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Multi-objective Optimal Power Flow

Using Improved Multi-objective

Multi-verse Algorithm

Muhammad Abdullah1, Nadeem Javaid1(B

), Annas Chand2,

Zain Ahmad Khan2, Muhammad Waqas1, and Zeeshan Abbas1

1COMSATS University Islamabad, Islamabad 44000, Pakistan

nadeemjavaidqau@gmail.com

2COMSATS University Islamabad, Abbottabad 22010, Pakistan

https://www.njavaid.com

Abstract. This study proposes an improved multi-objective multi-verse

optimization (IMOMVO) algorithm for solving multi-objective optimal

power ﬂow (MOOPF) problem with uncertain renewable energy sources

(RESs). Cross and self-pollination steps of ﬂower pollination algorithm

(FPA) along with crowding distance and non-dominating sorting app-

roach is incorporated with the basic MOMVO algorithm to further

enhance the exploration, exploitation and for well-distributed Pareto-

optimal solution. To conﬁrm the eﬀectiveness of the proposed IMOMVO

algorithm, modiﬁed IEEE 30-bus system with security constraints is uti-

lized by considering the total generation cost and active power loss min-

imization. The simulation results obtained with IMOMVO is compared

with MOMVO, NSGA-II, and MOPSO, which reveals the capability of

the proposed IMOMVO in terms of solution optimality and distribution.

1 Introduction

The optimal power ﬂow (OPF) is considered a vital optimization tool for eﬃcient

operation and planning of the power system. The OPF is used to determine the

control variables at a speciﬁc time for the power system operation in order

to optimize (maximize or minimize) diﬀerent objective functions (OFs) while

assuring the technical feasibility of the solution [1].

Limited reserves of fossil fuels and environmental protection issues have

raised the concern to integrate renewable energy resources (RESs) into the grid.

Wind power plants and solar photovoltaic are amongst the most common RESs.

A lot of research work has been done on classical OPF, which consider only

thermal generators. Classical OPF, which is by itself non-linear and non-convex

and large scale optimization problem because of diﬀerent OFs and constraints,

however, by incorporating uncertain RESs further increase the complexity of the

problem [2].

The OPF was ﬁrstly introduced in 1962 by Carpentier. Since the inception of

OPF, many optimization methods have been introduced to the solution of OPF

c

Springer Nature Switzerland AG 2019

L. Barolli et al. (Eds.): WAINA 2019, AISC 927, pp. 1071–1083, 2019.

https://doi.org/10.1007/978-3-030-15035-8_104

1072 M. Abdullah et al.

problems. These approaches can be divided into classical numerical analysis and

modern metaheuristics algorithms. The classical method includes linear pro-

gramming, quadratic programming, interior point, and dynamic programming

etc. These techniques are inadequate for large-scale problems because they are

comprised of complex, lengthy calculation and they are excessively dependent on

initial guess values. In the past few years, diﬀerent metaheuristic algorithms have

been developed to overwhelm the deﬁciencies of the classical methods [2,3]. Some

of these algorithms are; whale optimization algorithm (WOA), particle swarm

optimization algorithm (PSO), moth-swarm algorithm (MSA), modiﬁed ﬁreﬂy

algorithm (MODFA), non-dominating sorting genetic algorithm (NSGA), diﬀer-

ential evolution (DE), elephant herding optimization (EHO) technique, Teaching

learning-based optimization (TLBO) technique etc. In any case, due to the incon-

sistency of the OPF solution for diﬀerent objectives, no heuristic algorithm can

be proven optimal for solving all these objectives, Hence, there is always a room

for new techniques and algorithms that can eﬀectively solve multiple objectives

of OPF.

