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Tracing the complex power of generatorsRuay‐Nan Wu a & Hua‐Wen Tsai a

a Department of Electrical Engineering , National Taiwan University of Science andTechnology , Taipei, Taiwan 106, R.O.C.Published online: 03 Mar 2011.

To cite this article: Ruay‐Nan Wu & Hua‐Wen Tsai (2002) Tracing the complex power of generators, Journal of the ChineseInstitute of Engineers, 25:3, 309-316, DOI: 10.1080/02533839.2002.9670705

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Journal of the Chinese Institute of Engineers, Vol. 25, No. 3, pp. 309-316 (2002) 309

TRACING THE COMPLEX POWER OF GENERATORS

Ruay-Nan Wu* and Hua-Wen TsaiDepartment of Electrical Engineering

National Taiwan University of Science and TechnologyTaipei, Taiwan 106, R.O.C.

Key Words: deregulation, open transmission access, complex powerflow tracing, transmission loss, power flow contribution.

ABSTRACT

It is important to estimate the responsibility attributed to eachgenerator in a deregulated environment. As a consequence, methodol-ogy for estimating the usage of the transmission system is indispensable.This paper proposes a method for tracing the output current as well asthe output power of individual generators in the power grid. The methodis based on a principle of proportional sharing of complex power(PSCP), which is proven to be in conformity with the circuit theorem.By using a 4-bus system, the application of the proposed principle iselaborated step by step. The contributions of each generator on branchflows, branch losses, and bus flows can be calculated. The tracingresults, based on the principle of proportional sharing of active power(PSAP), are also evaluated for comparison. Taking the results of PSCPas a benchmark, a deviation of up to 27% from PSAP has been found.The deviation becomes significant when tested on the IEEE 118-bussystem. In addition, the proposed PSCP is feasible to implement. As aconsequence, it is obvious that these interesting points are likely tojustify the further application of PSCP.

*Correspondence addressee

I. INTRODUCTION

The trend toward deregulation makes it moreand more important to compute the contributions ofindividual generators and loads to line flows, andline loss. The effort facilitates the allocation of totaltransmission cost to each network user in a reason-able way.

The transmission network mesh structure pro-vides a variety of possible paths through which elec-trical power can flow from the generators to the loads.Many researchers worldwide have been expendingtheir valuable efforts on estimating the usage of trans-mission service by each network user. Generallyspeaking, these methods can be classified into threecategories. They are the OPF based methods (Happ,

1994; Shirmothammadi et al., 1996), the sensitivityfactor based methods (Rudnick et al., 1995; Claytonet al., 1990), and the load flow based methods(Macgueen and Irving, 1996; Bialek and Tam, 1996;Bialek, 1996a; 1996b; 1997; Kirschen et al., 1997;Wu et al., 2000).

In recent years, a number of novel electricitytracing methodologies, based on the principle of “pro-portional sharing” of active power and reactive power(Bialek and Tam, 1996; Bialek, 1996a; 1996b; 1997,Kirschen et al., 1997; Wu et al., 2000) have been welldeveloped. This principle assumes that each unit ofactive power or reactive power leaving the node con-tains the same proportion of the inflows as the totalnodal flow. The concept of “proportional sharing” isadopted and implemented by matrix operation (Bialek

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310 Journal of the Chinese Institute of Engineers, Vol. 25, No. 3 (2002)

