In2515231531

P.Poornachanra Rao, L.Rajagopal Reddy/ International Journal Of Engineering Research And

Applications (IJERA) ISSN: 2248-9622 www.ijera.com

Vol. 2, Issue 5, September- October 2012, pp.1523-1531

1523 | P a g e

TRANSMISSION NETWORK COST ALLOCATION USING

IMPEDANCE MATRIX METHOD

P.POORNACHANRA RAO *

Assistant Professor

Department of Electrical & Electronics Engineering

Sri Venkatesa Perumal College of Engineering & Technology

L.RAJAGOPAL REDDY ** PG Student

Department of Electrical & Electronics Engineering

Sri Venkatesa Perumal College of Engineering & Technology

ABSTRACT This paper addresses the problem of

allocating the cost of the transmission network

to and demands. A physically-based network

usage procedure is proposed. This proceeds

exhibits desirable apportioning properties and is

easy to implement and understand. A case study

based on the IEEE 24-bus system is used to

illustrate the working of the proposed technique.

Some relevant conclusions are finally drawn.

INDEX TERMS—Network usage, transmission

cost allocation Zavgbus

NOTATION

The notation used throughout this paper is stated

below for quick reference

Data

C jk Cost of line jk ($/h)

I j Nodal current i (A)

I jk Current through line jk (A)

n numbers Of buses

P Di Active power consumed by the

demand located at bus i (W)

P Gi Active power produced by the generator

located at bus i (W)

P jk Active power flow through line jk (W) S jk Complex power flow jk through line

calculated at bus (VA). V jk Nodal voltage at bus j (V).

Z bus Impedance matrix

Z jk Element jk of the impedance

matrix

Zavgbus average impedance matrix

B. Results a

jjk Electrical distance between bus i and line

jk

C Di Total transmission cost allocated to the

demand located at bus i ($/h)

C Gi Total transmission cost allocated to the

generator located at bus i ($/h)

CDijk Transmission cost of line jk allocated to

the demand located at bus i ($/h)

CGijk Transmission cost of line jk allocated to

the generator located at bus i ($/h)

P ijk Active power flow through line ik

associated with the nodal current i (W).

r jk Cost rate for line jk ($/W or H)

U jk Usage of line jk (W)

U Di

jk Usage of line jk allocated to the demand

located at bus i (W)

U Gi jk Usage of line allocated jk n to the

generator located at bus i (W)

U ijk Usage of line jk associated with nodal

current i (W)

CIGI The cost of contribution of generator I

using all lines in the network

interchanging „from bus‟ and „to bus

CIDI The cost of contribution of load I using all

lines in the network interchanging „from bus‟ and „to bus

CGIavg the average cost contribution of generator

i using the line jk

CDIavg the average cost contribution of generator

i using the line jk

Cavg the total average cost contribution of

generator i using the line jk

Cavg the total average cost contribution of load

i using the line jk

1. INTRODUCTION A. Motivation and Approach

THIS PAPER provides a methodology to

apportion the cost of the transmission network to

generators and demand that use it. How to allocate

the cost of the transmission networks an open

research issue as available techniques embody

important simplifying assumptions (see the literature review below),which may render




1524 | P a g e

controversial results. This paper contributes to seek

an appropriate solution to this allocation problem

using a usage-based procedure that relies on circuit

theory. The proposed technique consists of the

following steps.

1) The active power flow of any transmission line

is apportioned among all nodal currents. 2) Based on the above apportioning, the cost of any

line is allocated to all generators and demands.

3) The procedure is repeated for all lines.

B. Literature Review

A brief description of the most significant

proposals reported in the technical literature on the

allocation of the cost of the transmission network

among generators and demands follows. In the

traditional pro rata method, reviewed in [1] and [2],

both generators and loads are charged a flat rate per

megawatt-hour, disregarding their respective use of individual transmission lines.

Other more elaborated methods are flow-

based [3]. These methods estimate the usage of the

lines by generators and demands and charge them

accordingly. Some flow-based methods use the

proportional sharing principle [4], [5], which

implies that any active power flow leaving a bus is

proportionally made up of the flows entering that

bus; such that Kirchhoff‟s current law is satisfied.

Other methods that use generation shift distribution

factors [6], are dependent on the selection of the slack bus and lead to controversial results.

