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This article was downloaded by: [Tripura University]On: 22 February 2013, At: 02:06Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK
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A New Approach to EconomicDispatching Using Load Bus
Elimination TechniquesJawad Talaqa
University of Bahrain Department of Electrical
Engineering P.O. Box 32038, Bahrain
Version of record first published: 30 Nov 2010.
To cite this article: Jawad Talaq (2000): A New Approach to Economic Dispatching
Using Load Bus Elimination Techniques, Electric Machines & Power Systems, 28:8,
723-734
To link to this article: http://dx.doi.org/10.1080/07313560050082712
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Electric Machines and Power Systems, 28:723734, 2000
Copyright cs 2000 Taylor & Francis
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A New A pproach to Economic Dispatching UsingLoad Bus Elimination Techniques
JAWAD TALAQ
University of BahrainDepartment of Electrical EngineeringP.O. Box 32038, Bahrain
This paper presents a new technique to solving the economic dispatch problembased on a reduced order model of the original system. Loads are rst mod-
eled by their appropriate voltage-dependent models as load admittances. Loadadmittances are then added to the bus admittance matrix and their respectivebuses are eliminated. The obtained model is a reduced model of the originalsystem. The admittance matrix is of the same order as the number of voltage-controlled buses in the system. The variables of the reduced model are thevoltage-controlled buses angles and active power generations. Newton Raph-son method is used to calculate the angles and active generations of the reduced
model while minimizing the operational cost. Load bus voltages and angles ofthe original system are then calculated by a direct method and load admittancesare modied. The process is repeated until convergence is achieved. The simu-lation is carried out on IEEE 118 bus test system. A comparison between thenew approach and the penalty factors method has been made. It is shown thatoperational cost is improved and solution time is signicantly reduced whencompared to the penalty factors method of economic dispatch.
1 Introduction
Economic dispatch is used in real time to allocate the total generation producedamong all units in the power system in such a way as to minimize the operationalcost. The equal incremental cost criterion for economic dispatch was rst imple-mented by utilities due to its simplicity when transmission losses are neglected.With the developments of power systems, transmission losses could not be ignoredand had to be included in the power balance equation. Transmission losses couldhave been modeled as a constant percentage of total system demand and the equalincremental cost criterion still could be implemented; however, transmission lossesdepend on the allocation of generation among units, and the treatment of losses in
the power balance equation needed to be improved. This led researchers to nd suit-able transmission loss models, and therefore the B-coecients method had emerged.From then, researchers concentrated in nding methods that can be implementedin real time and still meet the required accuracy in modeling transmission lossesand penalty factors of units. References [15] review recent advances in classic eco-nomic dispatch. The optimal power ow is an exact formulation of the networkactive and reactive power mismatches at all buses, and hence led researchers to
Manuscript received in nal form October 7, 1999.Address correspondence to Jawad Talaq.
723
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further investigate the possibility of real time implementation. Reference [6] con-tains a survey of the optimal power ow literature. Now, optimal power ow is usedin power system planning, and researchers are going on for further improvementthat hopefully it can replace the economic dispatch in real time implementation.
Until then, however, utilities still will rely on the classical economic dispatch for itssimplicity, and further developments that can improve both operational cost andsolution time should be encouraged.
2 T he Equal Incremental Cost C riterion
Traditionally, active power generation is distributed among all generators in such away as to minimize the total operational cost of all generators in the power systemwhile meeting system demand and transmission losses.
The Lagrangian function formulated in this case is:
L =X
i
Ci +
PD + PL o s s
Xi
Pi
!, (1)
where
Ci = a0i + a1iPi + a2iP2
i+ a3iP
3i (2)
is a function of third order representing operational cost of generator i, Pi is theactive generation of unit i, PL o s s is the transmission losses, PD is the total systemdemand, and is the Lagrangian multiplier associated with the power balanceequation.
If transmission losses are ignored, then the familiar equal incremental cost cri-terion for minimum operational cost is obtained:
@Ci@Pi
=@Cj@Pj
= . (3)
3 T he Penalty Factors Method
If transmission losses are included in equation (1), then the following criterion isobtained:
@Ci@Pi
P Fi =@Cj@Pj
P Fj = , (4)
where
P Fi = 1=(1 @PL o s s=@Pi ) (5)
is the penalty factor of unit i.Transmission losses may be related to active generation through the familiar
B-coecients as follows:
PL o s s = PT BP + B0 P + B 00, (6)
where P is the active power generation vector, B is a square matrix, B0 is a rowvector, and B 00 is a constant.
