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RESEARCH ARTICLE
Developing chaotic Bonobo optimizer for optimal power
flow analysis considering stochastic renewable energy
resources
Mohamed H. Hassan
1
| Salah K. Elsayed
2
| Salah Kamel
1
|
Claudia Rahmann
3
| Ibrahim B. M. Taha
2
1
Department of Electrical Engineering,
Faculty of Engineering, Aswan University,
Aswan, Egypt
2
Department of Electrical Engineering,
College of Engineering, Taif University,
Taif, Saudi Arabia
3
Department of Electrical Engineering,
University of Chile, Santiago, Chile
Correspondence
Salah K. Elsayed, Department of Electrical
Engineering, College of Engineering, Taif
University, P.O. Box 11099, Taif 21944,
Saudi Arabia.
Email: salah_kamal1982@yahoo.com
Funding information
Taif University, Grant/Award Number:
TURSP-2020/61
Summary
During the last years, the electricity networks worldwide have rapidly devel-
oped, especially with integrating many types of renewable energy sources
(RESs). The optimal operation is an opportunity to increase the penetration
level of stochastic RESs into the power grid to maximize energy efficiency.
Generally, the optimal power flow (OPF) problem is a highly complex, non-
convex, and non-linear optimization problem. The complexity of the OPF
problem is further increased as stochastic RESs are incorporated into the net-
work. This paper presents an effective solution to the OPF problem for a tradi-
tional power generation with stochastic RESs. For solving this problem,
chaotic Bonobo optimizer (CBO) is proposed in this paper based on the Chaos
Theory to avoid the stuck in the local minimum by applying the original
Bonobo optimization (BO). The performance of BO is enhanced using the cha-
otic maps sequences technique to enhance its global search capability and pre-
vent getting stuck into local solutions. Uncertainty of the output power
generated by RESs is forecasted based on probabilistic models. To minimize
the total operating cost, the direct, underestimation, and overestimation costs
of RESs are considered. Three different objective functions are considered,
minimizing total operating cost, emissions, and power losses. Moreover, a car-
bon tax is incorporated in the objective function problem to minimize carbon
emissions. The proposed OPF model and CBO technique are verified on the
modified IEEE-30 and IEEE-57 bus test systems to confirm the superiority and
effectiveness of the proposed CBO to achieve the optimal solution. The simula-
tion results prove the efficiency and robustness of CBO for finding the best
solution to the OPF problem with stochastic RESs.
KEYWORDS
Bonobo optimizer algorithm, carbon emission, chaotic sequences, optimal power flow,
stochastic renewable energy sources
Received: 2 October 2021 Revised: 19 March 2022 Accepted: 23 March 2022
DOI: 10.1002/er.7928
Int J Energy Res. 2022;1–35. wileyonlinelibrary.com/journal/er © 2022 John Wiley & Sons Ltd. 1
1|INTRODUCTION
The continuous growth in energy consumption, the
deregulation of the electricity markets, the need to reduce
gas emissions, and recent privileged prices of renewable
energy sources (RESs) have become motivated to the
rapid expansion of utilizing RESs in recent decades. It is
proven that RESs are the preferable alternatives to fossil
fuels for generating power. Because of enhanced technol-
ogies of generating power from RESs, the rapid increase
of utilizing RESs has occurred, leading to decreased sys-
tem installation costs.
1
As concluded in Reference 2, the
output power from wind turbine (WT) generators and
solar photovoltaic (SPV) generators will soon be lower-
priced than obtained by fossil fuels.
The essential objective of optimal power flow (OPF) is
to achieve an objective function as optimizing fuel cost,
minimization real power loss, improving voltage profile,
and reducing gas emissions through obtaining the appro-
priate optimal control variables with satisfying different
operational constraints.
3
Generally, the mathematical
modeling of conventional OPF just contains traditional
fossil-fuel-based power generation sources is a compli-
cated nonconvex, nonlinear, large-scale, and constrained
problem.
4,5
It is confirmed from recent literature that
adding many RESs to the power grid has a positive
impact at the operational and planning phase. However,
the integration of multiple stochastic RESs into power
systems raises the complexity of the OPF problem due to
the uncertainties associated with RESs such as WT and
SPV generators.
6,7
In the previous literature, a considerable amount of
research has been done for solving the OPF problem and
was utilized for the optimal operation with traditional
generating units and/or RESs. Several optimization
approaches are employed to solve a single or multi-
objective function, such as minimizing the fuel cost, real
power loss, and gas emission. However, this led to an
increase in the complexity of the OPF problem as well as
the computational cost. Traditional optimization
methods such as quadratic programming were used for
optimal economic dispatch problems and for solving the
OPF problem in References 8 and 9. An interior-point
algorithm was used in Reference 10 to find the solution
for non-linear operational problems of the power net-
work model. Anyway, the traditional optimization
methods are constrained by the requirement of deriva-
tive, dimensionality of the considered problem, and sea-
rch stagnation without guarantee of a global solution.
Several heuristic optimization algorithms have been
proposed during the last years to address the OPF prob-
lem such as Crow Search Algorithms,
11
Teaching-Learn-
ing-Based Optimization (TLBO) algorithm,
12
Modified
Jaya Algorithm,
13,14
Hybrid Firefly Bat Algorithm with
the constraints-prior object-fuzzy sorting technique,
15
Rao-3 algorithm,
16
Hybrid Firefly with Particle Swarm
Optimization (PSO), and hybrid PSO with Artificial Phys-
ics Optimization technique to reach to the optimum solu-
tion for the security-constrained OPF problem as well as
improve the system security.
17
In Reference 18, a Moth Swarm Algorithm (MSA)
was applied to solve the constrained OPF problem for
optimizing the control variables of bus voltages, load tap
changer ratios, real power generations, and shunt capaci-
tance values. A black-hole-based optimization technique
was investigated in Reference 19 to solve the OPF prob-
lem with different objectives to reduce fuel cost and total
emissions and improve voltage stability. In the multi-
objective problem of OPF formulation fuel cost and real
loss; fuel cost and total voltage deviation; fuel cost, real
loss, and total voltage deviation are minimized simulta-
neously, the gravitational search algorithm (GSA) was
applied to solve the multi-objective problem of OPF as
presented in Reference 20.
The incorporation of RESs into the power systems
will cause several issues related to the economic opera-
tion of the grid, active power losses of the transmission
system, the emissions from thermal plants, and the volt-
age of all buses.
21
Usually, RESs are managed by private
operators, where the grid system operator announcement
an agreement to purchase scheduled power. The main
challenge of incorporating RESs such as WT and SPV
generators is their intermittent nature. Furthermore, the
generated power from RESs is uncertain, and it might be
less than the scheduled power resulting in an over-
estimation of the available quantity. Therefore, power
system operators need to have a spinning reserve, which
increases the total operating cost.
In contrast, the generated power of RESs might be
larger than the scheduled power resulting in underesti-
mation. Therefore, the operators will pay a penalty cost
as additional power goes wasted if not consumed.
22,23
Accordingly, in the event of neglecting uncertainty
related to RESs when solving the OPF problem, this will
lead to inaccurate results in the system's total operating
cost. To model the stochastic nature of solar irradiance,
most authors use beta probability distribution function
(PDF)
24
or lognormal PDF,
25
whereas the stochastic
behavior of wind speed can be described by Weibull
PDFs.
26
Further research into OPF is required for various net-
works consisting of the conventional thermal generation
with WT and SPV generations, and these types of RESs
generations cannot be scheduled in the same way as tra-
ditional thermal generations because they depend on cli-
mate factors such as solar irradiation and wind velocity.
27
2HASSAN ET AL.
Many optimization algorithms have been implemented
to solve the OPF problem incorporating RESs. A fuzzy
logic optimization model was added to the PSO in Refer-
ence 28 for solving bi-objective functions (fuel cost and
gas emissions) for a power network connected to RESs.
The authors employed the Moth Swarm Optimization
(MSO) algorithm in Reference 18 and MSO with (GSA)
in Reference 29 to solve the OPF problem for a power
system network incorporated with wind power
generation.
A robust model was presented in Reference 30 to
overcome the issues of uncertain parameters for solving
the OPF problem with the inclusion of WT generators
where the uncertainty of the wind power and demand
system have been categorized, including restricted
periods forming a polyhedral uncertainty set. A novel
method based on modified PSO with the evidence theory
framework was presented in Reference 31 to find the best
solution to the OPF problem considering the uncer-
tainties of both wind generation and the several factors
in the power system. The Modified Cuckoo Search (MCS)
algorithm was proposed in Reference 32 to solve the OPF
problem, including the wind power generation costs in
the objective function. Shabanpour-Haghighi et al
33
pro-
posed a self-learning optimization approach based on the
wavelet mutation method and fuzzy clustering to solve
the OPF problem. In Reference 34, an improved differen-
tial evolution algorithm, namely DEa-AR, was investi-
gated to find the best solution for the OPF problem of the
power system integrated with RESs.
Uncertainty conditions for finding the solution for the
OPF problem, a modified version of MSA was proposed
in Reference 35 to solve the OPF problem of a power net-
work incorporating stochastic WT generators to decrease
the total operating cost transmission power l improve the
voltage profile. Nusair and Alasali
36
presented Golden
Ratio Optimization Method to achieve the optimum solu-
tion for the OPF problem for a power system incorporat-
ing stochastic RESs to enhance the power system's
energy, reliability, and environmental performance. To
address the stochastic behavior of RESs for solving the
OPF problem in Reference 37, the uncertainty of the fore-
casted output power from SPV and WT generators were
addressed with the support of PDFs, then a Grey Wolf
Optimization algorithm (GWO) was implemented for
optimizing the generation cost.
Kotur and Stefanov
38
applied optimal power con-
verters to solve the OPF problem to minimize real power
losses incorporated offshore WT generators. Phasor PSO
and GSA called hybrid (PPSO-GSA) was proposed in Ref-
erence 1 to solve OPF with integrated PV and WT genera-
tors. Prediction of WT and SPV generators' generated
power was formulated according to the probabilistic
models and real-time measurements of solar irradiance
and wind speed. Moreover, the proposed techniques in
Reference 39 need exact future knowledge of the RESs at
the operation cycle of the power system generation.
Bonobo optimization (BO) algorithm is one of the
recent population-based optimization techniques pro-
posed in Reference 40. BO was utilized for solving the
complex and non-convex optimization problems based
on the social behavior of a group of bonobo and repro-
ductive strategies of the bonobo. The BO algorithm
begins with an initial population for random solutions
namely bonobos. The optimal variables are introduced as
a bonobo; all bonobo has fitness value via the main objec-
tive function; the best solution is then signed as the alpha
bonobo, and then the produced new values as imitation
bonobos community's a mating strategy. This procedure
is repeated for each iteration until reaching the end of
the maximum iteration or the convergence criteria. How-
ever, the BO and similar metaheuristic algorithms are
often stuck in locally optimal solutions. To overcome this
shortcoming, different modifications can be
implemented.
Chaos theory has been widely utilized in numerous
metaheuristics and in a broad spectrum of applications to
improve the metaheuristics performance to provide better
convergence speed and avoid stuck in a local minimum.
41
In the literature, examples of the metaheuristics that
employed the chaos theory include Differential Evolution
(DE),
42
Imperialist Competitive Algorithm (ICA),
43
GWO,
44
GSA,
45
Cuckoo Search Algorithm,
46
Whale Opti-
mization Algorithm,
41
Salp Swarm Algorithm,
47,48
and
the Bat Algorithm (BA).
49
The utilization of the chaotic
theory to improve the global search ability of the BO
technique is proposed in this paper.
The main objective of this paper is to improve the
original BO algorithm. The improved BO algorithm,
called CBO is applied to achieve the optimal solution of
the OPF problem for power system network incorporat-
ing stochastic RESs. The proposed CBO technique was
inspected by a modified IEEE-30 and IEEE-57 bus test
systems. The essential contributions of this paper can be
summarized as follows:
•Proposing an improved version of the classical BO
based on chaos theory, called CBO, to improve the
original BO's performance by avoiding being stuck in
local optimums.
•A new application for the classical BO and proposed
CBO for solving OPF considering the uncertainty
modeling of WT and SPV generators.
•Proposing a novel objective function is a problem that
includes the generating fuel cost of traditional thermal
generators along with the direct, reserve, and penalty
HASSAN ET AL.3
costs of RESs. Further, the impacts of varying reserve
and penalty costs on optimal scheduling are analyzed.
•Due to the harmful gas emitted from conventional
thermal generators, a CT was imposed in many coun-
tries.
50
A CT is embedded in the considered objective
function to explore its impact on generated power
scheduling.
•The proposed algorithm's obtained statistical results
are compared with the original BO algorithm and the
other recent techniques.
•The superiority and reliability of the CABO5-based
methodology in solving the OPF problem considering
the uncertainty modeling of WT and SPV generators
are proved.
The article is presented in the following manner:
Section 2 introduces the mathematical formulation of the
OPF and operational constraints. The uncertainty of WT
and SPV generators output models are presented in
Section 3. In Section 5, the CBO technique is introduced.
Section 6 applies the new proposed CBO for solving the
OPF problem with stochastic RESs, presenting the
obtained results and comparisons. Section 7 concludes
the work.
2|RESEARCH GAPS
Formulation and solution for constrained OPF problems
with the integration of stochastic RESs such as WT and
SPV generators in power systems need more research as
a few pieces of literature could be traced related to the
same problem. This study proposes multi objectives of
both economic and environmental OPF problem formu-
lation with WT and SPV generators. However, wind and
solar power are modeled using Weibull and lognormal
PDFs, respectively. Penalty and reserve costs of these
intermittent sources to represent both underestimation
and overestimation of these sources are suitably added to
the generation cost.
Recently, diverse population-based metaheuristic
techniques have been implemented for solving the vari-
ous power system problems. However, there are some
problems with stucking into the local optimal solutions.
