007 BLE TB2 ; Branch if ACl THl 005634 012705 MOV #0 , R5 ; R5<r0
000000 005640 020537 CMP R5,'CT1 ; (CTl :
007156 005644 001027 BNE TB3 ; Branch if CTl ^ 0 005546 000002 CI: RTI ; Return from interrupt 005650 012704 TB2; MOV #1, R4 ; R4<rl
000001 005654 060437 ADD R4, CT2 ; CT2 = CT2 + 1
007164 005660 173537 CMPF ACl, TH2 ; (TH2 ; ACl)
007174 005664 003010 BGT TB3 ; Branch if AC1> TH2 005666 060437 ADD R4, CT3 ; CT3 = CT34-1
007162 005672 173537 CMPF AC 2, TH3 ; (ACl : TH3)
007176 005674 003003 BGT TB3 ; Branch if ACl> TH3 005676 060437 ADD R4, CT4 ; CT4=CT4+1
007164 005702 060437 TB3: ADD R4, CTl ; CTl = CTl + 1
007158 005706 012703 MOV #12, R3 ; R3 12
000014 005712 020337 CMP R3, CTl ; (R3 : CTl)
007156 005716 001150 BNE CI : Branch if CTl ^ 12 005720 012703 MOV #3, R3 ; R3 3
000003 005724 020337 CMP R3, CT2 ; (R3 : CT2)
007160 005730 002012 BGE TB5 ; Branch if CT2 3 005732 010527 TB4: MOV R5, CTl ; CTl R5
007156 005736 010527 MOV R5, CT2 ; CT2<- R5
007160 005742 010527 MOV R5, CT3 J CT3 "f- R5
007162 005746 010527 MOV R5, CT4 ; CT4 <- R5
007164 005752 000002 RTI ; Clear interrupt
179
005754 012703 TB5: MOV# 6, R3 ; R3 6 000006
005760 020337 CMP R3, CT2 ; (R3 : CT2) 007160
005764 002415 BLT TB9 ; Branch if CT2 Z.6 005766 012703 MOV #10, R3 ; R3 <^10
000012 005772 020337 CMP R3, CT2 ; (R3 : CT2)
007160 005776 002411 BLT TB7 ; Branch if CT2 10 006000 012703 MOV #3, R3 ; R3^3
000003 006004 020337 CMP R3, CT4 ; (R3 : CT4)
007164 006010 002411 BLT TB8 ; Branch if CT4 < 3 006012 010437 TB6: MOV R4, RESULT ; RESULT<r 1
007200 006016 000543 BR TB4 ; Branch 006020 012703 TB7: MOV 4^2, R3 ; R3 <-2
000002 MOV 4^2, R3 ; R3 <-2
006024 020337 CMP R3, CT4 ; CR'3 ; CT4 ) 007176
006030 002071 B§E TB6 ; Branch if CT4 > 2 006032 010337 TB8: MOV R3, RESULT ; RESULT 2
007212 008036 012703 TB9; MOV 2, R3 ; R3 2
000002 006042 020337 CMP R3, CT3 ; (R3 ; CT3)
007162 006046 002562 BGE TB6 ; Branch if CT3 i 2 006050 060437 ADD R4, CT5 ; CT5 = CT5fl
007170 006054 020437 CMP R4, CT5; (R4;CT5)
007170 006060 001005 BNE TB10 ; Branch if CT5 1 006062 010537 MOV R5, CT6; CT6<-0
007174 006066 000534 BR TB4 ; Branch 006070 012703 TB10: MOV # 176, R3 ; R3«- l76
00022f 006074 020337 CMP R3, CT6 ; (R3; CT6)
007170 006100 000526 BLT TB4 ; Branch 006102 012703 MOV # 228, R3 Î R3«- 228 ,
00032 2 006106 024337 CMP R3, Cr6; (R3 : CT6)
007170
180
#61# #3(#5 BGT TBll ; Branch if CT6 228 006112 012703 MOV #2, R3 ; R3^ 2
000002 #2, R3 ; R3^ 2
006116 010337 MOV R3, RESULT ; RESULT^- 2 007200
006120 010537 TBll: MOV R5, CT5 ; CTS^ 0 007166
R5, CT5 ; CTS^ 0
006124 000515 BR TB4 ; Branch