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AppendicesAppendix ATransmission Lines
Power system’s main power corridor: Transmission lines and transformers with atwo-port network.
Referring to Fig. A.1a approximated two port transmission network and(b) phasor diagram:
V receiving end phase voltageE sending end phase voltageP single-phase real powerQ single-phase reactive power
BCj j ¼ XIcosu ¼ E sind
Hence Icosu ¼ EX sind
ACj j ¼ XIsinu ¼ Ecosd� V
Hence Isinu ¼ EX cosd� V
XReal power, P ¼ VIcosu ¼ EV
X sind
) P dð Þ ¼ EVX
sind power-angle characteristics
d is known as the load angle or power angle
Since the real power P depends on the product of phase voltages and the sine ofthe angle d between their phasors. In power networks, node voltages must be withina small percentage of their nominal values. Hence such small variations cannotinfluence the value of real power. Large changes of real power, from negative topositive values, correspond to changes in the sin d. The system can operate only inthat part of the characteristic which is shown by a solid line in Fig. A.1c. The angle
© Springer Nature Singapore Pte Ltd. 2018T. Bambaravanage et al., Modeling, Simulation, and Control of a Medium-ScalePower System, Power Systems, https://doi.org/10.1007/978-981-10-4910-1
159
d is strongly connected with system frequency f; hence the pair ‘P and f’ is alsostrongly interrelated.
Reactive power,
Q ¼ EVX
cos d� V2
X
cos d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� sin2 d
p
Q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEVX
� �2
�P2
s
� V2
X
Due to stability considerations, the system can operate only in that part of thecharacteristic which is shown by a solid line. The smaller the reactance X, thesteeper the parabola; even for small changes in V, cause large changes in reactive
(a)
(b)
(c)
Fig. A.1 a Two-port p equivalent circuit corresponding to an approximated transmission line.b Corresponding phasor diagram. c Real power and reactive power characteristics
160 Appendix A: Transmission Lines
power. Obviously the inverse relationship also takes place: a change in reactivepower causes a change in voltage.
Hence the three factors that can affect the stability of PS can be identified as:
• Load angle, d• Frequency, f• Nodal voltage magnitude, V
Appendix A: Transmission Lines 161
Appendix BComposite Loads
Usually each composite load represents a relatively large fragment of the systemtypically comprising
• low- and medium-voltage distribution networks,• small power sources operating at distribution levels,• reactive power compensators,• distribution voltage regulators,• a large number of different component loads such as motors, lighting and
electrical appliances [6].
In the steady state the demand of the composite load depends on the bus-barvoltage V and the system frequency f. The functions describing the dependence ofthe active and reactive load demand on the voltage and frequency P(V, f) and Q(V,f) are called the static load characteristics.
The characteristics P(V) and Q(V), taken at constant frequency, are called thevoltage characteristics while the characteristics P(f) and Q(f), taken at constantvoltage, are called the frequency characteristics. The slope of the voltage or fre-quency characteristic is referred to as the voltage (or frequency) sensitivity of theload. Figure B.1, illustrates this concept with respect to voltage sensitivities.
Voltage sensitivities kPV and kQV and the frequency sensitivities kPF and kQF areusually expressed in per units with respect to a given operating point:
kPV ¼ DP=P0
DV=V0
kQV ¼ DQ=Q0
DV=V0
kPf ¼ DP=P0
Df =f0
© Springer Nature Singapore Pte Ltd. 2018T. Bambaravanage et al., Modeling, Simulation, and Control of a Medium-ScalePower System, Power Systems, https://doi.org/10.1007/978-981-10-4910-1
163
kQf ¼ DQ=QP0
Df =f0
where,P0, Q0, V0, f0, DP and DQ, are: real power, reactive power, voltage, frequency,
real power change, and reactive power change at a given operating point.A load is considered to be stiff, if at a given operating point, its voltage sensi-
tivities are small.If,
• kPV ’ 0kQV ’ 0, the load is considered to be ideally stiff. The power demand of thatload does not depend on the voltage.