However, there is still a lot that those metaheuristic
methods can solve the complex and nonlinear problems
of the power system networks by discovering new meta-
heuristic methods or making improvements for the exis-
ting metaheuristic methods to overcome those problems.
Chaos theory has been widely used to overcome these
problems and improve the metaheuristic performance, as
indicated in the references of the previous section. There-
fore, this paper will investigate the improvement of the
original BO algorithm performance by applying chaos
theory to provide better convergence velocity and avoid
tripping into the local minimum. Therefore, the main
goal here is investigating the implementation of the mod-
ified CBO to solve the proposed objective function that
consists of the fuel cost of thermal generators with the
direct, reserve, and penalty costs of RESs.
3|THE MATHEMATICAL
MODEL OPF
It is worth noticing that the OPF problem is a highly
complex nonlinear problem. The solution of OPF is deter-
mining the optimal control variables which minimize the
objective function, considering different operational con-
straints. The main objective function in this study is
reducing the total operating cost that includes the fuel
cost of the conventional thermal generators, with or
without the direct reserve and penalty costs resulting
from the nature of intermittency in the outputs of WT
and SPV generators. Further, the optimal control vari-
ables of the OPF problem are generators' active/reactive
power and voltages, transformer tap ratio, and shunt
VAR capacitors.
3.1 |Cost model for the conventional
thermal generators
The conventional thermal generators are operated based
on fossil fuels. The fuel cost can be expressed as
13
:
FP
cg
¼XNg
i¼1αiP2
cg,iþβiPcg,iþγi,ð1Þ
where Fis the fuel cost of generated power, Ngis the
total number of the conventional thermal generators,
Pcg,irepresents the real power generated from unit i, and
αi,βi, and γiare the cost coefficients related to the ith
conventional thermal generators. However, considering
the valve point loading effect, the quadratic cost function
becomes precise and more realistic.
51
Figure 1 shows a
multi-valve loading effect on the fuel cost function.
37
The
valve-point loading effect occurs because the valves of
thermal generation units are opened in case of steam
admission and these cause the losses to increase suddenly
and lead to ripples in the cost function curve, as shown
in Figure 1. Consequently, the cost function can be
described as follows:
FP
cg
¼XNg
i¼1αiP2
cg,iþβiPcg,iþγi
þeisin giPmin
cg,iPcg,i
,ð2Þ
4HASSAN ET AL.
where eiand giare valve point cost coefficients of unit i,
while Pmin
cg,irepresents the minimum real power that the ith
conventional generators produce when it is in operation.
3.2 |Direct cost of WT and SPV
generators
WT and SPV generators need no fuel for operation. Only
the initial maintenance or spending cost of the RESs is
assigned.
52
The scheduled power produced from RES is
charged according to the mutually agreed contract. A
direct cost function of WT and SPV generators belongs to
private parties and is given in Equation (3)
6
:
CWjPWs,j
¼gWdPWs,j,ð3Þ
where CWjis the direct cost of the jth WT generator, gWd
denotes the direct cost coefficient of the WT generator,
and PWs,jrepresents the scheduled power of the jth WT
generator. In the same manner, the direct cost of the kth
SPV generator in terms of scheduled power can be
expressed as follows:
CSkPSs,k
¼gSdPSs,k,ð4Þ
where gSddenotes the direct cost coefficient of the SPV
generator and PSs,krepresents the scheduled power of the
kth SPV generator.
3.3 |Cost assessment of uncertainties
for WT generators
In terms of the output power of RES, two cases may occur.
The first case arises when the output power of RES is greater
than the expected value (underestimated output power). In
this case, the surplus power can be potentially lost. System
operators aim to reduce produced power from conventional
thermal generators to avoid this. However, the associated
cost is referred to as penalty cost. The penalty cost is paid by
the system operators commensurate with the additional
powerfromWTgeneratorsisexpressedasfollows:
CUw,jPWa,jPWs,j
¼pW,jPWa,jPWs,j
¼pW,jZPWr,j
PWs,j
PW,jPWs,j
FWPW,j
dPW,j,
ð5Þ
where PWa,j,PWs,j, and PWr,jare the available, schedule,
and rated output power from jth WT generator, respec-
tively, pW,jdenotes the penalty cost coefficient of the jth
WT generator and FWPW,j
is the PDF for the generated
power of the jth WT generator.
ThesecondcaseiswhentheoutputpowerofRESisless
than the estimated value (overestimated output power). To
face this situation, system operators need to allocate spinning
reserves on conventional generators to compensate for over-
estimated RES power and thus ensure uninterrupted power
to consumers. The cost related to this power reserve is called
reserve generation cost
53
and is described as follows:
COw,jPWs,jPWa,j
¼RW,jPWs,jPWa,j
¼RW,jZPWs,j
0
PWs,jPWa,j
FWPW,j
dPW,j,
ð6Þ
where RW,jrepresents the reserve cost coefficient of the
jth WT generator. Further, the generated power probabil-
ity determination of various WT generators at different
wind speeds is presented in the later section.
3.4 |Cost assessment of uncertainties
for SPV generators
Also, The generated power from the SPV system in the power
network is uncertain in nature. The procedure to solve under
and overestimation of SPV generators is similar to that used
for WT generators except that the solar radiation is expressed
by a lognormal PDF. In this study, the penalty and reserve
costs models are introduced as in Reference 54. Anyway, the
penalty cost for the kthSPVgeneratorsiswrittenasfollows:
CUS,KPSa,KPSs,K
¼pS,KPSa,KPSs,K
¼pS,KFSPSa,K>PSs,K
EP
Sa,K>PSs,K
PSs,K
,
ð7Þ
FIGURE 1 Valve point loading effect on a fuel cost function
37
HASSAN ET AL.5
where PSa,Kand PSs,jare the available and schedule output
power from the Kth SPV generator, respectively, pS,Kis
the penalty cost coefficient pertaining to the Kth SPV
generators, FSPSa,K>PSs,K
denotes the probability of
excess power generated by the kth SPV generator com-
pared to PSa,K, and EP
Sa,K>PSs,K
is the predictable sur-
plus output power. In case of overestimation, the reserve
cost is evaluated as follows:
COS,KPSs,KPSa,K
¼RS,KPSs,KPSa,K
¼RS,KFSPSa,K<PSs,K
PSs,KEP
Sa,K<PSs,K
,
ð8Þ
where RS,Krepresents the reserve cost coefficient of the
Kth SPV generator, FSPSa,K<PSs,K
defines the shortage
probability of the SPV generators, and EP
Sa,K<PSs,K
is
the expected power of the SPV generator less than PSs,K.
3.5 |Carbon tax model
The conventional thermal generators release gasses to
the environment. When the generation from thermal
generators increases, the emission gasses such as SO
x
and
NO
x
also increase. The harmful emissions are represen-
ted in tons per hour (ton/h) as follows:
E¼XNG
i¼1aiP2
cg,iþbiPcg,iþci
0:01 þωiePcg,iμi
ðÞ
,ð9Þ
where ai,bi,ci,ωi, and μidenote the emission coefficients
of the conventional thermal generators. These coeffi-
cients have values are like those in Reference 55.
Recently, several countries are forcing a CT on harm-
ful gas emissions to safeguard the environment, produce
clean energy, and address global warming risks.
56
More-
over, the energy production companies are subjected to
enormous pressure to generate clean energy from RES
and/or to reduce their emissions. In this paper, a CT is
imposed on the gas emissions model. The carbon emis-
sion cost ($/h) is formulated as follows:
CEC ¼CTE,ð10Þ
where CTcarbon tax per unit value of gasses.
3.6 |Objective functions
In this paper, the objective functions of the OPF prob-
lem are expressed from different model components
presented above in the previous subsections. In this
paper, there are two considered objective functions as
follow:
•Minimization of total operating costs
The first objective function in this paper is based
on minimizing the sum of all costs without consid-
ering the emissions, while the second objective
includes the emissions. So, the first objective will
minimize:
F1¼FP
cg
þXNW
j¼1CWjPWs,j
þCUw,jPWa,jPWs,j
h
þCOw,jPWs,jPWa,j
iþXNP
K¼1CSkPSs,k
h
þCUS,KPSa,KPSs,K
þCOS,KPSs,KPSa,K
i,
ð10Þ
where NWand NPare the number of WT and SPV gener-
ators, respectively.
Considering CT's modeling, the second objective function
is formulated by adding the carbon emission cost. So,
from Equations (10) and (11).
F2¼FP
cg
þXNW
j¼1CWjPWs,j
þCUw,jPWa,jPWs,j
h
þCOw,jPWs,jPWa,j
iþXNP
K¼1CSkPSs,k
h
þCUS,KPSa,KPSs,K
þCOS,KPSs,KPSa,K
iþCEC:
ð11Þ
Further, the two above objective functions are sub-
jected to equality and inequality constraints as
described below.
•Minimization of real power losses
The minimizing of real power losses is considered as
one of the most major issue targets for system opera-
tors that can be described as follows
13
:
PL¼Xnl
k¼1GkV2
iþV2
j2ViVjcos θiθj
hi
,ð12Þ
where PLis the real power loss, Gkis the conductance
of kth line, Vi,Vj,θi, and θjindicate the voltage
magnitudes and their angles at buses iand j,
respectively.
•Minimization of voltage deviations at load buses
To guarantee systems security, one of the most impor-
tant indicators is decreasing the voltage deviations of
load buses to get a sound voltage profile. The voltage
deviations can be represented as follows
13
:
6HASSAN ET AL.
VD¼XNL
j¼1Vk1
jj
,ð13Þ
where a nominal value is 1 p.u. which is taken as a ref-
erence value. NLis the total number of load buses.
•Equality constraints
The equality constraints illustrate the typical load flow
equations that are used for power balancing as follows:
PGi PDi ViXNL
j¼1VjGijcosθij þBij sinθij
¼0, ð14Þ
QGi QDi ViXNL
j¼1VjGijsinθij Bij cosθij
¼0, ð15Þ
where PGi and QGi are the real and reactive power gen-
erated, respectively. While PDi and QDi are demand
real and reactive power of bus j, respectively. Gij is the
transfer conductance between two buses and Bij is the
susceptance between two buses.
•Inequality constraints
The inequality constraints are the operating limits for
the components of the power system and the security
constraints related to the lines and load buses.
•Generator constraints
The voltages and active power at all generating buses
should be limited within their upper and lower values:
Vmin
Gi ≤VGi ≤Vmax
Gi ,i¼1,2, 3…,NT,ð16Þ
Pmin
cg,i≤Pcg,i≤Pmax
cg,i,i¼1,2, 3…,Ng,ð17Þ
Pmin
Ws,j≤PWs,j≤Pmax
Ws,j,j¼1,2,3…,NW,ð18Þ
Pmin
Ss,K≤PSs,K≤Pmax
Ss,K,k¼1,2,3…,NP,ð19Þ
Qmin
cg,i≤Qcg,i≤Qmax
cg,i,i¼1,2, 3…,Ng,ð20Þ
Qmin
Ws,j≤QWs,j≤Qmax
Ws,j,j¼1,2,3…,NW,ð21Þ
Qmin
Ss,K≤QSs,K≤Qmax
Ss,K,k¼1,2,3…,NP,ð22Þ
where NTthe total number of generator buses.
Equation (17) represents the voltage limits on the
generator buses. Equations (18 to 20) represent
active power limits for the conventional thermal gen-
erators, WT, and SPV generators. Equations (21 to
23) denote reactive power capabilities for all generat-
ing buses.
•Constraints of line and load bus voltages
Sli ≤Smax
li ,i¼1,2, 3…,Nl,ð23Þ
Vmin
Li ≤VLi ≤Vmax
Li ,i¼1,2, 3…,NL,ð24Þ
where Sli is the apparent power of the line ith. Smax
li
denotes the maximum limit of the apparent power of
line ith. Vmin
Li ,Vmax
Li are the minimum and maximum
voltage magnitude at load buses, respectively. Nl
denotes the total number of transmission lines.
4|THE STOCHASTIC WIND
POWER AND SOLAR POWER
UNCERTAINTY MODELS
fννðÞ¼
K
C:ν
C
K1:eν
C
ðÞ
K
:ð25Þ
To determine the mean power of WT generators, Weibull
PDFs are used
57
as follows:
where νis the wind speed m/s, while Kis the Weibull
distribution shape parameter or the slope parameter,
which is a kind of numerical parameter of a parametric
family of probability distributions, and Cis the Weibull
distribution scale parameter or characteristic life that is
also a special kind of numerical parameter of probability
distributions.
The mean of Weibull distribution is defined.
The gamma function is computed as:
Mwt ¼C:Γ1þ1
=
K
ðÞ:ð26Þ
Γx¼Z∞
0
tx1etdt,x>0:ð27Þ
The output of WT generators essentially depends on the
wind speed and the power curve of WT
41
and is expressed
as follows:
PwνðÞ¼
0ν≤νci ν>νco
ν2ν2
ci
ν2
nom ν2
ci
:PWrνci <ν≤νnom,
PWrνnom <ν≤νco
8
>
>
<
>
>
:
ð28Þ
where νci,νco ,and νnom denote the cut-in, cut-out, and
rated wind speed of the WT, respectively. While PWrrep-
resent the rated power from the WT. With reference to
Equation (29), one can observe that if νis above νco and
below νci, there is no output power. Also, the WT pro-
duces power when wind speed between νnom and νco.