and Tam, 1996; Bialek, 1996a; 1996b; 1997). Bialek(1997) combined the concepts of the power flow trac-ing method with the MW-MILE method to estimatethe share of a particular generator or load in any lineflow. The techniques, such as domain, common, linkand state graph are used for large-scale power sys-tem application (Kirschen et al., 1997). A graphtheory based procedure is utilized to allocate thepower transfer between generators and loads (Wu etal., 2000). All the literature mentioned above adoptsthe intuitive concept of “proportional sharing” forseparately tracing active and reactive power. How-ever, the principle itself is left unproven theoretically.This causes doubt as to the correctness of the appli-cation of the principle in the deregulated powermarket. In order to overcome this deficiency, a pro-portional sharing of complex power (PSCP), insteadof active power, is proposed. This proposed prin-ciple is proven to indicate its theoretical correctness.Then, a simple procedure for implementing the PSCPis demonstrated. A simple 4-bus system is adoptedto illustrate the application of the proposed principleon the tracing of complex power. The contributionof each generator on the bus flow, bus load, branchflow, as well as branch losses is also presented. Inaddition, the tracing results of power flow based onthe principle of proportional sharing of active power(PSAP) are determined.

II. DERIVATION OF THE PSCP

Consider the one-line diagram of a simple

system as shown in Fig.1. Assume that the powerflow solutions have been obtained from state estima-tor or other means. The arrows in the figure are thedirection of power flow and current, which is adoptedas the direction of active power flow.

Use the π-circuit model, as shown in Fig. 2, torepresent branch k−n. The outflow current of bus k,by way of branch k−n, can be formulated as:

Ik, kn=Vk×yk+(Vk−Vn)×ykn

=Vk×[yk+ykn×(1− Vn

Vk)]

=Vk× Y kkn (1)

The admittance Y kkn is called the Miller’s equiva-

lent admittance (Millman and Grabel, 1987). In otherwords, each outflow branch of a specific bus can berepresented by a shunt admittance. Based on thisobservation, outflow branches k−m and k−n are re-spectively represented with Y k

km and Y kkn to result in

Fig. 3. The current arriving at bus k by way of branchi−k is expressed as:

Ik, ik =

Sk, ik

Vk

*

(2)

Since Y kkm and Y k

kn are parallel connected, it fol-lows that the portion Ik,ik contributed to outflow cur-rent Ik,kn of bus k can then be derived as:

Ik, knik = Ik, ik × Yk

kn

Ykkm + Yk

kn (3)

Therefore, the portion of Sk,kn coming through branchi−k is formed as follows.

S k, kn

ik = Vk × (Ik, knik )* = Vk × (Ik, ik)* ×

Vk2 × (Y k

kn)*

Vk2 × (Y k

km + Y kkn)*

= Sk, ik ×

Sk, kn

Sk, km + Sk, kn= Sk, kn ×

Sk, ik

Sk(4)

where the bus flow Sk is the total power passing

Fig. 1 The simple system for the demonstration of bus k

Fig. 2 The π-circuit model of branch k−n

Fig. 3 The equivalent circuit for Fig. 1

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R.N. Wu and H.W. Tsai: Tracing the Complex Power of Generators 311

through bus k, and is defined as:

Sk = S k, jkΣj∈ Uk

= S k, kmΣm∈ Dk

(5)

Equation (4) can be considered since all the in-flow power is fully mixed before flowing out to aspecific bus. As a consequence, the portion of eachoutflow power from the same source shares the sameratio. This verifies that the proportional sharing prin-ciple is valid for complex power, aside from the ac-tive power mentioned in the literature.

Since Sk,kn is the only inflow power of branch k−n, the portion of Sn,kn which is arriving at bus n in Fig.2 comes from power injected through branch i−k, re-tains the same ratio as that in (4). This can be writ-ten as:

S n, kn

ik = Sn, kn ×S k, ik

Sk(6)

By incorporating (4) and (6), the complex trans-mission loss of branch k−n, attributed to Sk,ik, can beexpressed as:

L kn

ik = S k, knik – S n, kn

ik = (Sk, kn – Sn, kn) ×Sk, ik

Sk

= L kn ×

Sk, ik

Sk(7)

where Lkn is the complex transmission loss of branchk−n.

It can be observed from (4), (6), and (7) that thecontribution ratios of branch i−k to the qualities ofSk,kn, Sn,kn, and Lkn are all the same, which is given as

Sk, ik

Sk.

