Graph theory application to deregulated power system

Starting from a power load flow solution, the method first identifies the buses whether are load buses or generation buses by getting the net power.. To charge effectively, it has bec

GRAPH THEORY APPLICATION TO DEREGULATED POWER SYSTEM

Department of Electrical & Computer Engineering

Tennessee Technological University Cookeville, Tennessee 38505

Absfmet - This paper presents a methodology using

graph theory to determine the specific generator

contributions to different loads and the transmission

system usage of different utilities Starting from a

power load flow solution, the method first identifies the

buses whether are load buses or generation buses by

getting the net power Then a directed graph is formed

with vertices represent buses and the edges represent

transmission lines Second constructs all possible

directed and rooted subgraphs of the system from its

generator vertices Notice that the union of these

subgraphs is the original directed graph Using

proportionality and average assumptions, it is possible

to calculate the specific generator contributions to

different loads and the transmission system usage of

different utilities

I INTRODUCTION

In many countries, the power supply industry is

undergoing rapid changes in structure and operation For

years, the regulated monopoly structure of power supply

industry allowed very little option to customers in

choosing the electricity supplier They have to buy their

power from the local utility [I] The process of

deregulating this industry is affording choices to all the

stakeholders The new electricity market has some

common features with other markets: product (electricity),

sellers (generators), buyers (loads), and transportation

system (electrical transmission grid) While the market is

decomposed into components, many fonns of charges in

each component have been introduced To charge

effectively, it has become very important to determine

which generators are supplying a particular load, how

much use each generator is making of a transmission line

and what is each generator’s contribution to the system

Unfortunately, no expert or utility company in the

world power industries can ever answer these questions

They all know the advantages and benefits of these

matters However, no one is taking any action due to the

complexity of this issue One says, “it is too expensive to

find out the answers when the impact on the current

system is very trivial”

In North America, the current practices of selling

buying power between two security coordinators are based

loss [2]

on the straightforward contracts or the cost of wheeling methodologies When a selling buying tag is done, the buying security-coordinators are making sure the amount

of power flowing into the system through the tie lines A

charge is imposed on the buyer with the fixed agreement prices or the current-hour-market prices; regardless the generator location, the distance of transmission and the actual line losses Those charging methods are very unsecured, and not accurate at all but they serve the purpose very well

A paper [2] published in 1997 proposed a method to

solve this issue The authors described a method to calculate the contribution of each generator to a group of

common loads However, the contributions of individual loads are not clearly specified

This paper presents a method for determining the contribution of each generator to each load in the system Besides, the proposed method is not restricted to the incremental changes of active and reactive powers; it will show the possible distance of power transmission through the network Therefore, the “M W-miles” pricing method can be implemented accurately Starting from a power load

flow solution, the method first identifies the buses as load buses or generation buses by getting the net power in the generation load buses Then a directed graph is formed With vertices representing busses and the edges represent transmission lines Second construct all possible directed- rooted subgraph of the system from its generator vertices Notice that the union of these subgraphs is the original directed graph Using proportionality and average assumptions, it is possible to calculate the specific generator contributions to different loads and the transmission system usage of different utilities

The proposed method is described in detail in the following section with the help of a simple example The proposed method can be applied to standard IEEE test systems

11 CONCEPTS

From the solved power flow computation, the proposed technique identifies whether each bus is a generation bus or load bus by taking the net power

between generation power and load power at that bus

“0-7803-6661 -1/01/$10.00 ~ 2 0 0 1 IEEE”

Then, a directed graph can be drawn to represent the

system With each generation vertex as a root, a number of

possible directed-rooted subgraphs can be constructed

Applying the solved branch flows into each subgraph, the

amount of power consumed by each load bus can be

determined By using the proportionality assumption, the

contribution of that generator to each load can be

calculated for each possible subgraph Finally, the overall

contribution of the generator to each load is obtained by

using the average assumption This technique not only can

answer the question of "which generators are supplying

this load and how much power?", it can show the average

distance of transmission through the system

This method can be used independently to both active

and reactive power flows To simplify the example, the

following description will use active power flow and a

lossless system

Based on a solved power load flow, all the branch

flows and generation powers are known If a bus i s

generating power and supplying local load, the bus is

determined whether as a generation bus or a load bus by

subtracting the local load power from the generation

power If the net power is positive, the bus is a generation

bus and if the net power is negative, the bus is a load bus

A 6-bus system as shown in Figure 1 is used to

demonstrate the concepts After running a power flow

computation, all the values are shown in the Figure 1 as

well

150 MW

2nn

1 t 5 O M W l I fOM5Wl f

MW

-

L J

100 MW

100 MW

Fig 1.6-bus system used to demonstrate the concepts

TABLE I

GENERATION-LOAD BUS DETERMINATION

1 n this 6-bus system, there are four generators supplying the power to the system By taking the net power at buses 1, 3, 4, and 6, the generation-load bus determination can be finalized and is shown in Table I

The net power formulation as following:

Net Power = Generation Power - Load Power (MW) IFrom the result above, buses 1 , 3 , and 6 are generation buses and bus 4 is a load bus In order to have power only flowing out of all generation buses, the tie line (line 1-3) of

two generation buses should be eliminated by lumping the power into Generator A It is seem like Generator A is buying 50 MW fiom Generator B This 50 MW will tie corrected to the respective contribution in the later section After this procedure, the 6-bus system can be modified is

shown in Figure 2

150 MW

n

200MW

1

MW

tloo MW I 100 MW I c)

