ICBO
ICBO
ICBO
a r t i c l e i n f o a b s t r a c t
Article history: This paper proposes Improved Colliding Bodies Optimization (ICBO) algorithm to solve efficiently the
Received 24 October 2015 optimal power flow (OPF) problem. Several objectives, constraints and formulations at normal and pre-
Received in revised form 9 January 2016 ventive operating conditions are used to model the OPF problem. Applications are carried out on three
Accepted 22 January 2016
IEEE standard test systems through 16 case studies to assess the efficiency and the robustness of the
Available online 4 February 2016
developed ICBO algorithm. A proposed performance evaluation procedure is proposed to measure the
strength and robustness of the proposed ICBO against numerous optimization algorithms. Moreover, a
Keywords:
new comparison approach is developed to compare the ICBO with the standard CBO and other well-
Colliding Bodies Optimization
Optimal power flow
known algorithms. The obtained results demonstrate the potential of the developed algorithm to solve
Security-constrained optimal power flow efficiently different OPF problems compared to the reported optimization algorithms in the literature.
Power system optimization © 2016 Elsevier B.V. All rights reserved.
Metaheuristics
http://dx.doi.org/10.1016/j.asoc.2016.01.041
1568-4946/© 2016 Elsevier B.V. All rights reserved.
120 H.R.E.H. Bouchekara et al. / Applied Soft Computing 42 (2016) 119–131
its performances are compared to CBO, ECBO and other well-known optimization Hence, u can be expressed as:
algorithms.
The main contributions of this paper can be summarized as follows: uT = PG2 . . .PGNG , VG1 . . .VGNG , QC1 . . .QCNC , T1 . . .TNT (7)
1. Development of an improved version of the CBO algorithm. where NG, NT and NC are the number of generators, the number
2. Implementation of ICBO, CBO, ECBO and other well-known optimization algo- of regulating transformers and the number of VAR compensators,
rithms for solving realistic OPF problems. respectively.
3. Implementation of a complete set of tests in order to assess optimization algo- It is worth mentioning that, transformer tap settings and shunt
rithms using different OPF problems, test systems, objective functions and
constraints.
devices settings are discrete in nature. In many works reported
4. Resolution of the OPF problem using practical constraints like prohibited zones in literature addressing the OPF, these settings are considered as
and using non-smooth objective functions by including valve point effect and continuous variables for simplicity. Then, the discrete variables are
multi-fuels options for a more realistic OPF. set to their nearest discrete value after the optimization has been
5. Resolution the OPF problem considering security constraints for more challeng-
done. The results have shown that this approach leads to acceptable
ing conditions.
6. Implementation of a new comparison method based on best and average values. results without incurring the exponential complexity as reported
7. Utilization of nonparametric statistics for the validation of the comparative by [38]. This last approach is adopted in this paper.
method.
2.4. State variables
The remainder of this paper is organized as follows. In Section 2, the OPF problem
is formulated. In Section 3, the proposed ICBO algorithm along with the standard
and enhanced versions of the CBO are described. The applications and results are The set of state variables for the OPF problem formulation are:
presented in Section 4. Finally, the conclusions are drawn in Section 5.
PG1 : active power generation at slack bus.
2. Problem formulation VL : voltage magnitudes at PQ buses or load buses.
QG : reactive power output of all generator units.
As previously mentioned, generally, the objective of the OPF Sl : transmission line loadings (or line flow).
problem is to identify or adjust a set of control variables that opti-
mize predefined power system objectives while satisfying system Hence, x can be expressed as:
and practical constraints [36,37]. In this paper, two formulations of
xT = PG1 , VL1 . . .VLNL , QG1 . . .QGNG , Sl1 . . .Slnl (8)
the OPF are considered. These are the classical OPF formulation and
the security constrained optimal power flow (SCOPF) formulation. where NL and nl are the number of load buses and the number of
transmission lines, respectively.
