<<

. 14
( 17)



>>

P0Di ¼ PDi ‚ CPmk ;
0
°8:19Þ
P0G ¼ PG ‚ CPmk ;
0
°8:20Þ
P0mk
0
¼ ; °8:21Þ
CPmk
PD 1 þ P D 2 þ Á Á Á þ PD n þ PG

where i ¼ 1; 2; . . . ; n, and single primes indicate the sending end.
As an extension, the contribution of the n in¬‚ows at the receiving end of branch mk is
determined by the following expressions:

P00 k ¼ P00 1 þ P00 2 þ Á Á Á þ P00 n þ P00 ; °8:22Þ
m D D D G
P00 i ¼ PDi ‚ CPmk ;
00
°8:23Þ
D
P00 ¼ PG ‚ CPmk ;
00
°8:24Þ
G
P00
00
¼ ; °8:25Þ
mk
CPmk
PD 1 þ P D 2 þ Á Á Á þ PD n þ PG

where i ¼ 1; 2; . . . ; n, and double primes indicate the receiving end.
In these expressions, PDi are the power contributions of dominion Di to bus m. The
contribution of each dominion will contain in¬‚ows from every one of its branches. If bus m
is the starting point then the bus in¬‚ow will be PG as opposed to PDi . CPm are contribution
coef¬cients.
319
DOMINIONS

8.4.2 Dominion Contributions to Reactive Power Flows

In each branch, the reactive power contribution of each dominion, and any source of reactive
power connected directly to the bus, is also determined by using the proportional sharing
principle. The circuit representation in Figure 8.6 is suitably modi¬ed to accommodate the
slightly more involved situation prevailing in reactive power applications, where sources of
reactive power may come in a variety of forms: namely, as synchronous generators, shunt
and series compensation, FACTS equipment, and long transmission lines. This situation is
illustrated in Figure 8.7.
In Figure 8.7, QL is a reactive power load, and QS is a reactive power sink. The sink caters
for the possibility of generator or FACTS equipment connected at bus m and drawing
reactive power from the bus. Also, QTL is the reactive power contribution from the
capacitive effects of transmission lines, and QD1 ; . . . ; QDn are the reactive power
contributions of dominions 1; . . . ; n to bus m. If bus m is the starting point of the dominion
then the bus in¬‚ow will be QG or QF as opposed to QD1 ; . . . ; QDn . QG would correspond to a
generator, and QF to FACTS equipment.
Based on Figure 8.7, the contributions from the dominion are obtained by using the
following equations.




QD
1




Q′mk Q′′mk
QD
2
¦




QCk
QCm
QDn k


QG

QF

QTL
QL QS

m
Contribution of reactive power dominions to branch mk
Figure 8.7
320 POWER FLOW TRACING

Sending end of line mk:
Q0mk ¼ Q0D1 þ Q0D2 þ Á Á Á þ Q0Dn þ Q0G þ Q0F þ Q0TL ; °8:26Þ
Q0Di ¼ QDi ‚ CQ mk ;
0
°8:27Þ
Q0G ¼ QG ‚ CQ mk ;
0
°8:28Þ
Q0F ¼ QF ‚ CQ mk ;
0
°8:29Þ
Q0TL ¼ QTL ‚ CQ mk ;
0
°8:30Þ
Q0mk
0
CQ mk ¼ ; °8:31Þ
QD1 þ QD2 þ Á Á Á þ QDn þ QG þ QF þ QTL
where i ¼ 1; 2; . . . ; n.
Receiving end of line mk:
Q00 ¼ Q00 1 þ Q00 2 þ Á Á Á þ Q00 n þ Q00 þ Q00 þ Q00 ; °8:32Þ
mk D D D G F TL
Q00 i ¼ Q0Di ‚ CQ km ;
00
°8:33Þ
D
Q00 ¼ Q0G ‚ CQ km ;
00
°8:34Þ
G
Q00 ¼ Q0F ‚ CQ km ;
00
°8:35Þ
F
Q00 À QCk
00
CQ mk ¼ ; °8:36Þ
mk
QD1 þ QD2 þ Á Á Á þ QDn þ QG þ QF þ QTL À QCm
À Á
Q00 ¼ Q0TL þ QCm ‚ CQ mk þ QCk ;
00
°8:37Þ
TL

where i ¼ 1; 2; . . . ; n.


8.4.3 Dominion Contributions to Loads and Sinks

The proportional sharing principle is also used for ¬nding the dominion and source
contributions to the load connected at bus m. Based on Figure 8.8, the following equations
apply for the case of active power:

PL ¼ P0D1 þ P0D2 þ Á Á Á þ P0Dn þ P0G ; °8:38Þ
P0Di ¼ PDi ‚ CP L ; °8:39Þ



PG

PD PD PD n
1 2

m


PL
Pij

Active dominions contributions to load L
Figure 8.8
321
TRACING ALGORITHM

P0G ¼ PG ‚ CP L ; °8:40Þ
PL
CP L ¼ ; °8:41Þ
PD1 þ PD2 þ Á Á Á þ PDn þ PG
where i ¼ 1; 2; . . . ; n. For the case of reactive power, the variable Q replaces P in
Equations (8.38)“(8.41). Note also that reactive power contributions from QF and QTL may
exist.


8.5 TRACING ALGORITHM

The general algorithm for tracing power ¬‚ows is summarised in Figure 8.9. Note that the
algorithm differs slightly for the cases of active and reactive powers.



Run the base power flow case



Based on power flows as given by the
power flow solution, determine the
source dominions


Find all the branches that belong to
more than one dominion (i.e. the
common branches)


In each branch, find the power
contribution of the relevant dominions
and/or local source to the total branch
flow and associated nodes



In each node, find the power
contribution of the relevant dominions
and/or local source to the node™s load


Account for the power losses in each
dominion


Figure 8.9 Flowchart for the tracing algorithm
322 POWER FLOW TRACING

8.6 NUMERICAL EXAMPLES

This section is concerned with the application of the power tracing algorithm to solve a
number of test cases of varying degrees of complexity. The ¬rst example (Section 8.6.1)
corresponds to a simple radial network, which serves rather well the purpose of illustrating
the application of the theory to active power concerns. The second example (Section 8.6.2)
addresses the case of active power in a meshed network, which is still a fairly simple
network. The motivation for solving this test case is that it enables a direct comparison
between the power tracing methodology presented in this chapter and an alternative tracing
methodology (Bialek, 1996, 1997, 1998). The third test case (Section 8.6.3) deals with
reactive power, as opposed to active power, and includes the contribution of FACTS
equipment to reactive power generation. The tracing of reactive power in a large power
network is quanti¬ed in the fourth test case (Section 8.6.4). The last case (Section 8.6.5)
corresponds to the tracing of active power contributed by one wind generator and one
conventional generator in a meshed network.