Probabilities are described for these discrete zones as
follows
23
:
HASSAN ET AL.7
FwPw
ðÞPw¼0
fg
¼1exp νci
C
k
þexp νco
C
k
:ð29Þ
FwPw
ðÞPw¼Pwr
fg¼exp νnom
C
k
exp νco
C
k
:ð30Þ
In contrast to the mentioned discrete zones, the output
power of WT is continuous between rated and cut-in
speeds of wind. So, the probability for this zone can be
described as follows
23
:
FwPw
ðÞ¼
Kνnom νci
ðÞ
CKPwr
νci þPw
Pwr
νnom νci
ðÞ
k1
exp νci þPw
Pwr νnom νci
ðÞ
c
!
k
0
@1
A:ð31Þ
Similarly, when the weather conditions are more disper-
sive, the lognormal function is used to describe the fre-
quency distribution quite better. The mean and SD of the
global irradiation are utilized to compute the parameters
for the lognormal distribution function. The output of
SPV generators is directly proportional to the solar irradi-
ance (I) that follows the lognormal PDF. The solar irradi-
ance probability is described as follows:
fIIðÞ¼ I
Iμffiffiffiffiffi
2π
p:eln Xσ½
2
2μ
,I>0, ð32Þ
where μand σare the mean and SD, respectively. The
lognormal distribution mean is expressed as follows:
Mld ¼eσþμ2
2
ð33Þ
The direct relationship between solar irradiance and
energy for the SPV system is described as:
58
Psr IðÞ¼
Psr
I2
Isr Ic
;0< I<Ic
Psr
I
Isr
;I> Ic
8
>
>
>
<
>
>
>
:
ð34Þ
where Psr is the rated output from the SPV generator, Ic
define a specific irradiance point, and Isr is the solar irra-
diance at the rated environment. It is significant to notice
that scheduled power does not have a constant value
rather there is a mutually known power between the sys-
tem operators and the private side that sells solar power.
The following equations are utilized to calculate the
under and overestimation costs of the SPV generators.
6
CUSPSaPSs
ðÞ¼pSPSaPSs
ðÞ¼pSXNþ
N¼1PSSþPSS
fpsþ,
ð35Þ
COSPSsPSa
ðÞ¼RSPSsPSa
ðÞ¼RSXN
N¼1PSSPSS
½fps,
ð36Þ
where PSSþand PSSdenote the surplus power and short-
age power. fpsþand fpsrepresent the relative frequen-
cies for the occurrence of PSSþand PSS.
5|MATHEMATICAL MODEL OF
THE OPTIMIZATION ALGORITHMS
5.1 |The Bonobo optimizer
Bonobo optimizer (BO) is a well-known optimization
technique that was developed by Amit Kumar Das et al
40
Bonobos show four different mating strategies: promiscu-
ous, restrictive, consort ship, and extra-group mating.
These mating strategies are artificially modeled in the
original BO algorithm. The reproductive manner of
Bonobos depends on the fission-fusion approach. The
fission-fusion approach is the method that the Bonobo
society follows to obtain its nourishment and others. The
concept of the fission-fusion way is that the society is
divided into small groups of various sizes for a few days
to find the nourishment (fission process) then they are
gathering another time at nighttime for sleep (fusion
behavior), as shown in Figure 2. According to the phase
condition, the mating behavior of the bonobo community
is chosen. There are two phases considered in the lifestyle
of this society. The positive phase (PP) represents the
availability of all living conditions such as food, security,
and a successful mating procedure. On the other side, the
negative phase (NP) shows the absence of such condi-
tions. Promiscuous, restrictive mating strategies have
higher probabilities in the PP case while consort ship
mating and extra-group mating strategies are very famil-
iar in the NP case.
1. Promiscuous and restrictive mating strategies
Either promiscuous or restrictive mating produces a
new bonobo, and it is given as follows
60
:
new_bonoboj¼bonoboi
jþr1scab αj
bonobo bonoboi
j
þ1r1
ðÞscsb flag
bonoboi
jbonoboP
j
,ð37Þ
8HASSAN ET AL.
where new_bonobojand αj
bonobo are the jth variables of
the new offspring and the alpha bonobo, respectively,
and jvaries from 1 to dim, where dim is the total
number of variables. r1is a random number in the
range of [0,1]. scab and scsb are the sharing coeffi-
cients for the αj
bonobo and chosen Pth-bonobo, respec-
tively. flag gives either 1 or 1.
2. Consortship and extra-group mating strategies
The consortship and extra-group mating strategies are
randomly created, based on phase probability (pp), and
formed through the extra-group mating probability:
β1¼er2
4þr42
r4
,ð38Þ
β2¼er2
4þ2r42
r4
,ð39Þ
new_bonoboj¼bonoboi
jþβ1var_maxjbonoboi
j
,
ð40Þ
new_bonoboj¼bonoboi
jβ2bonoboi
jvar_minj
,
ð41Þ
where β1and β2are the two intermediate measured
values utilized to define the value of new_bonobo. r4
is a random number generated in between [0 1].
var_maxjand var_minjare the lower and upper limits
corresponding to the jth-variable, respectively.
5.2 |The proposed chaotic Bonobo
optimizer
The proposed algorithm, named chaotic Bonobo opti-
mizer (CBO) algorithm, is based on the BO algorithm
plus the incorporation of chaotic maps. The proposed
CBO technique improves the low convergence speed and
the weak local optimum of the original BO algorithm
thus enhancing the performance of the proposed algo-
rithm to achieve the best value for the OF. This modifica-
tion is based on chaotic theory, which is one of the most
effective methods to overcome those issues. Instead of
using random parameters, a set of chaotic equations
61
is
used to enhance the convergence characteristics and
strengthen the performance and robustness of the
original BO.
The chaotic matrix is used instead of the random
number r4in Equations (39) and (40) using chaotic
maps. Therefore, Equations (39) and (40) can be rewrit-
ten as follows:
β1¼eChaotic2
4þChaotic42
Chaotic4
:ð42Þ
β2¼eChaotic2
4þ2Chaotic42
Chaotic4
:ð43Þ
Table 1 tabulates the 10 chaotic maps are applied for the
original BO algorithm to improve the parameter of explo-
ration qas follows
61
:
q¼ykþ1,ð44Þ
where y
k+1
is the chaos map selected for solving the prob-
lem and is tabulated in Table 1. The flowchart of the pro-
posed CBO algorithm is displayed in Figure 3.
6|SIMULATION RESULTS AND
DISCUSSION
The effectiveness of the proposed CBO and the optimiza-
tion framework are shown in this section using both the
standard and modified IEEE-30 and IEEE-57 bus test sys-
tems. The simulation results using 10 different CBO vari-
ants, compared with that obtained by the standard BO,
based on the standard IEEE-30 bus test systems. The
potential for CBO is to minimize both the cumulative
FIGURE 2 Bonobos' fission-fusion
social groups
59
HASSAN ET AL.9
cost and optimization convergence rates. Twelve scenar-
ios are evaluated, including different types of RES and
CT. The application of standard BO and 10 versions of
the proposed CBO algorithm based on the 10 chaotic
maps to OPF framework have been run on 16GB RAM
PC, an I7-8700 CPU, 2.8GHz, and MATLAB 2016a and
MATPOWER.
62
6.1 |Test system 1
The standard IEEE 30-bus system has 6 power generating
units and 24 load buses. Also, 41 branches link the gener-
ators and load busses. The total connected loads of both
active and reactive power are 2.834 and 1.262 p.u., respec-
tively. Bus 1 was selected as the slack bus. The voltage
magnitude limits for both generating units and load bus-
ses are between 0.95 and 1.1 p.u. For the tap changing
transformer, the tap setting is varied between 0.9 and 1.1
p.u. Furthermore, the VAR compensators are supposed
to change between 0 and 0.05 p.u. In more detail, all line
and bus data can be found in Reference 37. The standard
system was modified in this paper as described in Refer-
ence 37 to accommodate the conventional thermal gener-
ators units WT and SPV generators. The conventional
thermal generators units are placed at buses 1, 2, and
8. However, one SPV generator is installed at bus 13. Two
WT generators are placed at bus No. 5 and 11 with reac-
tive power capability. Moreover, the modified test system
network that comprises three different power generation
sources is shown in Figure 4. It is worth mentioning that
the WT and SPV outputs contain variations, which must
be balanced by reserve and outputs of another generator
collectively. Therefore, the total generation cost com-
prises the total operational cost of the conventional ther-
mal generators, along with the penalty and reserve costs
resulting from intermittency in the SPV and WT outputs.
The mentioned PDF parameters in the previous
section are used in the proposed case studies to compute
wind speed. The wind frequency distribution by Weibull
fitting
6
is shown in Figure 5A,B at buses 5 and 11, respec-
tively. The requirements of WT design are introduced in
Reference 63. The power curve is accomplished after run-
ning 8000 Monte-Carlo scenarios. The Weibull PDF
parameters of WT generator at bus 5 are: c=9, k=2,
while at bus 11 are: c=10, k=2.
64,65
The Weibull mean
is Mwt =7.976 and 8.862 m/s at buses 5 and 11, respec-
tively. A 3 MW WT was utilized at νci ¼3m=s, νco ¼
25 m=s and νnom ¼16 m=s based on the manufacture
datasheet of Enercon E82-E4.
TABLE 1 Ten chaotic maps
No. Name Chaotic map formula
CBO1 Chebyshev ykþ1¼cos kcos1yk
ðÞðÞ
CBO2 Circle ykþ1¼mod ykþb1b2
2π
sin 2πyk
ðÞ,1
b1¼0:5,b2¼0:2
CBO3 Gauss/mouse
ykþ1¼
1yk¼0
1
mod yk
ðÞ
otherwise
8
<
:
CBO4 Iterative ykþ1¼sin bπ
yk
,b¼0:7
CBO5 Logistic ykþ1¼byk1yk
ðÞ,b¼4
CBO6 Piecewise
ykþ1¼
yk
H0≤yk≤H
ykH
0:5HH≤yk≤0:5
1Hyk
0:5H0:5≤yk≤1H
1yk
H1H≤yk≤1
,H¼0:4
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
CBO7 Sine ykþ1¼b
4sin πyk
ðÞ,b¼4
CBO8 Singer ykþ1¼u7:86yk123:31yk12
28:75yk313:302875yk4
,u¼1:07
CBO9 Sinusoidal ykþ1¼by1
ksin πyk
ðÞ,b¼2:3
CBO10 Tent
ykþ1¼
yk
0:7yk<0:7
10
31yk
ðÞyk≥0:7
8
>
<
>
:
10 HASSAN ET AL.
In the same manner to describe the output of PV gen-
erators, the parameters of the lognormal PDF are chosen
based on the mean and SD of the global irradiation,
where σ¼6, μ¼0:6, and lognormal mean equal
I=483 W/m. After running the Monte-Carlo technique
with a sample size of 8000, the frequency distribution
and lognormal fitting of solar irradiance are shown in
Figure 6. Figure 7 shows the histogram for the SPV gen-
erators output; it is seen that the SPV output has stochas-
tic nature due to the variance in solar irradiance.
1. Case A
This case aims to investigate the effect of both scheduled
power and PDF parameters of WT and SPV generators
on the total generating cost.
a. Effect of scheduled wind and solar power on the
total cost
Using the Weibull PDF parameters besides a discussed
WT parameter selection in Section 3, the impact of wind
parameters on the cost by the proposed CBO5 is investi-
gated in the present case. The direct cost coefficients of
WT generators at buses 5 and 11 are 1.6 and 1.75, respec-
tively. Further, the coefficient of penalty cost and reserve
cost for utilizing wind power at these buses are assumed
1.5 at both buses, while the coefficient of reserve cost is
assumed 3 at both buses. It is worth noting that the direct
cost of RES is lower than the average cost of conventional
thermal generators. Likewise, the penalty cost for not uti-
lizing full the available wind power is lower than the
FIGURE 3 Flowchart of the proposed chaotic Bonobo optimizer technique
HASSAN ET AL.11
direct cost of RES.
50
In this case, scheduled available
wind power is varied from 0 to the rated power, where
the direct, penalty, reserve, and total wind power costs
are computed. Figure 8 shows the variation of different
cost components for wind power vs the scheduled wind
power. The results show a linear relationship between
the scheduled wind power and the direct cost. In addi-
tion, a larger spinning reserve is required, which leads to
an increase in the reserve cost then consequently, the
generation costs move upwards. In addition, the penalty
FIGURE 4 Modified IEEE 30-bus test system
FIGURE 5 Wind speed distribution for wind turbine generators at buses 5 and 11
12 HASSAN ET AL.
cost rightly decreases with a lower rate with increased
scheduled power from wind. As a result, the total wind
power cost is increased.
Similarly, cost variations appear due to related pen-
alty and reserve costs coefficients when SPV output is
underestimated/overestimated from scheduled power. To
estimate the total cost of SPV generators, it is required to
analyze the operation and maintenance cost. From Refer-
ence 66, it is seen that selected the cost ranges for the pre-
sent study are like those of onshore wind power plants.
So, the direct, reserve, and penalty cost coefficients are
assumed in Reference 6 with 1.6, 3, and 1.5, respectively.
Variation of solar power generation cost vs scheduled
power is illustrated in Figure 9. It is significant to observe
that the total cost of SPV generators does not rise uni-
formly with PDF parameters of solar irradiance. In addi-
tion, the minimum cost is accomplished when scheduled
power from SPV generators is set to 22 MW.
2. Effect of PDF parameters on the total generating cost
of wind and solar power
The cost of the WT generator is directly affected by the
Weibull distribution scale parameter C. This present case
evaluates how Weibull distribution scale parameter varia-
tions affect WT generator cost while a fixed scheduled
power. The shape parameter for both utilized WT genera-
tors at two specified buses is equal to K=2. Also, the
Rayleigh distribution is equal to a Weibull distribution
with the same value of K=2. Considering the same
values of the cost coefficients used in the previous case
and the output power of two WT generators are 25 and
20 MW, respectively. Figure 10 shows the relationships
between WT generators cost and the Weibull distribution
scale parameter. It is seen that; the lowest value of costs
is accomplished when the scale parameter's value is in
the center of a certain range. Scale parameter with higher
value denotes the propagation of higher wind speeds with
a particular probability. This occurs since the scheduled
power is still the same for a constant interval that
increases penalty costs and raises the overall costs.