2.1. Classical OPF formulation
2.5. Constraints
The classical OPF problem can be formulated as follows [25,30]:
2.5.1. Equality constraints
Minimize F(x, u) (1)
The equality constraints of the OPF reflect the physics of the
Subject to g(x, u) = 0 (2) power system which is represented by the typical power flow equa-
tions [39,40]. The equality constrains can be represented as follows:
and h(x, u) ≤ 0 (3)
NB
where u is the vector of independent variables or control variables. PGi − PDi − Vi Vj [Gij cos (ij ) + Bij sin (ij )] = 0 (9)
x is the vector of dependent variables or state variables. F(x,u): j=1
objective function. g(x,u): set of equality constraints. h(x,u): set of
inequality constraints.
NB
QGi − QDi − Vi Vj [Gij sin (ij ) − Bij cos (ij )] = 0 (10)
2.2. SCOPF formulation j=1
The set of control variables in the OPF problem formulation are: PGmin ≤ PGi ≤ PGmax , i = 1, . . ., NG (12)
i i
NG
F(x, u) = ai + bi PGi + ci PG2 (19)
i
i=1
with Multi-fuels f
NG
i=1
2.6.5. Valve-point effect where F is the objective function and n is the population size.
For more realistic and precise modeling of fuel cost function, Step 3: sort the population where the best CB is ranked first. After
the valve-point effect has to be considered. The generating units that, the population is divided into two groups (with equal size).
with multi-valve steam turbines exhibit a greater variation in the The first group starts from the first (best) CB to the one of the
fuel-cost functions. The valve opening process of multi-valve steam middle and the second group starts from the middle to the end of
turbines produces a ripple-like effect as illustrated in Fig. 3 [43]. the population.
The significance of this effect is that the real cost curve function The CBs of the first group (i.e. the best ones) are considered as
of a large steam plant is not continuous but more important it is stationary whilst the CBs of the second group (the worst ones)
non-linear [43]. move toward the first group. Therefore, the velocities before col-
lision of the first group (stationary CBs) are given by:
n
vi = 0, i = 1, . . ., (27)
2
without valve point effect
and for the second group (moving CBs) they are given by:
with valve point effect
n
vi = xi − xi−(n/2) i = + 1, . . ., n (28)
2
e
Generation Cost ($/h)
where vi and xi are the velocity and the position of the ith CB,
respectively.
Step 4: calculate the new velocities after the pairwise collision
between the members of the first group and the ones of the second
d group. Therefore, the velocities after collision of the first group are
given by:
Start
Select constraints
Which
ICBO algorithm is ECBO
selected
Stop
Print opmal
soluon
Fig. 4. Flowchart of the proposed OPF solution using ICBO, CBO and ECBO.
124 H.R.E.H. Bouchekara et al. / Applied Soft Computing 42 (2016) 119–131
Table 1 Table 6
The main characteristics of the IEEE 30-bus test system. The main characteristics of the IEEE 118-bus test system.
1 1 0 2 0.00375 18 0.037
2 2 0 1.75 0.0175 16 0.038 and the positions of stationary CBs are updated as follows:
3 5 0 1 0.0625 14 0.04
4 8 0 3.25 0.00834 12 0.045 n
5 11 0 3 0.025 13 0.042
xinew = xi + rand × vi , i = 1, . . ., (33)
2
6 13 0 3 0.025 13.5 0.041
where xinew is the new position of the ith CB after collision.
Steps from 2 to 5 are repeated until a termination criterion is
Table 3
met.
Cost coefficients for generators 1 and 2 for the IEEE 30-bus test system when consid-
ering multi-fuels.
Generator Bus a b c PGmin PGmax a b c PGmin PGmax 3.2. Enhanced Colliding Bodies Optimization (ECBO)
i1 i1 i2 i2
Table 7
Summary of the studied cases.
CASE # 1 OPF Cost minimization using (19) Equality and non-equality constraints –
CASE #2 SCOPF Cost minimization using (19) Equality and non-equality constraints 1 and 6
for normal and selected N − 1
conditions.
CASE #3 OPF Cost minimization using (19) Equality, non-equality and prohibited –
IEEE 30-bus zones constraints
test system CASE #4 SCOPF Cost minimization using (19) Equality, non-equality and prohibited 1 and 6
zones constraints for normal and
selected N − 1 conditions.
CASE # 5 OPF Cost minimization and voltage profile Equality and non-equality constraints –
improvement using (21)
CASE #6 SCOPF Cost minimization and voltage profile Equality and non-equality constraints 1 and 6
improvement using (21) for normal and selected N − 1
conditions.