III. PROPOSED ALGORITHM

Since the outflow power is related to the inflowpower, tracing the complex output on the power gridall the way back to the generators becomes possible.The generator-to-branch contribution ratio (GBR),

Akn

g =S k, kn

g

S k, kn=

S n, kng

S n, kn=

L kng

L kn, is defined as the ratio of

power flow through branch k−n attributed to the gen-erator at bus g. And, the generator-to-node contribu-

tion ratio (GNR), Akg =

S kg

Sk, is defined as the ratio of

power through bus k attributed to the generator g.From (4), (6), and (7), it can be seen that Akn

g = Akg .

For the purpose of implementing the PSCP, the com-putational procedure can be summarized as follows.Step 1 Obtain the system power flow solution from

the state estimator or other means. Thegenerator is treated as the inflow source and

marked as “determined”. The load is consid-ered as the outflow sink. According to the di-rection of active power flow, a directed graph(Bialek and Tam, 1996; Bialek, 1996a; 1996b;1997) is formed.

Step 2 Find an unmarked bus k. If the bus containsonly the source of generator g as inflow, go tostep 3.If the GBR’s of all the upstream branches ofthe bus have been determined, go to step 4.Otherwise, go to step 6.

Step 3 Initialize the contribution ratios by setting Ak

g = Akng =1 for n∈ Dk and Ak

h = Aknh = 0 for h≠g.

Go to step 5.Step 4 Compute all the GNR’s and GBR’s by the fol-

lowing formula for n∈ Dk

Ak

g = Akng =

Sk, ik

Sk× Aik

gΣi∈ Uk

Step 5 Mark the bus. Go to step 2.Step 6 Multiply the obtained GNR’s and GBR’s by

the corresponding quantities to obtain the con-tribution of each generator to the bus flows,branch flows, branch losses and bus load.

IV. EXAMPLE AND DISCUSSION

To illustrate the application of the PSCP withthe proposed algorithm, a simple system (Bialek,1997) as shown in Fig. 4, is adopted. The system hasfour buses, five branches, two generators and threeloads. The numbers indicated in the circles are thebus numbers.

1. The Implementation of Proposed Algorithm on4-Bus System

For the sake of demonstration, the followingimplementation is expressed step by step.

Fig. 4 A 4-bus power system

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312 Journal of the Chinese Institute of Engineers, Vol. 25, No. 3 (2002)

Step 1:The directed graph of step 1 is formed as shown

in Fig. 4.Step 2:

Since the generator G1 is the only inflow, bus 1is the first bus to be taken care of.Step 3:

The GNR and GBR’s of bus 1 are initialized as:

A1G 1 = A1 – 2

G 1 = A1 – 3G 1 = A1 – 4

G 1 = 1

A1G 2 = A1 – 2

G 2 = A1 – 3G 2 = A1 – 4

G 2 = 0

Mark bus 1 and go back to step 2.Step 2:

Since all the up-stream inflows have beendetermined, bus 2 should then be processed. Go tostep 4.Step 4:

The GNR’s and GBR’s of bus 2 can be deter-mined as:

A2

G 1 = A2 – 4G 1 =

S 2, 1 – 2

S 2× A1 – 2

G 1 = 0.2313 + j0.0370

= 0.2342 ∠ 9.0841°

A2G 2 = A2 – 4

G 2 =S G2

S 2= 0.7687 – j0.0370

= 0.7696 ∠ – 2.7542°

Since the bus flow comes from the generators,it can be checked that A2

G 1 + A2G 2 = 1. Mark bus 2 and

go to step 2.Step 2:

Bus 4 is the third bus to be handled.Step 4:

The GNR’s and GBR’s of bus 4 can be deter-mined as:

A4

G 1 = A4 – 3G 1 =

S 4, 1 – 4

S 4× A1 – 4

G 1 +S 4, 2 – 4

S 4× A2 – 4

G 1

= 0.5383 + j0.0290 = 0.5391 ∠ 3.0860°

A4

G 2 = A4 – 3G 2 =

S 4, 1 – 4

S 4× A1 – 4

G 2 +S 4, 2 – 4

S 4× A2 – 4

G 2

= 0.4617 – j0.0290 = 0.4626 ∠ – 3.5973°

Similarly, the relation A4G 1 + A4

G 2 = 1 holds.Step 2:

Bus 3 is the last bus to be processed.Step 4:

The GNR of bus 3 can be obtained as:

A3

G 1 =S 3, 1 – 3

S 3× A1 – 3

G 1 +S 3, 4 – 3

S 3× A4 – 3

G 1

= 0.8669 – j0.0092 = 0.8669∠ – 0.6109°

A3

G 2 =S 3, 1 – 3

S 3× A1 – 3

G 2 +S 3, 4 – 3

S 3× A4 – 3

G 2

= 0.1331 + j0.0092 = 0.1335∠ 3.9711°

Again, it is obvious that A3G 1 + A3

G 2 = 1. Markbus 3 and go to step 2.Step 2:

Since there is no unmarked bus left, go to step6.Step 6:

The power contributions from each generator tothe bus flows, branch flows, and losses are obtainedby multiplying the GNR’s and GBR’s with the spe-cific quantities to render the results as listed in Table1 to Table 4.

2. Results of 4-bus System Based on the PSCP

Table 1 summarizes the GNR’s between genera-tors and bus flows. It is noted that the GBR’s of thedown-stream branches are the same as the specificGNR. By multiplying the GNR by the bus flow, thecontribution of a particular generator to the bus flowcan then be obtained. For instance, the bus flowof bus 3 is of 300+j100 and A3

G 2=0.1335 ∠ 3.97°.Therefore, the contribution of G2 can be found as thevalue of 39.02+j16.09. All these contributions are

Table 1 The PSCP based GNR’s

Bus1 2 3 4

Gen.

0.2342 0.8669 0.5391G1 1 ∠ 9.0841° ∠ -0.6109° ∠ 3.0860°

0.7696 0.1335 0.4626G2 0 ∠ -2.7542° ∠ 3.9711° ∠ -3.5973°

Sum 1 1 1 1

Table 2 The bus flow distribution based on PSCP

Bus1 2 3 4

Gen.

400 59 260.98 149.33G1 +j125 +j36 +j83.91 +j64.20

214 39.02 133.67G2 0

+j76 +j16.09 +j39.80

400 273 300 283Sum

+j125 +j112 +j100 +j104

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R.N. Wu and H.W. Tsai: Tracing the Complex Power of Generators 313

computed and tabulated in Table 2. In Table 2, thelast row presents the bus flow.

By utilizing the GBR’s between generators andbranches, the flow on a specific branch can be di-vided into the contribution attributed to eachgenerator. Take branch 4-3 as an example. The in-flow and outflow of branch 4-3 are of the value of83+j24 and 82+j40, respectively. The contributionratio A4– 3

G 1 is 0.5391∠ 3.09°. Then, the complex trans-mission loss of this branch attributed to G1 is evalu-ated by (7).

[83+j24−(82+j40)]×0.5391∠ 3.09°

=1.0027−j8.5844

The same processes are conducted for all thepossible combinations of generators and branches toresult in Table 3. As shown in Table 2 and Table 3,the contribution of generator to bus flow and branchflow can be considered as the usage of the specificfacility. This observation makes the allocation offixed costs of buses and lines feasible. Besides, thefar right-hand side column lists the total transmissionloss attributed to each generator. This loss is veryessential in determining operation costs for wheelingcharges.

Based on PSCP, the source distribution of loadcan be evaluated by multiplying the load demand bythe specific GNR. This distribution is tabulated in

Table 4. By precisely knowing the contribution ofeach generator to each load, bilateral transactions canbe carried out in a fair environment.

