APPENDIX A: 10 Machine System Data

The single line diagram, bus data, line, generator and excitation system
data are given in the following pages. They are taken from Athay et a1 [ 19791.
TABLE A.1
BUS DATA
Bus Type Volts Load Load Gen Gen
(O=P-Q) MW MVAR Load MVAR
(l=P-V)
1 0 0.0 0.0 0.0 0.0
2 0 0.0 0.0 0.0 0.0
3 0 322.0 2.40 0.0 0.0
4 0 500.0 184.0 0.0 0.0
5 0 0.0 0.0 0.0 0.0
6 0 0.0 0.0 0.0 0.0
7 0 233.8 84.0 0.0 0.0
8 0 522.0 176.0 0.0 0.0
9 0 0.0 0.0 0.0 0.0
10 0 0.0 0.0 0.0 0.0
11 0 0.0 0.0 0.0 0.0
12 0 7.5 88.0 0.0 0.0
13 0 0.0 0.0 0.0 0.0
14 0 0.0 0.0 0.0 0.0
15 0 320.0 153.0 0.0 0.0
16 0 329.0 32.3 0.0 0.0
17 0 0.0 0.0 0.0 0.0
18 0 158.0 30.0 0.0 0.0
19 0 0.0 0.0 0.0 0.0
20 0 628.0 103.0 0.0 0.0
21 0 274.0 115.0 0.0 0.0
22 0 0.0 0.0 0.0 0.0
23 0 247.5 84.60 0.0 0.0
24 0 308.6 -92.20 0.0 0.0
25 0 224.0 47.20 0.0 1.0
26 0 139.0 17.0 0.0 0.0
27 0 281.0 75.5 0.0 0.0
28 0 206.0 27.6 0.0 0.0
29 0 283.5 26.9 0.0 0.0
30 1 1. 0475 0.0 0.0 250.0
31 1 .982 9.2 4.60
32 1 .9831 0.0 0.0 650.0
33 1 .9972 0.0 0.0 632.0
34 1 1. 0123 0.0 0.0 508.0
35 1 1. 0493 0.0 0.0 650.0
36 1 1.0635 0.0 0.0 560.0
37 1 1.0278 0.0 0.0 540.0
38 1 1.0265 0.0 0.0 830.0
39 1 1.03 1104.0 250.0 1000.0
 N
N
.".

@ 26 I I 28 I 29
~t 25 I 1
+30 I ] +-38
2 i Q0
18 27 24
1 17
I I I
~--.....,

-I---_---,r--1_6 ~ I
3 (Ei) I

"i) 15 35~ I

+ ~ 21 22-
39
---,_4+- I 14 I 1
5
6 12 19 23

7 - 4 - --r---3-1 T"""" TTL 13 l i 20*1. i36

---1-..;;....8.J....
1 tl2
L--=r 11
10
-L
~ 34 33
0) t'l
Z
t'l

9 ~
><
'oz:I
c::
Z
C'l
o ....;
H
o
Z
Fig. A.1 10 unit 39 bus New England test system.
!l;
~
><
(Il
H
(Il
APPENDIX A 225

TABLE A.2
LINE DATA FOR 10 UNIT SYSTEM
Line Data Transformer Tap
Bus Bus Resistance Reactance Susceptance Magnitude Angle
1 2 .0035 .0411 .6987 .000 .00
1 39 .0010 .0250 .7500 .000 .00
2 3 .0013 .0151 .2572 .000 .00
2 25 .0070 .0086 .1460 .000 .00
3 4 .0013 .0213 .2214 .000 .00
3 18 .0011 .0133 .2138 .000 .00
4 5 .0008 .0128 .1342 .000 .00
4 14 .0008 .0129 .1382 .000 .00
5 6 .0002 .0026 .0434 .000 .00
5 8 .0008 .0112 .1476 .000 .00
6 7 .0006 .0092 .1130 .000 .00
6 11 .0007 .0082 .1389 .000 .00
7 8 .0004 .0046 .0780 .000 .00
8 9 .0023 .0363 .3804 .000 .00
9 39 .0010 .0250 1.2000 .000 .00
10 11 .0004 .0043 .0729 .000 .00
10 13 .0004 .0043 .0729 .000 .00
13 14 .0009 .0101 .1723 .000 .00
14 15 .0018 .0217 .3660 .000 .00
15 16 .0009 .0094 .1710 .000 .00
16 17 .0007 .0089 .1342 .000 .00
16 19 .0016 .0195 .3040 .000 .00
16 21 .0008 .0135 .2548 .000 .00
16 24 .0003 .0059 .0680 .000 .00
17 18 .0007 .0082 .1319 .000 .00
17 27 .0013 .0173 .3216 .000 .00
21 22 .0008 .0140 .2565 .000 .00
22 23 .0006 .0096 .1846 .000 .00
23 24 .0022 .0350 .3610 .000 .00
25 26 .0032 .0323 .5130 .000 .00
26 27 .0014 .0147 .2396 .000 .00
26 28 .0043 .0474 .7802 .000 .00
26 29 .0057 .0625 1. 0290 .000 .00
28 29 .0014 .0151 .2490 .000 .00
12 11 .0016 .0435 .0000 1.006 .00
12 13 .0016 .0435 .0000 1.006 .00
6 31 .0000 .0250 .0000 1.070 .00
10 32 .0000 .0200 .0000 1.070 .00
19 33 .0007 .0142 .0000 1.070 .00
20 34 .0009 .0180 .0000 1.009 .00
22 35 .0000 .0143 .0000 1. 025 .00
23 36 .0005 .0272 .0000 1. 000 .00
25 37 .0006 .0232 .0000 1.025 .00
2 30 .0000 .0181 .0000 1.025 .00
29 38 .0008 .0156 .0000 1.025 .00
19 20 .0007 .0138 .0000 1.060 .00
226 ENERGY FUNCTION ANALYSIS