8.6.1 Simple Radial Network

The tracing algorithm is applied ¬rst to the case of active power in the test system shown in
Figure 8.1, which is a radial network. In addition to ¬nding the individual power
contributions of generators G1 and G2 to power ¬‚ows in transmission lines TL1 and TL2, and
to system loads L1, L2, and L3, the individual contributions of the generators to transmission
loss becomes readily available. If information exists on charges for use of line then it is a
straightforward matter to allocate charges to each generator per transmitted or lost
megawatt. It is assumed in this example that there is a charge of £1 per lost megawatt in TL1
and TL2.
The dominions of generators G1 and G2 were identi¬ed by inspection in Section 8.2;
however, for the purpose of computer implementation it is essential to have a systematic
approach. In this section the branch“bus incidence matrix is used for the purpose of
dominion identi¬cation. The branch“bus incidence matrix of this network is given in
Figure 8.10, together with the branch searches for the dominions of generators G1 and G2.


b b
1’2 2’3 1’2 2’3
n n

1 +1 1 +1


2 2
’1 ’1
+1 +1


’1 ’1
3 3


(a) (b)

Figure 8.10 Branch“bus incidence matrices and branch searches for the dominions of (a) generator
G1 and (b) generator G2
323
NUMERICAL EXAMPLES

Generator G1 is connected to bus 1. This entry provides the starting point for establishing
the dominion of G1. The þ 1 in location (1, 1) of the matrix indicates that line 1“2 belongs
to the dominion of G1. One additional line belongs to this dominion and, from the branch“
bus incidence matrix, it is found as follows: the sending end of line 1“2 is bus 1, and the
receiving bus is bus 2, as given by þ 1 and À 1 in locations (1, 1) and (2, 1) of the matrix,
respectively. There is a þ 1 entry in the row corresponding to the receiving end of line 1“2.
This indicates that bus 2 contains in¬‚ows, and the search is moved from column 1 to column
2 of the matrix. This makes line 2“3 part of the dominion of G1. Thee is no þ 1 found in the
row corresponding to the receiving end of line 2“3. Hence, bus 3 contains no out¬‚ows, and
the search stops at bus 3.
Using the same line of reasoning, G2 is connected to bus 2. This entry provides the
starting point for establishing the dominion of G2. The þ 1 in location (2, 2) of the matrix
indicates that line 2“3 belongs to the dominion of G2. There is no þ 1 found in the row
corresponding to the receiving end of line 2“3. Hence, bus 3 contains no out¬‚ows, and the
search stops at bus 3.
The dominion contributions to system loads and power losses in transmission lines are
determined quite straightforwardly by using Equations (8.18)“(8.25).
In transmission line TL1:

P0G1 ¼ PG1 ‚ C12 ¼ 160 ‚ 0:6875 ¼ 110;
0

P00 1 ¼ PG1 ‚ C12 ¼ 160 ‚ 0:625 ¼ 100;
00
G

with
P12 110
0
C12 ¼ ¼ ¼ 0:6875;
PG1 160
P0 100
00
¼ 12 ¼ ¼ 0:625:
C12
PG1 160
In transmission line TL2:
P0D1 ¼ PD1 ‚ C23 ¼ 100 ‚ 0:75 ¼ 75;
0

P00 1 ¼ PD1 ‚ C23 ¼ 100 ‚ 0:7 ¼ 70;
00
D
P0G2 ¼ PG2 ‚ C23 ¼ 100 ‚ 0:75 ¼ 75;
0

P00 2 ¼ PG2 ‚ C23 ¼ 100 ‚ 0:7 ¼ 70;
00
G

with
P23 150
0
C23 ¼ ¼ ¼ 0:75;
PD1 þ PG1 100 þ 100
P023 140
00
¼ ¼ ¼ 0:7:
C23
PD1 þ PG2 100 þ 100
This information is summarised in Table 8.1, and Table 8.2 presents a breakdown of the
power loss and charge for use of line for generators G1 and G2.
From this simple example, it is interesting to note that generator G2 contributes no power
¬‚ow to transmission line TL1 and, quite correctly, is not charged for the use of TL1.
Similarly, both generators contribute in equal measure to power ¬‚ow and power loss in
transmission line TL2 and are charged equally.
324 POWER FLOW TRACING

Table 8.1 Contributions to active power ¬‚ows
Sending end Receiving end

C0 (%) P0D1 (MW) P0D2 (MW) C00 (%) P00 1 (MW) P00 2 (MW)
Branch D D

1“2 68.75 110 0 62.5 100 0
2“3 75 75 75 70 70 70



Table 8.2 Contributions to power loss and charges for use of line
for dominions D1 and D2
Power loss (W) Charge for use of lines (£)

Branch D1 D2 D1 D2
1“2 10 0 10 0
2“3 5 5 5 5
Total 15 5 15 5
Note: The charge for use of line is set at £1 per megawatt.



8.6.2 Simple Meshed Network: Active Power

The test system used by Bialek (1996, 1997, 1998) forms the basis of the example presented
in this section. As shown in Figure 8.11, it is a simple power network comprising four buses,
linked together by ¬ve transmission lines. Generation is available at buses B1 and B2, and
loads are connected at buses B3 and B4.