The usage-based method reported in [7]

and [8] uses these-called equivalent bilateral

exchanges (EBEs). To build theEBEs, each

demand is proportionally assigned a fraction of

Each generation, and conversely, each generation is

proportionally assigned a fraction of each demand,

in such a way as both Kirchhoff‟s laws are

satisfied.

The technique presented in this paper is

related to the allocation of the cost of transmission

losses based on the Zbus previously reported and explained in [9]. It should be emphasized that all

transmission lines must be modeled including

actual shunt admittances. Doing so the impedance

matrix Z bus presents an appropriate numerical

behaviour.

A salient feature of the proposed

technique is its embedded proximity effect, which

implies that a generator/demand uses mostly the

lines electrically close to it. This is Zbus not

artificially imposed but a result of relying on circuit

theory. This proximity effect does not take place if the EBE principle is used [7], as this principle

allocates the production of any generator/demand

proportionally to all loads/generators, which

implies treating similarly close by” and “far away”

lines.

Other techniques require stronger

assumptions, which diminish their practical

interest. Applying the proportional sharing

principle implies imposing that principle, and using

the pro rata criterion implies disregarding

altogether network locations.

Particularly, it should be noted that the proposed

methodology simply relies on circuit laws while the

proportional sharing technique, on top of circuit laws, relies on the proportional sharing principle,

not needed by the Zbus methodology

C. Contributions

The contributions of this paper are stated

below. The proposed technique:

1) Uses the contributions of the nodal currents to

line power flows to apportion the use of the lines;

2) Shows a desirable proximity effect; that is, the

buses to a line retain a significant share of the cost

of using that line;

3) Is slack independent. 4) Does not require an a priori definition of the

proportion in which to split transmission costs

between generators and demands. Specifically, the

main contribution of this paper is a physical-based

technique to identify how much an individual

power injection “uses” the network.

D. Paper Organization

The remaining of this paper is organized

as follows. Section II formulates the problem and

describes the proposed apportioning technique. A simple four-bus example is used to

Illustrate how the proposed technique works and to

show it salient features. Section III provides and

analyzes results from a case study based on the

IEEE 24-bus Reliability Test System (RTS).

Finally, Section IV gives some relevant

conclusions

II. Zavg

BUS NETWORK COST

ALLOCATION METHODOLOGY A. Problem Statement

The purpose of the methodology presented

in this paper is to allocate the cost pertaining to the




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transmission lines of the network to all the

generators and demands. Once a load flow solution

is available, the proposed method determines how

line flows depend on nodal currents. This result is

then used to allocate network costs to generators

and demands.

B. Background

Consider the complex power flow S jk

computed at bus and flowing through the line

connecting j bus to k bus as shown in Fig. 1 [10].

As the power flow solution is known, we select the

direction of the complex power flow so that Pij >0

.The complex power flow is S jk

*

jkS j jkV I (1)

Using the matrix, Z bus the voltage at node is given

by

(2)

Where is the element jk of Z bus and „n‟is

the total number of buses? Current through the line

jk can be written as

(3)

Substituting (2) in .3) and rearranging

(4)

At this stage, we wish to make equ (4) as

dependent on Pgen, Qgen, Pload and Qload of the bus-i. This would help in building up the relevant

mathematical support in identifying the

contribution of each generator and load on the line

flow jk.this aspect is considered in proposing new

technique From the load flow analysis, the nodal

current can be written as a function of active and

reactive power generations at bus i ( and

respectively) and the active and reactive

load demands at bus i ( and

respectively ) as

(5)

Note that the first term of the product in (2.4) is

constant, as it depends only on network parameters. Thus, (2.4) can be written as

i

N

I

i

jkJKIaI

1 (6)

Where sh

jkjijkkiji

i

jk yzyzza (7)

Observe that the magnitude of parameteri

jka

provides a measure of the electrical distance

between buses i and line jk.

