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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.
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