Network and Generator Constrained Economlc Dispatch Using Hybrid PSO-QP Algorithm
Keerati Chayakutkheeree, Member, IEEE and Wattanapon Kamklar
considered as powerfi.r1 method to obtain the solutions in power system optimization problems [2], [6-8].
PSO t9l is one of the modem heuristic algorithms which gained lots of attention in various power system applications.
PSO can be applied to nonlinear and non-continuous optimization problems with continuous variables. It has been developed through simulation of simplified social models.
PSO is similar to the other evolutiorlary algorithms in that the system is initialized with a population of random solutions.
Generally, the PSO is characterized as a simple heuristic of well balanced mechanism with flexibility to enhance and adapt to both global and local exploration abilities [9]. It is a stochastic search technique with reduced memory requirement, computationally effective and easier to implement compared to other artificial intelligent technique.
A1so, PSO has a more global searching ability at the begiruring of the run and a local search near the end of the run.
Therefore, while solving problems with more local optima, there are more possibilities for the PSO to explore local optima at the end of the run.
In l2l, a modified PSO mechanism was suggested to deal with the equality and inequality constraints in the ED problems. A PSO method for solving the ED problem with generator prohibited zone and non-smooth cost function was proposed in [6]. However, the methods require large nurnber of population search in order to obtain global or near global optima. To overcome this drawback a hybrid method that integrates the PSO with sequential quadratic programming (SQP) was proposed in t7-81. Nonetheless, transmission line and transformer loading limit constraints were not included in the problem formulation.
In this paper, hybrid particle swarm quadratic programming based economic dispatch (PSO-QP-ED) with network and generator constrained algorithm is proposed. The transmission line and transformer loading and generator prohibited operating zones constraints are taken into accourft. In the proposed PSO-QP-ED algorithm, the sets of reai power generation at generator bus are used as particles in the PSO.
The QP based ED with transmission line limit and transformer loading constraints is performed every generation to obtain the best solution of each population search. The proposed PSO- QP-ED is tested with the IEEE 30 bus system and compared to the PSOED (without QP) under transmission line and transformer loading limit constraints and generators disconiinues operating cost functions.
The organization of this paper is as follows. Section 2 The aurhors are with Electrical Engineering Department, Faculty of addresses the PSO-QP-ED problem formulation. The Hybrid rngineering, Sripatum University, Bangkok 10900, Thailand (e-mail: PSO-QP algorithm for PSO-QP-ED problem is given in [email protected]).
ECII-GON 2(,06
The 2OOO ECTI lnternabional Conference Abstruct-- This paper proposes a hybrid particle swarm
optimization (PSO) and quadratic programming (QP) atgorithm for power system economic dispatch (ED) with transmission line limit and transformer loading and generator prohibited operating zones constraints. In the proposed PSO-QP-ED algorithm, the sets of real power generation at generator bus are used as particles in the PSO' The quadratic programming ED with transmission line limit and transformer loading constrained is performed every generation to obtain the best solution of each population search. The proposed PSO-QP-ED is tested with the IEEE 30 bus system and compared to the PSO-ED. The proposed PSO-QP-ED results in more probability to obtain global minimum total system operating cost in the constrained ED than the PSOED considering generator prohibited operating zones constraints.
Index Terms-- economic dispatch, parficle swarm optimizationo quadratic programming' generator prohibit operating mne.
I. INTRODUCTION
CONOMIC dispatch (ED) plays an importance role in power system operation. The principal objective of ED is to obtain the minimum operating cost satisfuing power balance constraint. Currently, the ED problem includes generator and network limits constraints. The cost fi'mction for each generator is usually approximately represented by a single quadratic function and solved by mathematical programming based on optimization techniques such as gradient method, linear programming or quadratic programming (QP) t1l. However, the input output characteristics of larye units are actually having discontinuously cost curves due to valve-point and multi-fuel effects [2-5]. These discontinuously cost curves were usually ignored in the ED method leading to inaccurats dispatch result. To obtain accurate dispatch results, approaches without restriction on the shape of incremental fuel cost functions are needed.
The practical ED problems with valve-point and multi-fuel effects are represented as a non-smooth optimization problem with equality and inequality consffaints, and this makes the problem of finding the global optimum difficult. To solve this problem, many salient methods have been proposed such as genetic algorithm [3], evolutionary programming [4], Tabu search [5], and particle swann optimization (PSO) are
Section 3. Numerical results on the IEEE 30 bus test system are illustrated in Section 4. Lastly, the conclusion is given.