200 MW
300 MW
’4
’3
171 MW
218 MW
82 MW 83 MW


112 MW



115 MW



225 MW 173 MW
59 MW
60 MW


’2
’1
400 MW 114 MW



Power ¬‚ows in a small, meshed network
Figure 8.11
325
NUMERICAL EXAMPLES

Charges for use of line
Table 8.3

Line 1“2 1“3 1“4 2“4 4“3
Charge (£) 12.75 6 11.7 3.5 5.75



Table 8.3 gives information on applied charges for use of line, which have been taken to
be equal to the branch resistances (Bialek, 1996, 1997, 1998). The power ¬‚ows, as given by
a power ¬‚ow solution, are shown in Figure 8.11. Based on these trajectories, the two
domains of the network are determined, one domain per generator.
The branch“bus incidence matrix of this network is given in Figure 8.12 together with the
branch search for the dominion of generator G1. Generator G1 is connected to bus B1. This
entry provides the starting point for establishing the dominion of generator G1. A þ 1 at
locations (1, 1), (1, 2), and (1, 3) of the matrix indicates that lines 1“3, 1“2, and 1“4 belong
to the dominion of generator 1. Additional lines belong to this dominion, and they will be
found as follows:
 The sending end of line 1“3 is bus B1, and the receiving end is bus B3, as given by the þ 1
and À 1 in locations (1, 1) and (3, 1) of the matrix, respectively. There is no þ 1 entry in
the row corresponding to the receiving end of line 1“3. This indicates that bus B3 contains
no out¬‚ows. Therefore, the search stops at bus B3 for this route.
 The row corresponding to the receiving end of line 1“2 contains þ 1s. Thus, the search is
moved from column 2 to column 4 of the matrix. This makes line 2“4 part of the
dominion of generator G1. Applying the same line of reasoning, we move the search on
from column 4 to column 5, and line 4“3 is incorporated into the dominion of generator
G1. No þ 1 is found in the row corresponding to the receiving end of line 4 “3. Hence, bus
B3 contains no out¬‚ows, and the search stops at bus B3 for this route.
 The row corresponding to the receiving end of line 1“4 contains a þ 1 in column 5.
Accordingly, the search is moved on from column 3 to column 5 of the matrix. This
makes line 4“3 part of the dominion of generator G1. It must be noted that this line has


b
1’3 1’2 1’ 4 2’4 4’3
n

1 +1 +1 +1


2 ’1 +1


3 ’1 ’1


4 ’1 ’1 +1



Branch“bus incidence matrix and branch search for the dominion of generator G1
Figure 8.12
326 POWER FLOW TRACING


B3 B4




B1 B2




Dominion of generator G1
Figure 8.13

already been added to the dominion of generator G1, and it should not be incorporated
twice in the dominion. No þ 1 is found in the row corresponding to the receiving end of
line 4“3.
 Since we have exhausted all the þ 1 entries in row 1 of the matrix then we are satis¬ed
that we have completed the search for all branches belonging to the dominion of
generator G1. It must be noted that, in this case, the entire network belongs to the
dominion of generator G1.
The directed subgraph of dominion D1 is shown in Figure 8.13.
The branch“bus incidence matrix of the network shown in Figure 8.11 together with the
branch search process for the dominion of generator G2 are shown in Figure 8.14. Generator
G2 is connected to bus B2. This entry provides the starting point for establishing the
dominion of generator G2. A þ 1 entry in location (2, 4) of the matrix indicates that line 2“ 4
belongs to the dominion of generator G2. Additional lines belonging to this dominion will


b
1’3 1’2 1’4 2’4 4’3
n

1 +1 +1 +1


’1
2 +1


3 ’1 ’1


4 ’1 ’1 +1


Branch“bus incidence matrix and branch search for the dominion of generator G2
Figure 8.14
327
NUMERICAL EXAMPLES


B4
B3




B2



Dominion of generator G2
Figure 8.15

be found as follows:
 The row corresponding to the receiving end of line 2“ 4 contains þ 1 entries. Accord-
ingly, the search is moved on from column 4 to column 5 of the matrix. This makes line
4“3 part of the dominion of generator G2. We move the search on from column 4 to
column 5 of the matrix and line 4“3 is incorporated into the dominion of generator G2. No
þ 1 entries are found in the row corresponding to the receiving end of line 4“3. Bus B3
contains no out¬‚ows.
 We have exhausted all the þ 1 entries in row 2 of the matrix and this indicates that we
have completed the search for all branches belonging to the dominion of generator G2. At
this point we have also completed the search for all the dominions available in this
network.
The directed subgraph of dominion 2 is shown in Figure 8.15.
Branch 2“ 4 and branch 4 “3 are both common to dominions D1 and D2. Hence, the power
tracing algorithm is used to calculate the contributions of each dominion to common
branches 2“ 4 and 4 “3. This information is presented in Table 8.4.
By way of example, the power ¬‚ow contribution of dominion D1 at the sending and
receiving ends of transmission line 2“4 are calculated as follow:
173
0
¼ 1; P0D1 ¼ 1 ‚ 59 ¼ 59;
C24 ¼
59 þ 114
171
00
¼ 0:988444; P00 1 ¼ 0:988444 ‚ 59 ¼ 58:317919:
¼
C24
59 þ 114 D



Contribution of dominions D1 and D2 to branches 2“4 and 4“3
Table 8.4

Sending end Receiving end

C 0 (%) P0D1 (MW) P0D2 (MW) C 00 (%) P00 1 (MW) P00 2 (MW)
Branch D D

2“4 100 59 114 98.8439 58.3179 112.6821
4“3 29.3286 49.9519 33.0481 28.9753 49.3501 32.6499
328 POWER FLOW TRACING

System power losses (sending end) and charges for use of line for dominions D1 and D2
Table 8.5
Power loss (MW) Charge for use of lines (p.u.)

Branch D1 D2 Per line D1 D2
1“2 1 0 12.75 12.75 0
1“3 7 0 6 6 0
1“4 3 0 11.7 11.7 0
2“4 0.6821 1.3179 3.5 1.1937 2.3063
4“3 0.6018 0.3982 5.75 3.4604 2.2896
Total 12.2839 1.7161 N.A. 35.1041 4.5959
N.A. Not applicable.



