Deem-2011 Proceedings

International Conference on Deregulated Environment and Energy Markets, July22-23, 2011

1

Congestion Management based on Power Flow

Contribution Factors 1Vinod Kumar Pal,

2Ashwani Kumar

1,2NIT Kurukshetra

[email protected]

[email protected]

Abstract In this paper, an application of proportional sharing principle and power flow comparison method have

been presented for congestion management deciding

rescheduling of generators for congestion management.

Contribution formulae based approach has been

implemented and Power flow comparison method and

contribution formulae have been presented for congestion

management studies. The comparison of PFC and CF

methods has been tested on IEEE-57 bus test system for

congestion management. The comparison of PFC and CF

based approaches has been implemented for 6-bus test

system, IEEE-57 bus test system and 25-bus test system for

congestion management.

Keywords Congestion management, contribution formulae, proportional sharing principle, power flow comparison

method.

I. INTRODUCTION

Restructuring of the power industry aims at abolishing

the monopoly in the generation and trading sectors,

thereby, introducing competition at various levels

wherever it is possible. Generating companies may enter

into contracts to supply the generated power to the power

dealers/distributors or bulk consumers or sell the power in

a pool in which the power brokers and customers also

participate. In a power-exchange, the buyers can bid for

their demands along with their willingness to pay. Power

generation and trading will, thus, become free from the

conventional regulations and become competitive.

According to Phillipson and Willis [1], Deregulation is a

restructuring of the rules and economic incentives that

governing authority sets up to control and drive the

electric power industry.

In a restructured electricity market environment, when

the producers and consumers of electric energy desire to

produce and consume in amounts that would cause the

transmission network to operate at or beyond one or more

transfer limits, the system is said to be congested [2]. The

congestion in the system can not be allowed to persist for a

long time, as it can cause sudden rise in the electricity price

and threaten system security and reliability. Congestion

management is one of the most challenging tasks of the SO

in the deregulated environment.

In different types of market, the method of tackling the

transmission congestion differs. There are three different

ways to tackle the network congestion:

Price Area Congestion Management

Available Transfer Capability (ATC) based Congestion Management

Optimal Power Flow (OPF) based Congestion Management

The first method is used in Nordic pool; the second one

in US and the third is employed in UK. In Nordic pool,

which consists of Norway, Sweden, Denmark, and Finland,

when congestion is predicted, the system operator declares

that the system is split into price areas at the predicted

congestion bottlenecks. Spot market bidders must submit

separate bids for each price area in which they have

generation or loads. In case of no congestion, the market

will settle at one price and in case of congestion, the price

areas are separated settled at prices that satisfy

transmission constraints. Area with excess generation will

have lower prices, and those with excess load will have

higher prices.

The US Federal Energy Regulatory Commission

(FERC),[3] established a system, where each SO would be

responsible for monitoring its own regional transmission

system and calculating its ATC for potentially congested

paths entering, leaving, and inside its network. The ATC

values for next hour and for each hour in the future are

placed on OASIS, operated by SO. Anyone wishing to do

transaction would access OASIS web pages and use ATC

information available there to determine if system could

accommodate transaction. Details of the regional

congestion management methodologies applied in

Europian electricity market is presented in final report [21].

The report summarizes the congestion management

methods and procedures adopted in Europian region. A

report on congestion management in Nordic region details

out the rules for congestion management and nodal and

zonal pricing based methods [22].

In an optimal power flow, (OPF) is performed to

minimize generators operating cost subject to set of

constraints that represent a model of the transmission

system within which the generator operate. The generator

sends a cost function and those customers willing to

purchase power send a bid function to the SO. The SO has

a complete transmission model and performs OPF

calculation. OPF solution gives cost/MW at each node of

the system. In some countries zonal pricing method is

followed in which the system is divided into various zones


2

on geographical basis. The zone prices obtained from OPF

are used in the following manner:

Generators are paid zone price of energy The loads must pay the zone price of energy

In case of no congestion, the zone price will be same and

the generators are paid the same price for their energy as

the loads pay. When there is congestion, the zone prices

will differ, each generator is paid its zone price and the

loads also pay its zone price for the energy. Thus, the OPF,

through different zonal pricing, performs the function of

controlling the transmission flows and thus maintaining the

transmission security. The goal of deregulation is to

encourage lower electric utility rates by structuring an

orderly transition to competitive bulk power markets. The

key is to open up transmission services, the vital link

between sellers and buyers. To achieve the benefits of

robust, competitive bulk power markets, all wholesale

buyers and sellers must have equal access to the

transmission grid. Otherwise efficient trades cannot take

place, and ratepayers will bear unnecessary costs. Problems

arise because all transactions have to share the same

transmission network simultaneously. What portion of the

capacity of a particular transmission system is used by a

certain transaction? How can a fair charge be calculated

based on the capacity usage? Can some transactions be

adjusted, or even canceled, when transmission congestion

occurs? Which generator or transaction should be

considered first for adjustment in congestion management,

based on transmission capacity usage?

To answer these questions, an algorithm was developed

which can calculate the contribution from individual

generator units to the flows and losses trough the

transmission network and at the load centers. This is both

an essential and challenging task. Scholars from England

were the first to propose the Flow Based Proportional

Sharing Method (PS), based on a strong proportional

sharing assumption, which has not been proven either

correct or incorrect at this time [4-8]. Among them, D.

Kirschen proposed the famous Topological Trace

Algorithm; J. Bialek proposed the Upstream-Looking and

Downstream-Looking algorithms. R. Shoults, who

proposed the Circuits Based Method to challenge the

currently very popular Flow Based Proportional Sharing

Method, believed that the correct method should be

founded upon established circuits theories. The Circuits

Based Method is comprised of two sub-methods:

1. Current Division method

2. Voltage Division method.

A new method named the Power Flow Comparison Method (PFC) which makes an effort to conform to the physical concepts commonly understood and accepted by

power system engineers. These methods are applied to an

example of 6-bus power system, and the results are

presented, compared and discussed to verify their

correctness. Further applications in transmission charges

[9-11] and transmission congestion management [12, 13]

show the great practical value of tracing the flow of power

from source to load. The impact of corrective actions on

one group of lines to other heavily loaded lines is also

shown.

II.POWER FLOW CONTRIBUTION FACTORS METHOD

The Power Flow Comparison Method (PFC) is

comprised of the following procedure to find the

contribution of each generator to the line flows, losses and

loads:

i) Calculate the base case power flow.

ii) For the generator of interest, remove generation and a

corresponding load from the power system in even

quantities.

iii) Make this generator bus the swing bus and calculate

the power flow again

iv) Find the line flow difference on each transmission

line by comparing the two power flow results above.

An example 3-bus power system, shown in Fig 1, is

provided to illustrate this algorithm. Line resistance is

ignored, and distributed capacitance is considered. Part I

of Fig 1 is the base case power flow result. Part II of the

Fig 1 is obtained by removing the generation at bus A

(150 MW) and its corresponding loads at bus C (150MW)

from the base case power system. The contribution of

generator A to the line flows can be found by subtracting

the line flow results in Part II from those in Part I. Part III

of the Figis obtained by removing the generation at bus B

(50MW) and its corresponding loads at bus C (50MW)

from the base case power system. The contribution of

generator B to the line flows can be obtained by

subtracting the line flow results in Part IV from those in

Part I. The final results are listed in Part IV. From this

illustration, the reader may be reminded of the distribution

factors method, which shows the sensitivity of the line

flows to changes in generation. It is always easy to remove

a generator (in a simulation.) It is rather difficult to find

the corresponding loads of this generator, which are

different from the contract loads. The term

corresponding loads is used to mean the load served by

this particular generator. How they are calculated is

summarized in the following paragraphs. It is assumed

that the voltage at each network node will not change

much when a quantity of generation and its corresponding

loads are evenly removed from the network. This is the

constant voltage assumption used in the paper.

Fig.1 (a): Base case power flow

C

B 50MW 150MW A

200MW


3

Fig1 (b): Remove generator A and its load from base case power flow

Fig1(c): Remove generator B and its load from base case power flow.