II. PSO-QP-ED PROBLEM FoRMULATTON The PSO-QP-ED objective ftrrction can be expressed as, Minimize total system operating cost,
FC = ZF(PG), (1)
subject to the power balance constraints, ,\ts
P - p - \ ' l r z l l z l l . , l ^- Di Llr iy i1iil-os(9u - 6r),i=7,"'N8, Q )
.{?
Qo, -Qo, = -Ilrlln,lln rlsn@u - 51),i =1,...,N8, (3)
j=l
and the line flow limit and transformer loading constraints,
lf'l< "f,* , for l:1, ..., NL, (4)
and generator minimum, maximum, and prohibited operating zones limits constraints,
Plit s Po, < Ptij, i eBG,j : 7,...,t{zi,
p 1 , 1 _ p m i n r C i - a c i ,
p[:NZ' = p;;^', where
FC is the total system operating cost ($/h),
F(Pc,) is the operating cost ofthe generator connected to bus i (9,&),
Pc, is the real power generation ofthe generator connected to bus i (MW),
Pn, is the real power load at bus i (MW), It ) is the voltage magnitude of bus i (V),
BG is the set ofbuses connected with generators,
/-n' ir the MVA flow limit of line or transformer / (MVA) , NB is the total number of buses ,
PSU* is the maximum real power generation at bus i (MW), Pffi" is the minimum real power generation at bus i (MW), Qn is the reactive power demand at bus t (MVAR) ,
t l
ly,,l is the magnitude of the .V u eiement of 16* (mho), Qij is the angle of the y i; element of I5o. (radian), fi is the MVA flow of line or transformer / (MVA), Qc, is the reactive power generation at bus i (MVAR).
6u is the voltage angle difference between bus i and 7 (radian)
f [:l and, P(;i are the boundary of the generator prohibited operating zone. Pci , i e BG, is the output of the pSO-Qp-ED algoriihm. The method is intended to line flow and transformer loading limits constrained economic dispatch in
power system. The bus voltages and are not included in the paper.
II1. CFCOPD ArcoRTrHu
In a PSO system, particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience, and the experience of neighboring particies, making use of the best position encoultered by itself and its neighbors. The swarm direction of a particle is defined by the set ofparticles neighboring the particle and its history experience. In the proposed hybrid PSO-QP-ED algorithm, the set of particle is represenled as,
P", = [P[,,Pj,,..., Pll ], PLt = lPit, P12,..., Pl, *ol'
Where P61 is the matrix representing the set of individual searches. More specifically, it is the set of the generator real power generations. The sub-matrix P[, is the set of current position ofparticleT representing the real power generation of the generator connected to bus i ( PS). Each particle is used to solve the quadratic programming optimal power flox- (QPOPF) and the best previous position of the 7th particle i*
recorded and represented as,
p b e s t = l p b e s t l , p b e s l 2 , . . . , p b e s t * o f r { 1 0 i {he index of the best particle among all the particles in the group is represented by the gbestl . The rate of the velocin- for particleT is represented as,
f = f u l ,u l , . . . , u t o l ' . ( 1 1 :
The modified velocity and position of each particle can be calcu-lated using the current velocity and the distance from
pbesti to gbest, as shown in the following formulas:
= w.ui(D + c, . rand() . (pbest, - p|,ru) + C, . Rand0 .(gbest, - Pl:',), j = 7 , 2 , . . . . , M , i = \ , 2 , . . . , N G ,
D i ( + l ) - 1 y l ( + t ) 1 , 2 , . . . . , M , i = 7 , 2 , . u ! < u ! < t t \ *
- ' t
where
M is the number of particles in a group, r\G is the number of members in aparticle, / is the pointer ofiterations (generations), w is the inertia weight factor,
Cr,C, we the acceleration constants,
rand{),RandQ are the uniform random values in the range [0,1],
is the velocity of paticleT correspondin gto Po, iteration l, and
power controls
( 1 2 i ( 8 )
(5) (6) (1)
EGtt-coN 2('06
The 2OOO ECII lnternational Conference
at
Pll') is fte cunent positiorr of particle i conesponding to P6., at iteration ,.