The OPF problem is considered as single-objective if only one objective func-

tion is reckoned to be optimized. However, OPF problem needs to optimize

conﬂicting objectives in real world, simultaneously. For solving multi-objective

optimization (MOO) problems, two techniques are used in the literature: a pri-

ori and a posteriori [4]. In the ﬁrst MOO technique, the MOO problem is con-

verted into a single objective (SO) by assigning weight to diﬀerent objectives and

aggregating them. The importance of each objective is dictated by the weights

assigned to it, which is generally set by an expert. Solving the problem with this

approach does not require any modiﬁcation in the algorithm. In posteriori MOO

approach, there is no need to deﬁne the weights by an expert. In this method,

diﬀerent solutions are there for the decision maker as compared to the former

one. There is no need to deﬁne weights for each objective by an expert and the

Pareto frontier is approximated, even in a single run [5,6].

Increasing the participation of consumers in the electrical network operations

and accommodation of RESs are the most important features of smart grids. The

role of OPF in smart grids is very vital in order to decrease the transmission,

distribution losses, emission of fossil fuels, and the energy generation cost. Sub-

sequently, decreasing the electrical power consumption and prices, concurrently,

enabling consumers and electrical utilities to regulate the demand [7–11].

2 Problem Formulation of MOOPF

The main goal of solving OPF problems is to optimize diﬀerent objectives by

adjusting control parameters. In MOOPF problems, two are more objectives are

optimized simultaneously subject to the constraints. The MOOPF mathematical

formulation can be described as below [12,13].

Minimizing:

fobj =min

Nobj

j=1

fj(´x,´v),(1)

Multi-objective Optimal Power Flow 1073

subject to:

gk(´x, ´v)=0 k=1,2,...,G (2)

hl(´x, ´v)0l=1,2,...,L, (3)

where, fobj are the OFs to be minimized, gk(´x, ´v)isthekth equality constraint

(EC) which actually represent the load ﬂow equations [14]. The inequality con-

straints (IC) are expressed by h(´x, ´v). ´xand ´vare the vectors of state (dependent)

variables and control (independent) variables respectively [15].

The state variables are:

(1) Real power at swing bus (PG1).

(2) Reactive power output of generators (QG).

(3) Voltage at P-Q (Load) buses (VL).

(4) Loading of transmission lines (SL).

The vector of static variables can be written as:

´x=[PG1,Q

G1,...,Q

GNG,V

L1,...,V

LNB,S

L1,...,S

LNTL ]

The Control variables are:

(1) Voltages at generation buses (VG).

(2) Generation buses active power output (PG).

Besides these, tap settings of transformer, and compensation of shunt VAR (reac-

tive power) compensators can also be considered as control parameters.

Accordingly, vector ´vcan be expressed as:

´v=[PG2,...,P

GNG,V

G1,...,V

GN,...,Q

C1,...,Q

CNC ,T

1,...,T

NT ].

2.1 Objective Functions

Two OFs are considered in this paper, i.e. minimization of generation cost and

power loss, whose details are provided subsequently.

2.1.1 First Objective: Minimization of Generation Cost

Total generation cost comprises of the operation cost of thermal generators and

RESs along with penalty and reserve cost, which are described below.

Thermal Power Generator Cost Model

For realistic cost model of thermal generators, valve point eﬀect is considered

[16]. In a thermal generating unit steam is regulated by valves to run the turbine

through a separate nozzle group. To achieve maximum eﬃciency, every nozzle

group operate at its full output [21].For the required output, the valves are

1074 M. Abdullah et al.

operated in sequence, which results in the non-continuous cost curve. The cost

model of thermal generators considering valve loading eﬀect is given by [17]:

CTG(PTG)=

NTG

i=1

´ai+´

bi+´

diPTG

i

+

´

di∗sin(´ei∗(Pmin

TG

i−PTG

i)

,

(4)

where, NTG is the number of thermal generators. ´ai,´

bi, and ´ciare the cost

coeﬃcient of ith generator, producing output power PTG

i,´

diand ´eiare the

coeﬃcients of valve point loading eﬀect. The values of thermal units coeﬃcients

are listed in Table 1.