Fig1 (d): Power flow comparison methods

III. APPLICATION IN CONGESTION MANAGEMENT

With the open access of transmission networks and the

competitive bulk power market, transmission congestion

has become a more and more serious problem in the

deregulated world. However, based on the calculated

contribution of line flows from each generator, the PFC

Method and the Proportional Sharing (PS) Method can

provide guidance to relieve these congestion problems.

Test cases on the IEEE 57-bus system [17] are presented

to show the details.

Results of two test cases on the IEEE 57-bus system

show the capability of the Power Flow Comparison (PFC)

Method over the Proportional Sharing Method (PS) for

solving the transmission congestion management

problems. Key output data are given in the [18], which

includes:

Bus output of the load-flow study; Line flows of the load-flow study; Generator contributions to MW line flows based on the

PFC Method;

Generator contributions to MW line flows based on the PS Method.

Case #1: Line 8 is heavily loaded with a total line flow

of 174.0983 MW. Based on the MW contribution of each

generator to the line flow, transmission congestion

management methods are suggested by both the PS

Method and the PFC Method as shown in Table 1. The PS

Method predicts that decreasing G3 is the only way to

lower the line flow. The PFC Method, on the other hand,

predicts that the line flow can be decreased with more

flexibility. In tests 1, 2, and 6, the line flow is decreased

without changing G3. The TESTS portion of Table 2.1

shows the sensitivities of line

flow changes vs. the generation changes. Because line 15

is also heavily loaded, the sensitivities of line 15 are listed

in the TESTS table as well so that a management method,

with a minimum impact online 15, can be found. Tests 1

to 6 apply 1 MW to each generation change. Test 6 is able

to lower the line flows at both lines 8 and 15. Test 5 is

able to drop the line flow at line 8 dramatically while

keeping the line flow increase at line 15 to a minimum.

Case #2: Line 69 is loaded with a total line flow of

5.8089 MW. Based on the MW contribution of each

generator to the line flow, the transmission congestion

management suggested by the PS Method and the PFC 40

3.Method are quite different, as shown in the Table 2.The

PS Method predicts that decreasing G3 is the only way to

lower the line flow. Conversely, the PFC Method predicts

that the line flow will be decreased, by increasing G3 and

decreasing the power output of the other generators, or

without changing G3. The TESTS table shows the

sensitivities of line flow changes v/s the generation

changes. Tests 1 to 8 apply 1 MW to each generation

change. Tests 1 to 3 show that the PS Method leads to a

change in the wrong direction because the line flow at line

69 is increased when G3 is decreased. Tests 4 to 8 show

that the PFC Method provides more usable guides and

more alternatives for transmission congestion

management.

IV. REAL AND REACTIVE POWER CONTRIBUTION FACTORS

Algorithm for calculating the contribution factors of each

generator to the line flows, losses and loads as proposed in

[16] is given below:

(a) Perform the base case Newton-Raphson power flow.

(b) Compute the sensitivity Sijk of the real and reactive

power flow Pij and Qij of a line connected between

bus-i and bus-j to real and reactive power output PGk

and QGk of generator-k. The fast-forward/fast-

backward substitution method allows an efficient

computation of the sensitivity[15]:

kG

ij

kG

ij

kG

ijkijp

P

ggP

P

P

dP

dPS

1

(1)

C

B 50MW 0.0MW A

50MW

C

B 0.0MW 150MW A

150MW

C

B 50MW 150MW A A: 50.42MW

B: 16.56MW

A:

99.58MW

B: 16.56MW

A: 50.42MW

B: 33.44MW

200MW


4

kG

ij

kG

ij

kG

ijkijq

Q

ggQ

Q

Q

dQ

dQS

1

(2)

kijpkijp MJNS

1 (3)

kijqkijq MJNS

1 (4)

where Nijp and Nijq are the sparse block vector with sub-

vector [ -bij Vi Vj, 00] and [ bij Vi Vj, 00] and [ -gij Vi Vj,

00] and [ gij Vi Vj, 00] in the ith

and jth

position,

respectively. Mk is the sparse block vector with sub-vector

[-1, 0] in the kth

position. [J] is the Jacobian power flow

matrix. g is the power flow equation vector and is the

voltage angle vector.

(c) As proved in [16], the contribution factor of slack

generator to real and reactive power flow of line i-j is:

NG

k

kG

NG

k

Gk

Gkijpij

ijp

P

PPSP

CF

1

2

1)( (5)

NG

k

kG

NG

k

GkG

kijqij

ijq

Q

QQSQ

CF

1

2

1)(

(6)

where CFijp and CFijq are contribution factor of slack

generator to the real and reactive power flow of line i-j.

PG1 is the real power output of the slack generator.

(d) The contribution factor of the mth generator (except

slack generator-1) to the real power flow of line i-j is

NG

k

kG

mG

NG

k

NG

k

kG

mijp

kG

kijpij

mijp

P

PPSPSP

CF

1

2 1 (7)

NG

k

kG

mG

NG

k

NG

k

kG

mijq

kG

kijqij

mijq

Q

QQSQSQ

CF

1

2 1 (8)

Where CFijpm and CFijq

m is the contribution factor of

generator-m (except slack generator) to real and reactive

power flow of line i-j.

(e) The contribution of each generator-m to losses of each

line i-j can be calculated as: mjip

mijp

mlossij CFCFP , (9)

mjiq

mijq

mlossij CFCFQ , (10)

Where Pij,lossm and Qij,loss

m is the contribution factor of

generator-m to loss of line i-j.

(f) The contribution of generator-m to load at the bus-i, is

given as:

ui

j

mijp

mloadi CFP , (11)

ui

j

mijq

mloadi CFQ , (12)

where iu is the set of nodes which are directly connected

to node-i.

The total contribution of all the generators to line i-j is

equal to the real power flow of this line: NG

m

mijpij CFP

1

(13)

NG

m

mijqij CFQ

1

(14)

The contribution factors based on sensitivity of power

flow are used to find the share of each generator on each

line flow and consequently the generator share will help

ISO to identify the generators to re-dispatch their

generation in congestion management. This can be

computed in advance and the SO can post the contribution

of each generator to line flow, so that during the

congestion hours the SO can send the signal in the market

for rescheduling of real and reactive powers to manage

congestion. An example 3-bus system is provided to

illustrate the capability of the above formulas.

Resistance is 0.01 p.u. Reactance is 0. 1 p.u. and

the distributed capacitance is considered with a n-model in

all transmission lines. Figs. 2 and 3 show the base case

power flow result and contribution of each generator unit

to each line flow in the above system, based on proposed

CF formulas and the PFC method [20], respectively. The

results of Fig 3.2 are similar to those of Fig 3.1, because

the voltage at each network node does not change much

when the quantity of generation and its corresponding

loads arc 'evenly' removed from the network.

Fig 2: Illustration of CF formulas for 3 bus system

Fig.3: Illustration of PFC method by 3 bus system


5

The proposed CF formulas could have applications in,

computation of contribution of generation to a DC-line or

DC-link of an AC-DC power system [19], computation of

contribution of' reactive generation to reactive power

flows when the real power is replaced by reactive power i

n the above formulas, application to individual customers

for the apportionment of the use-of-the system charges,

calculation of the sensitivity regarding corrections, that

will affect the flows in the critical lines.

V. POWER FLOW CONTRIBUTION FACTORS METHOD

Test results for 6 bus system

An example 6-bus power system with three generators

and five load centers are shown to verify and compare the

proposed CF formulas with the PFC method. Fig 4 gives

the topology of the 6-bus test system and the line flow

results. The contribution of the generator units to line

flows, resulting from the implementation of CF formulas

and PFC method, is shown in Tables 3 and 4, respectively.

The contribution of the generators to load centers,

resulting from the implementation or CF formulas and the

PFC method, is shown in Tables 5 and 6, respectively.

Table 7 shows the voltage at each network node changes,

when the PFC method is enforced and the generation and

'corresponding load' are removed. These results indicate

that the 'constant voltage' assumption, which is applied in

the PFC method, is not acceptable as accurate.