lIre parameter Y,** determined the resolutiort, tx litness' with rn'hich regions are to be searched betw*n the present position and the target positio-n. If u,o'* is too high, pafiicles might fly past good solulions. If ufl*is too small. particles may not explore sufficiently beyond local solutions. In many experiences with PSO, xlf"'was often set at lU-20% of the dynrunic range of the variable on each dirncnsion [2]'
The constants C,and Crrepresent tlre weiglrting of the stochastic acceleration tenns that pull erch prrticlc toward the pbesti and glte.tf, positions. Low valuc* allow particles to roam far frorn the target regions before being tugged back. On the other hand, high value,s result in abrupt movement toward, or past, targer regions. Hence, the acceleration constants
crwd c, were often set to be 2.0 according to past experiences I2], [?]. Suitable selection of inertia weight tl provides a balance between global and local exploration$, thu$
requiring less iteration on average to find a sufficiently optimal solution. As originally developed, ?l2often decreases tinearly from about 0.9 to 0.4 during atwrf2l, [7]. In general' rhe inertia weight 1# is set according to the following equation:
'llhe
evaluation value is ntlrmalized into tlre range between 0 and I as.
EV =1|(li.x,, + Ppn")
(IF(Pc,)-F,i") lvhere 1a*, =f *obr3ffi
P,1* -- t.(hr",- t o, - ri,- )',
4* is the maximum generation cost among all individuals in the initial population, ud F-r is the minimum generation cost among all
individuats in the initial population'
In order to linrit the evaluation v&tue o'f each individual of the population wiihin a feasible range, before estimating the evaluation value ofan individual, the generation power oulput mu$t $atisry the constraints in (5). ff one individual satisfres all consrraints, then it is a feasible individual and {* has a small value. Otherwise, &e 4*r value of the indiYidual is penalized with a v€ry large positive constant' The computational procedure is shovm in Fig' l.
IV. NUMEruCA'LRESULT$
Tlre IEEE 30 bus system [10] is u-red as the test data. It network diagram is shorrn in Fig.2. The generator fuel cost fimctions andprohibited operating zones {ue given in Table i' The parame{ers of the proposed PSO-QP'ED and the PSOED are as follows;
Population size = 200, Generation (M) = I0,
Pmin =0.4, wr"" =0'9,
ul* =0.5'P;f , u,-'=-0.5'P;li'',
C t = 2 ' a n d C 2 = ! '
1 1 6 )
( r 7 )
( 1 8 )
w= lfro - Pe$:wtnin
.6. (15)
ECfl-CON 29g6
The 2OOO ECfl lntz''rna1"lonal Confercnae
Rardomly ecarvhing initial point for each population in Pt -Gec Palviolatc iF---
opcntingboes or its tnnximrm J
Solvo &e OPF for ttc f ponul{tion usiue QP Calculate8l', uring Bq (tA)
thrt gives th bedf,4 x Pbcs if67. >81'of G'bcsi tbcr Gbcsl = Pbcdt
FE. l. PSG,QP-ED C-ompuatisral Proccdure
F(Pa)= a,1 6, ' P*
+ c r ' 4 ,
TSle2
Sy$crn
Tinrcoflhc
Table 2 addresses the xmrnry re*hs of PSOED and rhe propccd PSO{P-ED fr\ra 50 trials- Tbc coot€rgcnc€
profrg of ric bcs $olurrffi of PSO-QP-ED and PSOED re shosr in Ffu 3 Figure 4 $rrrs 6c totd.sySem opa'ating cost from 50 tnals of ruSED and PSO-QP-ED. The results strow that tlrc total sysern ograting coss of the poposed PSO-QP- ED are loser than that of the PSOED. Thecomputational dme of PSO-QP-ED is longer than that of the PSOED due to PSO- QP-ED solves QPOPF for each individual search in the search space. The computational time of the proposed method couid be decreased by using parallel computation method Howevern the PSO-QP-ED gives more probabilrty to obtain the global mirdmum total system operating cost.