CASE #7 OPF Cost minimization and voltage stability Equality and non-equality constraints
enhancement using (22)
CASE #8 SCOPF Cost minimization and voltage stability Equality and non-equality constraints 1 and 6
enhancement using (22) for normal and selected N − 1
conditions.
CASE # 9 OPF Cost minimization considering Equality and non-equality constraints –
multi-fuels using (24)
CASE #10 SCOPF Cost minimization considering Equality and non-equality constraints 1 and 6
multi-fuels using (24) for normal and selected N − 1
conditions.
CASE #11 OPF Cost minimization considering valve Equality and non-equality constraints –
point effect using (25)
CASE #12 SCOPF Cost minimization considering valve Equality and non-equality constraints 1 and 6
point effect using (25) for normal and selected N − 1
conditions.
IEEE 57-bus CASE # 13 OPF Cost minimization using (19) Equality and non-equality constraints –
test system CASE # 14 SCOPF Cost minimization using (19) Equality and non-equality constraints 8 and 50
for normal and selected N − 1
conditions.
IEEE 118-bus CASE # 15 OPF Cost minimization using (19) Equality and non-equality constraints –
test system CASE # 16 SCOPF Cost minimization using (19) Equality and non-equality constraints 21 and 50
for normal and selected N − 1
conditions.
Therefore, in every iteration, three CBs are selected based on a b. calculate the new velocities of the three CBs after collision using
selection rule. The best CB among these three CBs is assumed to the following expressions:
be stationary and the remaining two CBs are assumed to be mov-
(mCB2 + εmCB2 )vCB2 (mCB3 + εmCB3 )vCB3
ing in the search space. The moving CBs collide the stationary one vCB1 = + (37)
and move in the search space with one more research direction mCB1 + mCB2 mCB1 + mCB3
than in CBO and ECBO. (mCB2 − εmCB1 )vCB2
The main steps of the proposed ICBO are: vCB2 = (38)
mCB2 + mCB1
each CB before collision are defined by: d. check the CBs, if a CB goes beyond the bounds of the search
space it is brought back inside it.
vCB1 = 0 (34) Step 5: diversify the population by exchanging some dimensions
of the CBs from the population with the ones from the archive pool.
vCB2 = xCB1 − xCB2 (35) Step 6: group both the CBs from the population and the ones from
the archive pool and select the best n ones to keep the same num-
vCB3 = xCB1 − xCB3 (36) ber of CBs.
126 H.R.E.H. Bouchekara et al. / Applied Soft Computing 42 (2016) 119–131
Table 8
Optimal results for CASE 1 through CASE 12.
CASE 1 CASE 2 CASE 3 CASE 4 CASE 5 CASE 6 CASE 7 CASE 8 CASE 9 CASE 10 CASE 11 CASE 12
PG1 177.0420 129.8879 179.1541 129.8979 176.1456 129.8709 176.6085 129.8745 139.9998 129.9232 198.9038 129.9606
PG2 48.6983 63.8534 44.9999 64.5881 48.7418 65.0360 48.4090 63.8713 54.9998 54.9872 44.0371 78.4900
PG5 21.3264 26.4452 21.4360 25.3971 21.6498 26.1814 21.4258 27.3903 24.4796 27.7688 18.6926 24.7823
PG8 21.0768 34.9958 22.1359 34.7964 22.4701 34.8561 21.6314 31.5182 34.6169 34.8721 10.0033 34.3127