Table 1. Thermal generators coeﬃcient

Cost coeﬃcients ´a ´

b´c ´

d´e

GTh10 2 0.00375 18 0.037

GTh201.75 0.0175 16 0.038

GTh803.25 0.00834 12 0.045

Cost Model of Renewable Energy Sources

Generation from RESs does not require any fuel source for its operation. There-

fore, cost function does not exist, except maintenance cost, if the RESs are owned

by independent system operators (ISO). However, in the case when RESs are

operated by private bodies, the ISO pays to them accordingly to the scheduled

power contractually agreed [18].

The direct cost function of the kth wind farm in terms of scheduled power is

given by [19]:

CW(PW)=gwPW,(5)

where PWis the scheduled power and gwis the direct cost coeﬃcient of the wind

power unit. Similarly, the direct cost for the solar PV unit with scheduled power

PPV , and cost coeﬃcient hpv is given by:

CPV (PPV )=hpvPPV .(6)

The third renewable energy resource that is considered in this study is the

combination of wind and run of river small hydro generation plant. The scheduled

power is jointly delivered by wind and hydro generators. The hydro generation

considered in this work is of 5 MW, which is proportional to the river’s ﬂow rate

[20]. The direct cost of the combined wind and hydro plant is described as:

CWH(PWH)=gwhPWH =gwPWH,w +ghPWH,h,(7)

Multi-objective Optimal Power Flow 1075

where, PWH represent the scheduled real power from the combined plant, PWH,w

and PWH,h are the electrical energy contribution from the wind and hydro units

respectively. gwis the direct cost coeﬃcient of wind unit and ghis that of hydro

unit.

As wind energy is stochastic in nature, the actual power produced may be less

or more than scheduled power. Therefore, the ISO must have reserve generating

capability to meet the demand. Reserve cost for the wind unit can be calculated

as [21]:

CRW,i (PWsh,i −PWac,i)=krw,i(PWsh,i −PWac,i)

=krw,i PWsh,i

0

(PWsh,i −pw,i )fw(pw,i)dpw,i.(8)

In the case when the output power of the wind generators is greater then the

scheduled power, the ISOs pay penalty cost if the surplus power is not utilized

by them by reducing the power of thermal generators.

The penalty cost of wind plant can be mathematically described as:

CPW,i (PWac,i −PWsh,i)=kpw,i(PWac,i −PWsh,i)

=kpw,i PWr,i

PWsh,i

(pw,i −PWsh,i)fw(pw,i)dpw,i,(9)

where, PWsh,i is the scheduled power, PWac,i is the available power from wind

plant, PWr,i is the rated power, and fw(pw,i)is the probability density function

of wind power. Similarly the reserve and penalty cost of solar and combined

wind and hydro can be calculated. The values of direct, reserve, and penalty

cost coeﬃcients for RESs are provided in Table 2.

The output power of wind power units is directly proportional to wind speed.

Wind speed probability is predominantly represented by Weibull, probability

distribution function (PDF) [17,21,22].

The power given by PV arrays depends on solar irradiance. The probability

distribution of solar irradiance Gsis represented by lognormal PDF [21,23].

Gumbel distribution correctly represent the river ﬂow rate [24,25]. Summary of

the number of turbines, the rated power output from RESs, and the values of

PDF parameters are listed in Table 3.

2.1.2 Second Objective: Minimization of Real Power Loss

The active power loss in the branches of the power system is unavoidable because

of inherent resistance of lines. The real (active) power loss can be mathematically

written as,

f2(x, u)=Ploss =

nL

i=1

nL

j=1,j=i

Gij [V2

i+V2

j−2ViVjcos(δij )] (10)

Where nL is the number of branches, Gij is the conductance of the branch which

linked together the ith and jth bus, and δij is the voltage angle between them.

Viand Vjare the voltages of bus iand bus j. The magnitude of these voltages

are calculated through newton-Raphson load ﬂow equation.