• A load is voltage sensitive if kPV and kQV are high• For a small DV change cause high change in the demand, DP .• Usually kPV\kQV
Fig. B.1 Illustration of thedefinition of voltagesensitivity
164 Appendix B: Composite Loads
Appendix CGeneration Characteristic
In the steady state, the idealized power–speed characteristic of an ith generating unitcan be written as:
Dxxn¼ �qDPm
Pn
DPm
Pn¼ �K Dx
xn
Dffn¼ �qi
DPmi
Pni
DPmi
Pni¼ �Ki
Dffn
In the steady state, all the generating units operate synchronously at the samefrequency. When,
Dx fraction of rated speedxn rated speedx turbine speedDf fraction of frequencyf system frequencyDPT the overall change in the total power generatedNG no. of generator unitsPm turbine powerPn nominal power output;
DPT ¼XNG
i¼1DPmi
�KiDffn¼ DPmi
Pni
© Springer Nature Singapore Pte Ltd. 2018T. Bambaravanage et al., Modeling, Simulation, and Control of a Medium-ScalePower System, Power Systems, https://doi.org/10.1007/978-981-10-4910-1
165
)DPT ¼ �Dffn
XNG
i¼1KiPni
) DPT ¼ �DfXNG
i¼1
KiPnifn
� �ðC:1Þ
Figure C.1, illustrates how the characteristics of individual generating units canbe added according to Eq. (C.1) to obtain the equivalent generation characteristic.This characteristic defines the ability of the system to compensate for a powerimbalance at a situation of a system frequency deviation from its rated value. For apower system with a large number of generating units, the generation characteristicis almost horizontal such that even a relatively large power change only results in avery small frequency deviation. This is one of the benefits due to combininggenerating units into one large system.
To obtain the equivalent generation characteristic of Fig. C.2, it has beenassumed that the speed–droop characteristics of the individual turbine-generatorunits are linear over the full range of power and frequency variations. In practice theoutput power of each turbine is limited by its technical parameters. The speed–droop characteristics of a turbine with an upper limit is shown in Fig. C.2.
If a turbine is operating at its upper power limit then a decrease in the systemfrequency will not produce a corresponding increase in its power output. At thelimit q = ∞ or K = 0 and the turbine does not contribute to the equivalent systemcharacteristic. Consequently the generation characteristic of the system will bedependent on the number of units operating away from their limit at part load; thatis, it will depend on the spinning reserve, where the spinning reserve is thedifference between the sum of the power ratings of all the operating units andtheir actual load.
Fig. C.1 Generation characteristic as the sum of speed–droop characteristics of all the generationunits
166 Appendix C: Generation Characteristic
The allocation of spinning reserve is an important factor in power systemoperation as it determines the shape of the generation characteristic. This isdemonstrated in Fig. C.3, with two generating units.
In Fig. C.3a, the spinning reserve is allocated proportionally to both units (whichoperate at a frequency off0) and the maximum power of both generators is reachedat the same operating frequencyf1. The sum of both characteristics is then a straightline (as given in Eq. (C.1)), up to the maximum powerPMAX ¼ PMAX1þ PMAX2.
Figure C.3b shows a situation where the total system reserve is the same (equalto the amount of the previous case), but it is allocated solely to the second gen-erator. That generator is loaded up to its maximum at the operating point (fre-quencyf2). The resulting total generation characteristic is nonlinear and consists oftwo lines of different slopes. The first line is formed by adding both inverse droops,KT1 6¼ 0 and KT2 6¼ 0, in Eq. (C.3). The second line is formed noting that the firstgenerator operates at maximum load and KT1= 0, so that only KT2 6¼0 appears in thesum in Eq. (C.3). Hence the slope of that characteristic is higher (Fig. C.3).
Generally, the number of units operating in a real system is large. Some of themare loaded to the maximum but others are partly loaded, generally in a non-uniformway, to maintain a spinning reserve. Adding up all the individual characteristicswould give a nonlinear resulting characteristic consisting of short segments withincreasingly steeper slopes. That characteristic can be approximated by a curve asshown in Fig. C.4. The higher the system load, the higher the droop until itbecomes infinite qT = ∞, and its inverse KT = 0, when the maximum power PMAX
is reached. If the dependence of a power station’s auxiliary requirements on fre-quency were neglected, that part of the characteristic would be vertical (shown as adashed line in Fig. C.4).