The B-coecients are functions of the operating point of the power system but
may be assumed constant for a certain range of operating points. Several methods
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have been proposed to nd the B-coecients by several authors. One of these isthe one described in [3]:
PL o s s = Re aV(I* ZI), (7)
where Z is the bus impedance matrix and I is the bus current injection vector.Simplifying equation (7) yields
PL o s s =NX
i= 1
NXj= 1
[D ij (PiPj + Q i Qj ) + Cij (Q iPj PiQj )], (8)
where Pi and Q i are active and reactive power injections at bus i, and N is thetotal number of buses in the system:
D ij = R ij cos(ij )=ViVj , (9)
Cij = R ij sin(ij )=Vi Vj . (10)
R ij in equations (9) and (10) are elements of the real part of the impedance matrix.Furthermore, if we assume that bus angle dierences are small, i.e., sinij ~= 0,
then the following simplied equation is obtained [3]:
@PL o s s=@Pi = 2N
Xj = 1
D ijPi . (11)
4 T he Proposed Technique
Load modeling has been applied to the power ow solutions [78]. Bus eliminationtechnique [8] has been applied to the power ow solution without considering theeconomic dispatch problem. The aim here is to apply load modeling and load buselimination techniques to the economic dispatch problem and treat transmission
losses as an exact model. Consider both active and reactive power of loads asexponentially voltage dependent models having the following forms:
PL = P0Va , (12)
QL = Q0Vb, (13)
S*
L= PL + j QL , (14)
where
P0 is the active power specied at nominal voltage.Q0 is the reactive power specied at nominal voltage.
a is the voltage dependent exponential constant of active power model.
b is the voltage dependent exponential constant of reactive power model.
Also we have
S*
L= V
*IL = YL V
2. (15)
This yields
YL = 1V2 [P
0Va j Q 0Vb]. (16)
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Equation (16) represents the load admittance model. The exponential constants ofthe load model usually are determined from eld tests. The algorithm is capable ofhandling dierent values of exponential constants for the dierent buses; however,three special cases may be obtained from equation (16):
1. Constant admittance load model (a = b = 2.0):
YL = (P0 j Q0 )
2. Constant complex power model (a = b = 0.0):
YL =1
V2(P0 j Q 0)
3. Constant current model (a = b = 1.0):
YL =1
V(P0 j Q0 )
The admittance matrix is modied by adding the load admittances to thediagonal elements that belong to load buses. The load buses are then eliminated toobtain a reduced model as follows:
" ILIG#
=" Y
L0Y
LGYG L YG# " V
LVG#
, (17)"0
IG
#=
"YL YLG
YG L YG
# "VL
VG
#, (18)
where
YL = YL0 + Yd (19)
and Yd is a diagonal matrix containing the load admittance models described by
equation (16).Solving equation (18) yields
VL = Y1
L YLGVG (20)
and
IG = YRVG , (21)
where
YR = YG YG LY1
L YLG (22)
is an admittance matrix of the same order as the number of voltage-controlled buses(NG).
The complex power injections of the reduced model are
S*
G i= V
*
G iIG i = V*
G i
N G
Xj= 1VG j Yij , (23)
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where Yij are the elements of the reduced bus admittance matrix YR and N G isthe number of the voltage-controlled buses in the system.
This yields the active power mismatches of the voltage-controlled buses
DPi = ViN GXj= 1
Vj[G ij cos(ij ) + B ij sin(ij )] Pi . (24)
The augmented Lagrangian function is formulated as follows:
L =N GXi= 1
Ci +N GXi= 1
pi DPi +X 1
2si (Pi Pi lim )
2, (25)
where Ci is the operational cost of unit i as dened in equation (2) and pi is aLagrangian multiplier associated with the active power mismatch of unit i. The lastterm of equation (25) is the sum of quadratic penalty functions imposed on activegeneration of units that exceed their maximum or minimum power limits and si isthe penalty factor corresponding to each penalty function.
The penalty factors, si , for the violating units are increased with iterationsuntil variables are within acceptable tolerances. If a violating unit goes back toits operational range, its si value is set to zero. The reactive power mismatchesneed not be included in the Lagrangian function if they do not violate their limits;
however, reactive generations of units are tested for violations, and if one violatesa limit, then the reactive generation is set to the limit and the voltage is released.Usually, in power ow solutions, PV buses are converted to PQ buses if they violatetheir reactive generation limits. This technique is not implemented here becauseactive generation of units that violate their reactive generation should still remaindispatchable. Instead, the reactive power mismatch of the violated unit is introducedto the Lagrangian function together with its Lagrangian multiplier. This adds two
variables to the Lagrangian function: the voltage and the Lagrangian multiplierassociated with the reactive power mismatch.
