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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
1fvinod.k.pal@rediffmail.com
2ashwa_ks@yahoo.co.in
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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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.
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management, A report, 2007.
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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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)
Contribution of G-20 (MW)
Contribution of G-30 (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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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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
( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW )
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
( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW )
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
( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW ) ( MW )
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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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Chaotic Particle Swarm Optimization for
Reduced Order Model of Automatic Generation
Control 1Cheshta Jain,
2H. K. Verma
1, 2SGSITS, Indore
1cheshta_jain194@yahoo.co.in
2vermaharishgs@gmail.com
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.
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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)
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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,
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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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).
Dataset2 (Tp=30, R=8% of freq, T12=0.145).
Dataset3 (Tp=30, R=4% 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.
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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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.
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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.
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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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
1salkuti.surenderreddy@gmail.com ,
2abhyankar@ee.iitd.ac.in,
3prbijwe@ee.iitd.ac.in
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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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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
International Conference on Deregulated Environment and Energy Markets, July22-23, 2011
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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)
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(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
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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.
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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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