Finally, it is of interest to observe that the realpart of the entry at the third row and the sixth columnof Table 3, is of negative value. The calculation ofthis figure is repeated hereafter for the investigationof the reason.

L 4 – 3G 2 = L 4 – 3 ⋅ A4 – 3

G 2 = (1 – j16) ⋅ 0.4626∠ – 3.5973°

= – 0.0027 – j7.4156

As you can see, the phase angle of branch loss L4-3

is −86.42°. After adding the phase angle of contri-bution ratio, the phase angle of L 4 – 3

2 becomes−90.0210°, which results in negative active power.This phenomenon states that the contribution of G2

to branch 4-3 has a negative real part projection sincethe contribution ratio is a complex quantity other thana scale quantity. Although no concrete explanationhas been reached, it is for sure that the phenomenonis worthy of further investigation.

3. Results of 4-bus System Based on PSAP

It is of significant meaning to understand thedeviation between the results based on PSCP andthose of PSAP. Since the application of PSAP putsemphasis on the allocation of active power, somemeans such as dc power flow technique, should betaken to obtain the distribution of active power in theentire power grid. However, using the same exampleof Fig. 4, the active power distribution results fromdirectly excluding the reactive power from all thequantities for simplification. Similarly, by exclud-ing the reactive power from the relations of PSCP,Eqs. (4) and (6) are respectively reformed as follows.

Pk, kn

ik = Pk, kn ×Pk, ik

Pk

Pn, kn

ik = Pn, kn ×Pk, ik

Pk

Table 3 The complex loss distribution based on PSCP

Branch1-2 1-3 1-4 2-4 3-4 Sum

Gen.

0.3886 1.0027 12.3913G1 1-j40 7+j44 3-j19

+j0.5365 -j8.5844 -j23.0479

1.6114 -0.0027 1.6087G2 0 0 0

+j1.4635 -j7.4156 -j5.9521

Sum 1-j40 7+j44 3-j19 2+j2 1-j16 14-j29

Table 4 The source distribution of loads based onPSCP

LoadW2 W3 W4 Sum

Gen.

21.28 260.98 105.35 387.61G1 +j15.26 +j83.91 +j48.87 +j148.05

78.72 39.02 94.65 212.39G2 +j34.74 +j16.09 +j31.13 +j81.95

100 300 200 600Sum

+j50 +j100 +j80 +j230

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314 Journal of the Chinese Institute of Engineers, Vol. 25, No. 3 (2002)

where P denotes active power.The above equations exactly imply the concept

of PSAP. The formula of GNR and GBR can also bemodified as

Akg = Akn

g =Pk, ik

PkΣ

i∈ Uk× Aik

g

By applying the proposed procedure, outlinedin section III, on PSAP, results similar to those ofPSCP are rendered and tabulated in Table 5-8.

4. Comparison between PSCP and PSAP

Since the PSCP fulfills the circuit theorem, theresults based on the PSCP can then be regarded asbenchmarks for comparison. By the comparison ofthe bus flow distribution between Table 2 and Table6, the deviations of PSAP are in the range of -0.4613%to 0.0776%. As a matter of fact, the obtained devia-tion is of no significance. This observation impliesthat both the principles of PSCP and PSAP share simi-lar characteristics.

Comparing the active loss of Table 3 and Table7, a deviation range of -3.49% to 26.9% can beobtained.

5. Simulation on the IEEE 118-bus System

The proposd algorithm with PSAP and PSCPwere also successfully tested on the IEEE 118-bussystem in which there are 54 generators, 118 buses,179 branches and 99 loads. Some of the results aretabulated as shown in Table 9. Based on the PSCP,the transmission loss of line 21 is allocated to fivegenerators. The value of zero in the second columnmeans that line 21 is not within the flow domain ofG4 and G8 based on PSAP. Besides, the active powerlosses attributed to G10 are of opposite sign. It is

noted that the allocation patterns are quite different.Since the deviation is significant, one should be cau-tious in estimating operation costs by PSAP usage.