The contributions of dominions D1 and D2 to active power losses in branch 2“ 4 become
readily available from the above result. Table 8.5 gives the power losses and charges for use
of line.
The charge, E, for use of line in line 2“ 4 is calculated as follows:
 
3:5
ED1 ¼ ‚ 0:6821 ¼ 1:1937 p:u:;
0:6821 þ 1:3179
 
3:5
ED2 ¼ ‚ 1:3179 ¼ 2:3063 p:u:
0:6821 þ 1:3179
The charge for use of line in line 4“3 is:
 
5:75
ED1 ¼ ‚ 0:6018 ¼ 3:4604 p:u:;
0:6018 þ 0:3982
 
5:75
ED2 ¼ ‚ 0:3982 ¼ 2:2896 p:u:
0:6018 þ 0:3982
It is important to remark that a number of methodologies exist for calculating charges for
use of line. For instance, Table 8.6 gives the charges for use of line as calculated by three


Table 8.6 Comparison of charges (p.u.) for use of line by three different methods for dominions
D1 and D2: (a) the tracing algorithm (presented in this chapter), (b) the generalised factor algorithm
(Ng, 1980), and (c) the topological factor method (Bialek, 1996, 1997, 1998)
(a) Tracing (b) Generalised factor (c) Topological factor

Branch D1 D2 D1 D2 D1 D2
1“2 12.75 0 12.75 0 12.75 0
1“3 6 0 5.22 0.78 6 0
1“4 11.7 0 11.7 0 11.7 0
2“4 1.1937 2.3063 1.77 1.73 1.21 2.29
4“3 3.4604 2.2896 3.06 2.69 3.48 2.27
Total 35.1041 4.5959 34.5 5.2 35.14 4.56
329
NUMERICAL EXAMPLES

different methods. The tracing algorithm presented in this chapter is compared with the
method of topological factors (Bialek, 1996, 1997, 1998) and the method of generalised
factors detailed by Ng (1980). It is brought to the reader™s attention that in this example
some of the generalised factors are negative and would produce negative charges for use of
line (i.e. a generator would be compensated for using a transmission facility; Bialek, 1996,
1997, 1998). In actual applications all negative factors and costs are set to zero (Ng, 1980).
The charges based on topological factors compare very well with the charges given by the
tracing algorithm. In contrast, some differences are observed with respect to the charges
given by the generalised factor algorithm. Perhaps the most suspect results are the charges
made to generator G2 for the use of branch 1“3, and the undercharge to generator G1. It must
be noted that branch 1“3 is not part of the dominion of generator G1. This fact is correctly
recognised by the topological factor algorithm. Also, important differences exist in the
charges made to generators G1 and G2 for the use of line 2“ 4.


8.6.3 Meshed Network with FACTS Controllers: Reactive Power

In this case, two FACTS devices are included: one uni¬ed power ¬‚ow controller (UPFC) in
branch Lake“Main and one static compensator (STATCOM) in node Elm. The reactive
power ¬‚ows throughout the network are shown in Figure 8.16. The dominions of the


1.4
15 5

Lake
North Main
UPFC
2.8
5.5 2.0
2.9 3.9
1.5
2.2
1.1
6.7



3.2
4.0
2.6
3.0

South Elm
10.6
10
STAT COM
10 2.1 3.2
5.7

Figure 8.16 Five-node system with uni¬ed power ¬‚ow controller (UPFC) and static compensator
(STATCOM)
330 POWER FLOW TRACING

Table 8.7 Reactive dominions of generators, FACTS, and transmission lines (TLs)
Transmission line Loads and sinks
Gen“North North“Lake Lake
Gen“South South“Lake South, Lake
UPFC Lake“Main, Main“South, South“Lake Lake, Main, South
STATCOM Elm“South, South“Lake Elm, South, Lake
TL: North“Lake None Lake
TL: South“Lake None Lake
TL: North“South North“Lake, South“Lake South, Lake
TL: Main“South South“Lake South, Lake
TL: Lake“Main Main“South, South“Lake Main, South, Lake
TL: Main“Elm Elm“South, Main“South, South“Lake Main, Elm, South, Lake
Note: UPFC, uni¬ed power ¬‚ow controller; STATCOM, static compensator.



reactive source™s dominions are given in Table 8.7. The dominions of generators and FACTS
equipments are as shown in Figure 8.17.
The reactive dominion of Gen“North reduces to line North“Lake, and this generator
contributes only to the reactive load connected at Lake. Six transmission lines become




(a) (b)




(d)
(c)

Figure 8.17 Reactive dominions: (a) Gen“North, (b) Gen“South, (c) uni¬ed power ¬‚ow converter,
and (c) static compensator
331
NUMERICAL EXAMPLES

Dominion contributions to system loads
Table 8.8
South Elm Main Lake
Gen“North 0.0 0.0 0.0 1.4
Gen“South 4.1 0.0 0.0 1.6
UPFC 0.3 0.0 1.6 2.9
STATCOM 1.2 8.0 0.0 0.5
TL: North“Lake 0.0 0.0 0.0 2.6
TL: South“Lake 0.0 0.0 0.0 2.7
TL: North“South 2.3 0.0 0.0 2.4
TL: Main“South 1.4 0.0 0.0 0.5
TL: Lake“Main 0.1 0.0 1.6 0.2
TL: Main“Elm 0.6 2.0 1.8 0.2
Total load at node 10 10 5 15
Note: STATCOM, static compensator; UPFC, uni¬ed power ¬‚ow controller; TL,
transmission line; TL: South“Elm is obsorbing MVAR.




sources of reactive power but they also form part of various dominions. Table 8.8 shows the
contributions of the various sources to the individual reactive system loads.


8.6.4 Large Network

In order to show how the tracing algorithm works with a larger power system, the New
Zealand South Island 220 kV system illustrated in Figure 8.18 is used to carry out this study.
The system data are given in Arrillaga and Watson (2001).
From the power ¬‚ow solution, it emerges that there are two machines that contribute
substantial reactive power injections into their connecting nodes. These synchronous
machines are the one connected at Islington“220 and the one connected at Benmore“016.
The synchronous machine™s dominion connected to Islington“220 is depicted
schematically in Figure 8.19 together with the system contribution to reactive power ¬‚ow
in this particular dominion.
In this case, the transformers complex taps have been set to nominal values (i.e. the
transformer equivalent circuits do not contain shunt admittances).
Information on the Islington“220 dominion is shown in Table 8.9. The reactive power
absorption of each line is depicted in Table 8.10.