1076 M. Abdullah et al.

Table 2. RESs direct, reserve, and penalty cost coeﬃcient

Direct cost coeﬃcient

Wind (at bus number: 5, 13) gw1=1.6,

gw1=1.75

Solar PV (at bus number 11) hs=1.6

Small hydro (at bus number 13) gh=1.5

Reserve cost coeﬃcient

Wind (at bus number 5) krw =3

Solar PV (at bus number 11) krs =3

Combined wind and hydro (at bus number 13) krwh =3

Penalty cost coeﬃcient

Wind (at bus number 5) kpw =1.5

Solar PV (at bus number 11) kps =1.5

Combined wind and hydro (at bus number 13) kpwh =1.5

2.2 System Constraints

MOOPF is a non-linear, non-concave optimization problem, which requires to

satisfy diﬀerent equality constraints (ECs) and inequality constraints (ICs).

These constraints are described as follow.

2.2.1 Equality Constraints

The ECs in MOOPF comprising of real and reactive power load ﬂow equations

[26], which stipulates that the real and reactive power generation must be equal

to the load demand and power losses in the network. The ECs can be mathe-

matically described as:

PG−PD−Vl

NB

j=1

Vl[Gij cos(δij +βij sin(δij ))] = 0 ∀iNB, (11)

and

QG−QD−Vl

NB

j=1

Vl[Gij cos(δij +βij sin(δij ))] = 0 ∀iNB. (12)

2.2.2 Inequality Constraints

ICs are actually the active, reactive power generation limits, security constraints

of lines, and voltage limits of generation and load buses [17,27].

The constraints of generators are:

Multi-objective Optimal Power Flow 1077

Table 3. PDF parameters for uncertain RESs.

Wind power generator Solar PV unit

Tota l w ind

turbines

Cumulative rated

power (MW)

Weibull PDF

parameters

Cumulative

rated power

(MW)

Lognormal

PDF

parameters

25 75 k=2, c=9 60 μ=6,

σ=0.6

Combined wind+small-hydro

Tota l w ind

turbines

Rated wind

power (MW)

Weibull PDF

parameters

Small hydro

rated power

(MW)

Gumbel PDF

parameters

15 45 k=2, c=10 5λ= 15,

γ=1.2

Pmin

TG,i PTG,i Pmax

TG,i i=1,...,N

TG,(13)

Pmin

WG,j PWG Pmax

WG ,(14)

Pmin

PV,k PPV Pmax

PV ,(15)

Pmin

WHG,j PWHG,j Pmax

WHG,j,(16)

Qmin

TG,i QTG,i Qmax

TG,i,i=1,...,N

TG,(17)

Qmin

WG QWG Qmax

WG,(18)

Qmin

PV QPV Qmax

PV ,(19)

Qmin

WHG QWHG Qmax

WHG,(20)

Vmin

G,i VG,i Vmax

G,i ,i=1,...,N

G,(21)

where, PTG,i is the active power generation of ith thermal unit, PWG,PPV,and

PWHG is the real power of wind farm, solar PV array, and combined wind and

hydro generators, respectively. Similarly, QTG,i,QWG,QPV,andQWHG are the

reactive power output of the thermal, wind, solar, and combined wind and hydro

generators. VG,i is the voltage of the ith generation bus.

Security constraints includes the voltage limits of P-Q buses and the trans-

mission lines loading capability, which are described as follow:

Vmin

Lp VLp Vmax

Lp p=1,...,N

LB ,(22)

and

Slq Smax

lq q=1,...,N

l.(23)

Equation 22 represents the voltage limits for NLB number of load buses and

the line loading constraint is described in Eq. 23 for Nlbranches.

1078 M. Abdullah et al.

Table 4. Best results of two-objective for Case 1 of 30 bus power system at 500

iteration.