Substituting(.6) in (1)

n

i i

i

jkj

n

i i

i

jkjjk IaVIaVS1

***

1 (8)

Then, the active power through line jk is

n

i i

i

jkjjk IaVP1

**

(9)

Or, equivalently

n

i i

i

jkjjk IaVP1

**

(10)

Note that the terms in the summation

represent contribution due to each bus - Ii Thus; the

active power flow through any line can be

identified as function of the nodal currents in a

direct way. Then, the active power flow through

line jk due to the nodal current Ii is

**

i

i

jkj

i

jk IaVP (11)

2.3. Transmission Cost Allocation

Following (1),we define the usage of line jk due to

nodal current as the absolute value of the active

power flow componenti

jkP ,i.e.,

i

jk

i

jk PU (12)

That is, we consider that both flows and counter-

flows do use the line. The total usage of line jk is

then power flow component

i

jk

N

ijk UU

1 (13)

Then, we proceed to allocate the use of

transmission line jk to any generator and demand.

Without loss of generality, we consider at most a

single generator and a single demand at each node

of the network.

Then, the usage of line jk apportioned to the

generator or demand located at bus is stated below.

If bus i contains only generation, the usage allocated to generation pertaining to line jk is

i

jk

Gi

jk UU (14)

On the other hand, if bus contains only demand, the

usage allocated to demand pertaining to line jk is

i

jk

Di

jk UU (15)

Else, if bus i contains both generation and demand,

the usage allocated to the generation at bus

pertaining to line jk is

i

jkDiGiGi

Gi

jk UPPPU (16)

And the usage allocated to the demand at bus

pertaining to line jk is

1

n

j ji i

i

V Z I

jiZ

( ) sh

jk j k jk j jkI V V y V y

1

( )n

sh

jk ji ki jk ji jk i

i

I Z Z y Z y I

i

genP

i

genQ

i

loadPi

loadQ

*

( ) ( )i i i i

gen load gen load

i

i

P P j Q QI

V




1526 | P a g e

i

jkDiGiDi

Di

jk UPPPU (17)

The complex power flow components

through line jk due to individual power generations

and load demands have been found out. Having

found the contributions of individual generators

and demands in each of the line flows and the

usage of line by those generations and demands,

allocation of transmission cost among generators

and demands can be found out. Let in $/h,

represents the total annualized line cost including

operation, maintenance and building costs [8].

Then the per unit usage cost rate can be

written as

(18)

Using the per unit cost rate, we can write, ,

the allocated cost of line to the generator „i'

located at bus „i' is

(19)

In the same way, we can write, , the allocated

cost of line to the demand „i' located at bus „i'

is

(20)

The total transmission network cost, allocated

to generator „i' is the sum of the individual cost

components of each line due to that generator.

(21)

Find the cost of contribution of generator

i using all the lines in the network interchanging

„from bus‟ and „to bus

GI

jk

GI CICI (22)

Where „n line‟ represents the set of all transmission

lines present in the system. Similarly, the total

transmission cost, allocated to the demand „i‟

is given as

(23)

Find the cost of contribution of load i using all the

lines in the network interchanging „from bus‟ and

„to bus‟

DI

jk

DI CICI (24)

Find the average Cost contribution of generator

i using the line jk

2

CCI GI

jk

GI

jk

avgc

(25)

Find the average cost contribution of

load i using the line jk

2

CCI DI

jk

DI

jk

avgc

(26)

Find the average cost contribution of generator

i using all the lines in the network

Gi

jkjk

GI CavglinesallforCavg (27)

Find the average costcontribution of load i using all

the lines in the network

Di

jkjk

DI CavglinesallforCavg(28)

D.Effect of Flow Directions Given a converged power flow, the

technique allows computing the actual usage that any generating unit or demand makes of any

network line. This is raw information that needs to

be processed to generate appropriate network usage

tariffs. A similar situation occurs with electric

energy LMPs, which have to be properly

aggregated to produce stable consumer prices.

In order to avoid hourly volatility,

network usage values can be aggregated by peak

and off-peak periods and perhaps also by season

within a yearly framework. Given the annualized costs of all network lines, usage charges are then

computed ex post for all considered periods within

the year. Alternatively, usages can be predicted and

aggregated, and tariff constructed ex ante. This

second procedure has to be complemented with

yearly adjustments due to prediction errors.

Using the above aggregating techniques,

generating units or demands face stable network

usage charges within a given year, which allows

them to properly develop strategies (pool bidding,

forward contract involvements, and the like) taking into account network usage costs.

F. Example

To illustrate the working of the Zbus and

the Zavgbus methods for transmission cost allocation,

we consider the four-bus system depicted in Fig. 2.