Taking the rst and second derivatives of the Lagrangian function with respectto the variables and applying Newton Raphson method yields the following matrixequations:
f = W D x, (26)
D x = [D, D p, D P]T , (27)
f = "@L
@
,@L
@p,@L
@P#
T
, (28)
W =
24H JT 0J 0 1
0 1 IIC
35 , (29)
where f is a vector representing the rst derivatives of the Lagrangian function, D xis the incremental variables vector, and W is the Jacobian matrix. Submatrix Hcontains the second derivatives of the Lagrangian function with respect to angles,and submatrix J contains the rst derivatives of the active power mismatches with
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Figure 1. Flow chart of the proposed algorithm.
respect to angles. IIC is a diagonal submatrix containing the second derivatives ofthe operational cost functions. T denotes the transpose:
H =@2(DP)@2
, (30)
J =@(D P)@
, (31)
IIC =@2C
@P2. (32)
Equation (26) is solved for D x using Newtons method. Once voltage-controlledbus angles are computed, equation (20) is solved for load bus voltages and angles.Load admittances are recalculated using equation (16), and YL of equation (22)
is modied using equation (19). Equation (26) is solved again, and the process isrepeated until convergence is achieved. A ow chart describing the implementationof the proposed algorithm is shown in Figure 1.
5 Results of Simulation
The proposed algorithm has been tested on IEEE 118 bus test system using UNIX-based Mips Millennium Computer System. A comparison between the equal incre-mental cost criterion, the penalty factors method, and the proposed method has
been made for three dierent exponential constants of load modeling. IEEE 118
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bus test system consists of 54 voltage-controlled buses, of which 18 are dispatch-able generators.
The proposed technique consists of major and minor iterations. A major iter-ation refers to the loop in which loads are modeled, eliminated, and the system is
reduced. A minor iteration refers to each Newtons iteration of the reduced model.In the proposed technique, operational cost is minimized while power mismatchesare satised. This means that transmission losses are exactly modeled compared tothe approximate models used for the penalty method. This fact assures least oper-ational cost solutions for the proposed technique compared to the penalty method,even with more updates of the B-coecients. Following are the four categories ofsimulation that have been compared with the proposed technique for each loadmodeling case:
1. The equal incremental cost criterion without losses: Transmission losses areneglected in the power balance equation. System demand assumed constantand units are dispatched followed by a normal power ow. The process con-sists of only one major iteration.
2. The equal incremental cost criterion with losses: First, transmission lossesare neglected and units are dispatched, assuming constant system demand,followed by a normal power ow. This is the rst major iteration. Systemdemand and transmission losses obtained from the rst major iteration areused in the power balance equation, and units are redispatched followed by
another power ow. This is the second major iteration.3. The penalty factors method (two iterations): First, units are dispatched
according to the equal incremental cost criterion and, assuming constantsystem demand, followed by a normal power ow. This is the rst majoriteration. B-coecients are calculated and units are redispatched throughthe penalty factors method followed by another power ow. This is the sec-ond major iteration. The process is repeated for another major iteration byupdating the B-coecients, system demand, and transmission losses.
4. The penalty factors method (three iterations): Same as in 3 above withanother extra update of the B-coecients, system demand, and transmissionlosses.
Operating cost, solution time, and iteration counts are shown in Table 1. Outputgeneration for the three cases, the constant complex power demand, the constantcurrent demand, and the constant load admittance demand are shown in Tables 2,3, and 4.
The operational cost for the constant complex power case of the equal incre-mental cost criterion is taken as basis for the comparison as shown in Table 1. Using
the proposed approach, operational cost has been reduced to 0.978824 pu (2.12%)instead of 0.980297 pu (1.97%) when the penalty factors method is used with twoiterations. The solution time has been considerably reduced to 9.70 seconds insteadof 31.80 seconds, which is just comparable to the ordinary equal incremental costcriterion (8.40 seconds). Operational cost may be further reduced using the penaltyfactors method with more iterations to values comparable with that obtained bythe proposed method, but this will be on the expense of solution time. This is ob-
vious from Table 1, as operational cost has been reduced to 0.978881 pu (2.11%),when the penalty factors method is used with three iterations but solution time in-
creased to 47.20 seconds. T he iteration counts shown in Table 1 refer to the required
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Table
1
IEEE118bustest
system
operatingcost,s
olutiontime
iterationco
unts
Constantcom
plex
Constantcurrent
Constanta
dmittance
power
loadcase
loa
dcase
loa
dcase
Operating
So
l.