250MW

60

40 50

20 10

30

250MW

350.145MW

50MW

100MW

200 MW

200MW

300MW

Fig 4: 6-bus system

Application in Congestion Management

In this Section, the implementation of the proposed CF

formulas and PFC method [20] in the transmission

congestion management are presented. The proposed CF

formulas can provide guidance to relieve congestion

problems. Results on the 25-bus test system show the

ability and accuracy of CF over PFC for solving

transmission congestion problems. In this test system,

line-22 (130 bus-230 bus) is heavily loaded with a total

line flow of 381.54MW. Based on the CF and PFC

methods, transmission congestion management solutions

are suggested as shown in Tables 8 and 9 respectively.

Tables 8 and 9 show the contribution ( P) of each

generator to line-22 based on the CF and PFC methods,

respectively. The results of these tests show the

sensitivities of line-22 flow changes vs. the generation

changes ( line.22/ G ).This tests shown are chosen to

decrease the flow of line-22 with more flexibility because

they present the maximum value in the above sensitivities.

Since line-21 (120 bus- 230 bus) is also heavily loaded

(366.51 MW), the sensitivities of line-21 ( line.21/ G )

are shown in Tables 7 and 8 as well, so that a management

method, with a minimum impact on line-21, can he found.

Tests 1 to 5 apply 1 MW to each generation change. In

tests 1, 3, 4 and 5, generators whose power output changes

(1 MW) have the same effect on the sensitivities ( P

/ G ) are indicated with '(or)'. Both methods predict

that the line flows can be decreased with more flexibility.

The difference between these methods is the proposed

scenario for solving the congestion problem, and that

occurs because of the inaccuracy of the 'constant voltage'

assumption of the PFC method [20].

IV. CONCLUSIONS

In this paper, the Power Flow Comparison Method (PFC)

and CF method has been analyszed and applied for

congestion management by rescheduling the power

available from the generators. Results of calculations

applied to an example 6 bus power system and IEEE-57

bus test systems are presented, compared and discussed to

verify the correctness of the methods. An application of

PFC and CF approaches has been carried-out and

compared for 25-bus system for congestion management

studies. In particular, the following conclusions are made

Changes in generation suggested by all the methods

will affect flows in other lines. Hence, other heavily

loaded lines should be considered as well when solving

a transmission congestion problem. The PFC Method

provides convenient facilities for doing this.

The sensitivities used in the congestion management

change very little when the generation is increased.

According to the PFC Method, all generators have

some contribution to each line flow. Based on their

contribution, the order in which the generators should

be adjusted can be determined. When a generator has a

positive contribution to the line flow, the output of this

generator needs to be decreased to lower the

transmission congestion and vice versa. When two

generators have conflicts in congestion management,

the one with higher MW contribution has more impact

on the line flow changes.

The Proportional Sharing Method sometimes may

lead to changes in the wrong direction.

The CF based approach expressing the contribution of

each generator unit to loads or flows or losses or lines has

been implemented based on [16].The proposed CF

formulas are applicable independently to active and


6

reactive loads, flows and losses replacing the real power

by reactive power in the load flow scheme. These

formulas and other state-of-the-art methods have been

tested on various test systems to compare results. The

results obtained from CF are similar as obtained from PFC

method. Also, the proposed CF formulas can provide

guidance to relieve the congestion problems

REFERENCES

[1] L.PHILIPSON and H.LEE WILLIS, Understanding Electric Utilities and Deregulation, Marcell Dekker Inc., NY 1998.

[2] R.S.FANG and A.K. DAVID, Transmission Congestion Management in Electricity Market, IEEE Transactions on Power Systems, Vol. 14, No. 3, August 1999, pp. 877-883.

[3] North American Electric Reliability Council (NERC), Available Transfer Capability Definition and Determination, NERC Report,

June 1996. [4] D. KIRSCHEN, R ALLEN and G. STRBAC, Contributions of

Individual Generators to Loads and Flows. IEEE Transactions on Power Systems, Vol. 12, No. 1, pp. 52-60, February 1997.

[5] G. STRBAC, D. KIRSCHEN, S. AHMED, Allocating Transmission System Usage on the Basis of Traceable Contributions of Generators and Loads to Flows, paper PE-222-PWRS-0-01-1997, IEEE, 1997.

[6] J. BIALEK, Topological Generation and Load Distribution Factors for Supplement Charge Allocation in Transmission Open

Access, IEEE Transactions on Power Systems, Vol. 12, No. 3, August 1997.

[7] J. BIALEK, Identification of Source-sink Connections in Transmission Networks, Power Systems Control and Management, 16-18 April 1996.

[8] R. SHOULTS and L.D. SWIFT, A comparison Between Circuits Based Methods and Topological Trace Methods for Determining

the contribution of Each Generator to Load and Line Flows, Proceedings of the workshop on Available Transfer Capability, pp.

167-182, University of Illinois at Urbana-Champaign, June 26-28,

1997. [9] R. A. WAKEFIELD, J. S. GRAVES, A. F. VOJDANI, A

Transmission Services Costing Framework, IEEE Transactions on Power Systems, Vol. 12, No. 2, May 1997.

[10] C. W. YU, A. K. DAVID, Pricing Transmission Services in the Context of Industry Deregulation, IEEE Transactions on Power Systems, Vol. 12, No. 1, February 1997.

[11] H. H. HAPP, Cost of Wheeling Methodologies, IEEE Transactions on Power System, Vol. 9, No. 1, February 1994.

[12] J. D. FINNEY and H. A. OTHMAN, Evaluating Transmission Congestion Constraints in System Planning, IEEE Transactions on Power System, Vol. 12, No. 3, August 1997.

[13] H. SINGH, S. HAO and A. PAPALEXOPOULOS, Transmission Congestion Management in Competitive Electricity Markets, paper PE-543- P WRS-2-06-2997, IEEE, 1997.

[14] R. A. WAKEFIELD, J. S. GRAVES, A. F. VOJDANI, A Transmission Services Costing Framework, IEEE Transactions on Power Systems, Vol. 12, No. 2, May 1997.

[15] A.G. BAKIRTZIS: Sensitivity computation of power flow control in electric power systems. Electric Power System. I991. 22. pp. 77-84

[16] J.G.VLACHOGIANNIS: Accurate model for contribution of generation to transmission system effect on charges and congestion management. IEEE proceeding generation transmission vol.147 no.6 November 2000, pp342-348.

[17] J. D. GLOVER and G. DIGBY, Software Manual, Power System Analysis and Design, Second Edition, PWS Publishing Company,

Boston, 1994.

[18] JIAN YANG, Power Systems Deregulation and Development of Powergraf, a PhD Dissertation, University of Missouri - Rolla, July, 1998.

[19] J.G. VLACHOGIANNIS: Control Adjustment in Fast Decoupled load flow, Electric Power System Res. I994. 31..pp. 185-194.

[20] J. YANG and M. D, ANDERSON,:Tracing the power flow of power in transmission networks foe use of transmission system charges and congestion management, IEEE Trans. Power System., 1998.13 (2), pp.399-405.

[21] www. Constec.de, Towards a common coordinated congestion management methods in Europe, Final Report, 2007, Directorate-General Energy Transport, 2007.

[22] www.nordiceenergyregulators.org, Congestion management in the Nordic region: A common regulatory opinion on congestion

management, A report, 2007.