The total system power generation is equal to the total system load (PL),including transmission losses.
XNG
i¼1Pmi ¼ PL
Fig. C.2 Speed–droop char-acteristic of a turbine with anupper limit
Appendix C: Generation Characteristic 167
Equation (C.1)/PL gives:
DPT
PL¼ �KT
Dffn
orDffn¼ �qT
DPT
PLðC:2Þ
(a)
(b)
Fig. C.3 Influence of the turbine upper power limit and the spinning reserve allocation on thegeneration characteristic
Fig. C.4 Static system gen-eration characteristic
168 Appendix C: Generation Characteristic
where,
KT ¼PNG
i¼1 KiPnið ÞPL
ðC:3Þ
qT ¼1KT
Equation (C.2) describes the linear approximation of the generation character-istic calculated for a given total system demand. Further, the coefficients inEq. (C.3) are calculated with respect to the total demand, not the sum of the powerratings, so that qT is the local speed-droop, of the generation characteristic anddepends on the spinning reserve and its allocation in the system as demonstrated inFig. C.4.
Appendix C: Generation Characteristic 169
Appendix D
Generation and transmission network of Sri Lanka as at 2011.
1130
PO
LPI-
1
132.
9
1170
SA
MA
N-1
134.
6
1210
BO
WA
T-1
131.
4
1770
KIR
IB-1
2220
KO
TMA
-2
224.
8
2230
VIC
TO-2
226.
8
2240
RA
ND
E-2
227.
3
2250
RA
NTE
-2
227.
3
32.9
32.8
3620
BA
DU
L-3
33.4
12.6
4252
RA
NTE
-G2
1790
RA
TMA
-1
130.
8
1300
KE
LAN
-1
130.
9
130.
8
4760
CO
L_F
-11
11.0
4750
CO
L_E
-11
129.
7
3500
KO
SG
A-3
33.3
33.3
3670
MA
TAR
A-3
133.
3
33.1
133.
4
3520
NU
WA
R-3
33.3
1520
NU
WA
R-1
132.
3
1620
BA
DU
L-1
131.
8
3540
OR
UW
A-3
32.7
1670
MA
TAR
A-1
132.
6
130.
5
1660
EM
BIL
-1
1100
LAX
-1
133.
3
130.
4
3200
UK
UW
E-3
3530
THU
LH-3
33.2
33.1
2300
KE
LAN
-2
217.
4
1540
OR
UW
A-1
130.
6
1530
THU
LH-1
130.
2
133.
7
217.
1
33.0
3860
MA
DA
M-3
133.
8
221.
2
1705
NE
WA
NU
-1
133.
2
1700
AN
UR
A-1
1850
PA
NA
D-1
127.
8
1800
MA
TUG
-1
129.
8
130.
4
1240
VA
VU
N-1
3240
VA
VU
N-3
3
33.3
33.1
32.8
134.
5
3400
HA
MB
A-3
3
32.5
3660
EM
BIL
-3
33.1
32.8
33.1
1160
ING
IN-1
127.
2
3160
ING
IN-3
125.
7
33.3
3700
AN
UR
A-3
A
217.
4
33.1
33.3
1710
TRIN
C-1
33.1
130.
6
32.9
33.0
134.
4
33.1
3690
HA
BA
R-3
33.1
1740
RA
TNA
P-1
133.
533
.0
1595
KH
D -
1
132.
4
132.
0
1600
BO
LAW
-1
131.
633
.4
132.
8
33.3
1840
JPU
RA
_1
130.
513
0.4
129.
9
129.
8
11.0
4435
CO
L_A
_11
3890
DE
HIW
_3
33.2
3800
MA
TUG
-3
33.2
134.
6
33.3
12.
0
0.8
12.
0
0.8
29.4
5.1 29.4
5.1
28.9 3.6
28.9 3.6
11.2
8.0
11.2
8.0
29.7
2.7
29.4
4.5
4.1 4.5
4.1
10.4
1.5
49.5
3.0
1.0
2.8
36.2
19.0
12.3
39.0
25.1
15.0
10.5
30.0
13.6
1.6
7.2
1.6
7.2
7.5
2.7 7.5
2.7
18.0
29.7
29.4
1.8
5.9
8.0
21.1
19.1 3.9
19.1 3.9
19.4
5.1
19.