Gan.Ettrn{smbar -
.t'' Fig. 3. Conv,rgcnce Jropcrtics of IFEE 30 bus trct s,,$sn
Fig.4. The reslts ftottl 50 trirls ofIFF? 3{t b{rs resrryst€&
V. Coxcrustox
In this pap€r, a hytrid particle swarTn optimization and quadratic pro{framming for economic dispatch (PSO-QP-ED) is efficiently minimizing the rotal system operating cost satisfying ransmission line limits and transformer loading conslrainls wi& the generstor prohibited operating zones constraints. The proposed PSO-QP-ED results in more probability to obtain global minimum total syste{n operatrng co$t in the constrained ED $'ith generator prohibited operating zones constraints than the PSOED.
VI- -Rrrgnurrcls
tll A. J. lYood ard B, F. Wollenberg. Power Generation, Op€ralion erd Corxrol. John Wiley & Sons, Cauada 1996,
121 O. Zne-lne, "?article Srtum Oplirnizalion to Solving tte fronomic DiE{clr Crxsidcring thc Censrator (bnskaints", IF-EE Tmns. fo*s Sysr, vol-18, no.3,.{ug.2003, pp. I 187-1 195.
t3l D. ,Srinirrasan and G.B. Slrcble, *Oonclic Algcillun Sslufion of Econ<rmic Dispatch rvith !'alvc-Point Loadfu4". IEEE Tnrns. Pos€
Syst. vol.8, noJ. 1993, pp. 1325-1331.
t4l I'1. T. Yang an4 ?. C. Yaug. aru! C. t,. Ilung, "Evohrriurary Prograruming Based Ecosomic Dispatch fi'r Unit rvith Non-Srrooth Incrernenral F,uel Cosl Functicns", IEEE I'rarx. Power Syst.. wl.ll- rrc.1, 1996, pp. t12-1IE.
W. M. Liu, F. S. Cheng, and T. M. Tsay, -An lrproved Tabu Sercb fa Economic Diryatch with :Vultiple trlinirnr", IEEE Trans. Poner Sys- rnl.17. nc. t, 2002. pp. 108-1 1.2.
[6] . J. Pdt, K. Lce, J. Shil, and K. Y. Lrc,'.rr Padicle SwarmOptrmizati,m for Ecsomic Dispatc*r rviilr Nonsmooilr Cod Functions". IEEE Trc- Po*'er Sys.. vol. 15, no.4, 2000, pp. 1232-1739.
F1 T. Aruldoss Albcrt Victoirea and A. Ibenciry Jqnkumarb, "Ilybrit PSO-SQP tbr F*oncmic Di$arc,lr rvith Vah'o-Point Etrect", Et€cric Powtr Sy$em Reseac$. Vol.7l, 2004, pp. 5l-59.
tE] T, Addos3 Albet Victoirea and A Ebenerzer Jeyakunud
"Deterninislislty Guided PSO for Dyrmmic Dispatch ConsieiB Valv+Poirt Effect", Elearic Porver System Research, Vol.?3, ?005. pp- 313-32?.
t9l J. Kernedy and R Ebertlurt" "Particle Sn'arm Opdmization", Prr"
IF'FE Int- Csrf. Ncrrral Networkq vol.W, pp.1942.1948, 1995 [0] O. .A,lsac ard B. Slott, "Optirnal load tlow $ith seady Sar. se{aritlf,
IEEE Trans. Pon'er Apparst. S)sLr ?ol. PAS93, no. 3, pp. 745-751. tt{ry 1913.
Brocna,priles
Dr. Kcereti Chryrkrlkheeree rrceinod thc M-Eng and D.Eng. degre in Eteclric Polr'€r Systan Managerrcnt ftom Asian hrstilute of Tcdrnolog ie 1999 ard 2004, rcryeclivcly. He is clrrvrtly a Hcad of EE dc.partrnca, Sripatrm Universiry. Thailand. His reserdr id€rests are io polr€r Ent€E arralFis, optiuriarion ard AI appticariors to power qysterrc, and power qrsl4{n fdtn cturing ard deregulation.
Mr. Wrttrnepon Xrmklrr receivetl the B-Edg (Honor) degrees in EE frr Sripalrrn University in 1999. He is nrrrently a lecturer of EE depadneu, Sripaftrn Unircsity, Tlailand.