7

TABLE. 1 : CASE #1 - LINE 8 IS HEAVILY LOADED, G3 MUST BE DECREASED

Method

LF_G 1

(MW)

LF_G2

MW)

LF_ G3

(MW)

LF_G4

(MW)

Production of

management

PS

PFC

0

-23.342

0

1.0689

174.0983

232.6123

0

-36.2409 G1 G2

G3 G4

G1 G2 G3 G4

Test

G1

G2

G3 G4

G

Pline

G

Pline 15

1

2

3

4

5

6

+1

0

0

+1

0

-1

-1

-1

0

0

+1

0

0

0

-1

-1

-1

0

0

+1

+1

0

0

+1

-0.0841

-0.1781

-0.7031

-0.090

-0.5250

-0.0941

+0.3715

+0.0343

+0.0523

+0.3795

+0.0181

-0.3372

Method

LF_G1(MW))

LF_G2(MW)

LF_G3(MW) LF_G4(MW) Production of management

PS

PFC

0

-23.342

0

1.0689

174.0983

232.6123

0

-36.24O9 G1 G2 G3 G4

G1 G2 G3 G4

TABLE. 2: CASE #2 - PS METHOD GIVES WRONG ANSWER AND PFC METHOD PROVIDES ALTERNATIVES

TABLE 3: CONTRIBUTION OF GENERATION OF 6-BUS TEST SYSTEM TO LINE FLOWS RESULTING FROM CF FORMULAS

Lines

Contribution Of G-10

(MW)

Contribution

Of G-20 (MW)

Contribution of G-30

(MW)

10-20

10-30 10-40

20-50

30-50 30-60

40-60

50-40

90.617

127.702 131.811

70.038

32.955 53.559

49.415

20.600

106.71 0 41.184 65.552

128.558

-19.321 31.099

06.723

50.409

14.695

-58.830 44.123

0.0104

73.524 88.230

-14.705

14.707

TEST G1MW G2MW G3MW G4 MW G

Pline 8

1

2

3

4

5

6

7

8

+1

0

0

-1

0

0

0

-1

0

+1

0

0

-1

0

+1

+1

-1

-1

-1

+1

+1

+1

0

0

0

0

+1

0

0

-1

-1

0

+0.0092

+0.0048

+0.0144

-0.0092

-0.0048

-0.0144

-0.0096

-0.0044


8

TABLE 4: CONTRIBUTION OF GENERATION OF 6-BUS TEST SYSTEM TO LINE FLOWS RESULTING FROM PFC METHOD [20]

Lines

Contribution

Of G-10 (MW)

Contribution

Of G-20 (MW) Contribution

Of G-30 (MW)

10-20 10-30

10-40

20-50 30-50

30-60

40-60 50-40

89.725 126.897

131.535

70.144 32.987

53.725

49.114 20.736

106.370 40.880

65.490

128.217 -19.051

31.258

6.657 50.315

14.346 -58.527

44.181

0.378 73.261

88.078

-1 4.643 14.811

TABLE 5: CONTRIBUTION OF GENERATION OF 6-BUS TEST SYSTEM TO LOAD CENTERS RESULTING FROM CF FORMULAS

(Load centers)/ load (MW)

Contribution of G-10 (MW)



120)/50

(30)/100 (40)/100

(50)/200

(60)/300

20.579

41.188 82.396

82.393

123.574

14.732

29.406 58.829

58.828

88.231

14.684

29.416 58.828

58.827

88.232

TABLE 6: CONTRIBUTION OF GENERATION OF 6-BUS TEST SYSTEM TO LOAD CENTERS RESULTING FROM PFC METHOD [20]

(Load centers)/

load (MW)

Contribution

of G-10 (MW)

Contribution

of G-20 (MW)

Contribution

of G-30(MW)

(20)/50 (30)/100

(40)/100

(50)/200 (60)/300

19.581 40.185

82.421

82.395 123.575

15.413 28.673

58.833

58.821 88.230

13.968 30.134

58.824

58.828 88.246

TABLE 7: VOLTAGE DIVERGENCES WHEN PFC METHOD [20] IS APPLIED IN 6-BUS TEST SYSTEM

Load bus

Base case Load flow solution

(Voltage p. u.)

bus voltage when Generation and

corresponding load

are removed

V ( voltage divergences)

(%s)

20

30

40 50

60

1.000

1.000

0.993 0.996

0.994

0.967

0.958

0.959 0.954

0.950

3.3

4.2

3.4 4.2

4.4

TABLE 8: APPLICATION OF CF FORMULAS IN CONGESTION MANAGEMENT OF 25-BUS TEST SYSTEM (LINE-22 IS HEAVILY LOADED)

Contribution of generation to flow of line-22

Method G10 G20 G70 G130 G150 G160 G180 G210 G220 G230

( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW )

CF 7.643 7.940 9.323 73,799 -19.957 -21.463 -50.060 -48.928 -54.131 -285.706


9

TABLE 9: APPLICATION OF PFC METHOD [20] IN CONGESTION MANAGEMENT OF 25-BUS TEST SYSTEM (LINE-22 IS HEAVILY LOADED)

Proposed output changes of generation in order to lower line flows in line-22 to be achieved

Test

G10 G20 G70 G130 G150 G160 G180 G210 G220 G230


1

2

3

4

5

(or)+1 (or)+1 (or)+1 -1

+1 -1 (or)+1 (or)+1 (or)+1 (or)+1 (or)+1 -1

+1 or)-1 (or)-1 (or)-1 (or)-1 (or)-1

(or)-1 (or)-1 (or)-1 +1

Results of tests line.22/ G line.21/ G

1

2

3

4

5

-0.29 -0.27

-0.45 -0.25 -0.14 -0.11

-0.31 -0.30

-0.16 -0.04

Contribution of generation to flow of line-22

Method G10 G20 G70 G130 G150 G160 G180 G210 G220 G230


PFC 8.168 8.309 10.632 73.501 -20.312 -21.971 -51.110 -49.536 -54.203 -285.018

Proposed output changes of generation in order to lower line flows in line-22 to be achieved

Test

G10 G20 G70 G130 G150 G160 G180 G210 G220 G230


1

2

3

4

5

(or)+1 (or)+1 (or)+1 -1

+1 -1

(or)+1 (or)+1 (or)+1 (or)+1 (or)+1 -1 +1 or)-1 (or)-1 (or)-1 (or)-1 (or)-1

(or)-1 (or)-1 (or)-1 +1

Results of tests line.22/ G line.21/ G

1

2

3

4

5

-0.24 -0.22 -0.48 -0.19

-0.26 -0.23

-0.37 -0.35 -0.07 +0.11


10

Chaotic Particle Swarm Optimization for

Reduced Order Model of Automatic Generation

Control 1Cheshta Jain,

2H. K. Verma

1, 2SGSITS, Indore

[email protected]

[email protected]

Abstract This paper presents a new approach to control frequency and tie line power changes of multi area

interconnected system. AGC is very important in power

system to maintain system frequency and tie-line power,

when system subjects to small load perturbations. Chaotic

particle swarm (CPSO) is used to optimize gains of PI

controller and bias frequency, to maintain system frequency

and tie-line power flow at scheduled values. In this paper,

model order reduction technique has been used to obtain

lower order model for study of automatic generation control

(AGC). Model order reduction method can preserve the

identity of each generating unit. With this technique, the

computational complexity and effective time has been

reduced by re-sorting the lower order generating unit

models. This paper also presents the selection of suitable

value for governor speed regulation parameter. The

proposed method shows its robustness under critical

conditions when conventional optimization methods fail.

Keywords chaotic particle swarm optimization, automatic generation control, reduced order of AGC.

NOMENCLATURE F = Frequency deviation. i = Subscript referring to area (i = 1,2,). Ptie (i,j) = Change in tie line power. Pdi = Load change of i

th area.

Di = PDi / Fi Ri = Governor speed regulation parameter for i

th area.

Thi = Speed governor time constant for ith area.

Tti = Speed turbine time constant for ith area.

TPi = Power system time constant for ith area.

KPi = Power system gain for ith area.

ACEi = Area control error of ith area.

Hi = Inertia constant of ith area.

Ui = Control input to ith area.

Bi = Frequency bias for ith area.

Us = Undershoot of ACE.

Mp = Overshoot of ACE.

ts = Settling time ACE.

tr = Rise time of ACE.

ess = Steady state error of ACE.

W = Inertia weight.

C1, C2 =acceleration coefficient.

I. INTRODUCTION

Automatic generation control is one of the most

important issues in power system design. The purpose of

AGC is fast minimization of area frequency deviation and

mutual tie-line power flow deviation of areas for stable

operation of the system.

The overall performance of AGC in any power system

depends on the proper design of speed regulation

parameters and gains of controller. Fixed linear feedback

controller fails to provide best control performance. There

are no well defined methods available to compute tie-line

constant for an area with non-coherent generators and

multiple tie-line. The complexity of computation of tie-

line increases with the increase in number of areas. This

paper presents a reduced order AGC simulation technique

that overcomes this problem. These reduced order systems

overcome the need of identifying the non-coherent set of

generators in order to control the areas. This technique is

based on some assumptions. These assumptions follow the

fact that, all areas are operating at same frequency and tie-

line flow can no longer be computed in actual system. Tie-

line flows are required to determine area control error [3].