The relationship between the power cost of the SPV
generator and the lognormal PDF mean (λ) is shown in
Figure 11. In this study, λis varied in the range of 2 to
7withanincrementof0.5.
37
The Scheduled power from
theSPVgeneratoris20MWwithSD0.6.Also,costcoef-
ficients remain with the same values used in the previ-
ous case. The lowest cost of solar power is achieved
when λ=5.5. In addition, the penalty cost and reserve
costvaluesarethesameatλ=5.8. It is observed that
the penalty costs increase sharply for higher values of λ,
which results in leading the overall cost to a higher
level. However, a direct relationship between SPV gen-
erator output and solar irradiance is achieved with λ.
When λis low, then the SPV output is also low. High
reserve powers are required to resist this status, which
increases the reserve cost. When λis high, the solar irra-
diance is also predicted high, so increasing SPV output
power. In such a status, the penalty costs introduce an
increase in the overall cost. Considering these two sta-
tuses, an appropriate value from the SPV output usually
needs to be scheduled.
FIGURE 6 Solar irradiance distribution for solar photovoltaic
generators at bus13
FIGURE 7 Active power distribution in (MW) from solar
photovoltaic generator at bus 13
HASSAN ET AL.13
2. CASE B
The proposed versions of CBO are utilized in this case to
solve the OPF framework with different objective func-
tions with/without stochastic RES for different objective
functions as follows:
I. Case 1: Optimizing total generation cost without
incorporating RES
In this section, the performance of the CBO technique is
investigated on the standard IEEE 30-bus test system
without incorporating RES, where the fuel cost function
is considered the objective function. For case
1, Equation (2) calculates the fuel cost function. The opti-
mal control settings obtained by suggested versions of
CBO algorithms and the original BO are presented in
Table 2, such as generator active power, generator volt-
age, transformer tap ratio, and reactive power from the
capacitor bank. Also, the total generating cost as the
main objective function, real power loss, emission, and
voltage deviation are tabulated in Table 2. This compari-
son shows that the fuel cost obtained by the CBO5
method is better than that of the standard BO algorithm
and other versions of CBO. However, the best fuel cost
calculated by CBO5 for this case is 832.0752 $/h without
any violation to the constraints, while the emission, real
power loss, and voltage deviation became 0.319543 ton/h,
10.68 MW, and 0.846898 p.u., respectively. The conver-
gence characteristics of the compared optimization tech-
niques of CBO versions and BO are shown in Figure 12.
Regarding the convergence graph, the CBO5 has a
smooth convergence curve to the optimal solution with-
out oscillations. Figure 13 presents the voltage profile of
all proposed versions of CBO and the original BO algo-
rithms. This figure illustrates that all voltages magnitudes
are within the specified limits (upper and lower bounds).
The comparison of CBO5 with the original BO algorithm
in terms of Best, Worst, Median, and SD of fuel cost is
presented in Table 3.
II. Case 2: Optimizing the real power loss without
incorporating RES
This case aims to reduce the real power loss that is taken
as the main objective function without Incorporating
FIGURE 8 Variation of wind power cost components vs scheduled wind power
FIGURE 9 Variation of solar power cost components vs
scheduled solar power
14 HASSAN ET AL.
RES. The results of optimal control variables obtained by
CBO5 and the original BO are tabulated in Table 4. From
this table, the power loss is minimized to be 3.08819 MW
while the fuel cost, the emission, and the voltage devia-
tion became 1027.332 $/h, 0.233633 ton/h, and 0.900771
p.u., respectively. The convergence characteristics of the
OPF framework for this case are illustrated in
Figure 14A. This figure shows the best convergence by
CBO5. As in case 1, the magnitude of voltage is within
their limits for all buses, as shown in Figure 15. The sta-
tistical results for the current case are tabulated in
Table 5. This table shows that the CBO5 also achieves the
smallest best, mean, median, and SD values than the
original BO.
III. Case 3: Optimizing total generation cost with
considering a carbon tax
In this case, the CBO5 algorithm is utilized to minimize
the cumulative cost with the imposition of the CT with-
out considering RES. In the present study, carbon tax CT
is assumed with a rate of $20/ton as in Reference 50.
Then, its results are compared with the standard BO
method. The results of this case are tabulated in Table 4.
For the minimization of the cumulative cost, it can be
noted that the CBO5 gives the best minimization of the
cumulative cost, which is 837.7655 $/h, while the real
power loss, the emission, CT, and the voltage deviation
became 10.66505 MW, 0.319507 ton/h, 17.83 $/h, and
0.848031 p.u., respectively. The comparison of conver-
gence characteristics of minimizing the cumulative cost
with CT is depicted in Figure 14B. From this figure, the
convergence curves from the CBO5 outperform than that
obtained from the BO. The voltages magnitudes of all
buses of the CBO5 are within the specified limits, as
shown in Figure 15. Table 5 illustrates the comparison of
statistical results yielded by CBO5 and the original BO
algorithm.
Furthermore, the proposed CBO algorithm's results
are compared with more than 10 well-known algorithms'
FIGURE 10 Variation of wind power cost components vs Weibull scale parameter
FIGURE 11 Variation of solar power cost components vs
lognormal mean
HASSAN ET AL.15
TABLE 2 Simulation results of the proposed method and the BO method for case 1
Min Max BO CBO1 CBO2 CBO3 CBO4 CBO5 CBO6 CBO7 CBO8 CBO9 CBO10
Generator active power
P
G1
(MW) 50 200 198.6254 198.6991 198.6857 198.6739 198.7024 198.7367 198.5716 198.7221 198.9047 198.6208 198.7147
P
G2
(MW) 20 80 45.21471 45.11906 44.77042 44.89704 44.56043 44.96156 44.60777 44.82217 45.24557 44.6228 44.66742
P
G5
(MW) 15 50 18.2239 18.20115 18.57578 18.48418 18.77675 18.36983 18.84186 18.52777 17.98837 18.76609 18.67897
P
G8
(MW) 10 35 10 10.04527 10 10 10.00599 10 10 10.00003 10 10.02919 10
P
G11
(MW) 10 30 10.00101 10.00614 10.03021 10.00876 10.00022 10 10 10.0019 10.00003 10.00006 10
P
G13
(MW) 12 40 12 12 12 12 12 12 12 12.0001 12.00029 12.00269 12.00001
Generator voltage
V
1
(p.u.) 0.95 1.1 1.0853 1.083108 1.082972 1.083702 1.082819 1.082705 1.085082 1.083031 1.080218 1.082642 1.082499
V
2
(p.u.) 0.95 1.1 1.082611 1.1 1.090361 1.060996 1.1 1.1 1.072198 1.1 1.097758 1.059832 1.099925
V
5
(p.u.) 0.95 1.1 1.029013 1.027678 1.027467 1.027521 1.027396 1.026886 1.027424 1.026828 1.022684 1.081059 1.027605
V
8
(p.u.) 0.95 1.1 1.034571 1.023295 1.035512 1.034883 1.036265 1.035888 1.035348 1.009771 1.031406 1.034781 1.036244
V
11
(p.u.) 0.95 1.1 1.095888 1.070882 1.069605 1.057137 1.097282 1.075685 1.077993 1.1 1.055232 1.041216 1.099988
V
13
(p.u.) 0.95 1.1 1.037147 1.056721 1.053777 1.054794 1.025306 1.056338 1.053553 1.052925 1.0984 1.065886 1.026484
Transformer tap ratio
T
11
(6–9) 0.9 1.1 1.015298 1.034039 1.05907 1.009557 1.024553 1.007703 1.001557 1.099974 0.981481 0.961371 1.033192
T
12
(6-10) 0.9 1.1 0.971798 0.920428 0.906353 0.957821 0.962566 0.959703 0.979863 0.9 0.981841 1.011796 0.962145
T
15
(4-12) 0.9 1.1 0.969049 0.989422 0.981566 0.983852 0.97416 0.984684 0.972721 0.984992 1.015082 0.996291 0.973741
T
36
(28-27) 0.9 1.1 0.974182 0.969703 0.973543 0.976203 0.984331 0.973367 0.97362 0.972554 0.979353 0.976633 0.970739
Capacitor bank
Q
C10
(MVAR) 0 5 7.23E09 0.004603 0.025006 0.03511 0 0.017517 0.05 0.005931 0.047326 0.05 0.044607
Q
C12
(MVAR) 0 5 0.049999 0.049983 0.035753 0.05 0.05 0.035957 0 0.049998 0 0.007414 0.05
Q
C15
(MVAR) 0 5 0.05 0.016845 0.038026 0.049601 0.045636 0.05 0.05 0.05 0.034575 0.04996 0.05
Q
C17
(MVAR) 0 5 0.023808 0.049535 0.049997 0.05 0.049975 0.032373 0.05 0.05 0.026928 0.047528 0.05
Q
C20
(MVAR) 0 5 0.042794 0.043324 0.039923 0.05 0.040803 0.032991 2.09E06 0.048477 0.049651 0.05 0.0429
Q
C21
(MVAR) 0 5 0.049984 0.05 0.05 0.05 0.049904 0.05 0.05 0.05 0.049981 0.05 0.038009
Q
C23
(MVAR) 0 5 2.89E19 0.05 0.05 0 0.034243 0.018485 0.033671 0.029605 0.03653 0.029278 0.00137
Q
C24
(MVAR) 0 5 0.049924 0.05 0.016185 0.05 0.05 0.05 0.048278 0.05 0.049987 0.05 0.049996
Q
C29
(MVAR) 0 5 0.021418 0.011182 0.022319 0.023411 0.037643 0.01673 0.019068 0.018267 0.027052 0.021731 0.017435
Objective function
Fuel cost ($/h) 832.1145 832.0974 832.1366 832.1252 832.0994 832.0752 832.1081 832.1367 832.1684 832.1431 832.1241
Emission (ton/h) 0.31938 0.319503 0.319488 0.319459 0.319514 0.319543 0.319334 0.319524 0.319751 0.319407 0.319521
Power loss (MW) 10.67697 10.68265 10.67403 10.66395 10.65769 10.68 10.6331 10.68595 10.75089 10.64715 10.67302
Voltage deviation (p.u.) 0.83498 0.864783 0.840011 0.825256 0.886409 0.846898 0.854665 0.845794 0.834759 0.840725 0.890702
Abbreviations: BO, Bonobo optimization; CBO, chaotic Bonobo optimizer.
16 HASSAN ET AL.
results for cases 1 and 2. This comparison for the two
cases is presented in Table 6. It is seen from this table
that the CBO achieved lower function values, which con-
firms the robustness and efficacy of the proposed tech-
nique and other recent optimization techniques such as
Ensemble of constraint handling techniques with DE
(ECHT-DE), feasibly solutions with DE (SF-DE), the self-
adaptive penalty with DE (SP-DE), and so on.
IV. Case 4: Optimizing total generation cost
with RES
According to Equation (11), optimization scheduling of
conventional thermal generators and RES for minimizing
the total generation cost is performed. The cost coeffi-
cients of WT generators at buses 5 and 11 are like case
A. The PDF parameters of WT and SPV generators as
given in Section 5 are utilized. Table 7 summarizes the
obtained optimal control variables for this case in the
same manner as case 1 but considering RES. The simula-
tion results show that the CBO5 is more efficient because
it gives better results than the BO algorithm for similar
OPF frameworks. Hence, for this case, the CBO5 outper-
forms the BO algorithm in terms of the total generation
cost with a value of 781.3524 $/h. The convergence char-
acteristics of the compared optimization algorithms are
presented in Figure 16A. As shown, the CBO5 has a
smooth with fast convergence curve. Figure 17 shows the
voltage profiles of the CBO5 and other proposed methods
for all buses. It is shown that all voltage magnitudes are
within the specified boundaries. However, the voltage
profile for CBO5 has the better profile at all buses when
compared to the BO algorithm. The statistical analysis is
performed and tabulated in Table 8. The results prove
that the CBO5 has a better minimum SD over other origi-
nal BO methods.
FIGURE 12 Convergence characteristic of all compared
methods for case 1
FIGURE 13 Voltage magnitude of all compared methods for
case 1
TABLE 3 Statistical analysis of the
OPF results obtained with different
methods for case 1
Method Best Mean Median Worst SD
BO 832.11448096 832.3159239 832.217977 832.8043547 0.213989434
CBO1 832.09738097 832.3483076 832.3223544 833.0282696 0.23393155
CBO2 832.13660869 832.3138153 832.2985818 832.5276863 0.127775697
CBO3 832.12515527 832.4124289 832.2463911 834.6937676 0.555892645
CBO4 832.09944517 832.3744332 832.2782335 833.3874066 0.300424186
CBO5 832.07517616 832.262603 832.2295532 832.4173286 0.106426901
CBO6 832.10812904 832.4118822 832.2868731 833.5786145 0.364941685
CBO7 832.13668677 832.2901096 832.2418044 832.7007641 0.140583761
CBO8 832.16839578 832.284684 832.2281307 832.6251917 0.117317661
CBO9 832.14305184 832.3872906 832.3449818 832.8088767 0.181949807
CBO10 832.12410053 832.5580215 832.3585978 835.95637 0.842610868
Abbreviations: BO, Bonobo optimization; CBO, chaotic Bonobo optimizer; OPF, optimal power flow.