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The 2OOO ECII lntamaional Conferenac Fig. 2. IEEE 30 Bus lest System
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Abstract-- This paper proposes a hybrid particle swarm optimization (PSO) and quadratic programming (QP) algorithm for power system economic dispatch (ED) with transmission line limit and transformer loading and generator prohibited operating zones constraints. In the proposed PSO-QP-ED algorithm, the sets of real power generation at generator bus are used as particles in the PSO. The quadratic programming ED with transmission line limit and transformer loading constrained is performed every generation to obtain the best solution of each population search. The proposed PSO-QP-ED is tested with the IEEE 30 bus system and compared to the PSO-ED. The proposed PSO-QP-ED results in more probability to obtain global minimum total system operating cost in the constrained ED than the PSOED considering generator prohibited operating zones constraints.
Index Terms-- economic dispatch, particle swarm optimization, quadratic programming, generator prohibit operating zone.
I. INTRODUCTION
CONOMIC dispatch (ED) plays an importance role in power system operation. The principal objective of ED is to obtain the minimum operating cost satisfying power balance constraint. Currently, the ED problem includes generator and network limits constraints. The cost function for each generator is usually approximately represented by a single quadratic function and solved by mathematical programming based on optimization techniques such as gradient method, linear programming or quadratic programming (QP) [1]. However, the input output characteristics of large units are actually having discontinuously cost curves due to valve-point and multi-fuel effects [2-5]. These discontinuously cost curves were usually ignored in the ED method leading to inaccurate dispatch result. To obtain accurate dispatch results, approaches without restriction on the shape of incremental fuel cost functions are needed.
The practical ED problems with valve-point and multi-fuel effects are represented as a non-smooth optimization problem with equality and inequality constraints, and this makes the problem of finding the global optimum difficult. To solve this problem, many salient methods have been proposed such as genetic algorithm [3], evolutionary programming [4], Tabu search [5], and particle swarm optimization (PSO) are
The authors are with Electrical Engineering Department, Faculty of Engineering, Sripatum University, Bangkok 10900, Thailand (e-mail:
considered as powerful method to obtain the solutions in power system optimization problems [2], [6-8].
PSO [9] is one of the modern heuristic algorithms which gained lots of attention in various power system applications.
PSO can be applied to nonlinear and non-continuous optimization problems with continuous variables. It has been developed through simulation of simplified social models.
PSO is similar to the other evolutionary algorithms in that the system is initialized with a population of random solutions.
Generally, the PSO is characterized as a simple heuristic of well balanced mechanism with flexibility to enhance and adapt to both global and local exploration abilities [9]. It is a stochastic search technique with reduced memory requirement, computationally effective and easier to implement compared to other artificial intelligent technique.
Also, PSO has a more global searching ability at the beginning of the run and a local search near the end of the run.
Therefore, while solving problems with more local optima, there are more possibilities for the PSO to explore local optima at the end of the run.
In [2], a modified PSO mechanism was suggested to deal with the equality and inequality constraints in the ED problems. A PSO method for solving the ED problem with generator prohibited zone and non-smooth cost function was proposed in [6]. However, the methods require large number of population search in order to obtain global or near global optima. To overcome this drawback a hybrid method that integrates the PSO with sequential quadratic programming (SQP) was proposed in [7-8]. Nonetheless, transmission line and transformer loading limit constraints were not included in the problem formulation.
In this paper, hybrid particle swarm quadratic programming based economic dispatch (PSO-QP-ED) with network and generator constrained algorithm is proposed. The transmission line and transformer loading and generator prohibited operating zones constraints are taken into account. In the proposed PSO-QP-ED algorithm, the sets of real power generation at generator bus are used as particles in the PSO.
The QP based ED with transmission line limit and transformer loading constraints is performed every generation to obtain the best solution of each population search. The proposed PSO- QP-ED is tested with the IEEE 30 bus system and compared to the PSOED (without QP) under transmission line and transformer loading limit constraints and generators discontinues operating cost functions.
The organization of this paper is as follows. Section 2 addresses the PSO-QP-ED problem formulation. The Hybrid PSO-QP algorithm for PSO-QP-ED problem is given in
Network and Generator Constrained Economic Dispatch Using Hybrid PSO-QP Algorithm
Keerati Chayakulkheeree, Member, IEEE and Wattanapon Kamklar
E
2
Section 3. Numerical results on the IEEE 30 bus test system are illustrated in Section 4. Lastly, the conclusion is given.