Note that all buses are similar in terms of

generation/demand. The five lines in the system

have the same values of series resistances and

reactance: 0.01275 and 0.097 p.u., respectively,

and the shunt admittance is identical for the five lines: 0.4611 p.u. Fig. 2 provides the active power

generated and consumed at each bus and the active

power flow through the five lines. Finally, note that

the cost of each line is considered to be

proportional to its series reactance; thus,

jkC

jkr

jk

jk

jk

Cr

U

Gi

jkC

jk

Gi Gi

jk jk jkC r U

Di

jkC

jk

Di Di

jk jk jkC r UGiC

( , )

Gi Gi

jk

j k nline

C C

DiC

( , )

Di Di

jk

j k nline

C C




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This four-bus system allows

visualizing the proximity effect, as it is expected

that buses directly connected to a line would be

apportioned most of the usage of that line.

Tables I –VI provide the results of the

transmission cost al-location to each bus. The

results obtained are compared with

TABLE I

TRANSMISSION COST ALLOCATION

OF LINE 1 (1, 2) FOR EACH BUS

TABLE II

TRANSMISSION COST ALLOCATION OF

LINE 2 (1,3) FOR EACH BUS

TABLE III


OF LINE 3 (1,3) FOR EACH BUS

TABLE IV



TABLE V



TABLE I V

TOTAL TRANSMISSION COST

ALLOCATION FOR EACH BUS

Those obtained using other methods, namely, EBE

[7], proportional sharing (PS) [4], and pro rata

(PR) [1].

Those obtained using other methods, namely, EBE

[7], proportional sharing (PS) [4], and pro rata

(PR) [1].

Observing Tables I –V, it can be noted

that, for all the lines, the zavg bus methods have the

property that they allocate a significant amount of

the cost of each line to the buses directly connected

to it. For lines 1, 2, 3, and 5, the two buses with the highest line usage are these at the ends of the

corresponding line. Taking into account that the

power injected and extracted at each bus is very

similar; the results reflect the location of each bus

in the network. Note that the behaviour of other

procedures is different. For instance, the zavg bus

allocate most of the usage of line 5 (between buses

1000jk jkC x




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3 and 4) to buses 3 and 4, while the EBE to buses

1, 2, and 3 and the PS to buses 2 and 4.Note also

that, for line 4 (between buses 2 and 4), the results

provided by the zbus method are somewhat

different, since the allocation to bus 1, not directly

connected to line 4, is also relevant. This happens,

mostly, because the power injected at bus 1 is greater than the power extracted at bus 4: 261.3 and

250.0 MW, respectively. In addition, the absolute

values of the electrical distance terms a1 and a4 are

identical, as well as the values of z12 andz24, which

makes buses 1 and 4 being at the same electrical

distance to line 2–4. Nevertheless, the cost

allocated to bus 4 is significant and similar to the

cost allocated to bus 1. It should also be noted that

for line 4, the results provided by the zavg variant

allocate the highest portion of line usage to

Comparing method Z bus and, it can be

concluded that the zavg methods mouths the trend of the zbus one (as well as of other methods) to allocate

a higher portion of usage to generating buses

versus demand buses. Finally, Table VI provides

The total cost allocated to each bus for the

use of the entire net-work using the different

methods considered. Note that results are

significantly different. Note also the similar pattern

of allocation provided by methods Zavgbus and EBE

We conclude this example stating that the above

results illustrate adequately the features of the Zbus

methodology in relation to other methods and show its appropriate behaviour.

III. CASE STUDY The IEEE 24-bus RTS [11] depicted in

Fig. 3 is considered for this case study. The same

five methods considered in the previous example

are used in this section. The converged power flow

corresponds with the IEEE RTS peak load, taking

place on the Tuesday of week 51 from 5 P.M. to 6 P.M. All required data pertaining to the IEEE RTS

can be found in [11]. Note also that the costs of the

lines are considered to be proportional to their

respective series reactance

A. Results

Tables VII–X provide the transmission cost

allocation to generators and demands for lines 23

(bus 14 to bus 16) and 11 (bus 7 to bus 8),

respectively. These lines, highlighted in Fig.3, are

selected for the two reasons below. In terms of

transmission cost allocation, line 23 behaves as most lines throughout the system do, thus being a

representative line of the network. Conversely, line

11, which is peripheral, exhibits clearly the

proximity effect discussed in the example above

TABLE VII

LINE 23 TRANSMISSION COST

ALLOCATION TO GENERATORS

TABLE VIII


ALLOCATION TO DEMANDS




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TABLE IX



TABLE X


ALLOCATON TO DEMANDS

Additionally, Tables XI and XII show

the total transmission cost allocation for all the

generators and demands, respectively.