PG11 11.8689 21.8744 12.2960 21.1735 12.0759 21.5512 11.9694 22.8264 18.1996 24.8975 10.0002 14.9040
PG13 12.0008 18.2776 12.0003 19.8971 12.0622 19.0480 12.0023 19.8798 17.4871 23.6521 12.0000 12.9360
VG1 1.1000 1.0997 1.1000 1.0876 1.0395 1.0462 1.0996 1.0982 1.0993 1.0636 1.1000 1.0963
VG2 1.0807 1.0845 1.0804 1.0724 1.0205 1.0299 1.0808 1.0831 1.0828 1.0475 1.0777 1.0813
VG5 1.0542 1.0600 1.0536 1.0485 1.0041 1.0065 1.0613 1.0606 1.0563 1.0214 1.0500 1.0519
VG8 1.0619 1.0503 1.0617 1.0388 1.0031 0.9991 1.0612 1.0467 1.0690 1.0202 1.0612 1.0484
VG11 1.1000 1.0631 1.1000 1.0621 1.0353 1.0320 1.0950 1.0994 1.0995 1.0597 1.0999 1.0990
VG13 1.1000 1.0635 1.1000 1.0263 1.0011 1.0018 1.1000 1.0966 1.0996 1.0566 1.1000 1.0870
T11(6–9) 1.0221 1.0760 1.0230 0.9876 1.0494 1.0402 0.9725 1.0123 0.9833 1.0201 1.0366 0.9641
T12(6–10) 0.9001 0.9119 0.9002 0.9980 0.9040 0.9007 0.9054 0.9088 0.9738 0.9046 0.9218 0.9724
T15(4–12) 0.9665 0.9448 0.9676 0.9774 0.9601 0.9683 0.9630 0.9271 0.9811 0.9183 0.9886 0.9314
T36(28–27) 0.9542 0.9799 0.9544 0.9853 0.9676 0.9671 0.9390 0.9353 0.9680 0.9688 0.9666 0.9576
QC10 4.9953 4.3870 4.9951 4.1421 4.9889 0.6189 4.4335 2.5856 4.8294 4.2266 4.9961 4.1170
QC12 4.9618 0.8513 4.9909 3.3363 0.5828 0.2966 4.8880 1.9035 2.9372 1.8160 4.9903 2.1736
QC15 4.9586 4.5079 4.7669 3.0336 4.9932 4.9827 4.7446 1.0663 4.0182 4.3858 4.9386 3.6649
QC17 4.9978 3.7012 4.9627 3.1257 0.1543 3.6664 3.9321 1.3616 4.0978 1.9859 4.9959 3.5345
QC20 4.3264 2.3512 4.3618 1.2970 4.9998 4.9831 4.2391 3.3712 4.4819 2.7092 4.4360 0.9915
QC21 4.9985 3.6837 4.9958 3.7637 4.9965 4.3942 4.9758 4.1988 3.8690 4.8731 5.0000 4.0513
QC23 2.6712 4.2058 2.6157 1.6389 4.9852 4.8205 1.7367 2.1157 3.4973 1.7431 3.0879 2.9981
QC24 4.9988 4.3439 4.9996 2.3082 4.9940 4.9650 1.7975 2.2735 4.6754 4.0602 4.9884 4.4103
QC29 2.3894 3.4305 2.4016 2.6300 2.5994 2.8754 0.8419 3.5901 2.1650 3.4776 2.5689 4.4294
Cost ($/h) 799.0353 823.5223 799.3079 824.2754 803.3978 826.3644 799.3277 824.6657 645.1668 678.5076 830.4531 868.4815
VD (pu) 1.9652 1.2261 1.9526 0.6039 0.1014 0.1187 1.9961 1.8000 1.8232 0.8595 1.7450 1.7116
Lmax 0.1261 0.1347 0.1263 0.1419 0.1490 0.1488 0.1252 0.1259 0.1282 0.1398 0.1289 0.1286
Ploss (MW) 8.6132 11.9342 8.6221 12.3501 9.7453 13.1435 8.6465 11.9605 6.3828 12.7008 10.2370 11.9856
Qloss (MVar) 36.3549 50.6337 36.3904 48.1407 42.3559 53.9316 34.8570 48.0963 26.1448 51.1102 41.9725 47.0434
Steps from 2 to 6 are repeated until a termination criterion is 57-bus test system. This system has a total generation capacity of
met. The termination criterion used for the ICBO is the same one 1975.9 MW and its main characteristics are given in Table 5. The
used for the CBO and the ECBO that is the maximum number of cost coefficients and the detailed data of this test system can be
iterations. derived from [48].
Table 9 Table 10
Optimal settings of control variables for CASE 13 and CASE 14. Optimal settings of control variables for CASE 15 and CASE 16.