HASSAN ET AL.17
V. Case 5: Optimizing real power losses with RES
Optimization scheduling of both conventional thermal
generators and RES to minimize the real power loss is
presented in this case. With the same cost coefficients
and the PDF parameters of WT and PV generators uti-
lized in case 4, the best results yielded by the proposed
CBO5 technique are presented in Table 6 and the results
TABLE 4 The Results of the proposed CBO5 method and BO method for cases 2 and 3
Min Max
Case 2 Case 3
BO CBO5 BO CBO5
Generator active power
P
G1
(MW) 50 200 51.48872 51.48857 198.7282 198.7056
P
G2
(MW) 20 80 80 80 44.85148 44.82563
P
G5
(MW) 15 50 50 50 18.48704 18.52142
P
G8
(MW) 10 35 35 34.99993 10 10
P
G11
(MW) 10 30 30 29.99984 10.00151 10.00039
P
G13
(MW) 12 40 39.99985 39.99984 12.00001 12.00006
Generator voltage
V
1
(p.u.) 0.95 1.1 1.06111 1.061396 1.082316 1.082865
V
2
(p.u.) 0.95 1.1 1.056967 1.057312 1.05966 1.1
V
5
(p.u.) 0.95 1.1 1.037555 1.037837 1.02634 1.027826
V
8
(p.u.) 0.95 1.1 1.044163 1.043956 1.033842 1.034909
V
11
(p.u.) 0.95 1.1 1.051582 1.041716 1.063828 1.061986
V
13
(p.u.) 0.95 1.1 1.054497 1.05654 1.057566 1.055204
Transformer tap ratio
T
11
(6–9) 0.9 1.1 1.019284 1.047293 1.013279 0.983736
T
12
(6–10) 0.9 1.1 0.932402 0.900001 0.938206 1.003173
T
15
(4–12) 0.9 1.1 0.991923 0.991199 0.993955 0.9872
T
36
(28–27) 0.9 1.1 0.974778 0.974566 0.97541 0.978693
Capacitor bank
Q
C10
(MVAR) 0 5 0.028518 0.045774 0.009227 0.049775
Q
C12
(MVAR) 0 5 0.020393 0.001478 0.049977 0.043561
Q
C15
(MVAR) 0 5 0.046537 0.046257 0.049785 0.042716
Q
C17
(MVAR) 0 5 0.05 0.05 0.05 0.049929
Q
C20
(MVAR) 0 5 0.042133 0.042052 0.04109 0.038074
Q
C21
(MVAR) 0 5 0.05 0.05 0.05 0.05
Q
C23
(MVAR) 0 5 0.031086 0.031389 0.032529 0.04989
Q
C24
(MVAR) 0 5 0.05 0.05 0.05 0.05
Q
C29
(MVAR) 0 5 0.018052 0.019602 0.022632 0.01851
Objective function
Fuel cost ($/h) 1027.334 1027.332 832.1032 832.0687
Emission (ton/h) 0.233633 0.233633 0.319534 0.319507
Total cost ($/h) ——837.8005 837.7655
Carbon tax ($/h) ——17.83 17.83
Power loss (MW) 3.088588 3.088195 10.66829 10.66505
Voltage deviation (p.u.) 0.905851 0.900771 0.864543 0.848031
Abbreviations: BO, Bonobo optimization; CBO, chaotic Bonobo optimizer.
18 HASSAN ET AL.
of the original BO algorithm. From Table 3, it can be seen
that the CBO5 yielded a real power loss value of
1.832164 MW compared with the result of 1.833118 MW
by the BO algorithm. Figure 16B compares the conver-
gence curves of real power loss for this case. This figure
shows that the convergence characteristics of real power
loss for the CBO5 technique outperform those from the
compared BO algorithm. As in case 4, the voltage profiles
of all buses obtained using the CBO5 and the BO are
illustrated in Figure 17. It is seen that the voltage profile
for CBO5 has a better voltage profile when compared to
the BO technique. The statistical results for the current
case are summarized in Table 7. According to Table 8, it
is established that the CBO5 also achieves the lowest
best, mean, median, and SD values than the BO
algorithm.
VI. Case 6: Optimizing the total generation cost by
considering a carbon tax and RES
As in case 3, by employing Equation (12) with incorpo-
rating RES, the CBO5 algorithm is applied to minimize
the cumulative cost with the imposition of the
CT. Further, the penetration of RES is forecasted to
increase, and the simulation results prove this. The last
two columns of Table 7 show the best results obtained
using the CBO5 alongside the BO algorithm for optimiz-
ing the power generation schedule of all relevant parame-
ters required for OPF considering the CT. From this
table, it is seen that higher penetration of RES is accom-
plished compared to case 4. In the optimal generation
schedule, the range of RES penetration depends on both
CT imposed and the emission volume. Moreover, the
comparison of convergence characteristics for case 3 is
yielded by CBO5 with the BO is illustrated in Figure 16C.
It is confirmed that CBO5 has the best performance to
minimize the cumulative cost. In the same manner,
Figure 17 illustrates the voltage profiles for the current
case. For all buses, the operating voltages are within the
specified range.
Finally, Table 8 sums up the statistical analysis per-
formed by various algorithms besides the proposed algo-
rithm, such as grasshopper optimization algorithm
(GOA), black widow optimization algorithm (BWOA),
FIGURE 14 Convergence characteristic of CBO5 and BO for cases 2 and 3. (A) Case 2. (B) Case 3. BO, Bonobo optimization; CBO,
chaotic Bonobo optimizer
FIGURE 15 Voltage profiles of compared methods for cases
2 and 3
HASSAN ET AL.19
GWO, ant lion optimizer (ALO), PSO, GSA, moth-flame
optimization (MFO), barnacles mating optimizer (BMO),
and the conventional BO algorithm for the three cases.
Obviously, the CBO5 technique outperforms these algo-
rithms from the previous literature and the original BO
for cases 4 to 6. It is shown from this table that the pro-
posed CBO5 algorithm presents a superior performance
in terms of accuracy and robustness. Furthermore, the
results of some recent techniques, including jellyfish
search optimizer (JS), artificial bee colony (ABC), Chaos
Game Optimization (CGO), Giza pyramids construction
(GPC), Flower pollination algorithm (FPA), genetic algo-
rithms (GA), crow search algorithms (CSA), success
history-based adaptive differential evolution-feasible
solutions (SHADE-SF), GWO, PSO, MFO, multi-operator
differential evolution algorithm (MODE), and hybridiza-
tion of PSO with GWO (HPSO-GWO) for these cases are
presented in Table 9. This table confirms the efficacy of
TABLE 6 Comparison of CBO5 algorithm with the original BO algorithm and previous studies for single objective cases of IEEE 30-bus
system
Case # Algorithm Fuel cost ($/h) Emission (t/h) Ploss (MW) VD (p.u.)
Case 1 CBO5 832.0752 0.319543 10.68 0.846898
BO 832.1145 0.31938 10.67697 0.83498
ECHT-DE
67
832.1356 0.43765 10.6772 0.80326
SF-DE
67
832.0882 0.43730 10.6387 0.84935
SP-DE
67
832.4813 0.43651 10.6762 0.75042
PSO
68
832.6871 ———
ICBO
68
830.4531
a
—10.2370 1.7450
a
BSA
69
830.7779
a
0.4377 10.2908 1.2050
a
APFPA
70
830.4065
a
—10.2178 1.8909
a
Case 2 CBO5 1027.332 0.233633 3.088195 0.900771
BO 1027.334 0.233633 3.088588 0.905851
MSA
18
—0.20727 3.1005 0.88868
DSA
71
—0.20826 3.0945 —
GPU-PSO
72
—— 3.2601 —
ARCBBO
73
—0.2073 3.1009 0.8913
EM
74
—— 3.1775 —
IEM
74
—— 2.8699
a
—
APFPA
70
—— 2.8463
a
2.0720
a
ABC
75
—— 3.1078 —
EGA-DQLF
76
—— 3.2008 —
GEM
69
—0.2072 2.8863
a
1.9755
a
Abbreviations: ABC, artificial bee colony; ARCBBO, adaptive real coded biogeography-based optimization; BO, Bonobo optimization; CBO, chaotic Bonobo
optimizer; ECHT-DE, Ensemble of constraint handling techniques with DE; ICBO, improved colliding bodies optimization; MSA, Moth Swarm Algorithm;
PSO, Particle Swarm Optimization; SF-DE, feasibly solutions with DE; SP-DE, self-adaptive penalty with DE.
a
Infeasible solution, constraint on load bus voltage is violated.
TABLE 5 Statistical analysis for cases 2 and 3
Case no. Method Best Mean Median Worst SD
Case 2 BO 3.08858773 3.142113765 3.113191618 3.420240129 0.077467375
CBO5 3.08819485 3.098197495 3.09751512 3.116976241 0.008002977
Case 3 BO 837.8004995 838.1715663 837.9152852 840.1457422 0.583449443
CBO5 837.7655243 837.9594622 837.9311724 838.1693743 0.114139838
Abbreviations: BO, Bonobo optimization; CBO, chaotic Bonobo optimizer.
20 HASSAN ET AL.
the proposed CBO5 technique and proves its supremacy
on more than 19 techniques in different cases of the OPF
problem.
6.2 |Test system 2
In this subsection, a medium-scale test system of the
modified IEEE-57 bus test system is implemented to
prove the suitability and scalability of the CBO for
medium-scale systems. The system consists of seven
generators including four conventional thermal genera-
tors, two WT generators, and one SPV generator; 50 load
buses; and 80 branches. The voltage magnitude limits
for all buses are between 0.95 and 1.1 p.u. The total con-
nected loads for active and reactive power are 12.508
and 3.364 p.u. at 100 MVA base, respectively. More
details of the data, emission, and cost coefficients of the
TABLE 7 Results of the proposed CBO5 method and BO method for cases 4 to 6
Min Max
Case 4 Case 5 Case 6
BO CBO5 BO CBO5 BO CBO5
Generator active power
P
G1
(MW) 50 200 134.908 134.9078 7.645585 7.634951 125.4761 124.7555
P
G2
(MW) 20 80 27.83719 28.14207 61.50441 61.47893 33.34418 31.51679
P
W1
(MW) 0 75 43.43267 43.54993 75 75 46.22567 45.26177
P
G8
(MW) 10 35 10 10 35 35 10 10
P
W2
(MW) 0 60 36.68972 36.77103 60 60 38.95007 38.1232
P
S1
(MW) 0 50 36.31506 35.8098 46.08312 46.11828 34.75265 39.0851
Generator voltage
V
1
(p.u.) 0.95 1.1 1.071532 1.071571 1.0538 1.054095 1.070459 1.069179
V
2
(p.u.) 0.95 1.1 1.056449 1.056559 1.054491 1.054778 1.056739 1.056593
V
5
(p.u.) 0.95 1.1 1.034488 1.034492 1.045138 1.045352 1.035747 1.03585
V
8
(p.u.) 0.95 1.1 1.090926 1.059025 1.094693 1.051321 1.068384 1.041066
V
11
(p.u.) 0.95 1.1 1.1 1.09827 1.098714 1.1 1.1 1.099998
V
13
(p.u.) 0.95 1.1 1.049476 1.049468 1.059987 1.062491 1.0497 1.060039
Objective function
Fuel cost ($/h) 781.7198 781.3524 958.5272 957.8938 790.6987 790.9503
Fuel thermal unit cost 438.3715 439.3802 358.1352 358.014 435.0614 427.1315
Wind generation cost 244.2747 244.9571 464.6296 464.6296 261.9756 255.6239
Solar generation cost 99.07357 97.0151 135.7624 135.2502 93.66172 108.1948
Emission (ton/h) 1.762253 1.76216 0.099828 0.099827 0.99819 0.957464
Total cost ($/h) ————808.4964 808.0219
Carbon tax ($/h) ————17.83 17.83
Power loss (MW) 5.782634 5.780656 1.833118 1.832164 5.348664 5.342363
Voltage deviation (p.u.) 0.454874 0.454963 0.531491 0.538215 0.458006 0.486436
Generator reactive power
Q
G1
(MVAR) 20 150 2.1961 2.29821 6.04861 5.84803 2.84362 5.65912
Q
G2
(MVAR) 20 60 11.47038 11.67913 5.905437 6.579832 10.97277 13.33378
Q
W1
(MVAR) 30 35 22.44562 22.33918 20.50373 20.80685 22.37656 23.1282
Q
G8
(MVAR) 15 40 40 40 40 37.81501 40 35.89478
Q
W2
(MVAR) 25 30 30 30 30 30 30 30
Q
S1
(MVAR) 20 25 15.31231 15.30408 18.61107 19.61333 15.36297 19.13591
Abbreviations: BO, Bonobo optimization; CBO, chaotic Bonobo optimizer.
HASSAN ET AL.21
IEEE-57 bus system are given.
82
In the following case
studies for solving the OPF framework in the IEEE-57
bus test system, the best results are obtained under
equality and inequality constraints given in
Equations (17 to 25).
Cases 7 to 9 are considered to solve the OPF frame-
work without RES to establish the efficacy of the pro-
posed CBO5. The optimal control variables with the
corresponding results for three considered objective func-
tions for minimizing the total generating cost without
imposing CT, minimizing the real power loss, and mini-
mizing the total generating cost with imposing CT are
presented in Table 10. The convergence characteristics of
the OPF framework for these cases are given in
Figure 18. In addition, Figure 19 shows that all voltage
magnitudes by CBO5 are within their limits, and CBO5
has a better voltage profile than the compared BO
FIGURE 16 Convergence characteristic of CBO5 and BO for cases 4 to 6. (A) Case 4. (B) Case 5. (C) Case 6. BO, Bonobo optimization;
CBO, chaotic Bonobo optimizer
FIGURE 17 Voltage profiles of compared methods for
cases 4 to 6
22 HASSAN ET AL.
algorithm for all cases. However, these cases can be pres-
ented as follows:
Case 7: The main objective function of this case study
is to decrease the total operating fuel cost. The fuel cost
achieved by the proposed CBO5 algorithm is 41 666.24
$/h, and this value is the best result compared with that
obtained using the BO algorithm. At the same time, the
emission level, real power loss, voltage deviation, and the
voltage stability index became 1.352409 ton/h,
14.85959 MW, 1.697282, and 0.278571 p.u., respectively.
From Table 10, it can be noted that the proposed CBO5
for solving the OPF framework gives the best results for
minimizing fuel costs compared with the BO algorithm.