B. Result Analysis

Table VII shows that all methods allocate

most of the costs of using line 23 to generators 21,

22, and 23. This is expected be-cause all these

generators are electrically close to that line, and

their productions are comparatively high. As a

result of eliminating counter-flows, the PS procedure does not allocate any cost of line 23 to

generator 23, which is a questionable result.

Table VIII shows that the Z bus zavg

bus EBE

and PS methods allocate most of the cost of line 23

to demand 14. This is also reasonable because that

demand is comparatively high and is directly

connected to line 23. However, observe the

significant allocation differences among methods.

Tables IX and X show that, for the zbus

and zavg methods, almost 100% of the cost of line

11 is allocated to bus 7, split between its generation

and demand. This happens because the only way in which bus 7 can inject to or extract power from the

network is through line 11, as it can be seen in Fig.

3. Regarding the other methods, the EBE method

splits the power generated at bus 7 proportionally

to all the demands of the system; thus, no

significant proximity effect takes place. Because of

the existence of counter-flows, the PS method

allocates no cost to demand 7. This last result is not

desirable, as demand 7 uses line 11.

Additionally, for the zbus and zavg bus

methods, it can be noted that a relatively small portion of the total network cost is allocated to bus

7, because this bus is placed at the network

boundary (see Tables XI and XII). Note also that

for the zbus and zavg methods, the amount of the cost

of line 11 allocate to bus 8 (0.000117 and 0.0784

, respectively, demand only) is much smaller

than that allocated to bus 7 (61.4 and 59.1 , respectively, demand plus generation). However,

total network usage allocated to bus 8 (174.5 and

179.96 , respectively, demand only) is almost

as high as the allocation to bus 7 (179.9 and 180.74

, respectively, demand plus generation). This can be considered a reasonable result and a

consequence of bus 8 being is a less isolated spot

of the network, which allows bus 8 a more

intensive use of the network. Table XI shows that

the zbus and zavg bus methods allocate most of the

total cost of the network to generators 21, 22, and

23, just like the other methods. Considering that

these generators are the highest producers in the

network and that they feed a significant amount of

the demand of the system, this is an appropriate result. For the demands, using the zbus and zavg

bus

methods, the net-work costs are mostly allocated to

demands 3 and 8, as shown in




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TABLE XI

TOTAL TRANSMISSION COST


TABLE XII

TOTALTRANSMISSION COST

ALLOCATION TO DEMANDS

XII. This happens because buses 3 and 8

have the highest demands, and they are located far

away from the main generators: 21, 22, and 23.

Therefore, buses 3 and 8 use many of the lines in the network.

Finally, observe that all methodologies

tend to allocate significantly higher usage to

generators. The proposed zbus technique follows

this trend being the average allocation of net

generating buses 177 , while the zavg method

smoothes this trend allocating on average 172

to generating buses. The EBE, PS, and PR

procedures result in average allocations of 151,

172, and 144, respectively. Average allocation for net demand buses of

Zbus zavg bus EBE, PS, and PR

procedures are respectively, 95, 99, 81,

114, and 83 .

IV. CONCLUSION Both the zbus and the zavg

bus procedures to

allocate the cost of the transmission network to

generators and demands are based on circuit theory. They generally behave in a similar manner

as other techniques previously reported in the

literature. However, they exhibit a desirable

proximity effect according to the under-lying

electrical laws used to derive them. This proximity

effect is more apparent on peripheral rather isolated

buses. For these buses, other techniques may fail to

recognize their particular locations. The zavgbus

variant smoothes the trend of the zbus method (as

well as of other techniques) to allocate a higher line

usage to generators versus demands.

We have performed extensive numerical

simulations and encountered neither numerical zbus

induced ill-conditioning nor unreasonable results.

Thus, we conclude that the proposed methods are

appropriate for the allocation of the cost of the

transmission network to generators and demands,

complement existing methods, and enrich the

available literature

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