8.6.5 Tracing the Power Output of a Wind Generator

With the ongoing deregulation of the electricity supply industry, the opportunity has arisen
for the widespread incorporation of renewable sources of electricity into the power network.
In the United Kingdom, for instance, wind generation is a form of renewable generation that
is set to experience unprecedented growth, in particular, offshore wind generation.
Among the pressing problems that the industry will have to solve, if electricity genera-
tion from the wind is to become commercially successful in a deregulated environment,
332 POWER FLOW TRACING




Tekapo’220 Islington’220



Tekapo’ 011
Twizel’220

Bromley’220

Ohau-system
Benmore’220



Benmore’016

Aviemore’ 220




Aviemore’011
Roxburgh’220
Roxburgh’011
Invercarg’220
Livingstn’220



Tiwai’220
Manapouri’014




Manapouri’220
Figure 8.18 The New Zealand South Island 220 kV system. Reproduced by permission of John
Wiley & Sons Ltd from J. Arrillaga and N.R. Watson, 2001, Computer Modelling of Electrical Power
System, 2nd edn
333
NUMERICAL EXAMPLES

Tekapo’220 Islington’220




Twizel’220
Bromley’220




Roxburgh’220


Invercarg’220
Livingstn’220




Tiwai ’220




Figure 8.19 Islington“220 reactive power dominion




Islington“220 dominion: general data
Table 8.9

Index Sending end Receiving end Q absorbed (MVAR)
TL1 Islington“220 Tekapo“220 42.5108
TL2 Islington“220 Twizel“220 40.7907
TL3 Islington“220 Bromley“220 0.8010
TL4 Bromley“220 Twizel“220 38.2211
TL5 Islington“220 Livingstn“220 19.1111
TL6 Livingstn“220 Roxburgh“220 74.0988
TL7 Roxburgh“220 Invercarg“220 5.6353
TL8 Invercarg“220 Tiwai“220 0.4756
TL9 Invercarg“220 Tiwai“220 0.4756
334 POWER FLOW TRACING

Table 8.10 Line contributions to reactive power absorption
Q °MVARÞ
””” ”””””””””””” ”””” Contribution
Index Out¬‚ow In¬‚ow Contributed coef¬cients
TL1:
dominion 17.0548 0.0000 17.0548 0.4012
À 7.9559
system 17.5000 25.4560 0.5988
TL2:
dominion 12.0992 0.0000 12.0992 0.2966
À 6.1915
system 22.5000 28.6915 0.7034
TL3:
dominion 62.3661 61.5959 0.7701 0.9615
system 2.5000 2.4691 0.0309 0.0385
TL4:
dominion 6.0533 0.0000 6.0532 0.1584
À 9.41658
system 22.7513 32.1678 0.8416
TL5:
dominion 25.3950 14.0807 11.3143. 0.5920
system 17.5000 9.70321 7.7968 0.4080
TL6:
dominion 7.78264 0.7990 6.9836 0.0942
system 26.6437 2.7355 23.9082 0.3226
TL7:
dominion 0.3232 0.2860 0.0372 0.0066
system 28.3145 25.0573 3.2572 0.5780
TL8:
dominion 0.0873 0.0863 0.0010 0.0021
system 22.0149 21.7640 0.2501 0.5277
TL9:
dominion 0.0873 0.0863 0.0010 0.0021
system 22.0149 21.7640 0.2501 0.5277




is to develop an understanding of the impact that large, random blocks of electricity will
have on the power network. For instance, how much electricity can a wind-generating
company, under obligation to supply, afford to contract to supply given its ˜fuel™ supply
uncertainty?
An equally critical issue that needs addressing concerns the ability to trace the power
output of one or more wind farms within an interconnected network. This has a direct
bearing on the aspirations of a growing number of consumers keen on being supplied with
electricity that has been produced with little damage to the environment. The power tracing
methodology offers a realistic possibility of achieving this goal and one that should
encourage providers of clean energy.
The numerical example presented in this section addresses one way in which the tracing
methodology can be applied in the area of delivery of clean power. This requires a
simulation environment similar to the one shown in Figure 8.20, where the interaction of the
335
NUMERICAL EXAMPLES


Generation and load forecasting algorithms



k = 1 , 2, ¦ , n




Newton’Raphson power flow

k=k+1

Tracing algorithm



k≥n End

Figure 8.20 Power ¬‚ow simulation environment



forecasting, power ¬‚ow, and power tracing algorithms is illustrated. This simulation
environment offers a simple and yet comprehensive way of modelling time-dependent
generators and loads.



8.6.5.1 The wind generator model

Wind generators slaved to the power network are mostly of the induction type. During
high winds, when the rotor speed supersedes the synchronous speed, active power is injected
into the grid. In the presence of low winds there is an automatic cutout to prevent motoring
from happening. During normal conditions, the turbine operates at nearly constant
frequency. The induction wind generator achieves its operation at the expense of consuming
reactive power. From the power ¬‚ow point of view, it makes engineering sense to treat the
generator bus as a PQ bus with a positive active power injection and a negative reactive
power injection.
However, these power injections must be time-dependent to re¬‚ect the stochastic nature
of the prime mover (i.e. the wind). Figure 8.21 shows the active power output of a typical
wind farm for a period of 54 hours, where very large variations between measurements are
observed; for example, the generator goes from zero power output at 16 hours, to 1.8 MW at
18 hours (Johansson et al., 1992). For cases of wind farms of low capacity, their reactive
power requirements can be met locally. Moreover, if suitable power electronics equipment is
336 POWER FLOW TRACING


1800




Power (kw)
1600
1200
800
400
3φ active power to
grid supply
2 6 10 14 18 22 26 30 34 38 42 46 50 54
Time (hours)
Wind generator model for power ¬‚ow studies that caters for time dependency
Figure 8.21



used in tandem with the wind generator set then the reactive power compensation can be
met adaptively.