Case 8: In this case, the real power loss minimization
is considered the main objective function. Utilizing the
OPF framework, the real power loss is minimized to be
9.955717 MW while the fuel costs, emission, voltage devi-
ation, and voltage stability index became 44 567.04 $/h,
1.105042 ton/h, 1.679816, and 0.279129 p.u., respectively.
From Table 10, it is observed that the best result for
TABLE 8 Statistical analysis for cases 4 to 6
Case no. Method Best Mean Median Worst SD
Case 4 BO 781.7197605 782.3550213 782.3092122 783.2470357 3.47E01
CBO5 781.3524007 782.2580216 782.2807345 782.732256 3.65E01
GOA
77
785.7109 804.0168 —823.4731 9.52E+00
BWOA
77
784.8148 788.2471 —795.4683 5.83E+00
GWO
77
781.6645 783.0412 —783.3359 2.75E01
ALO
77
781.6562 784.3253 —791.9234 2.49E+00
PSO
77
781.9047 784.9048 —794.4221 2.52E+00
GSA
77
782.2237 785.8603 —794.8996 2.43E+00
MFO
77
781.6928 782.492 —783.9305 4.77E01
BMO
77
781.6519 781.8187 —783.5284 3.44E01
Case 5 BO 1.833116197 1.87402079 1.859883433 1.923364852 3.13E02
CBO5 1.832163805 1.856660468 1.85210747 1.897404142 2.40E02
GOA
77
2.2043 3.619731 —11.92648 2.25E+00
BWOA
77
2.1403 2.364899 —2.734243 1.23E01
GWO
77
2.0616 2.168241 —2.680391 3.66E01
ALO
77
2.0845 2.323007 —2.762308 1.74E01
PSO
77
2.1572 2.253856 —3.053334 2.50E01
GSA
77
2.5464 3.258794 —5.410662 5.11E01
MFO
77
2.0694 2.108907 —2.19696 3.34E02
BMO
77
2.0646 2.095184 —2.142221 2.54E02
Case 6 BO 808.4946721 809.1820118 809.0531943 810.5598586 5.23E01
CBO5 808.0220664 808.8406762 808.8640024 809.2685809 3.09E01
GOA
77
822.3074 839.1499 —857.8005 9.32E+00
BWOA
77
821.4095 824.358 —830.89 2.66E+00
GWO
77
811.2516 811.4707 —811.6266 1.26E01
ALO
77
811.4334 811.7083 —814.2972 5.36E01
PSO
77
811.5916 812.7758 —818.8017 1.43E+00
GSA
77
811.5264 811.7708 —813.3572 4.97E01
MFO
77
811.4229 811.7398 —812.4613 3.42E01
BMO
77
810.7982 810.7739 —811.1199 1.45E01
Abbreviations: ALO, ant lion optimizer; BMO, barnacles mating optimizer; BO, Bonobo optimization; BWOA, black widow optimization algorithm; CBO,
chaotic Bonobo optimizer; GOA, grasshopper optimization algorithm; GWO, Grey Wolf Optimization; GSA, gravitational search algorithm; MFO, moth-flame
optimization; PSO, Particle Swarm Optimization.
HASSAN ET AL.23
minimizing real power loss of the OPF framework is
achieved by CBO5 when compared with the BO
algorithm.
Case 9: In this case, the objective function aims to
minimize fuel costs by considering imposing a carbon
tax. Using the proposed CBO5 to solve the OPF frame-
work, the total cost is minimized to be 41 690.94 $/h,
where the fuel costs, emission, CT, real power loss, volt-
age deviation, and voltage stability index became
41 666.86 $/h, 1.349873 ton/h, 17.83 $/h, 14.87548 MW,
1.648446, and 0.279868 p.u., respectively. From
Table10,itcanbeconfirmedthattheproposedCBO5
obtains the best results compared with the BO
algorithm.
Also, to estimate the robustness of the CBO5, a statis-
tical analysis is conducted. However, the best, the mean,
the median, the worst, and SD are computed for these
cases and shown in Table 11. One can observe that the
TABLE 9 Comparison of CBO5 algorithm with the original BO algorithm and previous studies for single objective cases 4 to 6 of IEEE
30-bus system
Case # Algorithm Fuel cost ($/h) Emission (t/h) Ploss (MW) VD (p.u.)
Case 4 CBO5 781.3524 1.76216 5.780656 0.454874
BO 781.7198 1.762253 5.782634 0.454963
JS
78
781.6387 1.761998 5.773893 0.448284
CGO
78
782.1950203 1.761969 5.685199 0.453825
FPA
78
782.8596 1.762312 5.863689 0.455137
GPC
78
782.4229 1.764097 5.843226 0.537061
GA
37
787.84 2.76 6.43 0.87
PSO
37
785.82 2.36 6.79 1.08
CSA
37
784.77 1.96 6.47 0.85
ABC
37
783.81 1.75 6.06 0.56
GWO
37
781.40 1.75 5.44 1.05
MODE-OPF
79
782.3592764 ———
SHADE-SF
6
782.503 1.762 5.77 0.463
Case 5 CBO5 957.8938 0.099827 1.832163805 0.538215
BO 958.5272 0.099828 1.833116197 0.531491
Case 6 CBO5 808.0219 0.957464 5.342363 0.486436
BO 808.4964 0.99819 5.348664 0.458006
JS
78
810.12101 0.89377 5.276 0.46884
CGO
78
811.4568 0.902467 5.377988 0.499635
FPA
78
811.6664 0.923031 5.307636 0.465664
GPC
78
810.324 0.916127 5.327113 0.507454
GA
37
814.72 1.36 5.63 0.64
PSO
37
811.49 0.98 5.46 0.48
CSA
37
811.53 0.92 5.44 0.49
ABC
37
811.26 0.89 5.31 0.47
GWO
37
809.93 0.86 4.99 1.07
SHADE-SF
6
810.346 0.891 5.276 0.469
MFO
80
809.969 0.898 5.010 1.067
HPSO-GWO
81
809.277 0.914 5.132 0.462
Abbreviations: ABC, artificial bee colony; BO, Bonobo optimization; CBO, chaotic Bonobo optimizer; CGO, Chaos Game Optimization; CSA, crow search
algorithms; FPA, Flower pollination algorithm; GA, genetic algorithms; GPC, Giza pyramids construction; GWO, Grey Wolf Optimization; HPSO-GWO,
hybridization of PSO with GWO; JS, jellyfish search optimizer; MFO, moth-flame optimization; MODE, multi-operator differential evolution algorithm; PSO,
Particle Swarm Optimization; SHADE-SF, success history-based adaptive differential evolution-feasible solutions.
24 HASSAN ET AL.
TABLE 10 Results of the proposed CBO5 method and BO method for cases 7 to 9
Min Max
Case 7 Case 8 Case 9
BO CBO5 BO CBO5 BO CBO5
Generator active power
P
G1
(MW) 0 576 142.7743 142.7915 174.9728 178.6854 143.0531 142.7197
P
G2
(MW) 30 100 90.19244 90.38392 30 30 92.20437 90.6409
P
G3
(MW) 40 140 44.9847 45.25163 140 134.846 45.10395 45.04671
P
G6
(MW) 30 100 70.45563 70.39143 99.99762 99.99974 72.94854 71.59962
P
G8
(MW) 100 550 460.1464 460.3926 305.9038 307.2246 457.5174 459.7178
P
G9
(MW) 30 100 96.63655 96.74015 99.96983 100 96.90143 96.39607
P
G12
(MW) 100 410 360.4599 359.7084 409.9307 410 357.9451 359.5547
Generator voltage
V
1
(p.u.) 0.95 1.1 1.068701 1.066665 1.066729 1.065168 1.064178 1.062095
V
2
(p.u.) 0.95 1.1 1.066424 1.064343 1.062514 1.061365 1.062116 1.059981
V
3
(p.u.) 0.95 1.1 1.058204 1.056268 1.063823 1.06388 1.054645 1.052765
V
6
(p.u.) 0.95 1.1 1.061668 1.059556 1.061691 1.061822 1.059528 1.061764
V
8
(p.u.) 0.95 1.1 1.074487 1.07622 1.068629 1.067475 1.075274 1.074041
V
9
(p.u.) 0.95 1.1 1.049872 1.050607 1.047356 1.047262 1.048379 1.047081
V
12
(p.u.) 0.95 1.1 1.05086 1.051465 1.050802 1.052756 1.047157 1.045979
Capacitor bank
Q
C18
(MVAR) 0 20 10.198 5.084277 8.755611 7.764916 12.21105 8.16114
Q
C25
(MVAR) 0 20 13.84537 12.81139 15.60203 13.53967 13.12904 13.81369
Q
C53
(MVAR) 0 20 11.70113 11.01722 12.75123 13.15801 12.64346 11.9624
Transformer tap ratio
T
19
(4–18) 0.9 1.1 1.069633 1.034583 1.0928 1.05031 0.900012 0.937933
T
20
(4–18) 0.9 1.1 0.999451 0.939249 0.930333 1.031141 1.099982 1.025207
T
31
(21-20) 0.9 1.1 1.045801 1.011281 1.001453 1.071854 1.00902 1.008322
T
35
(24-25) 0.9 1.1 1.00294 0.974787 1.1 0.908503 1.043702 1.1
T
36
(24–25) 0.9 1.1 1.031886 1.046757 0.946079 1.1 0.980871 0.932915
T
37
(24-26) 0.9 1.1 1.03434 1.032326 1.00506 0.98854 1.036381 1.029856
T
41
(7-29) 0.9 1.1 0.995134 0.994727 0.996236 0.996313 0.998887 0.995242
T
46
(34-32) 0.9 1.1 0.96096 0.961015 0.950025 0.951205 0.960265 0.969007
T
54
(11-41) 0.9 1.1 0.9162 0.913632 1.037943 0.90606 0.9 0.907877
(Continues)
HASSAN ET AL.25
TABLE 10 (Continued)
Min Max
Case 7 Case 8 Case 9
BO CBO5 BO CBO5 BO CBO5
T
58
(15-45) 0.9 1.1 0.981886 0.981072 0.982663 0.982779 0.978985 0.976813
T
59
(14-46) 0.9 1.1 0.968734 0.966545 0.973583 0.970881 0.963378 0.962864
T
65
(10-51) 0.9 1.1 0.97479 0.976017 0.977511 0.977373 0.972604 0.971138
T
66
(13-49) 0.9 1.1 0.940854 0.93782 0.943897 0.940648 0.936124 0.935383
T
71
(11-43) 0.9 1.1 0.975222 0.974105 0.95584 0.97761 0.978299 0.972614
T
73
(40-56) 0.9 1.1 0.97591 0.991325 1.047203 0.989166 0.997103 0.984115
T
76
(39-57) 0.9 1.1 0.970954 0.964976 1.007276 0.956589 0.970735 0.971606
T
80
(9-55) 0.9 1.1 1.002546 1.001638 0.98587 0.987936 1.001733 0.996861
Objective function
Fuel cost ($/h) 41 667.03 41 666.24 44 799.15 44 567.04 41 667.54 41 666.86
Emission (ton/h) 1.352963 1.352409 1.099789 1.105042 1.340179 1.349873
Total cost ($/h) ————41 691.45 41 690.94
Carbon tax ($/h) ————17.83 17.83
Power loss (MW) 14.84992 14.85959 9.974691 9.955717 14.87384 14.87548
Voltage deviation (p.u.) 1.636026 1.697282 1.72999 1.679816 1.639563 1.648446
L-index (max) 0.279554 0.278571 0.28068 0.279129 0.279054 0.279868
Generator reactive power
Q
G1
(MVAR) 140 200 48.68674 46.05551 41.43618 35.07896 46.55154 45.66637
Q
G2
(MVAR) 17 50 50.62978 50.27508 49.97858 49.99539 50.02996 50.02675
Q
G3
(MVAR) 10 60 33.49309 36.50477 29.98118 33.99622 34.20023 31.53403
Q
G6
(MVAR) 8254.87987 7.82605 6.99093 5.90412 8.00321 0.617702
Q
G8
(MVAR) 140 200 47.15352 54.26851 50.90107 46.27143 55.32088 51.13646
Q
G9
(MVAR) 3 9 8.999538 8.997678 8.952382 8.996319 8.984235 9.001633
Q
G12
(MVAR) 150 155 52.92708 56.67265 43.06637 50.5751 52.19551 53.0369
26 HASSAN ET AL.
best, the mean, and the worst values of the considered
objective functions obtained by the CBO5 are very close,
which reveals the capability of the CBO5 to achieve the
best solution or very near to it in each run.
Finally, according to the results in Table 12, the pro-
posed CBO5 algorithm provided a slightly better result
than the original BO algorithm in solving the problem
of case 7. Compared with the other techniques, the dif-
ference in fuel cost was further increased in the IEEE
57-bussystem,whichisalargerpowersystemthanthe
IEEE 30-bus system. The result from the CBO5 algo-
rithm,theoriginalBOalgorithm,MSA,DSA,improved
colliding bodies optimization (ICBO), enhanced collid-
ing bodies optimization (ECBO), differential evolution
(DE),PSO,ABC,biogeographybasedoptimization
(BBO), adaptive real coded biogeography-based optimi-
zation (ARCBBO), modified imperialist competitive
algorithm and teaching-learning algorithm (MICA-
TLA), TLBO, lévy mutation TLBO (LTLBO), and evolv-
ing ant direction differential evolution (EADDE) were
41 666.24, 41 667.03, 41 673.72, 41 686.82, 41 697.33,
41 702.66, 41 689.73, 42 386.37, 41 715.76, 41 841.85,
41 698.93, 41 686, 41 675.05, 41 695.66, 41 679.55, and
41 713.62 $/h, respectively. It is shown that the result of
FIGURE 18 Convergence characteristic of CBO5 and BO for cases 7 to 9. (A) Case 7. (B) Case 8. (C) Case 9
HASSAN ET AL.27
the proposed CBO5 algorithm was the best result for
case 7 compared to those of the other algorithms.