The conventional controller improves steady state error

(ess) but with small overshoot. PI controller has such

capability to improve transient performance with

minimum steady state error. The aim of this paper is to use

CPSO to find optimum gains of PI controller and

frequency bias with proper system parameter. CPSO is an

optimization approach based on the PSO with adaptive

inertia weight factor methods. It provides more precise

description of natural swarm behaviour.

In the view of the above, following are main objectives

of the proposed paper:

1. To reduced the order of the generating unit of AGC system.

2. To optimized the gains of PI controller and frequency bias coefficient using chaotic PSO algorithm in

MATLAB.

3. To examine the effect of speed regulation parameters on reduced system.

The rest of the paper is organized as follow: section I

presents AGC system model with proposed reduction

technique. In section II chaotic particle swarm

optimization is discussed in brief and an algorithm to

implement CPSO based PI controller is presented in

section III. Section VI shows the result with discussion

and conclusion is drawn in section V.


11

II. SYSTEM MODEL

A. Conventional AGC system

Automatic control system of close loop system means

minimizing the area control error (ACE) to maintain

system frequency and tie-line deviation are set at nominal

value [15]. Block diagram of two area system is shown in

Fig 1.

Fig 1: Linear model of two area system.

The ACE of each area is linear combination of biased

frequency and tie-line error.

(1)

Where, Bi is frequency bias coefficient of ith

area. Fi is frequency deviation and Ptie is tie-line error of i

th area.

Based on the ACE a suitable control strategy can be taken

up either continuously or discretely. A practical system

consists of a number of generating units, which increases

computational time and complexity. In this paper the

reduction of order is used to reduce above effort.

B. Reduced order generating unit

Several methods are available for reducing the order

of a system transfer function (TF) [14]. One way is to

delete a certain insignificance pole of a transfer function,

which has a negative real part that is much more negative

than the other poles [14]. An effective approach is to

match the frequency response of the reduced order transfer

function with the original TF frequency response. Suppose

the higher order system is described as:

(2)

This system has poles in the left hand s-plane and m n. The lower order approximate transfer function is:

(3)

Where, p q n. The gain K is the same for original and approximated system. This method is based on

selection of c and d in such a way that GL(s) has a

response very close to that of GH(s). These coefficients are

evaluated as:

(4)

And

(5)

Where, M(s) and (s) are the numerator and denominator polynomials of GH(s)/GL(s) respectively. To

determine total (cp + dq) no of unknowns, requires (cp + dq)

no of equations. Therefore define

(6)

g = 0, 1, 2, ---- up to number required to solve the

unknown coefficients. An analogous equation for 2g. the solution for c and d coefficient is obtained by equating

M2g = 2g. In this paper two area AGC system is considered.

This model has second order transfer function of

generating unit of each area as:

(7)

The value of AGC parameters (TH and TL) are given

in appendix. After reduction as above method the lower

order generating unit becomes:

(8)

III.OVERVIEW OF CHAOTIC PARTICLE SWARM OPTIMIZATON

a. General PSO method

Particle swarm optimization (PSO) first proposed by

Kennedy and Eberhart [5]. Like evolutionary algorithms,

PSO techniques conducts search using a population of

particles, corresponding to individuals. In PSO, particles

changes their position by flying around in a search space

until computation limitations are exceed. PSO is inspired

by the ability of flocks of birds to reach unknown

destination. In PSO each particle is defined as moving part

in hyperspace. PSO is inherently continuous and must be

modified to handle design variables.

The basic PSO algorithm requires three steps [5].

Step 1: Initialization

PSO is initialized with the group of random particle

positions (xki) and velocities (Vk

i) between upper and

lower bound of design variable values for N particles as

expressed in equation 9.

X0i = Xmin + rand*(Xmax Xmin) (9)

Velocity is initialized as:

(10)


12

i = 0,1,2,.N. Step II: Update velocity

Velocity of all particle updated for next (k+1)th

iteration

using the particles fitness values, which is function of

particle positions. These fitness function value determines

which particle has a global best (gbestk) value in the

current swarm (kth

iteration) and also determine the best

position (pbesti) of each i

th particles.

The global best value speed up the rate of convergence.

gbest value maintains only single best solution across the

entire particle in the search space. If all particles are

converging to this position, the further improvement of

each particle stops.

After finding the two best values, the particles update its

velocity using following equation:

(11)

Where w is inertia weight factor, c1 and c2 are self

confidence and swarm confidence respectively.

Combinations of these values usually lead to much slower

convergence or sometimes non-convergence at all.

Therefore proper selection of these particles is important.

Step III: Update positions

Now positions of particles are updated using

following equation:

(12)

a. Chaotic PSO

The parameters r1 and r2 in equation (11) are important

control parameters. The use of chaotic sequence in PSO

can be useful to escape from local minima than general

PSO [4].

Chaotic sequence based on Henon mapping is used for random value r1as:

(14)

Where, a and b are Henon map attractor. Another mapping uses the same equation for random value r2 to

generate z 2(t) in range [0, 1]. Other parameters are same

as in equation (10). Hence, velocity of particle is updated

as:

(15)

IV. IMPLEMENTATION OF CPSO-PI CONTROLEER

A. Fitness function

The gain of PI controller can be selected based on degree

of relative stability, minimum overshoot, undershoot and

settling time. To satisfy all requirements following

objective function is design.

(16)

B. Algorithm

Step1. Choose the population size (N) and number of Step2. iteration (Nmax). Set the value of inertia weight w. Step3. Generate initial population (eq. 9) and velocities

randomly as given in equation10:

Step4. Set counter k=1. Step5. Run model of reduced AGC system and

determine performance parameters for each

particle.

Step6. Calculate fitness function for each particle (eq.16). Calculate gbest and pbest position value.

Using these values update velocity of each

particle (eq.15).

Step7. Update position of each particle (eq. 12). Step8. Calculate new fitness function for each updated

particles position, if it is better than previous value of fitness function then the current value of

particle position is set to pbesti.

Step9. Set k = k + 1. Step10. If the last change of the best solution is greater

than a pre specified number or the number of

iteration reaches the maximum iteration, stop the

process, otherwise go to step 4.

V. RESULT AND DISCUSSION

A comparative study of CPSO algorithm on reduced

order and full order model is carried out in this paper. The

time response plots of area control error of area 1 of the

reduced order as well as full order AGC system is given in

Fig2-5. These Figures shows the variation in system

frequency, tie-line power flow and area control error for

1% step load perturbation on area two. From these Figures

it can be seen that the response of reduced order

generating unit model and actual model are approximately

identical.

Table I presents computational result for a input data set

1 (Tp=10, R= 8% frequency, T12=0.145). The results

clearly established that reduced order model has less

computational time than full order model with

approximately same performance parameters. Hence,

lower order model can be used to obtain desired response

with saving in computation time.

The gains of PI controller and frequency bias are

obtained by chaotic PSO algorithm for different value of

system parameters of reduced order AGC system. As Fig

6-13 shows, PI controller act too fast to the generator

inputs and also exhibits fast oscillations. In all cases, an

acceptable overshoot and settling time on frequency

deviation signal in each area is maintained.

Table-II gives transient response parameters of CPSO

based PI controller for different data set. Three sets of

Input data have been used in this paper, namely data set1

(Tp = 10, R = 8%, T12 = 0.145), data set2 (Tp=30, R=8%

of frequency, T12=0.145) and data set3 have (Tp=30,


13

R=4% of frequency, T12=0.145). Table-II and Figures 10-

13 shows that the large values of power system time

constant (Tp=30) and low value of T12 (0.145) yield large

value of undershoot, overshoot and settling time and hence

high value of fitness function. These values (undershoot,

overshoot, settling time for data set2) become lower as

speed regulation, R decreases from 8% to 4% of

frequency.

The initial sharpness of the response in undershoot and

overshoot lies due to choice of weighting factor (1000 and

100) in the fitness function

TABLE. I: COMPARISON OF ACTUAL AND REDUCED ORDER SYSTEM FOR DATA SET-I

TABLE II: COMPARISON OF EFFECT ON INPUT SYSTEM PARAMETERS ON

REDUCED ORDER AGC SYSTEM.

Dataset1 (Tp=10, R=8% of freq, T12=0.145).