Table 10 (Continued) 10 7
840 10
CASE 15 CASE 16
Cost
VG54 0.9959 0.9695
Penalty
VG55 0.9949 0.9638 830 8
V56 0.9954 0.9667
V59 1.0193 0.9803
VG61 1.0276 0.9893 820 6
Cost ($/h)
Penalty
VG62 1.0250 0.9898
VG65 1.0261 1.0040
VG66 1.0322 1.0159
810 4
VG69 1.0288 1.0264
VG70 1.0030 1.0154
VG72 1.0168 1.0291
800 2
VG73 1.0084 1.0331
VG74 0.9793 0.9840
VG76 0.9704 0.9630
VG77 1.0020 0.9885 790 0
VG80 1.0140 0.9957 1 125 250 375 500
Iterations
VG85 1.0136 0.9966
VG87 1.0355 1.0251
Fig. 5. Evolution of cost and penalty for CASE 1.
VG89 1.0248 1.0086
VG90 1.0078 0.9925
VG91 1.0115 0.9956
VG92 1.0114 0.9949 10 8
VG99 1.0126 0.9940 1000 4
VG100 1.0124 1.0026
VG103 1.0082 1.0039 Cost
VG104 1.0030 0.9985 Penalty
VG105 0.9998 0.9973
VG107 0.9932 0.9973
VG110 0.9989 1.0058
Cost ($/h)
Penalty
VG111 1.0077 1.0180
VG112 0.9892 0.9989 900 2
VG113 1.0083 1.0153
VG116 1.0236 0.9867
T8(8–5) 1.0324 0.9727
T32(26–25) 1.0741 1.0009
T36(30–17) 1.0270 1.0239
T51(38–37) 1.0134 0.9709
T93(63–59) 0.9821 1.0103
T95(64–61) 0.9947 1.0138 800 0
T102(65–66) 0.9849 0.9970 1 125 250 375 500
Iterations
T107(68–69) 0.9463 0.9296
T127(81–80) 0.9931 0.9912
QC5 3.6110 2.7316 Fig. 6. Evolution of cost and penalty for CASE 2.
QC34 3.9118 4.9990
QC37 0.2035 1.7433
QC44 4.9939 2.7843 900 0.4
QC45 4.9696 4.9801
QC46 0.4866 1.5183 Cost
QC48 4.8699 2.0490 VD
QC74 0.9253 3.3752
QC79 4.8979 1.8538
QC82 4.9221 0.8065
Cost ($/h)
B, A has to be better than B over all tested cases which is, once Fig. 7. Evolution of cost and VD for CASE 5.
again, not true. Therefore, in order to make reliable conclusions, in
this paper, a ranking procedure is applied on several algorithms to
detect the best one or more precisely to detect where the developed
ICBO stands or ranks among these algorithms. The developed ranking procedure is achieved as follows:
Ranking is given to each algorithm on every case based on the
following criteria: 1. Rank the algorithms based on their ‘best’ values.
2. Rank the algorithms based on their ‘average’ values.
- Cases: Total 16 cases are tested in this paper. 3. Calculate the overall rank, which is equal to the sum of ranks for
- Runs/case: 30 runs per case are done. each algorithm for all cases.
H.R.E.H. Bouchekara et al. / Applied Soft Computing 42 (2016) 119–131 129
CBO – –
Lmax
815 0.145 ECBO CM = 5 Colliding memory
Pro = 0.3 Regeneration
810 0.14 probability
DE F = 0.8 Differential weight
805 0.135 Cr = 0.8905 Crossover probability
PSO w = 0.5 Inertia factor
800 0.13 c1 = 1.5 Cognitive factor
c2 = 1.5 Social factor
795 0.125 ABC FoodNumber = Np/2 Number of food
1 125 250 375 500 sources
terations GA Mutrate = .05 Mutation rate
Selection = 0.75 Fraction of population
Fig. 8. Evolution of cost and Lmax for CASE 7. kept
BBO pmutate = 0 Mutation probability
pmodify = 1 Habitat modification
4. Calculate the Average Rank using the following formula: pmutate = 0.005 probability
Initial mutation
Total Rank probability
Average Rank = (43) Common Population size = 50 for cases: 1, 2, 3, 4, 5, 6, 7, 8, 11
2 × Number of CASES
parameters and 12.
5. Rank the methods based on their Average Rank. Population size = 90 for cases: 9, 10, 13, 14, 15 and
16.