Operating
So
l.
Operating
Sol.
cost
time
Iter.
cost
time
Iter.
cost
time
Iter.
(pu
)
(sec
)
counts
(pu
)
(sec
)
counts
(pu
)
(sec)
counts
Equa
lincrementalcost
1.0
00000
8.4
6
0.9
937
50
9.8
6
0.9
87985
11
.4
6
(no
losses)
Equa
lincrementalcost
1.0
00012
12
.7
6,4
0.9
938
72
15
.7
6,4
0.9
88191
18
.9
6,4
(withlosses
)
Pena
lty
factorsmetho
d
0.9
80297
31
.8
6,4,4
0.9
753
25
35
.0
6,4,5
0.9
70686
37
.8
6,4,5
(two
iterations
)
Pena
lty
factorsmetho
d
0.9
78881
47
.2
6,4,4,3
0.9
741
09
51
.4
6,4,5,3
0.9
69634
54
.7
6,4,5,4
(three
iterations
)
Propose
dapproac
h
0.9
78824
9.7
6,3,3,2,2,2,2
0.9
740
61
8.4
6,3,3,2,
2
0.9
69594
4.0
6
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Table
2
IEEE
118bustestsystem
econom
icdispatchresu
ltsfo
rtheconstantcomp
lexpower
loa
dcase
Equa
lincrementalc
ost
Pena
lty
factorsmethod
Propose
d
(no
losses)
(withlosses)
(two
iterations)
(three
iterations)
app
roac
h
Gen
Output
Out
put
Output
Pena
lty
Output
Pena
lty
Ou
tput
no
(pu
)
(pu
)
(pu
)
factor
(pu
)
factor
(pu
)
1
2.6
48308
2.78
9004
2.4
91929
0.9
79561
2.4
25627
0.9
80862
2.5
15780
2
3.0
00000
3.00
0000
3.0
00000
0.9
93083
3.0
00000
0.9
92414
3.0
00000
3
3.0
00000
3.00
0000
2.8
98995
1.0
08289
2.8
59776
1.0
08374
2.9
19261
4
4.7
24726
3.01
8913
1.9
05124
1.0
00000
2.8
07982
1.0
00000
2.8
87783
5
1.8
18647
1.93
1203
1.7
03315
0.9
78853
1.6
85921
0.9
77559
1.6
65509
6
2.2
37294
2.46
2407
2.9
64136
0.9
45336
2.9
27559
0.9
44105
2.9
07532
7
1.1
82206
1.27
6003
1.8
69217
0.9
15163
1.8
63007
0.9
13247
1.8
61620
8
3.0
00000
3.00
0000
3.0
00000
0.9
49162
3.0
00000
0.9
47706
3.0
00000
9
0.0
00000
0.00
0000
0.5
66501
0.9
29570
0.5
30774
0.9
28873
0.5
01174
10
2.0
60386
2.23
6255
2.8
45772
0.9
36016
2.7
73667
0.9
36635
2.7
31307
11
0.1
98770
0.36
1094
0.5
94663
0.9
40489
0.5
64251
0.9
40274
0.5
48163
12
1.8
64411
2.05
2006
2.6
25525
0.9
39073
2.5
37073
0.9
40169
2.4
97523
13
1.3
00752
1.43
1632
2.1
37909
0.9
21830
2.0
20119
0.9
25968
1.9
76541
14
2.6
37294
2.86
2407
0.7
53693
1.0
42669
0.7
37510
1.0
40614
0.7
46670
15
3.0
00000
3.00
0000
3.0
00000
1.0
00179
3.0
00000
1.0
07192
3.0
00000
16
1.5
16013
1.72
4451
1.7
75624
0.9
60674
1.5
56930
0.9
66464
1.5
75372
17
3.0
00000
3.00
0000
2.6
46669
0.9
98249
2.5
36044
1.0
00830
2.5
13478
18
1.2
15920
1.33
8265
0.6
86765
1.0
06544
0.6
42112
1.0
07069
0.6
30012
Totaloutput(pu
)
38
.404727
38
.42
3640
37
.465
838
37
.468353
37.4
77726
Losses
(pu
)
1.7
24727
1.74
3640
0.7
85
838
0
.788353
0.7977263
Totalloa
d(pu
)
36
.68
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Table
3
IE
EE118bustestsystem
econom
icdispatchresultsfortheconstantcurrentloa
dcase
Equa
lincrementalcost
Pena
lty
factorsmetho
d
Propose
d
(no
losses)
(with
losses)
(two
iter
ations)
(three
iterations)
app
roac
h
Gen
Output
Output
Output
Pena
lty
Output
Pena
lty
Ou
tput
no
(pu
)
(p
u)
(pu
)
factor
(pu
)
factor
(pu
)
1
2.6
48308
2.771172
2.4
80031
0.9
79685
2.4
13588
0.9
80996
2.5
02934
2
3.0
00000
3.000000
3.0