Fig 2: Variation in area control error in reduced order and full order

system

Fig 3: Comparison of tie-line power for reduced order and full order

AGC system

Fig 4: Comparison of area frequency 1 for reduced order and actual

system.

Fig 5: Comparison of area frequency 2 for reduced and actual system

Fig 6: Variation in area control error in dataset1 and dataset2

Fig 7: Variation in area frequency 1 for dataset 1 and dataset 2.


14

Fig 8: Variation in area frequency2 for dataset 1 and dataset 2.

0 5 10 15 20 25 30 35 40-2

-1

0

1

2

3

4x 10

-3

Step Response

Time (sec)

Am

plitu

de

p12_dataset1

p12_dataset2

Fig 9: Comparison of tie-line power flow.

0 5 10 15 20 25 30 35 40-4

-3

-2

-1

0

1

2

3x 10

-3

Step Response

Time (sec)

Am

plitu

de

ACE1_dataset3

ACE1_dataset2

Fig 10: Comparison of ACE for dataset 2 and dataset 3.

0 5 10 15 20 25 30 35-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

Step Response

Time (sec)

Amplitude

f1_dataset3

f1_dataset2

Fig 11: Variation in area frequency 1 for dataset 2 and dataset3.

0 5 10 15 20 25 30 35-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

Step Response

Time (sec)

Amplitu

de

f2_dataset3

f2_dataset2

Fig 12: Comparison of area frequency 2 for dataset 2 and dataset3.

0 5 10 15 20 25 30 35 40-2

-1

0

1

2

3

4x 10

-3

Step Response

Time (sec)

Am

plitu

de

p12_dataset3

p12_dataset2

Fig 13: Comparison of tie-line power flow for dataset 2 and dataset 3.

VI. CONCLUSION

In this paper CPSO method is used to obtain optimum

gains of PI controller and frequency bias coefficient.

CPSO is new variant of PSO with faster speed because of

strong selection principle. In simple PSO, after certain

iterations, the populations set are almost identical and no

further improvement is observed. Reduced order AGC

system has shown saving in computation time with

identical responses. Like any other algorithms, this

method also somewhat sluggish in nature but positive

aspect of this method is its reliability and the number of

required generation for convergence decreases with

increase of population size.

APPENDIX

Nominal parameters of two area test system [15]:

H1= H1= 5 seconds

D1= D2= 8.3310-3

P.U. MW/Hz

R1= R1=2.4 Hz/P.U. MW

Th1= Th2=80 ms

Tt1= Tt2=0.3 seconds

Kp1= Kp2=120HzP.U. MW

Parameters for CPSO:

Population size= 20

Number of iteration=100

Wmin=0.6

Wmax=1

REFERENCES

[1] Zhihua Cui, Xingjiuan Cai and Jianchao zeng, Chaotic performance dependent particle swarm optimization. Division of system simulation and computer application Taiyuan University of

science and technology.


15

[2] C.F. Chen and L.S. Sheieh, A novel approaches to linear model simplification, International journal of control, 8, 561-570, 1968.

[3] K.C. Divya and P.S. Nagendra Rao, A novel AGC simulation scheme based on reduced order prime mover models, IEEE transaction on control system and applications, 1099-1103, 2003.`

[4] X. J. Cai, Z. H. Cui, J. C.Zeng and Y. Tan, Performance-dependent adaptive particle swarm optimization, International Journal of Innovative Computing, Information and Control, Vol.3,

No.6B,pp.1697-1706, 2007.

[5] J. Kennedy and R. Eberhart, Particle swarm optimization, in Proc. IEEE Int. Conf. Neural Networks, vol. IV, Perth, Australia,

1995, pp. 19421948. [6] Z.-L. Gaing, A particle swarm optimization approach for

optimum design of PID controller in AVR system, IEEE Trans. Energy Conversion, vol. 19, pp. 384-391, June 2004.

[7] M. Clerc, The swarm and the queen: towards a deterministic and adaptive particle swarm optimization, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 1999), pp.

1951-1957, 1999. [8] .D.Goldberg, Genetic algorithm in search optimization and

machine learning, Addison-Wesley, 1989. [9] Rania Hassan, Babak Cohanim, Oliver de Week, A comparison

of Particle Swarm Optimization and the Genetic Algorithm American institute of aeronautics and astronautics.

[10] R. C. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya,

Japan, pp.39-43,1995 [11] K.P. Wong, Z. Y. Dong, Special issue on evolutionary

computation for system and control application international journal of system science, vol, 35, No. 13-14, 20 oct-15 Nov.2004, pp 729-730.

[12] R. C. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, pp.39-

43,1995.

[13] G. Yu, and R. Hwang, Optimal PID speed Control of brush less DC motors using LQR Approach, in Proc. IEEE Int. Conf. Systems, Maand Cybernetics, 2004, pp. 473-478.

[14] Richard C. Dorf and Robert H. Bishop, Modern control system, Pearson international edition 2003.

[15] O.I. Elgerd, Electric energy system theory an introduction, McGraw Hill Co., 2001.


16

Combined Economic/Environmental Dispatch

with Fuzzified Multi-Objective Particle Swarm

Optimization considering Voltage Stability 1S. Surender Reddy,

2A. R. Abhyankar,

3P. R. Bijwe

1, 2, 3Indian Institute of Technology Delhi

[email protected] ,

[email protected],

[email protected]

AbstractTraditional Economic Load Dispatch deals with minimizing generation cost while satisfying a set of equality

and inequality constraints. The fossil fuel plants pollutes

environment by emitting some toxic gases. Thus,

conventional minimum cost operation cannot be the only

basis for generation dispatch; emission minimization must

also be taken care of. The objective of reactive power

optimization problem can be seen as minimization of real

power loss over the transmission lines. Large integrated

power systems are being operated under heavily stressed

conditions which imposes threat to voltage stability. The

objectives economic dispatch, emission dispatch, loss

minimization and voltage stability are to be met for efficient

operation and control. The results of all the four objectives,

considering one objective at a time are conflicting and non

commensurable. Hence, an efficient control which meets all

the specified objectives is required. Initially each objective is

optimized individually using Particle Swarm Optimization

(PSO) and then all the four objectives are optimized

simultaneously using fuzzified PSO. The effectiveness of the

proposed approach is tested on IEEE 30 bus system.

Keywprds Economic load dispatch, emission dispatch, reactive power optimization, voltage stability, particle swarm

optimization (PSO), fuzzy min-max approach.

I. INTRODUCTION

The Power system should be operated in such a way that

both real and reactive power are optimized

simultaneously. Real power optimization problem is the

traditional economic dispatch which minimizes the real

power generation cost. Reactive power should be

optimized to provide better voltage profile as well as to

reduce total system transmission loss. Traditional

Economic Dispatch [1] aims at scheduling committed

generating units outputs to meet the load demand at

minimum fuel cost while satisfying equality and inequality

constraints. On the other hand, thermal power plants

(which contribute major part of electric power generation)

create environmental pollution by emitting toxic gases

such as carbon dioxide (CO2), sulphur dioxide (SO2),

nitrogen oxides (NOx). Initially, Combined Economic and

Environmental/Emission dispatch (CEED) problem was

solved by minimizing fuel cost considering emission as

one of the constraints. The CEED problem is one of the

fundamental issues in power system operation. The

operation and planning of a power system is characterized

to maintain a high degree of economy and reliability [2].