II. PSO-QP-EDPROBLEM FORMULATION
The PSO-QP-ED objective function can be expressed as, Minimize total system operating cost,
∑
∈=
BG i
PGi
F
FC ( ), (1) subject to the power balance constraints,
, 1,..., , ) cos(
1
NB i
y V V P P
NB
j
ij ij ij j i Di
Gi− =
∑
− ==
δ
θ (2) ,
1,..., , ) sin(
1
NB i
y V V Q
Q
NB
j
ij ij ij j i Di
Gi− =−
∑
− ==
δ
θ (3) and the line flow limit and transformer loading constraints,
max l
l f
f ≤ , for l=1, …, NL, (4) and generator minimum, maximum, and prohibited operating zones limits constraints,
j u Gi Gi j l
Gi P P
P, ≤ ≤ , , i∈BG, j = 1,…,NZi, (5)
min,
1 ,
Gi l
Gi P
P = (6)
max,
,
Gi NZ u
Gi P
P i = (7) where
FC is the total system operating cost ($/h), )
(PGi
F is the operating cost of the generator connected to bus i ($/h),
PGi is the real power generation of the generator connected to bus i (MW),
PDi is the real power load at bus i (MW), Vi is the voltage magnitude of bus i (V), BG is the set of buses connected with generators,
max
fl is the MVA flow limit of line or transformer l (MVA) , NB is the total number of buses ,
max
PGi is the maximum real power generation at bus i (MW),
min
PGi is the minimum real power generation at bus i (MW), QDi is the reactive power demand at bus i (MVAR) ,
yij is the magnitude of the yij element of Ybus (mho),
θ
ij is the angle of the yij element of Ybus (radian), fl is the MVA flow of line or transformer l (MVA), QGi is the reactive power generation at bus i (MVAR), δij is the voltage angle difference between bus i and j(radian)
j l
PGi, and PGiu,jare the boundary of the generator prohibited operating zone. PGi, i∈BG, is the output of the PSO-QP-ED algorithm. The method is intended to line flow and transformer loading limits constrained economic dispatch in
power system. The bus voltages and reactive power controls are not included in the paper.
III. CFCOPD A
LGORITHMIn a PSO system, particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience, and the experience of neighboring particles, making use of the best position encountered by itself and its neighbors. The swarm direction of a particle is defined by the set of particles neighboring the particle and its history experience. In the proposed hybrid PSO-QP-ED algorithm, the set of particle is represented as,
] [ Gi1 Gi2 GiM
Gi P ,P ,...,P
P = , (8)
T j
NG G j G j
G P P
P , ,..., ]
[ 1 2 ,
=
j
PGi . (9) Where PGi is the matrix representing the set of individual searches. More specifically, it is the set of the generator real power generations. The sub-matrix PGij is the set of current position of particle j representing the real power generation of the generator connected to bus i (PGij). Each particle is used to solve the quadratic programming optimal power flow (QPOPF) and the best previous position of the jth particle is recorded and represented as,
T
pbestNG
pbest
pbest , ,..., ]
[ 1 2
=
pbest . (10)
The index of the best particle among all the particles in the group is represented by the
gbest
j. The rate of the velocity for particle j is represented as,T j NG j
j
u u
u , ,..., ] [
1 2=
u
j . (11) The modified velocity and position of each particle can be calculated using the current velocity and the distance frompbest
i togbest
i as shown in the following formulas:, ,..., 2 , 1 , ,...., 2 , 1
), (
()
) (
() .
) ( 2
) ( 1
) ( ) 1 (
NG i
M j
P gbest Rand
C
P pbest rand
c u w u
t j Gi i
t j Gi i t
j i t
j i
=
=
−
⋅
⋅ +
−
⋅
⋅ +
=
+
(12)
, ,..., 2 , 1 , ,...., 2 , 1
),
1 ( ) ( ) 1
( P v j M i NG
PGijt+ = Gijt + ijt+ = = (13)
max min
i j i
i u u
u ≤ ≤ , (14) where
M is the number of particles in a group, NG is the number of members in a particle, t is the pointer of iterations (generations), w is the inertia weight factor,
2 1
, c
c
are the acceleration constants, ()(),Rand
rand are the uniform random values in the range [0,1],
) (t j
u
i is the velocity of particle j corresponding to PGi at iteration t, and3
) (t j
PGi is the current position of particle j corresponding to PGi at iteration t.