L

100 MW

w

250 MW

Fig 2 Modified 6-bus system

C Constructing Graph and Subgraphs

Based on the direction of the flows in all the branches

of the &bus system, a directed graph can be constructed as

shown in Figure 3 with solid dot represents generation

buses and hollow dot represents load buses

Next step is to separate the graph into directed

subgraphs with generation vertices as its roots These

subgraphs are constructed in such a way that the power can

reach to all vertices in that subgraphs Figure 4, Figure 5,

and Figure 6 are shown the directed, rooted subgraphs of

Generator A, Generator B, and Generator C, respectively

100 M W

200MW

100 MW

250 MW

Fig 3 Directed graph

150 MW

150 MW

Fig 4 Root subgraph of Generator A

200 MW

100 M W

2

1Gv !VI w

Fig 5 Root subgraph of Generator B

9

300 MW

200 MW

100 M b

250 MVV

Fig 6 Root subgraph of Generator C

D Contribution to the load of a generator

From these subgraphs, one can see that the power flows are supplied by respective generators Furthermore, these subgraphs can be separated into some possible power flow subgraphs Basically, the number of the possible subgraphs, N for each root subgraph is:

N = 2 ' where x is the number of edges that are not incident to the root vertex

From these possible subgarphs and holding the branch power flows, one can determine the amount of power consumed by each load Then, based on the proportionality assumption, the contribution to the load of the generator can be calculated for each possible subgraph The formula

is shown below:

Receiving Power Load Power

c, = where i is generator A, B and C j is load

For root subgraph of Generator A, the x = 0 because the only edge in the subgraph is incident to the root Then

N, = 2' = 1 possible subgraph which is its original root

subgraph One can see that the power received by load 2 is

150 MW Therefore, the contribution to load 2 of Generator A, CM is 0.5, C A ~ is 0.0, and CAS is 0.0

For root subgraph of Generator B, the x = 2 because

the lines 2-5 and 4-5 are not incident to the root So Nb =

22 = 4 possible subgraphs These possible subgraphs are shown in Figure 7% Figure 7b, Figure 7c, and Figure 7d

200 hlW

200 Fig 7a

200 MW

100 MW

50 MW

200 MW

Fig 7b

100 MW

2

200 MW

200 MW

100 MW

Fig 7c

2

200 MW

100 MW

200 MW

100 MW

Fig 7d

By assuming the branch power flows are held in each

case, the amounts of power received by the loads are

known, as shown in Table 11 Then, the contribution to the

load of Generator B for all possible subgraphs can be

calculated Finally, the overall contribution to the load of

Generator B is obtained by using average assumption The

result is shown in Table 111

Similarly, for root subgraph of Generator C , the x = 2

because the lines 2-5 and 4-5 are also not incident to the

root Therefore, Nb = 2* = 4 possible subgraphs These

possible subgraphs are shown in Figure 8% Figure 8b,

Figure 8c, and Figure 8d The contribution to the load of

Generator C is calculated and shown in Table IV

TABLE 11

POWER RECEIVED BY EACH LOAD FROM

GENERATOR B

Load 4 Load 5

TABLE 111

CONTRIBUTION TO THE LOAD OF GENERATOR B

Subgraph I Load2 I Load4 I Load 5

7a I 0.0000 I 1.0000 I 0.5000

7b I 0.0000 I 0.5000 1 0.7500

7c I 0.1667 I 1.0000 I 0.2500

100 MW P O 0MW

250 MW

Fig 8a ~

250 MW

Fig 8b

m

100 M 1 e o oMW

MW

100 M M

250 MW

Fig 8c

250 MW

Fig 8d

TABLE IV

CONTRIBUTION TO THE LOAD OF GENERATOR IC:

TABLE V

CORRECTED CONTRIBUTION FOR TIE LINE

Portion

Corrected

Generator A Generator B 100/150=0.6666 5011 50=0.3333 0.6666 x 0.5000 0.3333 x 0.500

I Contribution I = 0.3333 I = 0.1667

Generator A

Generator B

Generator C

E Tie-line Contribution Correction

0.2500 0.7500 0.5000

0.4 I67 0.2500 0.5000

Recall the tie line between bus 1 and bus 3, Generator

B has sent 50 MW to Generator A Therefore, the

contribution to the load 2 of Generator A and B should be

corrected proportionally The corrected contribution of

Generator A and B is shown in Table V and the corrected

values of Generator B will be added to its average

contribution

F The overall Contribution

The final overall contribution to each load of each

generator is shown in Table VI One can see clearly that

the contributions of each generator to the loads Now, it is

possible to conclude that load 2 receives 33.3% power

from Generator A, 25.0% power from Generator B, and

41.7% from Generator C Generator A does not contribute

any power to load 4 and 5 Load 4 receives 75.0% power

from Generator B and 25.0% from Generator C On the

other hand; load 5 receives 50% power from both

Generator B and C

111 CONCLUSION

A simple graph-theory based method for calculating

the contribution of each generator to a particular load has

been discussed and illustrated This method can be

implemented individually to active and reactive power

flows and is not restricted to incremental changes Perhaps,

this technique can resolve some of the difficult pricing and

costing issues in the power deregulated market

For hture improvement, a weight value can be added

into each possible subgraph’s contribution before

computing the average contribution of the generator These

weight values can be determined by using a sensitivity

analysis on the applied power system so that the

contribution calculation will be close to the actual value

TABLE VI

THE OVERALL CONTRIBUTION

IV REFERENCES

[ 13 Thomas J Overbye, “Reengineering the Electric Grid”,

American Scientist, Vol 88, May-June 2000, pp 220

[2] Daniel Kirschen, Ron Allan, Goran Strbac,

“Contribution of Individual Generators to Loads and Flows,” IEEE Transactions on Power System,

Vol 12, No 1, February 1997, pp 52 - 60

[3] Gary Chartrand, Ortrud R Oellermann, Applied and

- 229

1993, pp 1 - 37