8.6.5.2 Numerical example

This numeric example illustrates how the simulation environment of Figure 8.20 works. The
example relates to the power network shown in Figure 8.22, where only active power ¬‚ows
are shown. In this example all the power ¬‚ows are expressed in kilowatts. Generator G2 is a
wind generator with the power generation pro¬le shown in Figure 8.21. The output of
generator G3 and loads are taken to remain constant. Generator G1 is the slack generator. By



1300
1000
G5 G3
592.1 606.9
935.8 393.1
227.9

332
220.7
G1
247.2
800
987.4
383.6
253.7

G2 770
812.6 G4
1800
900


Power ¬‚ows (in kilowatts) when the wind generator injects maximum active power (i.e.
Figure 8.22
1.8 MW)
337
NUMERICAL EXAMPLES


G5

G5 G3

G1

G1

G2 G4
G1 G4
(a) (c)
(b)

Dominions of (a) generator G2, (b) generator G1, and (c) generator G3
Figure 8.23




way of example, two cases are considered below: (1) at 18 hours when the wind generator
is injecting maximum power (i.e. 1.8 MW) and (2) at 36 hours when it is injecting zero
power.


The wind generator injects maximum power

Figure 8.22 shows the power ¬‚ows for the case when the wind generator is injecting
1.8 MW. Based on these power ¬‚ows, three network dominions are determined. Figure 8.23
shows the dominions of generators G1“G3.
It can be observed in Figure 8.23 that branches 4“1 and 5“1 are common to the dominions
of generator 2 and generator 3. Using the tracing algorithm, the contribution of both
dominions to each element of the network are calculated. Active power losses and charges
for use of line associated with each dominion are then established. This information is
summarised in Table 8.11.
The charge for use of line to dominion k, in branch ij, is calculated as follows:
 
xij
ED k ¼ LDk ; °8:42Þ
LTotal

where xij is the company charge assigned to the use of branch ij.


The wind generator contributes no active power

Figure 8.24(a) shows the power ¬‚ows for the case when the wind generator contributes no
active power. The directed subgraphs for the two dominions are shown in Figures 8.24(b)
and 8.24(c). The contribution of dominions D1 and D3 to active power losses throughout
the network become readily available. Table 8.12 gives the power losses and charges for
use of line.
Table 8.11 System power losses and charges for use of line to dominions D2 and D3
Sending end Receiving end Power loss

C0 C 00
Branch Charge (p.u.)
D2 D3 ED2 (p.u.) ED3 (p.u.)
D D
P0D2 (KW) P0D3 (KW) P00 2 (KW) P00 3 (KW)
3“5 0.6069 0 606.9 0.5921 0 592.1 0 14.8 0
x35 x55
3“4 0.3931 0 393.1 0.3836 0 383.6 0 9.5 0
x34 x14
4“1 0.2199 169.3 84.4 0.2143 165.0 82.2 4.3 2.2 x41 0.6615x41 0.3384x41
5“1 0.1492 139.6 88.3 0.1444 135.2 85.5 4.4 2.8 x51 0.6111x51 0.3889x51
2“4 0.5486 812.6 0 0.5206 770.0 0 42.6 0 0
x24 x24
2“5 0.4514 987.4 0 0.4278 935.8 0 51.8 0 0
x25 x25
Note: xij , company charge assigned to use of branch ij.




Table 8.12 System power losses and charges for use of line to dominions D1 and D3

Sending end Receiving end Power loss

C0 C00
Branch D1 D3 Charge (p.u.) ED1 (p.u.) ED3 (p.u.)
D D
P0D1 (kW) P0D3 (kW) P00 1 (kW) P00 3 (kW)
3“5 0.6295 0 629.5 0.6134 0 613.14 0 16.1 0
x35 x35
3“4 0.3705 0 370.5 0.3620 0 362.0 0 7.5 0
x34 x34
4“1 0.2761 566.1 0 0.2809 576.1 0 10 0 0
x41 x41
5“1 0.3223 661.0 0 0.3290 674.6 0 13.6 0 0
x51 x51
2“4 0.0291 16.47 10.53 0.0303 17.14 10.96 0.67 0.43 x24 0.6091x24 0.3909 x24
2“5 1 16.47 10.53 0.9481 15.62 9.98 0.85 0.55 x25 0.6072x25 0.3928x25
Note: See Table 8.11.
339
SUMMARY

1300
1000
G5 G3
613.4 629.5
25.6 370.5
661

2050.7
674.6
G1
576.1
800
27.0
362.0
566.1

G2 28.1
27.0 G4
0
900


(a)




G5
G5 G3



G1



G4 G2
G2 G4
(c)
(b)

Figure 8.24 (a) Power ¬‚ows in the test network for the case when the wind generator, G2, contributes
no active power (i.e. 0 MW); directed subgraph for (b) dominion D1 and (c) dominion D3




8.7 SUMMARY

The relentless trend towards deregulation and unbundling of transmission services in the
electricity supply industry has provided the motivation for developing methodologies that
trace the output of each generator throughout a power system, whether it is a simple radial
network or an interconnected network of national or even continental dimensions. Over the
last few years a great deal of progress has been made in this direction, and methods based on
the principle of proportional sharing are well regarded in academic circles. Several
alternative algorithms have appeared in the open literature since 1996, with a large
proportion of these papers devoted to economic issues. However, other applications are
beginning to emerge such as the tracing of power contributed by ˜green™ generators and
distortion power contributed by harmonic sources.
340 POWER FLOW TRACING

The tracing algorithm we have detailed in this chapter is the one we developed at
Glasgow, but the application cases contained in this chapter can equally be solved by using
any of the alternative methodologies found in the open literature. The algorithm may serve
the purpose of auditing the individual generator contributions to system loading, power
¬‚ows, transmission losses, generation costs, and charges for use of lines. The algorithm is
independently applied to the tracing of active, reactive, and distortion powers. The
algorithm is accurate and comprehensive. In fact, power ¬‚ow tracing is only a mechanism
for tracing, for instance, generation costs and allocating charges for use of line. These two
basic capabilities of the algorithm have been compared with results corresponding to a
simple case available in the open literature. Also, a larger study involving a subsection of an
interconnected power network has been conducted.