The parameter Vimaxdetermined the resolution, or fitness, with which regions are to be searched between the present position and the target position. If
u
imax is too high, particles might fly past good solutions. Ifu
imaxis too small, particles may not explore sufficiently beyond local solutions. In many experiences with PSO,u
imaxwas often set at 10–20% of the dynamic range of the variable on each dimension [2].Randomly searching initial point for each population in .
Does PGij violate its
prohibited operating zones or its maximum or minimum limits?
Solve the OPF for the i population using QP Calculates EVi using Eq (16)
i = i + 1 i > NG
Selects that gives the best EVi as Pbest
Update ui and PGij if EVi > EV of Gbest then Gbest = Pbest
j = j + 1 j > M
PGi = Gbest
Initialize the real power generation indices in the search space (i = 1, j = 1)
EVi = 1e-12
No
No Yes
Yes No Yes
j PGi
j
PGi
Fig. 1. PSO-QP-ED Computational Procedure
The constants
c
1andc
2represent the weighting of the stochastic acceleration terms that pull each particle toward thepbest
i andgbest
i positions. Low values allow particles to roam far from the target regions before being tugged back. On the other hand, high values result in abrupt movement toward, or past, target regions. Hence, the acceleration constantsc
1andc
2 were often set to be 2.0 according to past experiences [2], [7]. Suitable selection of inertia weightw
provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution. As originally developed,
w
often decreases linearly from about 0.9 to 0.4 during a run [2], [7]. In general, the inertia weightw
is set according to the following equation:M t w w w
w= max− max− min ⋅ . (15)
The evaluation value is normalized into the range between 0 and 1 as,
) /(
1 Fcost Ppbc
EV= + (16) where
) (
) ) ( ( 1
min max
min
cos F F
F P F abs
F i BG
Gi
t −
− +
=
∑
∈ , (17)
2
1
1 ⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ − −
+
=
∑ ∑
∈BG =
i
loss NB
i Di Gi
pbc P P P
P , (18)
F
max is the maximum generation cost among all individuals in the initial population, andF
min is the minimum generation cost among all individuals in the initial population.In order to limit the evaluation value of each individual of the population within a feasible range, before estimating the evaluation value of an individual, the generation power output must satisfy the constraints in (5). If one individual satisfies all constraints, then it is a feasible individual and
F
costhas a small value. Otherwise, theF
costvalue of the individual is penalized with a very large positive constant. The computational procedure is shown in Fig. 1.IV. NUMERICAL RESULTS
The IEEE 30 bus system [10] is used as the test data. It network diagram is shown in Fig.2. The generator fuel cost functions and prohibited operating zones are given in Table 1.
The parameters of the proposed PSO-QP-ED and the PSOED are as follows;
Population size = 200, Generation (M) = 10,
4 .
min =0
w , wmax =0.9,
max max 0.5 Gi
i P
u = ⋅ , uimin =−0.5⋅PGimin,
1=2
C , and C2 =2.
Table 1
Generator operating cost functions and constraints
2
) (
Gi i
Gi i i Gi
P c
P b a P F
⋅ +
⋅ +
= min
PGi PGimax Generator Prohibited Operating Zone Gen
Bus
ai bi ci MW MW From MW
To MW
From MW
To MW 1 0 2 0.00375 50 200 100 120 150 160 2 0 1.75 0.0175 20 80 25 30 40 60
5 0 1 0.0625 15 50 20 25 40 45
8 0 3.25 0.00834 10 35 15 20 25 30 11 0 3 0.025 10 30 15 18 22 25 13 0 3 0.025 12 40 20 25 30 35
Table 2
Results of IEEE 30 Bus Test System
PSOED PSO-QP-ED
Min Aver. Max Min Aver. Max Total System
Operating Cost
($/h) 806.10 818.13 834.05 805.14 809.46 816.47 Computation
Time of the Best Trial Solution
(sec)
52.67 69.39
4
Table 2 addresses the summary results of PSOED and the proposed PSO-QP-ED from 50 trials. The convergence property of the best solution of PSO-QP-ED and PSOED are shown in Fig.3. Figure 4 shows the total system operating cost from 50 trials of POSED and PSO-QP-ED. The results show that the total system operating costs of the proposed PSO-QP- ED are lower than that of the PSOED. The computational time of PSO-QP-ED is longer than that of the PSOED due to PSO- QP-ED solves QPOPF for each individual search in the search space. The computational time of the proposed method could be decreased by using parallel computation method. However, the PSO-QP-ED gives more probability to obtain the global minimum total system operating cost.