Maximum number of iterations = 500 for cases: 1,
It is worth mentioning that, while ranking algorithms based on 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.
their average values, some precautions have to be taken. Because Maximum number of iterations = 1500 for cases:
in 30 runs and for a given case an algorithm can converge (sat- 13 and 14.
Maximum number of iterations = 2500 for cases:
isfy all the constraints) only for some cases and not for all cases.
15 and 16.
Table 12
The best objective function values obtained using each algorithm.
Table 13
The average objective function values obtained using each algorithm.
Table 14 Table 15
Comparison of algorithms and final ranking. Wilcoxon signed ranks test results.
[5] S. Rebennack, P.M. Pardalos, M.V.F. Pereira, N.A. Iliadis (Eds.), Handbook of [28] H.R.E.H. Bouchekara, Optimal power flow using black-hole-based optimization
Power Systems II, Springer-Verlag, Berlin/Heidelberg, 2010, http://dx.doi.org/ approach, Appl. Soft Comput. J. 24 (2014) 879–888, http://dx.doi.org/10.1016/
10.1007/978-3-642-12686-4. j.asoc.2014.08.056.
[6] M. Niu, C. Wan, Z. Xu, A review on applications of heuristic optimization algo- [29] H.R.E.H. Bouchekara, M.A. Abido, M. Boucherma, Optimal power flow using
rithms for optimal power flow in modern power systems, J. Mod. Power Syst. teaching-learning-based optimization technique, Electr. Power Syst. Res. 114
Clean Energy 2 (2014) 289–297, http://dx.doi.org/10.1007/s40565-014-0089- (2014) 49–59, http://dx.doi.org/10.1016/j.epsr.2014.03.032.
4. [30] H.R.E.H. Bouchekara, M.A. Abido, A.E. Chaib, R. Mehasni, Optimal power flow
[7] S. Sivasubramani, K.S. Swarup, Multiagent based differential evolution using the league championship algorithm: a case study of the Algerian power
approach to optimal power flow, Appl. Soft Comput. J. 12 (2012) 735–740, system, Energy Convers. Manag. 87 (2014) 58–70, http://dx.doi.org/10.1016/j.
http://dx.doi.org/10.1016/j.asoc.2011.09.016. enconman.2014.06.088.
[8] J.A. Momoh, M. El-Hawary, R. Adapa, A review of selected optimal power flow [31] N. Daryani, M.T. Hagh, S. Teimourzadeh, Adaptive group search optimization
literature to 1993. Part II: Newton, linear programming and interior point meth- algorithm for multi-objective optimal power flow problem, Appl. Soft Comput.
ods, IEEE Trans. Power Syst. 14 (1999) 105–111. 38 (2016) 1012–1024, http://dx.doi.org/10.1016/j.asoc.2015.10.057.
[9] K.S. Pandya, S.K. Joshi, A survey of optimal power flow methods, J. Theor. Appl. [32] M. AlRashidi, M. El-Hawary, Applications of computational intelligence
Inf. Technol. (2008) 450–458. techniques for solving the revived optimal power flow problem, Electr.
[10] S. Frank, I. Steponavice, Optimal power flow: a bibliographic survey Power Syst. Res. 79 (2009) 694–702, http://dx.doi.org/10.1016/j.epsr.2008.10.
I—formulations and deterministic methods, Energy Syst. (2012) 221–258, 004.
http://dx.doi.org/10.1007/s12667-012-0056-y. [33] S. Frank, I. Steponavice, Optimal power flow: a bibliographic survey II. Non-
[11] M. Ghasemi, S. Ghavidel, M. Gitizadeh, E. Akbari, An improved teaching – deterministic and hybrid methods, Energy Syst. (2012) 259–289, http://dx.doi.
learning-based optimization algorithm using Lévy mutation strategy for non- org/10.1007/s12667-012-0057-x.
smooth optimal power flow, Int. J. Electr. Power Energy Syst. 65 (2015) [34] A. Kaveh, V.R. Mahdavi, Colliding bodies optimization: a novel meta-
375–384, http://dx.doi.org/10.1016/j.ijepes.2014.10.027. heuristic method, Comput. Struct. 139 (2014) 18–27, http://dx.doi.org/10.