REFERENCES
Acha, E., 1998, ˜Tracing Wind Power in a Pooled Transmission System™, EPSOM™98, Zurich,
Switzerland, September 1998, pp. 23“25.
Acha, E., Fuerte-Esquivel, C.R., Chua C.S., 1996, ˜On the Auditing of Individual Generator
Contributions to Power Flows and Losses in Meshed Power Networks™, RVP 96-SIS-10,
´ ´
Reunion de Verano de Potencia, IEEE Seccion Mexico, Acapulco Gro., Mexico, July 1996,
pp. 170“173.
Acha, E. Ambriz-Perez, H., Fuerte-Esquivel, C.R., Chua, C.S., 1997, ˜On the Auditing of Individual
Generator Contributions to Optimal Power Flows, Losses and Costing of Large, Interconnected
Power Networks™, IPEC™97 Proc., Volume II, Nanyang University, Singapore, May 1997,
pp. 22“24.
Acha, E., Tortelli, O.L., Angeles-Camacho, C., Santos Jr, A., 2003, ˜Reactive Power Tracking in FACS
Upgraded Power Networks™, IASTED, Proceedings of The Third IASTED International Conference
on Power and Energy Systems, Marbella, Spain, September 2003.
Arrillaga, J., Watson, N.R., 2001, Computer Modelling of Electrical Power System. 2nd edn, John Wiley
& Sons, Chichester.
Bialek, J., 1996, ˜Tracing the Flow Electricity™, IEE Proc. 143(4) 313“320.
Bialek, J., 1997, ˜Topological Generation and Load Distribution Factors for Supplemental Charge
Allocation in Transmission Open Access™, IEEE Trans. on Power Systems 12(3) 1185“1193.
Bialek, J., 1998, ˜Allocation of Transmission Supplementary Charges to Real and Reactive Loads™,
IEEE Trans. on Power Systems 13(3) 749“754.
EPEW, 1993, ˜An Introduction to Pool Rules™, April 1993, Electric Pool of England and Wales,
London.
Johansson, T.B., Kelly, H., Reddy, A.K.N., Willians, R.H., 1992, Renewable Energy: Sources for Fuel
and Electricity, Earthscan Publications London.
Kirschen, D., Strbac, G., 1999, ˜Tracing Active and Reactive Power between Generators and Loads
Using Real and Imaginary Currents™, IEEE Trans. on Power Systems 14(4) 1312“1319.
Kirschen, D., Allan, R.N., Strbac, G., 1997, ˜Contributions of Individual Generators to Loads and
Flows™, IEEE Trans. on Power Systems 12(1) 52“60.
´
Laguna-Velasco, R., 2002, Asignacion de cargos por el porteo de ¬‚ujos de potencia activa y reactiva en
´ ´
los sistemas de transmision basada en el metodo de rastreo de la electricidad, MSc thesis (in
´ ´
Spanish), Centro de Investigacion Avarzada del Instituto Pol±tecnico Nacional, Unidad Guadalajara,
Mexico.
Laguna-Velasco, R., Fuerte-Esquivel, C.R., Acha, E., Ambriz-Perez, H., 2001, ˜On the Auditing of
Individuals Generator Contributions to Reactive Power Flows in Power Networks™, paper presented
at the IEEE Powertech Conference, Porto, Portugal, September 2001.
341
REFERENCES

Macqueen, C.N., 1993, Time-based Load Flow Analysis and Loss Costing in Electrical Distribution
System, PhD thesis, School of Engineering, University of Durham, Durham.
Ng, W.Y., 1981, ˜Generalised Generation Distribution Factors for Power Systems Security Evaluations™,
IEEE Trans. Power App. Systems PAS-100(3) 1001“1005.
˜Recovery of Cost under BETTA and Ofgem/DTI Conclusions Document™, July 2003, OFGEM (Of¬ce
of Gas and Electricity Markets), London.
Reta, R., Vargas, A., 2001, ˜Electricity Tracing and Loss Allocation Methods Based on Electric
Concepts™, IEE Proc. on Gener. Transm. Distrib. 148(6) 518“522.
Saunders, B., Boag M., 2001, ˜NETA: A Dramatic Change™, Conseil International des Grands Reseaux
´ ´
Electriques (CIGRE), Electra No. 199, 14“23.
Secretary of State for Energy, 1988, Privatising Electricity “ The Government™s Proposals for the
Privatisation of the Electric Supply Industry in England an Wales, February 1988, Her Majesty™s
Stationary Of¬ce, London.
Appendix A: Jacobian
Elements for FACTS
Controllers in Positive
Sequence Power Flow

A.1 TAP-CHANGING TRANSFORMER

The partial derivatives of the power equations with respect to the primary tap of the two
winding transformer are:
2V 2 T 2 ‚ À 2 Á Ã
qPk
Tv ¼ k v Rm Uv þ R1 þ Xm R2 À 2Gkk °Rm F1 þ Xm F2 Þ þ Vk Vm ½Gkm cos°1 Þ
qTv Á
!
2 3
4Tv 2Vk Vm Tv Uv
þ Bkm sin°1 ފ 1 À °Rm F1 þ Xm F2 Þ þ ½Xm sin°1 À 1 Þ
Á Á
À Rm cos°1 À 1 ފ;
À2 Á
qQk 2V 2 T 2
Tv ¼ k v ½2Bkk °Rm F1 þ Xm F2 Þ À Rm R2 þ Xm Um þ R1 Š þ Vk Vm ½Gkm sin°1 Þ
qTv Á
!
2
4Tv
À Bkm cos°1 ފ 1 À ° Rm F1 þ Xm F 2 Þ
Á
3
2Vk Vm Tv Uv
À ½Rm sin°1 À 1 Þ þ Xm cos°1 À 1 ފ:
Á
The partial derivatives of the power equations with respect to the secondary tap of the two
winding transformer are:
2Vk Uv ‚ À 2 Á Ã
qPk 22
Uv ¼ Rm Uv þ R1 þ Xk R2 þ F1 À 2Gkk °Rk F1 þ Xk F2 Þ
qUv Á
!
2
4Uv
þ Vk Vm ½Gkm cos°1 Þ þ Bkm sin°1 ފ 1 À ° Rk F 1 þ Xk F 2 Þ
3
2Vk Vm Uv Tv
þ ½Xk sin°1 À 1 Þ À Rk cos°1 À 1 ފ;
Á

FACTS: Modelling and Simulation in Power Networks.
´ ´
Enrique Acha, Claudio R. Fuerte-Esquivel, Hugo Ambriz-Perez and Cesar Angeles-Camacho
# 2004 John Wiley & Sons, Ltd ISBN: 0-470-85271-2
344 APPENDIX A: JACOBIAN ELEMENTS FOR FACTS CONTROLLERS