00000
0.9
93097
3.0
00000
0.9
92396
3.0
00000
3
3.0
00000
3.000000
2.8
91147
1.0
08247
2.8
52101
1.0
08321
2.9
10766
4
4.5
06134
3.014110
1.9
38106
1.0
00000
2.7
99179
1.0
00000
2.8
74410
5
1.8
18647
1.916938
1.6
93776
0.9
78977
1.6
76291
0.9
77689
1.6
55924
6
2.2
37294
2.433876
2.9
45675
0.9
45412
2.9
10642
0.9
44129
2.8
90394
7
1.1
82206
1.264115
1.8
60466
0.9
15298
1.8
54993
0.9
13326
1.8
53963
8
3.0
00000
3.000000
3.0
00000
0.9
49074
3.0
00000
0.9
47559
3.0
00000
9
0.0
00000
0.000000
0.5
53624
0.9
29654
0.5
19818
0.9
28864
0.4
90652
10
2.0
60386
2.213966
2.8
28636
0.9
36200
2.7
59700
0.9
36686
2.7
17809
11
0.1
98770
0.288125
0.5
85584
0.9
40612
0.5
56641
0.9
40290
0.5
40652
12
1.8
64411
2.028230
2.6
06553
0.9
39289
2.5
21934
0.9
40232
2.4
83118
13
1.3
00752
1.415044
2.1
16378
0.9
22487
2.0
04543
0.9
26302
1.9
62918
14
2.6
37294
2.833876
0.7
36971
1.0
42752
0.7
28309
1.0
40387
0.7
37329
15
3.0
00000
3.000000
3.0
00000
1.0
00916
3.0
00000
1.0
07248
3.0
00000
16
1.5
16013
1.698033
1.7
40770
0.9
61432
1.5
34594
0.9
66757
1.5
54360
17
3.0
00000
3.000000
2.6
29004
0.9
98497
2.5
24167
1.0
00808
2.5
02359
18
1.2
15920
1.322759
0.6
81477
1.0
06333
0.6
38166
1.0
06763
0.6
26485
Totaloutput(pu
)
38
.186134
38.200244
37
.288197
37
.294667
37.3
04076
Losses
(pu
)
1.7
05317
1.726982
0.75
5894
0.7
80144
0.7
89341
Totalloa
d(pu
)
36
.480817
36.473262
36
.512304
36
.514522
36.5
14735
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A New Approach to Economic Dispatching 733
Table
4
IEEE118bustestsystemec
onom
icdispatchresu
lts
fortheconstanta
dm
ittance
loa
dcase
Equa
lincrementalcost
Pena
lty
factorsmetho
d
Propose
d
(no
losses)
(with
losses
)
(two
iter
ations)
(three
iterations)
app
roac
h
Gen
Output
Output
Output
Pena
lty
Output
Pena
lty
Ou
tput
no
(pu
)
(p
u)
(pu
)
factor
(pu
)
factor
(pu
)
1
2.6
48308
2.754576
2.4
68611
0.9
79806
2.4
62075
0.9
81126
2.4
90715
2
3.0
00000
3.000000
3.0
00000
0.9
93111
3.0
00000
0.9
92387
3.0
00000
3
3.0
00000
3.000000
2.8
83650
1.0
08208
2.8
44777
1.0
08272
2.9
02705
4
4.3
02682
3.010645
1.9
66801
1.0
00000
2.7
90022
1.0
00000
2.8
61717
5
1.8
18647
1.903661
1.6
84980
0.9
79073
1.6
67481
0.9
77788
1.6
47095
6
2.2
37294
2.407321
2.9
28447
0.9
45471
2.8
94889
0.9
44140
2.8
74375
7
1.1
82206
1.253050
1.8
52280
0.9
15414
1.8
47502
0.9
13389
1.8
46782
8
3.0
00000
3.000000
3.0
00000
0.9
48981
3.0
00000
0.9
47413
3.0
00000
9
0.0
00000
0.000000
0.5
41589
0.9
29722
0.5
09556
0.9
28848
0.4
80749
10
2.0
60386
2.193220
2.8
12663
0.9
36360
2.7
46629
0.9
36725
2.7
05114
11
0.1
98770
0.276055
0.5
77113
0.9
40715
0.5
49524
0.9
40295
0.5
33597
12
1.8
64411
2.006101
2.5
88889
0.9
39476
2.5
07786
0.9
40282
2.4
69567
13
1.3
00752
1.399605
2.0
96512
0.9
23077
1.9
90097
0.9
26598
1.9
50152
14
2.6
37294
2.807321
0.7
22288
1.0
42779
0.7
19733
1.0
40164
0.7
28589
15
3.0
00000
3.000000
3.0
00000
1.0
01548
3.0
00000
1.0
07282
3.0
00000
16
1.5
16013
1.673446
1.7
08939
0.9
62103
1.5
13945
0.9
67014