[12] T. Niknam, M. rasoul Narimani, M. Jabbari, A.R. Malekpour, A modified shuf- 1016/j.compstruc.2014.04.005.
fle frog leaping algorithm for multi-objective optimal power flow, Energy 36 [35] A. Kaveh, M. Ilchi Ghazaan, Enhanced colliding bodies optimization for design
(2011) 6420–6432, http://dx.doi.org/10.1016/j.energy.2011.09.027. problems with continuous and discrete variables, Adv. Eng. Softw. 77 (2014)
[13] S.R. Paranjothi, K. Anburaja, Optimal power flow using refined genetic algo- 66–75, http://dx.doi.org/10.1016/j.advengsoft.2014.08.003.
rithm, Electr. Power Components Syst. 30 (2002) 1055–1063, http://dx.doi.org/ [36] M. Ghasemi, S. Ghavidel, S. Rahmani, A. Roosta, H. Falah, A novel hybrid
10.1080/15325000290085343. algorithm of imperialist competitive algorithm and teaching learning algo-
[14] M.Z.L.L. Lai, J.T. Ma, R. Yokoyama, Improved genetic algorithms for optimal rithm for optimal power flow problem with non-smooth cost functions, Eng.
power flow under both normal and contingent operation states, Electr. Power Appl. Artif. Intell. 29 (2014) 54–69, http://dx.doi.org/10.1016/j.engappai.2013.
Energy Syst. 19 (1997) 287–292. 11.003.
[15] M.A. Abido, Optimal power flow using tabu search algorithm, Electr. Power [37] M. Balasubbareddy, S. Sivanagaraju, C.V. Suresh, Multi-objective optimization
Components Syst. (2002) 469–483. in the presence of practical constraints using non-dominated sorting hybrid
[16] M.A. Abido, Optimal power flow using particle swarm optimization, Int. cuckoo search algorithm, Eng. Sci. Technol. Int. J. 18 (2015) 603–615, http://dx.
J. Electr. Power Energy Syst. 24 (2002) 563–571, http://dx.doi.org/10.1016/ doi.org/10.1016/j.jestch.2015.04.005.
S0142-0615(01)00067-9. [38] B.A. Robbins, H. Zhu, A.D. Dominguez-Garcia, Optimal tap settings for volt-
[17] A.C. Roa-Sepulveda, B.J. Pavez-Lazo, A solution to the optimal power flow using age regulation transformers in distribution networks, North Am. Power Symp.
simulated annealing, Int. J. Electr. Power Energy Syst. 25 (2003) 47–57, http:// (2013) 1–6, http://dx.doi.org/10.1109/NAPS.2013.6666885.
dx.doi.org/10.1016/S0142-0615(02)00020-0. [39] H.R.E.-H. Bouchekara, M.A. Abido, Optimal power flow using differential search
[18] A.A. Abou El Ela, M.A. Abido, S.R. Spea, Optimal power flow using differential algorithm, Electr. Power Components Syst. 42 (2014) 1683–1699, http://dx.doi.
evolution algorithm, Electr. Power Syst. Res. 80 (2010) 878–885, http://dx.doi. org/10.1080/15325008.2014.949912.
org/10.1016/j.epsr.2009.12.018. [40] T. Niknam, M.R. Narimani, R. Azizipanah-abarghooee, A new hybrid algo-
[19] A.J. Ghanizadeh, G. Mokhtari, M. Abedi, G.B. Gharehpetian, Optimal power rithm for optimal power flow considering prohibited zones and valve point
flow based on imperialist competitive algorithm, Int. Rev. Electr. Eng. 6 (2011) effect, Energy Convers. Manag. 58 (2012) 197–206, http://dx.doi.org/10.1016/
1847–1852. j.enconman.2012.01.017.
[20] M. Ghasemi, S. Ghavidel, M.M. Ghanbarian, H.R. Massrur, M. Gharibzadeh, [41] G. Xiong, D. Shi, X. Duan, Multi-strategy ensemble biogeography-based opti-
Application of imperialist competitive algorithm with its modified techniques mization for economic dispatch problems, Appl. Energy 111 (2013) 801–811,
for multi-objective optimal power flow problem: a comparative study, Inf. Sci. http://dx.doi.org/10.1016/j.apenergy.2013.04.095.