In the same manner, as studied in the modified
IEEE-30 bus system, the following cases are presented
with incorporating WT generators into the OPF frame-
work to minimize the fuel cost with and without car-
bon gas emission and real power loss. Utilizing the
same cost coefficients of WT generators presented in
case A. Further, The PDF parameters of WT generators
have the same values as given in Section 5. The optimal
variables of results for all cases are summarized in
Table 13. Figure 20 compares the convergence charac-
teristics of CBO5 and the BO algorithm. The CBO5
algorithm converges faster for most cases, but the com-
pared BO technique demonstrates stable and fast con-
vergence characteristics. Figure 21 shows that all
voltages values are within specified limits for all con-
sidered cases.
FIGURE 19 Voltage profiles of compared methods for
cases 7 to 9
TABLE 11 The statistical analysis for cases 7 to 9
Case no. Method Best Mean Median Worst SD
Case 7 BO 41667.03409 41675.22151 41671.70027 41693.16048 8.527563174
CBO5 41666.23663 41668.57336 41667.13323 41684.00307 4.187697142
Case 8 BO 9.974691103 10.21021949 10.13157982 10.72890627 0.222606525
CBO5 9.95571668 10.1684184 10.16496739 10.55189755 0.163241655
Case 9 BO 41691.44782 41701.52668 41697.38732 41723.78741 9.331295153
CBO5 41690.94253 41696.52955 41695.39938 41707.59411 4.54549932
TABLE 12 Comparison of CBO5 algorithm with the original BO algorithm and previous studies for single objective cases of IEEE
57-bus system
Case # Algorithm Fuel cost ($/h) Emission (t/h) Ploss (MW) VD (p.u.)
Case 7 CBO5 41666.23663 1.352409 14.85959 1.697282
BO 41667.03409 1.352963 14.84992 1.636026
MSA
18
41673.72 1.9526 15.0526 1.5508
DSA
71
41686.82 ——1.0833
ICBO
68
41697.33 —15.5470 1.3173
ECBO
68
41702.66 ———
DE
68
41689.73 ———
PSO
68
42386.37 ———
ABC
68
41715.76 ———
GA
68
41841.85 ———
BBO
68
41698.93 ———
ARCBBO
73
41686 —15.3769 —
MICA-TLA
83
41675.05 —15.0149 1.6161
TLBO
84
41695.6626 —15.7469 —
LTLBO
84
41679.5451 —15.1589 —
EADDE
85
41713.62 —16.09 1.0977
28 HASSAN ET AL.
TABLE 13 Results of the proposed CBO5 method and BO method for cases 10 to 12
Min Max
Case 10 Case 11 Case 12
BO CBO5 BO CBO5 BO CBO5
Generator active power
P
G1
(MW) 0 576 136.9883 136.7953 180.0127 177.1987 137.2891 136.7083
P
G2
(MW) 30 100 49.76187 50.82713 30 30.00082 52.76404 51.52651
P
G3
(MW) 40 140 42.49689 42.54706 118.4475 126.0842 42.32723 42.45348
P
W1
(MW) 0 150 150 150 126.3598 112.7824 150 150
P
G8
(MW) 100 550 424.192 424.0137 275.5855 284.309 419.5832 423.5313
P
W2
(MW) 0 120 119.9995 120 120 120 120 120
P
G12
(MW) 100 410 343.5703 342.8145 410 409.9983 344.8822 342.7789
Generator voltage
V
1
(p.u.) 0.95 1.1 1.051309 1.0561 1.061479 1.066815 1.038682 1.051575
V
2
(p.u.) 0.95 1.1 1.047748 1.052773 1.05661 1.063062 1.035356 1.048515
V
3
(p.u.) 0.95 1.1 1.046776 1.050719 1.055962 1.065742 1.034619 1.048564
V
6
(p.u.) 0.95 1.1 1.06492 1.065558 1.058594 1.06387 1.061643 1.067197
V
8
(p.u.) 0.95 1.1 1.071408 1.071301 1.063574 1.070212 1.073628 1.07255
V
9
(p.u.) 0.95 1.1 1.043148 1.044862 1.044477 1.052424 1.039691 1.044847
V
12
(p.u.) 0.95 1.1 1.037199 1.042317 1.047987 1.058003 1.028484 1.041361
Capacitor bank
Q
C18
(MVAR) 0 20 6.49E07 13.81544 13.90482 0.001871 20 20
Q
C25
(MVAR) 0 20 13.93044 14.53243 10.51095 15.25083 10.73774 11.46375
Q
C53
(MVAR) 0 20 13.69509 10.92787 12.98852 10.89218 13.19914 13.27246
Transformer tap ratio
T
19
(4-18) 0.9 1.1 0.911746 1.091384 1.018504 0.98321 0.983096 0.96747
T
20
(4-18) 0.9 1.1 1.003614 0.980493 0.990904 0.974761 1.069372 1.055328
T
31
(21-20) 0.9 1.1 1.012165 1.039816 1.009528 1.023598 1.031582 1.00373
T
35
(24-25) 0.9 1.1 0.945103 0.95227 0.989373 1.075884 0.962985 0.900001
T
36
(24-25) 0.9 1.1 1.0998 1.1 0.977979 0.962627 1.026048 1.1
T
37
(24-26) 0.9 1.1 1.037795 1.034689 1.016129 1.009844 1.035478 1.032377
T
41
(7-29) 0.9 1.1 0.995671 0.99536 0.988057 0.996203 0.995254 1.002594
T
46
(34-32) 0.9 1.1 0.969285 0.967943 0.959248 0.957015 0.962784 0.9643
T
54
(11-41) 0.9 1.1 0.9 0.909532 0.903629 1.032369 0.9 0.90598
(Continues)
HASSAN ET AL.29
TABLE 13 (Continued)
Min Max
Case 10 Case 11 Case 12
BO CBO5 BO CBO5 BO CBO5
T
58
(15-45) 0.9 1.1 0.968821 0.972759 0.978494 0.985957 0.958744 0.968416
T
59
(14-46) 0.9 1.1 0.956681 0.960838 0.963655 0.971972 0.946251 0.962975
T
65
(10-51) 0.9 1.1 0.96382 0.967708 0.971413 0.983411 0.958182 0.96633
T
66
(13-49) 0.9 1.1 0.932834 0.933202 0.932271 0.948075 0.921064 0.933461
T
71
(11-43) 0.9 1.1 0.96785 0.96739 0.973522 0.963961 0.957102 0.966489
T
73
(40-56) 0.9 1.1 0.978994 0.989572 0.99139 1.01385 0.992495 0.989265
T
76
(39-57) 0.9 1.1 0.986788 0.96151 0.962282 0.995067 0.997444 0.96186
T
80
(9-55) 0.9 1.1 1.003125 0.997629 0.996618 0.997187 0.993963 1.001047
Objective function
Fuel cost ($/h) 31 603.21 31 602.45 34 848.87 35 561.46 31 608.56 31 603.21
Fuel thermal unit cost ($/h) 30 673.95 30 673.19 34 021.6 34 789.9 30 679.3 30 673.95
Wind generation cost ($/h) 929.2571 929.2592 827.2723 771.561 929.2592 929.2592
Emission (ton/h) 1.176602 1.174147 0.96663 0.989715 1.161806 1.172094
Total cost ($/h) ————31 629.28 31 625.54
Carbon tax ($/h) ————17.83 17.83
Power loss (MW) 16.20888 16.19773 9.605545 9.573578 16.04583 16.19843
Voltage deviation (p.u.) 1.540082 1.545702 1.65516 1.685101 1.417123 1.509428
L-index (max) 0.280615 0.280737 0.278044 0.280487 0.27986 0.280526
Generator reactive power
Q
G1
(MVAR) 140 200 44.76024 44.55739 41.22398 30.32486 137.2891 39.07301
Q
G2
(MVAR) 17 50 49.92548 51.64599 49.9736 49.86503 39.48904 49.99684
Q
G3
(MVAR) 10 60 44.47831 35.18091 20.12369 39.47751 50.00524 30.12627
Q
G6
(MVAR) 8250.98574 7.2014 7.53608 5.88456 19.49379 6.23201
Q
G8
(MVAR) 140 200 51.56645 47.1454 51.44385 47.87953 6.61966 48.99044
Q
G9
(MVAR) 3 9 8.991763 8.999695 9.181663 8.571043 71.1253 9.000454
Q
G12
(MVAR) 150 155 53.58612 58.66146 47.96259 53.42812 8.972786 62.25965
30 HASSAN ET AL.
Case 10: This case is like case 4 in the IEEE-30 bus
system to reduce total fuel cost by incorporating WT gen-
erators without considering the emission. The results in
Table 13 show that the proposed CBO5 algorithm is more
efficient in finding the optimal global solution when
compared to the BO algorithm. This table shows that the
fuel cost is 31 602.45 $/h, while the emission, real power
loss, voltage deviation, and the voltage stability index are
1.174147 ton/h, 16.19773 MW, 1.545702, and 0.280737
p.u., respectively.
Case 11: This case considers real power loss minimi-
zation as the main objective function. Using the proposed
CBO5 for solving the OPF framework with WT genera-
tors, the real power loss is reduced to 9.573578 MW,
which is more than case 8 (without RES) by 3.8383%.
Furthermore, the fuel cost, emissions, voltage deviation
and voltage stability index become 35 561.46 $/h,
0.989715 ton/h, 1.685101, and 0.280487 p.u., respectively.
FIGURE 20 Convergence characteristic of CBO5 and BO for cases 10 to 12. (A), Case 10. (B), Case 11. (C), Case 12
FIGURE 21 Voltage profiles of compared methods for cases 10 to 12
HASSAN ET AL.31
Case 12: In this case, the aim of the objective func-
tions in case 9 but to incorporate WT generators. How-
ever, the total cost is decreased to be 31 625.54173 $/h,
while the fuel costs, emission, carbon tax, real power loss,
voltage deviation and voltage stability index became
31 603.21 $/h, 1.172094 ton/h, 17.83 $/h, 16.19843 MW,
1.509428, and 0.280526 p.u., respectively. With higher
penetration of RES, the value of fuel cost and emission in
CBO5 are reduced by 24.1526% and 9.7565% compared to
case 9 (without RES).
The statistical results are conducted and tabulated for
cases 10 to 12, as seen in Table 14. Again, the CBO5 is
more robust than the standard BO algorithm from the
table.
7|CONCLUSIONS
The proposed CBO5 algorithm has been modified to
achieve the best solution for the non-linear OPF problem
in this research. Deterministic OPF solutions for two
standard systems (IEEE 30-bus and 57-bus) with only
thermal generators have been performed. Then, stochas-
tic OPF solution in case of considering time-varying load,
the uncertainty of wind and solar PV units have been
achieved for the modified IEEE 30-bus and 57-bus sys-
tems. Uncertain wind and solar PV are modeled with
Weibull PDF and lognormal PDF. We considered three
different objectives to minimize fuel cost, active power
loss, and total emission costs. In case 1, it was compared
the results of the 10 modifications of the BO algorithm
based on the chaos maps and the original BO algorithm.
The proposed CBO5 technique gave the optimal solution
for this case. Therefore, the proposed CBO5 was used in
the rest of the article. The results demonstrated the supe-
riority of the proposed CBO5 technique for achieving the
optimal solution to minimize the total fuel cost, the
power loss, and the fuel cost with emission in 12 cases
compared with the obtained results of several recent algo-
rithms such as MSA, DSA, ICBO, ECBO, DE, PSO, ABC,
BBO, ARCBBO, MICA-TLA, TLBO, LTLBO, EADDE,
and other recent algorithms for solving the OPF problem.
This comparison with other recently published algo-
rithms is executed to confirm the effectiveness and accu-
racy of the proposed CBO5. Finally, statistical measures
have been conducted which proved the stability and
robustness of the CBO5 technique.
In future work, the authors will be done to enhance
the robustness of the CBO5 technique, especially for
large-scale power systems. Moreover, the proposed algo-
rithm will be employed to solve the OPF considering the
FACTs devices and other RES such as hydro generation.
ACKNOWLEDGEMENT
The authors would like to acknowledge the financial sup-
port received from Taif University Researchers
Supporting Project Number (TURSP-2020/61), Taif Uni-
versity, Taif, Saudi Arabia.
ORCID
Salah K. Elsayed https://orcid.org/0000-0003-2463-8175
Salah Kamel https://orcid.org/0000-0001-9505-5386
REFERENCES
1. Ullah Z, Wang S, Radosavljevic J, Lai J. A solution to the opti-
mal power flow problem considering WT and PV generation.
IEEE Access. 2019;7:46763-46772. doi:10.1109/ACCESS.2019.
2909561
2. IRENA. Renewable Energy Statistics 2018. Abu Dhabi: The
International Renewable Energy Agency; 2018.
3. Lin J, Li VOK, Leung K-C, Lam AYS. Optimal power flow with
power flow routers. IEEE Trans Power Syst. 2017;32(1):531-543.
doi:10.1109/TPWRS.2016.2542678
4. Pourbabak H, Alsafasfeh Q, Su W. Fully distributed AC opti-
mal power flow. IEEE Access. 2019;7:97594-97603. doi:10.1109/
ACCESS.2019.2930240
5. Taher MA, Kamel S, Jurado F, Ebeed M. An improved moth-
flame optimization algorithm for solving optimal power flow
problem. Int Trans Electr Energy Syst. 2019;29(3):e2743. doi:10.
1002/etep.2743
6. Biswas PP, Suganthan PN, Amaratunga GAJ. Optimal power
flow solutions incorporating stochastic wind and solar power.
