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 Þ 1 À °Rm F1 þ Xm F2 Þ þ ½Xm sin°1 À 1 Þ

Á Á

À Rm cos°1 À 1 Þ;

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 þ Vk Vm ½Gkm sin°1 Þ

qTv Á

!

2

4Tv

À Bkm cos°1 Þ 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 Þ:

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 Þ 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 Þ;

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 Þ 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 Þ;

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

X¼

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°2Þ þ C2 sin°2Þf$ tan½$° À Þ À tan g

q

& '

cos2 ° À Þ

þ C2 $2 À1 :

cos2 ½$° À Þ

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 Þ;

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 Þ;

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 Þ;

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 Þ;

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 Þ;

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 Þ;

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 Þ;

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 Þ:

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 Þ;

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 Þ;

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 Þ;

qcR

qQk

¼ ÀNkcR ;

qcR

qPk

VcR ¼ Vk VcR ½Gkm cos°k À cR Þ þ Bkm sin°k À cR Þ;

qVcR

qQk

VcR ¼ HkcR ;

qVcR

qPk

¼ Vk VvR ½GvR sin°k À vR Þ À BvR cos°k À vR Þ;

qvR

qQk

¼ ÀNkvR ;

qvR

qPk

VvR ¼ Vk VvR ½GvR cos°k À vR Þ þ BvR sin°k À vR Þ;

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 Þ;

qk

qQm

¼ ÀNmk ;