Among the options available to the power system

operators to operate the generation system, the most

significant is the economic dispatch. Traditionally electric

power plants are operated on the basis of least fuel cost

strategies and only little attention is paid to the pollution

produced by these plants. The generation of electricity

from the fossil fuel releases several contaminants, such as

sulphur oxides (SO2), nitrogen oxides (NOx) and carbon

dioxide (CO2) into the atmosphere. But, if pollution is

decreased by suitably changing the generation allocation,

the cost of generation increases deviating from economic

dispatch. The characteristics of emissions of various

pollutants are different and are usually non-linear. This

increases the complexity of the CEED problem. Large

integrated power systems are being operated under heavily

stressed conditions which imposes threat to voltage

stability. Voltage collapse occurs when a considerable part

of the system attains a very low voltage profile or

collapses. Hence, voltage stability of the system is also an

important consideration and need to be taken care of

simultaneously along with economic dispatch. The four

objectives of minimization of fuel cost, minimization of

emission, minimization of losses and minimization of

system stability index are conflicting and non

commensurable. Hence, trade off solution using fuzzy

min-max approach is proposed in this paper. Assuming the

decision maker (DM) has imprecise or fuzzy goals of

satisfying each of the objectives, the multi-objective

problem can be formulated as a fuzzy satisfaction

maximization problem which is basically a minmax

problem. PSO is an unconstrained optimization method. A

PSO method that includes the constraints penalty factor

approach is used to convert the constrained optimization

to an unconstrained optimization problem [3]. In PSO, the

search for an optimal solution is conducted using a

population of particles, each of which represents a

candidate solution to the optimization problem. Particles

change their position by flying round a multidimensional

space by following current optimal particles until a

relatively unchanged position has been achieved or until

computational limitations are exceeded. Each particle

adjusts its trajectory towards its own previous best

position and towards the global best position attained till

then. PSO is easy to implement and provides fast

convergence for many optimization problems and has


17

recently gained lot of attention in power system

applications. In [4], it is demonstrated that PSO gets better

results in a faster, cheaper way compared with other

methods. In PSO, the potential solutions called particles,

fly through the problem space by following the current

optimum particles. Compared to GA, the advantages of

PSO are that PSO is easy to implement and there are few

parameters to adjust. PSO is initialized with a group of

random particles (solutions) and then searches for optima

by updating generations [5]. The description and

applications of PSO are presented in [6, 7]. The main aim

of this paper is to investigate the applicability of PSO to

the various optimization problems and prove that this

algorithm can be used effectively, to determine solutions

to various complex problems. Fuzzified PSO approach is

tested on several case studies which are extremely difficult

to solve by standard techniques due to the non-convex,

non-continuous and highly nonlinear solution space of the

problem. The paper is organized as follows. Sections II,

III, IV and V presents economic load dispatch with fuel

cost minimization, emission dispatch with emission

minimization, reactive power dispatch with active power

loss minimization, voltage stability maximization with L-

index minimization as single objective optimization

subproblems using PSO. Section VI describes multi-

objective optimization problem using fuzzified PSO.

Finally, brief conclusions are presented in Section VII.

II. ECONOMIC DISPATCH (ED) USING PSO

The ED problem is to determine the optimal combination

of power outputs of all generating units to minimize the

total fuel cost while satisfying the load demand and

operational constraints [8].

A. Objective Function:

The economic dispatch problem is a constrained

optimization problem and it can be mathematically

expressed as follows:

(1)

where ai, bi and ci are fuel cost coefficients, subjected to

number of power system network equality and inequality

constraints.

B. Equality Constraint:

The power balance constraint is an equality constraint that

reduces the power system to a basic principle of

equilibrium between total system generation and total

system loads. Equilibrium is met only when the total

system generation equals to the total system load (PD) plus

the system losses (PLoss).

(2)

where,

(3)

(4)

Bij are constants called B coefficients or loss coefficients.

C. Inequality Constraint:

The maximum active power generation of a source is

limited by thermal consideration. Unless, we take a

generator unit offline it is not desirable to reduce the real

power output below a certain minimum value Pmin.

Pi,min Pi Pi,max (5) Where Pi,min is minimum power output limit, Pi,max is

maximum power output limit of ith generator (MW).

D. Representation of individual:

For an efficient evolutionary method [9], the

representation of chromosome strings of the problem

parameter set is important. The proposed approach uses

the equal system incremental cost cost as individual (particles) of PSO [2]. Each individual within the

population represents a candidate solution for solving the

economic dispatch problem. The advantage of using

system Lambda instead of generator units output is that, it

makes the problem independent of the number of the

generator units and also number of iterations for

convergence decreases drastically. This is particularly

attractive in largescale systems.

E. Evaluation Function:

We must define the evaluation function for evaluating the

fitness of each individual in the population. In order to

emphasize the best chromosome and speed up

convergence of the iteration procedure, the evaluation

value is normalized into the range between 0 and 1. The

evaluation function [5] adopted is

(6)

where, k is a scaling constant (k = 50 in this study).

III. EMISSION DISPATCH (ED) USING PSO

Fossil-fuel fired electric power plants use coal, gas or

combinations as the primary energy resource, and produce

atmospheric emissions whose nature and quantity depend

on the fuel type and its quality. Coal produces particulate

matter such as ash and gaseous pollutants such as carbon

oxides, sulphur oxides and oxides of nitrogen. The thermal

energy dissipated in cooling water raises its temperature

and may be considered as a pollutant. Hydro-plants

produce no such emissions. Nuclear power produces

radiation emissions, which are well contained. Most of the

research work summarized aims at reducing oxides of

sulphur SO2 and oxides of nitrogen NOx. A major effort


18

has been devoted in reducing one type of pollutants or a

mixture of pollutants. The approaches are designed either

to reduce the total production of emissions or to reduce the

concentration of pollution at ground level at certain areas

which depends on both emissions and meteorological

factors.

A. Objective function:

The objective of emission dispatch is to minimize the total

environmental degradation or the total pollutant emission

due to the burning of fuels for production of power to

meet the load demand [10]. Dispatch the power generation

to minimize emissions instead of the usual cost objective

of economic dispatch is economic and easy in operation

[11]. The emission function can be expressed as the sum

of all types of emissions such as NOx, SO2, particulate

materials and thermal radiation with suitable pricing for

each pollutant emitted. In this paper only NOx emission is

taken into account, since it is more harmful than other

pollutants. The NOx emission can be approximated as a

quadratic function of the active power output from the

generating units and it is shown in Fig 1.

Fig 1: NOx Emission Function

B. Objective Function:

The emission dispatch problem is a constrained

optimization problem and it can be mathematically

expressed as [12],

(7)

where, E is total emission release (Kg/hr) and i, i, i are emission coefficients of the ith generating unit subject to

demand constraint (8) and generating capacity limits (9).

(8)

Pi,min Pi Pi,max (9)

C. Representation of Individual:

The proposed approach uses the equal system incremental

emission release emission as individual (particles) of PSO [13]. Each individual within the population represents a

candidate solution for solving the emission dispatch

problem.

IV. REACTIVE POWER OPTIMIZATION USING PSO

The purpose of Reactive Power Dispatch (RPD) is mainly

to improve the voltage profile in the system and to

minimize the real power transmission loss while satisfying

the unit and system constraints [14]. This goal is achieved

by proper adjustment of reactive power control variables

like generator bus voltage magnitudes (Vgi), transformer

tap settings (tk), reactive power generation of the capacitor

bank (Qci) [15].

A. Objective Function:

The objective of RPD is to identify the reactive power

control variables, which minimizes the real power loss

(Ploss) of the system. This is mathematically stated as [16],

(10)

subjected to the following constraints.

B. Equality Constraints:

These are load flow constraints such as

The power balance constraints includes real and reactive

power balances.

(11)

(12)

Where i, j are the bus indices, PGi and QGi are active and

reactive power generations at bus i respectively.

C. Inequality Constraints:

These constraints represent the system operating

constraints. Load bus voltages (Vload), reactive power

generation of generator (Qgi) and line flow limit (Sl) are

variables, whose limits are satisfied by adding a penalty

terms in the objective function. These constraints are

formulated as,

(i) Voltage limits

(13)

(ii) Generator reactive power capability limit

(14)

(iii) Capacitor reactive power generation limit

(15)


19

(iv) Transformer tap setting limit

(16)

(v) Transmission line flow limit

(17)

D. Evaluation Function:

Each particle consists of voltages, taps and shunts encoded

in it. The size of each particle is equal to sum of number

of voltages excluding slack bus voltage, taps, and shunts.

The fitness function [17] employed is

(18)

V. VOLTAGE STABILITY IMPROVEMENT USING PSO

Voltage stability is concerned with the ability of a power

system to maintain acceptable voltages at all nodes in the

system under normal condition and after being subject to a

disturbance [15]. A power system is said to have a

situation of voltage instability when a disturbance causes a

progressive and uncontrollable decrease in voltage level.

An accurate knowledge of how close the actual systems operating point is from the voltage stability limit is crucial

to operators [18]. Hence, to find a voltage stability index

has become an important task for many voltage stability

studies. These indices provide reliable information about

proximity of voltage instability in a power system.