1.5
34761
17
3.0
00000
3.000000
2.6
12813
0.9
98701
2.5
13128
1.0
00775
2.4
91959
18
1.2
15920
1.308327
0.6
76597
1.0
06120
0.6
34488
1.0
06467
0.6
23178
Totaloutput(pu
)
37
.982682
37.993328
37
.122172
37
.131633
37.1
41055
Losses
(pu
)
1.6
88615
1.711393
0.76
6963
0.7
72540
0.7
81589
Totalloa
d(pu
)
36
.294067
36.281930
36
.355209
36
.359093
36.3
59466
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734 Talaq
Newtons iterations for each major iteration. For the constant complex power case,it took seven major iterations for the proposed approach solution. The requiredNewtons iterations for this case are 6, 3, 3, 2, 2, 2, and 2.
For the constant current load model, the reduction in operational cost is
0.974061 (2.59%) compared to 0.975325 (2.46%) of the penalty factors method.The solution time has also been considerably reduced. For the constant admittanceload model case, the solution time, as expected, is lowest (4.0 seconds) becausethere is no need for updating load admittances as they are constants.
6 Conclusions
A new technique to solving the economic dispatch problem has been proposed.The technique uses a reduced order model of the original system. The reducedorder model is obtained by eliminating all load buses from the system after beingmodeled appropriately. The variables of the model are the voltage-controlled busesangles and active generation. The eect of dierent load modeling on generatoractive generation and operational cost has been studied. A comparison between theproposed technique and the classic economic dispatch has been made. Results ofthe simulation on IEEE 118 bus test system shows that improvement in operationalcost and solution time is achieved by using the proposed technique.
References
[1] Happ, H. H., 1977, Optimal Power Dispatch: A Comprehensive Survey, IEEE Trans.on Power App. and Syst., Vol. PAS-96, No. 3, pp. 841854.
[2] Aoki, K., and Satoh, T ., 1984, New Algorithms for Classic Economic Dispatch, IEEETrans. on Power Appt. and Syst., Vol. PAS-103, No. 6, pp. 14231431.
[3] Wenyuan, L., 1985, An On-Line Economic Power Dispatch Method With Security,Electric Power Systems Research, Vol. 9, pp. 173181.
[4] Lin, C. E., Hong, Y. Y., and Chuko, C. C., 1987, Real-Time Economic Dispatch,IEEE Trans. on Power Syst., pp. 968972.
[5] Chowdhury, B. H., and Rahman, S., 1990, A Review of Recent Advances in EconomicDispatch, IEEE Trans. on Power Syst., Vol. 5, No. 4, pp. 12481259.
[6] Huneault, M., and Galiana, F. D., 1991, A Survey of the Optimal Power Flow Liter-ature, IEEE Trans. on Power Syst., Vol. 6, No. 2, pp. 762770.
[7] El-Hawary, M. E., and Dias, L. G., 1987, Incorporation of Load Models in Load FlowStudies: Form of Model Eects, IEE Proceedings, Vol. 134, Part C, No. 1, pp. 2730.
[8] Talaq, J. H., 1995, Modeling and Elimination of Load Buses in Power Flow Solutions,IEEE Trans. on Power Syst., Vol. 10, No. 3, pp. 11541158.
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