~ ~
~
1
28 3
9 8
6 11
7 5 4
2
15 14
12
18 19
13
16
17
20 23 24
30 10
27 29 25 26
22 21
~
~
~
Fig. 2. IEEE 30 Bus test System
1 2 3 4 5 6 7 8 9 10
804 806 808 810 812 814 816
Generation Number
Total System Operating Cost ($/h)
PSOED PSO-QP-ED
Fig. 3. Convergence properties of IEEE 30 bus test system
0 5 10 15 20 25 30 35 40 45 50
800 805 810 815 820 825 830 835 840 845 850
Trial Number
Total System Operating Cost ($/h)
PSOED PSO-QP-ED
Fig. 4. The results from 50 trials of IEEE 30 bus test system
V. CONCLUSION
In this paper, a hybrid particle swarm optimization and quadratic programming for economic dispatch (PSO-QP-ED) is efficiently minimizing the total system operating cost satisfying transmission line limits and transformer loading constraints with the generator prohibited operating zones constraints. The proposed PSO-QP-ED results in more probability to obtain global minimum total system operating cost in the constrained ED with generator prohibited operating zones constraints than the PSOED.
VI. REFERENCES
[1] A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control, John Wiley & Sons, Canada 1996.
[2] G. Zwe-Lee, “Particle Swarm Optimization to Solving the Economic Dispatch Considering the Generator Constraints”, IEEE Trans. Power Syst., vol.18, no.3, Aug. 2003, pp. 1187-1195.
[3] D. Srinivasan and G.B. Sheble, “Genetic Algorithm Solution of Economic Dispatch with Valve-Point Loading”, IEEE Trans. Power Syst., vol.8, no.3, 1993, pp. 1325-1331.
[4] H. T. Yang and, P. C. Yang, and C. L. Huang, “Evolutionary Programming Based Economic Dispatch for Unit with Non-Smooth Incremental Fuel Cost Functions”, IEEE Trans. Power Syst., vol.11, no.1, 1996, pp. 112-118.
[5] W. M. Lin, F. S. Cheng, and T. M. Tsay, “An Improved Tabu Search for Economic Dispatch with Multiple Minima”, IEEE Trans. Power Syst., vol.17, no.1, 2002, pp. 108-112.
[6] J. Park, K. Lee, J. Shin, and K. Y. Lee, “A Particle Swarm Optimization for Economic Dispatch with Nonsmooth Cost Functions”, IEEE Trans.
Power Syst., vol.15, no.4, 2000, pp. 1232-1239.
[7] T. Aruldoss Albert Victoirea and A. Ebenezer Jeyakumarb, “Hybrid PSO-SQP for Economic Dispatch with Valve-Point Effect”, Electric Power System Research, Vol.71, 2004, pp. 51-59.
[8] T. Aruldoss Albert Victoirea and A. Ebenezer Jeyakumarb,
“Deterministically Guided PSO for Dynamic Dispatch Considering Valve-Point Effect”, Electric Power System Research, Vol.73, 2005, pp.
313-322.
[9] J. Kennedy and R. Eberthart, “Particle Swarm Optimization”, Proc.
IEEE Int. Conf. Neural Networks, vol.IV, pp.1942-1948, 1995
[10] O. Alsac and B. Stott, “Optimal load flow with steady state security”, IEEE Trans. Power Apparat. Syst., vol. PAS93, no. 3, pp. 745-751, May 1973.
VII. BIOGRAPHIES
Dr. Keerati Chayakulkheeree received the M.Eng and D.Eng. degrees in Electric Power System Management from Asian Institute of Technology in 1999 and 2004, respectively. He is currently a Head of EE department, Sripatum University, Thailand. His research interests are in power system analysis, optimization and AI applications to power systems, and power system restructuring and deregulation.
Mr. Wattanapon Kamklar received the B.Eng (Honor) degrees in EE from Sripatum University in 1999. He is currently a lecturer of EE department, Sripatum University, Thailand.