(Ny.) 281 (2014) 225–247, http://dx.doi.org/10.1016/j.ins.2014.05.040. [42] P. Kessel, H. Glavitsch, Estimating the voltage stability of a power system, IEEE
[21] P.K. Roy, S.P. Ghoshal, S.S. Thakur, Biogeography based optimization for multi- Trans. Power Deliv. 1 (1986) 346–354, http://dx.doi.org/10.1109/TPWRD.1986.
constraint optimal power flow with emission and non-smooth cost function, 4308013.
Expert Syst. Appl. 37 (2010) 8221–8228, http://dx.doi.org/10.1016/j.eswa. [43] H. Hardiansyah, A modified particle swarm optimization technique for eco-
2010.05.064. nomic load dispatch with valve-point effect, Int. J. Intell. Syst. Appl. 5 (2013)
[22] A. Bhattacharya, P.K. Chattopadhyay, Application of biogeography-based opti- 32–41, http://dx.doi.org/10.5815/ijisa.2013.07.05.
misation to solve different optimal power flow problems, IET Gener. Transm. [44] A. Kaveh, V.R. Mahdavi, Colliding Bodies Optimization method for optimum
Distrib. 5 (2011) 70–80, http://dx.doi.org/10.1049/iet-gtd.2010.0237. design of truss structures with continuous variables, Adv. Eng. Softw. 70 (2014)
[23] A. Bhattacharya, P.K. Roy, Solution of multi-objective optimal power flow using 1–12, http://dx.doi.org/10.1016/j.advengsoft.2014.01.002.
gravitational search algorithm, IET Gener. Transm. Distrib. (2012) 751–763, [45] A. Kaveh, R. Mahdavi, Colliding Bodies Optimization Extensions and Applica-
http://dx.doi.org/10.1049/iet-gtd.2011.0593. tions, Springer, 2015.
[24] A.R. Bhowmik, A.K. Chakraborty, Solution of optimal power flow using non [46] R.D. Zimmerman, C.E. Murillo-Sanchez, R.J. Thomas, MATPOWER: steady-state
dominated sorting multi objective opposition based gravitational search algo- operations, planning, and analysis tools for power systems research and edu-
rithm, Int. J. Electr. Power Energy Syst. 64 (2015) 1237–1250, http://dx.doi.org/ cation, IEEE Trans. Power Syst. 26 (2011) 12–19, http://dx.doi.org/10.1109/
10.1016/j.ijepes.2014.09.015. TPWRS.2010.2051168.
[25] N. Sinsuphan, U. Leeton, T. Kulworawanichpong, Optimal power flow solu- [47] R.D. Zimmerman, C.E. Murillo-Sánchez & Deqiang (David) Gan, MATPOWER,
tion using improved harmony search method, Appl. Soft Comput. J. 13 (2013) http://www.pserc.cornell.edu/matpower/#docs (n.d.).
2364–2374, http://dx.doi.org/10.1016/j.asoc.2013.01.024. [48] 2014 OPF PROBLEMS. https://www.uni-due.de/ieee-wgmho/competition2014
[26] A. Khorsandi, S.H. Hosseinian, A. Ghazanfari, Modified artificial bee colony (n.d.).
algorithm based on fuzzy multi-objective technique for optimal power flow [49] J. Derrac, S. García, D. Molina, F. Herrera, A practical tutorial on the use of non-
problem, Electr. Power Syst. Res. 95 (2013) 206–213, http://dx.doi.org/10.1016/ parametric statistical tests as a methodology for comparing evolutionary and
j.epsr.2012.09.002. swarm intelligence algorithms, Swarm Evol. Comput. 1 (2011) 3–18, http://dx.
[27] K. Ayan, U. Kılıç, B. Baraklı, Chaotic artificial bee colony algorithm based solution doi.org/10.1016/j.swevo.2011.02.002.
of security and transient stability constrained optimal power flow, Int. J. Electr. [50] U. Can, B. Alatas, Physics based metaheuristic algorithms for global optimiza-
Power Energy Syst. 64 (2015) 136–147, http://dx.doi.org/10.1016/j.ijepes.2014. tion, Am. J. Inf. Sci. Comput. Eng. 1 (2015) 94–106.
07.018.