2V 2 U 2 ‚ À2 ÁÃ
qQk
Uv ¼ k v 2Bkk °Rk F1 þ Xk F2 Þ À Rk R2 þ Xk Uv þ R1 þ F2
qUv Á
!
2
4Uv
þ Vk Vm ½Gkm sin°1 Þ À Bkm cos°1 ފ 1 À ° Rk F1 þ Xk F2 Þ
Á
3
2Vk Vm Uv Tv
À ½Rk sin°1 À 1 Þ þ Xk cos°1 À 1 ފ;
Á
where
F1 ¼ T 2 Rs þ Uv Rp þ Req1 ;
2

F2 ¼ Tv Xs þ Uv Xp þ Xeq1 ;
2 2

Req1 ¼ °Rp Rs À Xp Xs ÞGo À °Rp Xs þ Rs Xp ÞBo ;
Xeq1 ¼ °Rp Rs À Xp Xs ÞBo þ °Rp Xs þ Rs Xp ÞGo ;
R1 ¼ Rs Go À Xs Bo ;
R2 ¼ Rs Bo þ Xs Go ;
R3 ¼ Rp G0 À Xp B0 ;
R4 ¼ Rp B0 þ Xp G0 ;
1 ¼ tv À uv ;
2 ¼ uv À tv ;
Á ¼ F1 þ F2 ;
2 2

1 ¼ k À m :



A.2 THYRISTOR-CONTROLLED SERIES COMPENSATOR

Partial derivatives of the variable series impedance model are:
qPk
X ¼ ÀVk Vm Bkm sin°k À m Þ;
qX
qQk
X ¼ Vk Bkk þ Vk Vm Bkm cos°k À m Þ;
2
qX
qPX qPk

km
X:
qX qX
Partial derivatives of the ¬ring angle model:
qXTCSC°1Þ
qPk
¼ Pk BTCSC°1Þ ;
q q
qXTCSC°1Þ
qQk
¼ Qk BTCSC°1Þ ;
q q
qBTCSC°1Þ qXTCSC°1Þ
¼ B2 ;
q q
TCSC°1Þ

qXTCSC°1Þ
¼ À2C1 ½1 þ cos°2 ފ þ C2 sin°2 Þf$ tan½$° À ފ À tan g
q
& '
cos2 ° À Þ
þ C2 $2 À1 :
cos2 ½$° À ފ
345
UNIFIED POWER FLOW CONTROLLER

A.3 STATIC SYNCHRONOUS COMPENSATOR

Partial derivatives for the static compensator (STATCOM) model are:
qPk
¼ ÀQk À Vk GvR ;
2
qk
qPk
¼ Vk VvR ½GvR sin°k À vR Þ À BvR cos°k À vR ފ;
qvR
qPvR
¼ ÀQvR À VvR BvR ;
2
qvR
qPvR
¼ VvR Vk ½GvR sin°vR À k Þ À BvR cos°vR À k ފ;
qk
qPk
Vk ¼ Pk þ Vk GvR ;
2
qVk
qPk
VvR ¼ Vk VvR ½GvR cos°k À vR Þ þ BvR sin°k À vR ފ;
qVvR
qPvR
VvR ¼ PvR þ VvR GvR ;
2
qVvR
qPvR
Vk ¼ VvR Vk ½GvR cos°vR À k Þ þ BvR sin°vR À k ފ;
qVk
qQk
¼ Pk À Vk GvR ;
2
qk
qQk
¼ ÀVk VvR ½GvR cos°k À vR Þ þ BvR sin°k À vR ފ;
qvR
qQvR
¼ PvR À VvR GvR ;
2
qvR
qQvR
¼ ÀVvR Vk ½GvR cos°vR À k Þ þ BvR sin°vR À k ފ;
qk
qQk
Vk ¼ Qk À Vk BvR ;
2
qVk
qQk
VvR ¼ Vk VvR ½GvR sin°k À vR Þ À BvR cos°k À vR ފ;
qVvR
qQvR
VvR ¼ QvR À VvR BvR ;
2
qVvR
qQvR
Vk ¼ ÀVvR Vk ½GvR sin°vR À k Þ À BvR cos°vR À k ފ:
qVk


A.4 UNIFIED POWER FLOW CONTROLLER

Partial derivatives for the uni¬ed power ¬‚ow controller (UPFC) at bus k are:
qPk
¼ ÀQk À Vk Bkk ;
2
qk
qQk
¼ Pk À Vk Gkk ;
2
qk
346 APPENDIX A: JACOBIAN ELEMENTS FOR FACTS CONTROLLERS


qPk
¼ Vk Vm ½Gkm sin°k À m Þ À Bkm cos°k À m ފ;
qm
qQk
¼ ÀNkm ;
qm
qPk
Vk ¼ Pk þ Vk Gkk ;
2
qVk
qQk
Vm ¼ Hkm ;
qVm
qPk
Vm ¼ Vk Vm ½Gkm cos°k À m Þ þ Bkm sin°k À m ފ;
qVm
qQk
Vk ¼ Qk À Vk Bkk ;
2
qVk
qPk
¼ Vk VcR ½Gkm sin°k À cR Þ À Bkm cos°k À cR ފ;
qcR
qQk
¼ ÀNkcR ;
qcR
qPk
VcR ¼ Vk VcR ½Gkm cos°k À cR Þ þ Bkm sin°k À cR ފ;
qVcR
qQk
VcR ¼ HkcR ;
qVcR
qPk
¼ Vk VvR ½GvR sin°k À vR Þ À BvR cos°k À vR ފ;
qvR
qQk
¼ ÀNkvR ;
qvR
qPk
VvR ¼ Vk VvR ½GvR cos°k À vR Þ þ BvR sin°k À vR ފ;
qVmR
qQk
VvR ¼ HkvR :
qVvR
Partial derivatives at the receiving bus m are:
qPm
Hmk ¼ ¼ Vm Vk ½Gmk sin°m À k Þ À Bmk cos°m À k ފ;
qk
qQm
¼ ÀNmk ;

<<

. 14
( 17)



>>