TABLE A.3

DETAILED MODEL UNIT DATA 10 UNIT SYSTEM

Unit No. H R x' x' xd x T' T' xR.

a d q q do qo
(sec)

1 500.0 0 .006 .008 .02 .019 7.0 .7 .003

2 30.3 0 .0697 .170 .295 .282 6.56 1.5 .035
3 35.8 0 .0531 .0876 .2495 .237 5.7 1.5 .0304
4 28.6 0 .0436 .166 .262 .258 5.69 1.5 .0295
5 26.0 0 .132 .166 .67 .62 5.4 .44 .054
6 34.8 0 .05 .0814 .254 .241 7.3 .4 .0224
7 26.4 0 .049 .186 .295 .292 5.66 1.5 .0322
8 24.3 0 .057 .0911 .290 .280 6.7 .41 .028
9 34.5 0 .057 .0587 .2106 .205 4.79 1. 96 .0298
10 42.0 0 .031 .008 .1 .069 10.2 .0 .0125

TABLE A.4

DETAILED MODEL UNIT EXCITATION SYSTEM DATA 10 UNIT SYSTEM

Unit No. KA TA VRMIN VRMAX I): TE KF TF C1 C2

1 0 0 0 0 0 0 0 0 0 0
2 6.2 .05 -1. 0 1.0 -.633 .405 .057 .5 .66 .88
3 5.0 .06 -1. 0 1.0 -.0198 .5 .08 1.0 .13 .34
4 5.0 .06 -1. 0 1.0 -.0525 .5 .08 1.0 .08 .314
5 40.0 .02 -10.0 10.0 1.0 .785 .03 1.0 .07 .91
6 5.0 .02 -1. 0 1.0 -.0419 .471 .0754 1.246 .064 .251
7 40.0 .02 -6.5 6.5 1.0 .73 .03 1.0 .53 .74
8 5.0 .02 -1.0 1.0 -.047 .528 .0854 1. 26 .072 .282
9 40.0 .02 -10.5 10.5 1.0 1.4 .03 1.0 .62 .85
10 5.0 .06 -1. 0 1.0 -.0485 .25 .04 1 .08 .26

NOTE: A & B in Figure A.2 are computed as follows:

V
_ RMAX
EX 2 EX 1 = .75 EX 2
- ~+C2

B = R.n(C2/Cl)/ (EX 2 - EX 1 ) , A= C2/e B· EX2

Unit #1 has constant excitation.
APPENDIX A 227

Figure A.2 - IEEE Type 1 rotating excitation system model

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INDEX

A
Structure preserving, 35
Acceleration area, 71 Time-scale, 74
Acceleration method, 147 Quasi-analytical, 123
Advanced machines, 196 Energy integral, 44
Artificial intelligence, 221 Equal area criterion, 14
Extended equal area criterion, 70,206
B Excitation system, 127
Exit point, 154
Bus injected power, 128
F
C
Fast energy function, 79
Center of inertia, 27 First integral of motion, 29
Center of speed, 53 Flux decay model, 88,96
Closest u.e.p, 151 Frequency dependent loads, 41
Classical machine Frequency equilibrium, 192
representation, 45
Contingency analysis, 208 G
Controlling u.e.p, 8,196
Corner point, 196 Generalized potential energy, 119
Conservative system, 27 Global center of angle, 74
Critical cluster, 70 Gradient system, 184
Critical clearing time, 7
Critical energy, 117,152 H
Critical machine, 196
Cutset stability criterion, 60 Hessian, 142,176
Hyperplane, 147,159
D Hyperbolic equilibrium point, 146
Hypersurface, 159
Deceleration area, 71
Decomposition-aggregation, 73 I
Distribution factor, 216
Dynamic security assessment, 2,206 Internal nodes, 22
Internodal angles, 59,139
E
J
Eigenvalues, 166
Eigenvector, 168 Jacobian matrix, 150
Energy function Jacobian, load flow, 216
Cutset, 58
Individual machine, 31 K
Group, 51
Reduced order, 49 Kinetic energy, 16
Single machine infinite bus, 15
240 INDEX

L Region, 146
Stable equilibrium point, 54,192
Loads Stable manifold, 147
Constant impedance, 22 Step-by-step simulation, 100
Frequency dependent, 41
Voltage dependent, 133 T
Local center of angle, 76 Tangent hyperplane, 155
Lyapunov function, 29,140 Transient energy margin, 215
Transient stability index, 2
M Transient kinetic energy, 46
Transient reactance, 22,49
Machine model, 114 Transfer conductance, 27,106
Mode of instability, 195,201 Transversality condition, 147
Trapezoidal rule, 183
N
U
Normalized energy margin, 198
Numerical Lyapunov function, 117 Unstable equilibrium point, 8

p V

Parallel processing, 221 Vulnerable cutset, 62

Parameter variations method of, 92
Path dependent integral, 30
Potential energy, 40
Potential energy boundary
surface, 83,172,178
Potential energy margin normalized
Potential energy "well", 174
Pseudo-corner point, 196

Quasi-unstable equilibrium point, 72

Region of attraction, 7
Relevant u.e.p, 8
Rotor angle, 26

Saddle-type e.p, 174

Sensitivity, 209,210,217
Separatrix, 167
Slow energy function, 79
Slow coherency, 75
Stability
First swing, 21
Boundary, 146