Control variables MOMVO MOPSO NSGAII IMOMVO

Best

cost

Best

power

loss

Best

cost

Best

power

loss

Best

cost

Best

power

loss

Best

cost

Best

power

loss

PTG

2(MW) 23.6459 23.8473 20.0000 57.9310 20.6786 29.3117 24.8195 31.9106

PWG

5(MW) 42.5207 63.1317 46.3870 75.0000 44.7182 101.0292 44.7238 94.7263

PTG

8(MW) 10.0000 10.1696 10.0000 34.6162 10.1694 48.4191 10.1382 39.9864

PPV

11 (MW) 41.4415 54.7171 44.9287 60.0000 43.0685 66.2612 40.5211 68.7966

P(W+H)G13(MW) 36.4134 35.9545 32.5331 48.6378 34.8491 36.4088 32.1291 35.9207

VTG

1(p.u) 1.1000 1.1000 1.1000 1.1000 1.1137 1.0808 1.3422 1.2794

VTG

2(p.u) 0.9500 1.0921 1.0913 1.1000 1.0747 1.0813 1.2014 1.1614

VWG

5(p.u) 1.1000 1.0731 1.0989 1.0995 1.1697 1.0735 1.3208 1.2728

VTG

8(p.u) 1.1000 1.1000 1.1000 1.0958 1.1882 1.1506 1.3124 1.2888

VPV

11 (p.u) 1.1000 1.0975 1.0822 1.1000 1.1006 1.1565 1.3652 1.3513

V(W+H)G13 (p.u) 1.1000 1.0780 1.0999 1.0982 20.6786 1.0881 1.4777 1.2716

Cost ($/hr) 776.400 818.017 777.777 942.171 776.882 979.451 770.917 954.880

Power los s (MW) 5.526 3.515 5.275 1.760 5.166 1.376 3.572 0.999

3 IMOMVO Algorithm and Its Application to MOOPF

Problem

Many real-world optimization problems entail optimizing diﬀerent objectives

simultaneously while meeting the constraints. These objectives are often con-

ﬂicting in nature and the optimal value of all these objectives cannot be achieved

at the same time.

3.1 Basic MOMVO Algorithm

MOMVO optimization algorithm is based on the theory of the existence of multi-

verse, recently proposed by Mirjalili [28]. The inspiration of MOMVO is the inter-

action between universes through black, white, and worm wholes. The objects

from one universe move to another via tunnels (black, white holes), and move

within the universe or even from one universe to another through worm hole

[28,29].

3.2 IMOMVO Algorithm

The PF obtained by the MOMVO algorithm at higher times (iteration) is not

well distributed. Therefore, to further increase the exploration and exploitation

of MOMVO, levy ﬂight and self-pollination steps of ﬂower pollination algorithm

[30], is used to create a new set of universes. After ﬁnding the inﬂation rates of

both sets of universes, they are merged together. The solutions are sorted and

selected on the bases of domination level, and crowding distance approach is

Multi-objective Optimal Power Flow 1079

used to maintain the diversity of the non-dominated solution. The solutions are

truncated to the size of the archive to reduce computation time. The new solu-

tions are compared to the stored results in the archive and the better solutions

are replaced in the archive.

4 Simulation Results and Discussions

Modiﬁed IEEE 30-bus test system is considered to varify the eﬀectiveness of the

proposed IMOMVO for the solution of MOOPF and its performance is compared

with the results of MOPSO, MOMVO, and NSGA-II optimization algorithm.

The codes of these algorithms are written in the MATLAB environment and

MATPOWER (MATLAB package) is used for power ﬂow calculation. All the

simulations are performed on Intel Core i7, 2.00 GHz processor with an 8 GB

RAM personal computer. For the fair comparison of the proposed IMOMVO

with MOPSO, MOMVO, and NSGAII in terms of obtaining the best PF. The

size of the population (Np = 50) and numbers of iterations are taken the same

for all of these algorithms. The summary of the adapted 30-bus is provided in

Table 5.

Two conﬂicting objectives, total generation cost minimization, and power

loss minimization are solved simultaneously using proposed IMOMVO and its

performance is compared with other optimization algorithms. The PF obtained

from the simulation of these algorithms for solving MOOPF problem of modiﬁed

Fig. 1. Pareto optimum front obtained at 20 iterations.

1080 M. Abdullah et al.

Fig. 2. Pareto optimum front obtained at 100 iterations.

Fig. 3. Pareto optimum front obtained at 500 iterations.

Multi-objective Optimal Power Flow 1081

Table 5. Summary of the modiﬁed IEEE-30 bus system.