2
7.3
28.8
11.2 28.8
17.0
72.8
38.4
47.
0
24.
347
.0
24.3
60.5
35.0
60.
5
5.9
8.0
12.0
11.0
14.0
4.3
0.6
10.8 0.6
10.8
0.0
0.0
0.0
0.0
81.9
77.2
81.2
85.7
81.9
77.2
81.2
85.7
7.1
1.5
7.1
1.5
17.3
0.3
17.
2
2.0
59.
1
44.
7
59.
1
44.
7
19.9
14.8
14.8
2.0
0.3
1.8
24.
8
20.
7 2
4.8
20.
7
49.5
41.3
11.0
3.6
3.3
1.0
0.0
0.0
0.0
5.2
0.0
0.0
6.9
14.
16.
9
14.
1
0.0
0.5
0.0 0.0
81.4
79.9
10.
911
.0
11.6
3.0
5.2
9.0
1.7
3.0
5.2
24.0
16.1
25.2
13.340.0
125.4
51.7
125.9
125.4
51.7
125.9
54.1
32.2
4.3
21.
6
21.
6
9.8
18.8
12.0
18.7
13.0
0.1
0.9
0.1
0.9
0.2
6.1
89.
0
63.
8
89.3
89.
0
63.
8
89.3
63.3
63.0
0.0
0.0
46.
0
42.
1
26.0
12.0
25.
7
12.
926
.0
12.0
51.0
35.6
12.7
129.
6
47.8
129
.2
46.
312
9.6
47.8
129
.2
46.
3
97.6
23.7
97.6
23.7
21.1
28.
0
12.
0 2
8.0
12.
0
6.5
22.8
16.4
28.
6
49.3
34.5
28.6
8.3
28.
6
0.0
10.1
0.0
78.0
36.7
117.
1
58.2
25.9
26.0
13.9 1
2.0
1.1
9.2
9.3
1.6
18.4
6.6
33.9
7.4
33.0
28.0
6.0
18.
6
5.6
18.7
3.6
37.0
14.5
18.5
7.3
18.5
7.3
1.0000
49.5
2.4
30.3
30.2
32.4
5.8
17.3
0.6
17.3
0.6
3.3
8.6
5.3 8.6
5.3
47.2
35.2
47.2
37.2
47.2
35.2
47.2
37.2 97.5
5.5
5.5
2.5
3.7
4.5
11.
7
1 27.1
12.4R
32.3
17.0H
1
40.0
2.5R
1
2.0
1
2.7
1 38.0
13.0
1
1
15.0
2.9R
1
38.0
17.4
1
46.0
36.0
1 15.0
10.0
1 39.0
22.0
1 15.0
5.0
1 18.0
7.0
1
1
51.0
1
11.0
4.4
1
14.0
1
0.0
1
30.0
12.0
2.0
0.3
1
49.5
37.1
1
1 0.0
0.0
1 36.2
19.0
1 45.8
1 11.0
3.2
SW
0.0
1 3.3
1
0.0
1
18.4
6.1
27.1
12.4R
2 32.0
2
40.0
2.5R
28.5
3.0
L
2
22.5
2.0
L
1
25.0
0.0L
1
11.0
48.0
20.0
H
2
1
20.0
10.0
H
2
1
47.0
30.0
SW
9.9
1 24.0
14.9
1
29.0
13.0
1
0.2
6.0
SW
1 63.0
1 48.61 0.0
0.01
51.0
31.6
1
63.3
39.2
1
1 21.1
13.1
1
11.7
5.7
1
60.5
35.0
H
1
25.0
H
1
16.5
7.3
1
28.6
15.2
1
39.0
17.8
1
49.3
19.9
1
26.0
12.6
2
12.0
H
1
1
26.0
12.0
H
1
33.9
21.3
SW
1 33.0
15.0
1
30.0
H
1 37.0
1
1
5.8
1
32.2
20.0
1
8.0
1
80.3
SW
5.1
0.0
33.0
4.0
33.0
6.8
1.0
0.0
3121
WIM
AL-
3B
1.5
8.6
3780
VA
LAC
H_3
32.9
132.