6. Computational Efficiency

All the simulations are tested in the MATLABenvironment with a Pentium II-300 personalcomputer. The evaluated computational time is tabu-lated in Table 10. Despite the sizes of the examplesystems, the computational efficiencies of the PSAPand the PSCP are of the same order. Therefore, theproposed algorithm with PSCP is feasible in the de-regulated environment.

V. CONCLUSION

The principle of PSCP for tracing the output ofgenerators has been proposed in this paper. It statesthat the sharing of power flow should be proportionalto the complex power. The principle, conforming tothe circuit theorem, is proven.

A simple procedure for implementing the PSCP

Table 5 The PSAP based GNR’s

Bus1 2 3 4

Gen.

G1 1 0.2161 0.8705 0.5263G2 0 0.7839 0.1295 0.4737

Sum 1 1 1 1

Table 6 The bus flow distribution based on PSAP

Bus1 2 3 4

Gen.

G1 400 59 261.16 148.96G2 0 214 38.84 134.04

Sum 400 273 300 283

Table 7 The active loss distribution based onPSAP

Branch1-2 1-3 1-4 2-4 3-4 Sum

Gen.

G1 1 7 3 0.4322 0.5263 11.9585G1 0 0 0 1.5678 0.4736 2.0414

Sum 1 7 3 2 1 14

Table 8 The source distribution of loads based onPSAP

LoadW2 W3 W4 Sum

Gen.

G1 21.61 261.16 105.27 388.04G2 78.39 38.84 94.73 211.96

Sum 100 300 200 600

Table 9 Test results on IEEE 118-bus system

Energy Loss in Line 21

Generator PSCP Based PSAP BasedP + Qi P

G4 -0.0001 0G6 -0.0002i 0.0001G8 0.0001 - 0.0003i 0G10 -0.0021 - 0.0015i 0.0007G12 0.0046 - 0.0049i 0.0018Sum 0.0025 - 0.0068i 0.0025

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R.N. Wu and H.W. Tsai: Tracing the Complex Power of Generators 315

is also provided and elaborates on a 4-bus examplesystem step by step. It is noted that the allocation ofthe transmission line loss results in a negative valueof active power although it seldom happens. This isbecause the contribution ratio used here is a complexnumber, rather than a real number, traditionally usedin PSAP. The physical meaning of this phenomenonis surely worthy of further investigation before realapplication.

Based on the observation of the bus flow distri-bution results and implementation methodology,PSCP and PSAP exhibit many similarities. It is likelythere will be no difficulties in adopting the proposedPSCP to replace PSAP. In addition, a deviation ofup to 27% on line loss allocation is obtained by PSAPcompared to PSCP results. This deviation becomessignificant when tested on the IEEE 118-bus system.Therefore, one should be cautious in estimating theoperation cost when using PSAP. All the above su-perior results are believed to make the proposed PSCPattractive for application in the deregulated environ-ment of the present day power industry.

NOMENCLATURE

Akg the generator-to-node contribution ratio (GNR)

of generator g to bus k. Akn

g the generator-to-branch contribution ratio(GBR) of generator g to branch k−n.

Dk the set of down-stream buses of bus k.i−k the branch i−k connecting bus i and bus k.Ii the current, that, passing through bus i, is called

the bus current.Ik,ik inflow current arriving at bus k, which origi-

nated from bus i, by way of branch i−k.Ik,kn outflow current from bus k, toward bus n, by

way of branch k−n. I k, kn

ik the portion of Ik,kn that comes from Ik,ik. I n, kn

ik the portion of In,kn that comes from Ik,ik.Lkn the complex power loss on branch k−n.

L kng the contribution of generator g to the branch

loss Lkn. L kn

ik the contribution of Sk,ik to the branch loss Lkn.Si the complex power, that, passing through bus

i, is called the bus flow.SGi the generation of generator i.SWi the power demand of load i.Sk,ik inflow complex power arriving at bus k, which

originated from bus i, by way of branch i−k.