Item Quantity Details

Buses 30 6 generators buses and 24 load

buses

Branches 41

Thermal generators Buses (TG

1,

TG

2,TG

3)

3 At bus number: 1,2, and 8. Bus

number 1 is a slack bus

Wind generator 1At bus number 5

Solar PV array 1At bus number 11

Wind generator+small hydro unit 1 At bus number 13

Control variables 11 The real power of generator buses

except for the slack bus and

voltages of all generator buses

Allowed range of load bus voltage 24 0.95 to 1.05 p.u.

Connected load 283.4 MW, 126.2 MVAr

IEEE 30-bus system is shown in Figs. 1,2,and3for 20, 100, and 500 runs. From

the simulation results, it is obvious that the PF obtained from IMOMVO per-

formed better than MOPSO, MOMVO, and NSGA-II, both at lower and higher

iteration (times) in term of solution optimality and distribution. The values of

the control variables for the best cost, best power loss, and the resultant OFs are

provided in Table 4. The PF obtained by MOMVO is better than MOPSO and

NSGA-II at lower iteration, however, at higher iteration, the PF obtained by

MOMVO is not well distributed. The proposed IMOMVO eliminated the limi-

tation of PF distribution of MOMVO algorithm by incorporating the crowding

distance and non-dominating sorting approaches and displayed the best results

both at lower and higher iteration.

4.1 Conclusion

In this study, IMOMVO algorithm has been proposed and applied to solve

MOOPF problem. The proposed algorithm was successfully implemented on

modiﬁed IEEE 30-bus power systems. The simulation results reveal the superi-

ority of the IMOMVO over NSGA-II, MOPSO and MOMVO algorithms. The

IMOMVO eliminated the limitation of PF distribution of MOMVO algorithm

and displayed the better results both at lower and higher iteration. Therefore,

the IMOMVO provides better PF. The PF obtained by multi-objective algo-

rithms helps the decision maker to take a better informed-decision, concerning

the compromise between the conﬂicting objectives.

1082 M. Abdullah et al.

References

1. Abdi, H., Beigvand, S.D., La Scala, M.: A review of optimal power ﬂow studies

applied to smart grids and microgrids. Renew. Sustain. Energy Rev. 71, 742–766

(2017)

2. Attia, A.-F., El Sehiemy, R.A., Hasanien, H.M.: Optimal power ﬂow solution in

power systems using a novel Sine-Cosine algorithm. Int. J. Electr. Power Energy

Syst. 99, 331–343 (2018)

3. Bai, W., Eke, I., Lee, K.Y.: An improved artiﬁcial bee colony optimization algo-

rithm based on orthogonal learning for optimal power ﬂow problem. Control Eng.

Pract. 61, 163–172 (2017)

4. Branke, J., Kaußler, T., Schmeck, H.: Guidance in evolutionary multi-objective

optimization. Adv. Eng. Softw. 32(6), 499–507 (2001)

5. Tan, K.C.: Advances in evolutionary multi-objective optimization. In: Soft Com-

puting Applications. Springer, Heidelberg, pp. 7–8 (2013)

6. Mirjalili, S., et al.: Salp swarm algorithm: a bio-inspired optimizer for engineering

design problems. Adv. Eng. Softw. 114, 163–191 (2017)

7. Javaid, N., et al.: Energy eﬃcient integration of renewable energy sources in the

smart grid for demand side management. IEEE Access 6, 77077–77096 (2018)

8. Javaid, N., et al.: An intelligent load management system with renewable energy

integration for smart homes. IEEE Access 5, 13587–13600 (2017)

9. Khan, M., et al.: Game theoretical demand response management and short-term

load forecasting by knowledge based systems on the basis of priority index. Elec-

tronics 7(12), 431 (2018)

10. Awais, M., et al.: Towards eﬀective and eﬃcient energy management of single home

and a smart community exploiting heuristic optimization algorithms with critical

peak and real-time pricing tariﬀs in smart grids. Energies 11(11), 3125 (2018)

11. Ahmad, A., et al.: An optimized home energy management system with integrated

renewable energy and storage resources. Energies 10(4), 549 (2017)