0
1780
VA
LAC
H_1
6.5
15.
2
1.0
11.0
131.
4
54.1
41.
7
15.1
3560
PA
NN
I-3
26.0
1590
SA
PU
GA
-1
13.4
3830
VE
YA
N-3
3
1250
RA
NTE
-1
40.
3
105
.8
40.
3
12.3
0.8
3.9
2.6
20.6
0.8
1.8 2.4
2.2
29.0
1
5.1
70.4
35.8
26.
3
70.4
24.
7
16.
5
3600
BO
LAW
-3
1
1
1
SW
10.1
5.3
28.9 28.2
6.6
12.
0
1.1
11.7 11
.6R
22.2
R
33.4
48.6
13.7
13.7
1
7.6
1
1
1
1
1
28.0
10.
2
3.7
1
128.
4
129.
6
3880
AM
BA
LA
32.8
19.8
11.2
9.9
5.6
9.9
1
19.8
10.4
1 33.8
9.5
10.1
1900
PN
NA
LA
13.9
24.4
7.9
9.4
19.
2
1910
AN
IYA
3910
AN
IYA
33.8
18.
0
30.
0
1
30.0
18.0
133.
0
32.8
1 40.0
24.8
SW 0.0
SW
0.0
38.0
6.0
15.0
6.8
2.1
0.0
9.0
2.7
5.4
3710
TRIN
C-3
SW
16.0
H90
.0
50.0
H
135.
0
42.5
135.
0
42.5
9.8
3630
BA
LAN
-3
0.0
33.2
57.5
14.2
2.0
19.4
3565
PA
NN
I-C
23.0
10.
1
43.2
39.
2
33.0
132.
7
5.8
1.6
1
5.8
1.5
33.5
3.1
3.1
1.3
42.3
133.
5
49.5
23.7
49.5
23.7
3440
KA
TUN
A-3
33.3
1
19.8
21.8
SW
0.0
1
11.0
0.0
90.0
50.0
H
3811
CE
ME
NT
33.1 11
.0
130.
3
16.1
11.4
11.4
122
.1
49.
8
122
.1
49.
8
123.
5
39.8
123.
5
39.8
7.7
1
1
15.3
2580
KO
TUG
-2
19.5
8.6
SW
0.0
1
218.
3
33.3
12.3
214.
9
33.2
3.9
14.8
1
4.0
12.0
7.4
1920
SU
B-C
130.
9
4920
SU
B C
-11
11.1
1
10.9
1760
CO
L_F
-1
24.2
33.3
0.0
6.2
0.0
0.024.2
14.5
11.0
1500
KO
SG
A-1
17.0
97.5
28.2
17.4
27.8
18.2
11.3
5.5
45.2
2.3
44.
1
2.4
2.3
20.1
1
275.
0
1.0000
247
.0
79.
6
4811
PU
TT C
OA
L-2
20.1
1
0.0
0.0
63.3
1
28.0
25.0
132.
6R
15.
2
19.7
13.0
1.6
26.
4
180
.0
46.
2
38.
7
90.
0
30.2
5.1
33.2
1
25.4
3.6
0.4
1
1
99.0
90.0
H
1
54.0
35.0
H
1
99.
0
76.
8
54.
0
29.
6
0.0
0.0
0.0
0.0
0.0
4310
SA
PU
G-P
11.0
1
12.5
8.0H
11.0
1
25.5
10.0
H
2
18.0
10.0
H 1
2.5
7.7
43.
5
16.
2
1.3
33.4
1 8.0
1.8
10.8
18.9
5.0
10.
8
6.4
8.0
2560
PA
NN
I-2
12.9
1550
KO
LON
-1
46.1
30.3
19.9
11.5
1650
GA
LLE
-1
39.4
32.2
1
5.4
3405
HA
MB
A-3
3
1400
HA
MB
A-1
134.
9
44.3
6.3
1.9
1340
BE
ATT
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© Springer Nature Singapore Pte Ltd. 2018T. Bambaravanage et al., Modeling, Simulation, and Control of a Medium-ScalePower System, Power Systems, https://doi.org/10.1007/978-981-10-4910-1
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