Sk,kn outflow complex power from bus k, toward busn, by way of branch k−n.

S kg the portion of Sk that comes from generator g.

S k, kng the portion of Sk,kn that comes from generator

g. S n, kn

g the portion of Sn,kn that comes from generatorg.

S k, knik the portion of Sk,kn that comes from Sk,ik.

S n, knik the portion of Sn,kn that comes from Sk,ik.

Uk the set of up-stream buses of bus k.Vi voltage of bus i (or node i).yk he shunt admittance of the π circuit of branch

k−n, on the bus k side.ykn the series admittance in the π-circuit of branch

k−n. yk

kn the Miller’s admittance at bus k, to representthe effect of branch k−n.

REFERENCES

1. Bialek, J., and Tam, D. B., 1996, “Tracing TheGenerators’ Output,” Proc. of International Con-ference on (Conf. Publ. No. 419) Opportunitiesand Advances in International Power Generation,IEE, pp. 133-136.

2. Bialek, J., 1996a, “Identification of Source-SinkConnections in Transmission Networks,” FourthInternational Conference on (Conf. Publ. No.421) Power System Control and Management,IEE, pp. 200 -204.

3. Bailek, J., 1996b, “Tracing the Flow of Elec-tricity,” IEE Proceedings-Generation, Transmis-sion and Distribution, Vol. 143, pp. 313-320.

4. Bialek, J., 1997, “Topological Generation andLoad Distribution Factors for Supplement ChargeAllocation in Transmission Open Access,” IEEETransactions on Power Systems, Vol. 12, pp. 1185-1193.

5. Clayton, J. S., Erwin, S. R., and Gibson, C. A.,1990, “Interchange Costing and Wheeling LossEvaluation by Means of Incrementals,” IEEETransactions on Power Systems, Vol. 5, pp. 759-765.

6. Happ, H. H., 1994, “Cost of Wheeling Metho-dologies,” IEEE Transactions on Power Systems,Vol. 9, pp. 147-156.

7. Kirschen, D., Allan, R., and Strbac, G., 1997,“Contributions of Individual Generators to Loadsand Flows,” IEEE Transactions on Power System,Vol. 12, pp. 52-59.

8. Macqueen, C. N., and Irving, M. R., 1996, “AnAlgorithm for The Allocation of Distribution Sys-tem Demand and Energy Losses,” IEEE Trans-actions on Power Systems, Vol. 11, pp. 338-343.

9. Millman, J., and Grabel, A., 1987, Microelec-tronics, 2nd ed., McGraw-Hill International

Table 10 Computational time

4-bus IEEE IEEE IEEESystem system 14-bus 24-bus 118-busPSCP < 0.01 sec 0.05 sec 0.16 sec 6.48 secPSAP < 0.01 sec 0.06 sec 0.22 sec 6.75 sec

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316 Journal of the Chinese Institute of Engineers, Vol. 25, No. 3 (2002)

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Editions, Singapore.10. Rudnick, H., Palmar, R., and Fernandez, J. E.,

1995, “Marginal Pricing and Supplement Cost Al-location in Transmission Open Access,” IEEETransactions on Power Systems, Vol. 10, pp.1125-1142.

11. Shirmothammadi, D., Filho, X. V. F., Gorenstin,B., and Pereira, M. V. P., 1996, “Some Funda-mental Technical Concepts About Cost BasedTransmission Pricing,” IEEE Transactions on

Power Systems, Vol. 11, pp. 1002-1008.12. Wu, F. F., Ni, Y., and Wei, P., 2000, “Power

Transfer Allocation for Open Access Using GraphTheory - Fundamentals and Applications in Sys-tems without Loopflow,” IEEE Transactions onPower Systems, Vol. 15, pp. 923-929.

Manuscript Received: Nov. 13, 2001Revision Received: Feb. 06, 2002

and Accepted: Apr. 02, 2002

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