Problem Formulation:

Each particle consists of voltages, taps and shunts encoded

in it. The size of each particle is equal to sum of number

of voltages, taps, and shunts. The fitness function [17]

employed is

(19)

where, index is the stability index value. Stability index

of power system is computed [19] based on the solution of

the power flow equations. Here, L-index is used, which is

a quantitative measure for the estimation of the distance of

the actual state of the system to the stability limit [20].

The L index describes the stability of the complete system

and is formulated as,

where ng is number of generators, n is number of buses

in the system. The Lindex value varies in a range between

0 (no load) and 1 (voltage collapse) [21]. Stability Index

of the system is computed as

VI. FUZZIFIED PSO FOR MULTI-OBJECTIVE CEED

The real power optimization sub-problem minimizes fuel

cost by controlling controllable generator outputs while

keeping the generator bus voltages unchanged. The system

losses, stability index and emission computed at this

power dispatch are very high compared with the results

obtained when respective ones are taken as objective.

Similarly the reactive power sub-problem deals with

minimization of total transmission loss of the system by

controlling all the reactive power sources such as taps,

shunts etc. When loss minimization is taken as objective

total system losses reduces but cost, emission and stability

indices are high. The emission dispatch sub problem

minimizes total emission output from the fossil fuel plants

by controlling the generator outputs. At this power output

of generator, the cost, total system losses and stability

index are high [22]. In the same way Stability index sub

problem minimizes the index by controlling the PV bus

voltages and thus improves the system stability limit. But

the cost, emission and system losses are very high. Thus,

results of all the four sub problems are conflicting with

one other. Hence, multiobjective PSO technique is used to

find the best compromise solution. In this section results

of all the four sub problems are fuzzified using fuzzy min-

max approach and then PSO is used to determine the final

trade off solution from all these fuzzified values.

A. Problem Formulation:

Each particle consists of active power generations,

voltages, taps and shunts excluding slack bus voltages

encoded in it. The size of each particle is equal to sum of

active power generations, number of voltages excluding

slack bus, taps, and shunts.

Assuming the decision maker (DM) has imprecise or

fuzzy goals of satisfying each of the objectives, the multi-

objective problem can be formulated as a fuzzy

satisfaction maximization problem which is basically a

min-max problem [23]. The task here is to determine the

compromise solution for all the four optimization sub

problems. The goal is to minimize G(X) = compromised

solution of G1(X1), G2(X2), G3(X3), G4(X4) [24], while

satisfying the set of constraints Ax < B. where G1(X) is

Fuel cost minimization sub problem, G2(X) is Loss

minimization sub problem, G3(X) is emission

minimization sub problem, and G4(X) is L-index

minimization sub problem.

Let F1(Xi) be fuel cost in $/hr for ith control vector, F2(Xi)

be losses in p.u. for ith control vector, F3(Xi) be stability

index for ith control vector, and F4(Xi) be emission release

in kg/hr for ith control vector. Let the individual optimal


20

control vectors for the sub problems be X1 , X2 , X3

,X4 respectively. To find, the global optimal control

vector X such that,

The imprecise or fuzzy goal of the DM for each of the

objective functions is quantified by defining their

corresponding membership functions i as a strictly monotonically decreasing function with respect to the

objective function f.

Where i=1 to 4. In case of a minimization problem,

i =0 or tends to zero, if fi > fmaxi and i =1 or tends to 1, if fi < fmini . where fmaxi and f mini are the unacceptable and desirable

level for respectively. In the proposed approach, we have

considered a simple linear membership function for f i

because none of the objectives have very strict limits. The

membership function, i for ith objective function is shown in Fig 2.

VII. RESULTS AND DISCUSSION

The proposed approach is tested on IEEE 30 bus system

[25]. The PSO parameters used in CEED case studies are:

number of particles 60, learning factors C1=2.05, C2=2.05,

weight factor w=1.2, constriction factor K=0.7925.

Maximum number of iterations=100. First, each objective

is optimized individually at a time and then four objectives

are optimized simultaneously using fuzzified PSO.

1) Economic Dispatch using PSO : Table I shows all

objective function values considering fuel cost

minimization as single objective optimization subproblem

and the optimum value obtained is 806.498 MW. The total

system generation is 293.984 MW.

2) Emission Dispatch using PSO : Table II gives all

objective function values considering emission dispatch as

single objective problem and the optimum value obtained

is 229.145 (Kg/hr). The total system generation is 287.804

MW.

3) Reactive Power Optimization using PSO :

The lower voltage magnitude limits at all buses are 0.95

p.u. and the upper limits are 1.1 for all the PV buses and

1.05 p.u. for all the PQ buses and the reference bus. The

lower and upper limits of the transformer tapping are 0.9

and 1.1 p.u. respectively. Table III presents all objective

function values considering loss minimization as single

objective optimization problem. Table IV and V gives the

positions of tap changing transformers and shunt

susceptance values respectively.

4) Voltage Stability Improvement using PSO :

Table VI gives all objective function values considering

L-index minimization as single objective optimization

problem. Table VII and VIII gives the positions of tap

changing transformers and shunt susceptance values

respectively.


21

5) Fuzzified PSO for Multi-objective Optimization :

From the above single objective optimization techniques it

is observed that when one objective is optimized it will

give optimum value for that objective but other objective

function values are deviating from optimum. Hence, there

is a

conflicting behavior between one objective with reference

to other objective. To overcome this, multi-objective

optimization using Fuzzified PSO technique is proposed in

this paper. The compromise solution is given in Tables IX,

X and XI. The total system generation is 288.035 MW,

total load 283.4 MW.

VIII. CONCLUSIONS

In this paper, an approach to solve multi-objective

optimization problem which aims at minimizing fuel cost,

real power loss, emission release and improving stability

index of the system has been proposed. The proposed

algorithm has been tested on IEEE 30 bus system. The

four objectives economic dispatch, emission dispatch,

stability index and reactive power optimization are solved

individually and the results from these individual

optimizations are fuzzified and final best compromise

solution is obtained. The multi-objective problem

is handled using the fuzzy decision satisfaction

maximization technique which is an efficient technique to

obtain trade off solution in multi-objective problems. The

proposed approach satisfactorily finds global optimal

solution within a small number of iterations.

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[3] O. Alsac and B. Stott, Optimal Load Flow with Steady State Security, IEEE Trans. Power Apparatus and Syst., vol. 93, no.3, pp. 745-751, May/June 1974.

[4] M. R. AlRashidi and M. E. E. Hawary, A Survey of Particle Swarm Optimization Applications in Electric Power Systems, IEEE Trans.Evolutionary Computation., vol. 13, no. 4, pp. 913-918, Aug. 2009.

[5] J. Kennedy and R. Eberhart, Particle swarm optimization, in Proc. IEEE Int. Conf. Neural Networks (ICNN95), vol. 6, pp. 942-1948, Perth, Australia, 1995.

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[8] Z. L. Gaing, Particle Swarm Optimization to Solving the Economic Dispatch Considering the Generator Constraints, IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1187-1195, Aug. 2003.

[9] J. B. Park, K. S. Lee, J. R. Shin, K. Y. Lee, A Particle Swarm Optimization for Economic Dispatch with Non-smooth Cost Functions, IEEE Trans. Power Syst., vol. 20, no. 1, pp. 34-42, Feb. 2005.

[10] Keib, H. Ma, J. L. Hart, Economic dispatch in view of the Clean Air Act of 1990, IEEE Trans. Power Syst., vol. 9, no. 2, pp. 972- 978, May 1994.

[11] J. H. Talaq, F. E. Hawary, and M. E. E. Hawary, A Summary of Environmental/Economic Dispatch algorithms, IEEE Trans. Power Syst., vol. 9, no. 3, pp. 1508-1516, Aug.

1994. [12] S. Kumar, K. Dhanushkodi, J. J. Kumar, C. K. C. Paul,

Particle Swarm Optimization Solution to Emission and Economic Dispatch Problem, IEEE TENCON, 2003.

[13] P. Dutta , A. K. Sinha, Environmental Economic Dispatch constrained by voltage stability using PSO, IEEE Electrical Engineering, IIT Kharagpur., 2006.

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