12. Nuaekaew, K., et al.: Optimal reactive power dispatch problem using a two-archive

multi-objective grey wolf optimizer. Expert Syst. Appl. 87, 79–89 (2017)

13. Kumar, A.R., Premalatha, L.: Optimal power ﬂow for a deregulated power system

using adaptive real coded biogeography-based optimization. Int. J. Electr. Power

Energy Syst. 73, 393–399 (2015)

14. Chen, G., et al.: Quasi-oppositional cuckoo search algorithm for multi-objective

optimal power ﬂow. IAENG Int. J. Comput. Sci. 45(2), 255–266 (2018)

15. Ghasemi, M., et al.: Multi-objective optimal power ﬂow considering the cost, emis-

sion, voltage deviation and power losses using multi-objective modiﬁed imperialist

competitive algorithm. Energy 78, 276–289 (2014)

16. Abdelaziz, A.Y., Ali, E.S., Abd Elazim, S.M.: Flower pollination algorithm to

solve combined economic and emission dispatch problems. Eng. Sci. Technol. Int.

J. 19(2), 980–990 (2016)

17. Biswas, P.P., Suganthan, P.N., Amaratunga, G.A.J.: Optimal power ﬂow solutions

incorporating stochastic wind and solar power. Energy Convers. Manag. 148, 1194–

1207 (2017)

18. Chen, C.-L., Lee, T.-Y., Jan, R.-M.: Optimal wind-thermal coordination dispatch

in isolated power systems with large integration of wind capacity. Energy Convers.

Manag. 47(18–19), 3456–3472 (2006)

19. Wijesinghe, A., Lai, L.L.: Small hydro power plant analysis and development. In:

2011 4th International Conference on Electric Utility Deregulation and Restruc-

turing and Power Technologies (DRPT). IEEE (2011)

Multi-objective Optimal Power Flow 1083

20. Tanabe, R., Fukunaga, A.: Success-history based parameter adaptation for diﬀer-

ential evolution. In: 2013 IEEE Congress on Evolutionary Computation (CEC).

IEEE (2013)

21. Reddy, S.S., Bijwe, P.R., Abhyankar, A.R.: Real-time economic dispatch consider-

ing renewable power generation variability and uncertainty over scheduling period.

IEEE Syst. J. 9(4), 1440–1451 (2015)

22. Reddy, S.S.: Optimal scheduling of thermal-wind-solar power system with storage.

Renew. Energy 101, 1357–1368 (2017)

23. Chang, T.P.: Investigation on frequency distribution of global radiation using dif-

ferent probability density functions. Int. J. Appl. Sci. Eng. 8(2), 99–107 (2010)

24. Mujere, N.: Flood frequency analysis using the Gumbel distribution. Int. J. Com-

put. Sci. Eng. 3(7), 2774–2778 (2011)

25. Cabus, P.: River ﬂow prediction through rainfall runoﬀ modelling with a

probability-distributed model (PDM) in Flanders, Belgium. Agric. Water Manag.

95(7), 859–868 (2008)

26. Ghasemi, M., et al.: Solving non-linear, non-smooth and non-convex optimal power

ﬂow problems using chaotic invasive weed optimization algorithms based on chaos.

Energy 73, 340–353 (2014)

27. Mohamed, A.A.A., et al.: Optimal power ﬂow using moth swarm algorithm. Electr.

Power Syst. Res. 142, 190–206 (2017)

28. Mirjalili, S., et al.: Optimization of problems with multiple objectives using the

multi-verse optimization algorithm. Knowl.-Based Syst. 134, 50–71 (2017)

29. Mirjalili, S., Mirjalili, S.M., Hatamlou, A.: Multi-verse optimizer: a nature-inspired

algorithm for global optimization. Neural Comput. Appl. 27(2), 495–513 (2016)

30. Yang, X.-S.: Flower pollination algorithm for global optimization. In: International

Conference on Unconventional Computing and Natural Computation. Springer,

Heidelberg (2012)