Energy Conver Manage. 2017;148:1194-1207. doi:10.1016/j.
enconman.2017.06.071
TABLE 14 The Statistical analysis for cases 10 to 12
Case no. Method Best Mean Median Worst SD
Case 10 BO 31603.21195 31616.46108 31608.68458 31642.28651 13.85787214
CBO5 31602.44771 31612.03763 31608.44284 31632.41326 9.11786124
Case 11 BO 9.605545418 9.86487765 9.794855414 11.03740007 0.313604968
CBO5 9.573578081 9.761324494 9.702683982 10.07715062 0.176327123
Case 12 BO 31629.2796 31651.46155 31637.08036 31727.50628 29.22940237
CBO5 31625.54173 31642.24821 31636.50259 31686.23477 17.32578888
32 HASSAN ET AL.
7. Pagnetti A, Ezzaki M, Anqouda I. Impact of wind power pro-
duction in a European optimal power flow. Electr Power Syst
Res. 2017;152:284-294. doi:10.1016/j.epsr.2017.07.018
8. Glavitsch H, Spoerry M. Quadratic loss formula for reactive dis-
patch. IEEE Trans Power Appar Syst. 1983;PAS-102(12):3850-
3858. doi:10.1109/TPAS.1983.317899
9. Burchett RC, Happ HH, Wirgau KA. Large scale optimal power
flow. IEEE Trans Power Appar Syst. 1982;PAS-101(10):3722-
3732. doi:10.1109/TPAS.1982.317057
10. Lin W-M, Huang C-H, Zhan T-S. A hybrid current-power opti-
mal power flow technique. IEEE Trans Power Syst. 2008;23(1):
177-185. doi:10.1109/TPWRS.2007.913301
11. Hassanien AE, Rizk-Allah RM, Elhoseny M. A hybrid crow
search algorithm based on rough searching scheme for solving
engineering optimization problems. J Ambient Intell Humaniz
Comput. 2018. doi:10.1007/s12652-018-0924-y
12. Bouchekara HREH, Abido MA, Boucherma M. Optimal power
flow using teaching-learning-based optimization technique.
Electr Power Syst Res. 2014;114:49-59. doi:10.1016/j.epsr.2014.
03.032
13. Elattar EE, ElSayed SK. Modified JAYA algorithm for optimal
power flow incorporating renewable energy sources considering
the cost, emission, power loss and voltage profile improvement.
Energy. 2019;178:598-609. doi:10.1016/j.energy.2019.04.159
14. Premkumar M, Jangir P, Sowmya R, Elavarasan RM,
Kumar BS. Enhanced chaotic JAYA algorithm for parameter
estimation of photovoltaic cell/modules. ISA Trans. 2021;116:
139-166. doi:10.1016/j.isatra.2021.01.045
15. Chen G, Qian J, Zhang Z, Sun Z. Multi-objective optimal power
flow based on hybrid firefly-bat algorithm and constraints-prior
object-fuzzy sorting strategy. IEEE Access. 2019;7:139726-
139745. doi:10.1109/ACCESS.2019.2943480
16. Hassan MH, Kamel S, El-Dabah MA, Khurshaid T,
Dominguez-Garcia JL. Optimal reactive power dispatch with
time-varying demand and renewable energy uncertainty using
Rao-3 algorithm. IEEE Access. 2021;9:23264-23283. doi:10.1109/
ACCESS.2021.3056423
17. Teeparthi K, Vinod Kumar DM. Multi-objective hybrid PSO-
APO algorithm based security constrained optimal power flow
with wind and thermal generators. Eng Sci Technol Int J. 2017;
20(2):411-426. doi:10.1016/j.jestch.2017.03.002
18. Mohamed A-AA, Mohamed YS, El-Gaafary AAM,
Hemeida AM. Optimal power flow using moth swarm algo-
rithm. Electr Power Syst Res. 2017;142:190-206. doi:10.1016/j.
epsr.2016.09.025
19. Bouchekara HREH. Optimal power flow using black-hole-
based optimization approach. Appl Soft Comput. 2014;24:879-
888. doi:10.1016/j.asoc.2014.08.056
20. Bhattacharya A, Roy PK. Solution of multi-objective optimal
power flow using gravitational search algorithm. IET Gener
Transm Distrib. 2012;6(8):751. doi:10.1049/iet-gtd.2011.0593
21. Fang X, Hodge B-M, Jiang H, Zhang Y. Decentralized wind
uncertainty management: alternating direction method of mul-
tipliers based distributionally-robust chance constrained opti-
mal power flow. Appl Energy. 2019;239:938-947. doi:10.1016/j.
apenergy.2019.01.259
22. Reddy SS. Optimal scheduling of thermal-wind-solar power
system with storage. Renew Energy. 2017;101:1357-1368. doi:10.
1016/j.renene.2016.10.022
23. Dubey HM, Pandit M, Panigrahi BK. Hybrid flower pollination
algorithm with time-varying fuzzy selection mechanism for
wind integrated multi-objective dynamic economic dispatch.
Renew Energy. 2015;83:188-202. doi:10.1016/j.renene.2015.04.034
24. Kayal P, Chanda CK. Optimal mix of solar and wind distrib-
uted generations considering performance improvement of
electrical distribution network. Renew Energy. 2015;75:173-186.
doi:10.1016/j.renene.2014.10.003
25. Biswas PP, Suganthan PN, Mallipeddi R, Amaratunga GAJ.
Optimal reactive power dispatch with uncertainties in load
demand and renewable energy sources adopting scenario-based
approach. Appl Soft Comput. 2019;75:616-632. doi:10.1016/j.
asoc.2018.11.042
26. Güçyetmez M, Çam E. A new hybrid algorithm with genetic-
teaching learning optimization (G-TLBO) technique for opti-
mizing of power flow in wind-thermal power systems. Electr
Eng. 2016;98(2):145-157. doi:10.1007/s00202-015-0357-y
27. Reddy SS. Optimal power flow with renewable energy
resources including storage. Electr Eng. 2017;99(2):685-695. doi:
10.1007/s00202-016-0402-5
28. Hazra J, Sinha AK. A multi-objective optimal power flow using
particle swarm optimization. Eur Trans Electr Power. 2011;
21(1):1028-1045. doi:10.1002/etep.494
29. Shilaja C, Arunprasath T. Optimal power flow using moth
swarm algorithm with gravitational search algorithm consider-
ing wind power. Future Gener Comput Syst. 2019;98:708-715.
doi:10.1016/j.future.2018.12.046
30. Attarha A, Amjady N, Conejo AJ. Adaptive robust AC optimal
power flow considering load and wind power uncertainties. Int
J Electr Power Energy Syst. 2018;96:132-142. doi:10.1016/j.
ijepes.2017.09.037
31. Luo J, Shi L, Ni Y. A solution of optimal power flow incorpo-
rating wind generation and power grid uncertainties. IEEE
Access. 2018;6:19681-19690. doi:10.1109/ACCESS.2018.2823982
32. Mishra C, Singh SP, Rokadia J. Optimal power flow in the pres-
ence of wind power using modified cuckoo search. IET Gener
Transm Distrib. 2015;9(7):615-626.
33. Shabanpour-Haghighi A, Seifi AR, Niknam T. A modified
teaching–learning based optimization for multi-objective opti-
mal power flow problem. Energy Conver Manage. 2014;77:597-
607. doi:10.1016/j.enconman.2013.09.028
34. Awad NH, Ali MZ, Mallipeddi R, Suganthan PN. An efficient
differential evolution algorithm for stochastic OPF based
active–reactive power dispatch problem considering renewable
generators. Appl Soft Comput. 2019;76:445-458. doi:10.1016/j.
asoc.2018.12.025
35. Elattar EE. Optimal power flow of a power system incorporating sto-
chasticwindpowerbasedonmodifiedmothswarmalgorithm.IEEE
Access. 2019;7:89581-89593. doi:10.1109/ACCESS.2019.2927193
36. Nusair K, Alasali F. Optimal power flow management system
for a power network with stochastic renewable energy
resources using Golden ratio optimization method. Energies.
2020;13(14):3671. doi:10.3390/en13143671
37. Khan IU, Javaid N, Gamage KAA, Taylor CJ, Baig S, Ma X.
Heuristic algorithm based optimal power flow model incorpo-
rating stochastic renewable energy sources. IEEE Access. 2020;
8:148622-148643. doi:10.1109/ACCESS.2020.3015473
38. Kotur D, Stefanov P. Optimal power flow control in the system
with offshore wind power plants connected to the MTDC
HASSAN ET AL.33
network. Int J Electr Power Energy Syst. 2019;105:142-150. doi:
10.1016/j.ijepes.2018.08.012
39. Kumar S, Chaturvedi DK. Optimal power flow solution using
fuzzy evolutionary and swarm optimization. Int J Electr Power
Energy Syst. 2013;47:416-423. doi:10.1016/j.ijepes.2012.11.019
40. Das AK, Pratihar DK. A new Bonobo optimizer (BO) for real-
parameter optimization. Paper presented at: 2019 IEEE Region
10 Symposium (TENSYMP); 2019: 108–113. doi: 10.1109/
TENSYMP46218.2019.8971108
41. Kaur G, Arora S. Chaotic whale optimization algorithm.
J Comput Design Eng. 2018;5(3):275-284. doi:10.1016/j.jcde.
2017.12.006
42. Jia D, Zheng G, Khurram Khan M. An effective memetic differ-
ential evolution algorithm based on chaotic local search. Inf
Sci. 2011;181(15):3175-3187. doi:10.1016/j.ins.2011.03.018
43. Talatahari S, Farahmand Azar B, Sheikholeslami R,
Gandomi AH. Imperialist competitive algorithm combined with
chaos for global optimization. Commun Nonlinear Sci Numer
Simul. 2012;17(3):1312-1319. doi:10.1016/j.cnsns.2011.08.021
44. Kohli M, Arora S. Chaotic grey wolf optimization algorithm for
constrained optimization problems. J Comput Design Eng.
2018;5(4):458-472. doi:10.1016/j.jcde.2017.02.005
45. Mirjalili S, Gandomi AH. Chaotic gravitational constants for
the gravitational search algorithm. Appl Soft Comput. 2017;53:
407-419. doi:10.1016/j.asoc.2017.01.008
46. Wang G-G, Deb S, Gandomi AH, Zhang Z, Alavi AH. Chaotic
cuckoo search. Soft Comput. 2016;20(9):3349-3362. doi:10.1007/
s00500-015-1726-1
47. ZhangQ,ChenH,HeidariAA,etal.Chaos-inducedandmutation-
driven schemes boosting Salp chains-inspired optimizers. IEEE
Access. 2019;7:31243-31261. doi:10.1109/ACCESS.2019.2902306
48. Ateya AA, Muthanna A, Vybornova A, et al. Chaotic salp
swarm algorithm for SDN multi-controller networks. Eng Sci
Technol Int J. 2019;22(4):1001-1012. doi:10.1016/j.jestch.2018.
12.015
49. Mugemanyi S, Qu Z, Rugema FX, Dong Y, Bananeza C,
Wang L. Optimal reactive power dispatch using chaotic bat
algorithm. IEEE Access. 2020;8:65830-65867. doi:10.1109/
ACCESS.2020.2982988
50. Yao F, Dong ZY, Meng K, Xu Z, Iu HH-C, Wong KP. Quan-
tum-inspired particle swarm optimization for power system
operations considering wind power uncertainty and carbon tax
in Australia. IEEE Trans Ind Inform. 2012;8(4):880-888. doi:10.
1109/TII.2012.2210431
51. Adhvaryyu PK, Chattopadhyay PK, Bhattacharya A. Dynamic
optimal power flow of combined heat and power system with
valve-point effect using Krill Herd algorithm. Energy. 2017;127:
756-767. doi:10.1016/j.energy.2017.03.046
52. Qazi A, Hussain F, Rahim NABD, et al. Towards sustainable
energy: a systematic review of renewable energy sources, tech-
nologies, and public opinions. IEEE Access. 2019;7:63837-
63851. doi:10.1109/ACCESS.2019.2906402
53. Panda A, Tripathy M. Security constrained optimal power flow
solution of wind-thermal generation system using modified
bacteria foraging algorithm. Energy. 2015;93:816-827. doi:10.
1016/j.energy.2015.09.083
54. Shi L, Wang C, Yao L, Ni Y, Bazargan M. Optimal power flow
solution incorporating wind power. IEEE Syst J. 2012;6(2):233-
241. doi:10.1109/JSYST.2011.2162896
55. Chaib AE, Bouchekara HREH, Mehasni R, Abido MA. Optimal
power flow with emission and non-smooth cost functions using
backtracking search optimization algorithm. Int J Electr Power
Energy Syst. 2016;81:64-77. doi:10.1016/j.ijepes.2016.02.004
56. Jahid A, Islam MS, Hossain MS, Hossain ME, Monju MKH,
Hossain MF. Toward energy efficiency aware renewable energy
management in green cellular networks with joint coordina-
tion. IEEE Access. 2019;7:75782-75797. doi:10.1109/ACCESS.
2019.2920924
57. Roy R, Jadhav HT. Optimal power flow solution of power sys-
tem incorporating stochastic wind power using Gbest guided
artificial bee colony algorithm. Int J Electr Power Energy Syst.
2015;64:562-578. doi:10.1016/j.ijepes.2014.07.010
58. Surender Reddy S, Bijwe PR, Abhyankar AR. Real-time eco-
nomic dispatch considering renewable power generation vari-
ability and uncertainty over scheduling period. IEEE Syst J.
2015;9(4):1440-1451. doi:10.1109/JSYST.2014.2325967
59. Das AK, Nikum AK, Krishnan SV, Pratihar DK. Multi-objective
bonobo optimizer (MOBO): an intelligent heuristic for multi-
criteria optimization. Knowl Inf Syst. 2020;62(11):4407-4444.
doi:10.1007/s10115-020-01503-x
60. Abdelghany RY, Kamel S, Sultan HM, Khorasy A, Elsayed SK,
Ahmed M. Development of an improved bonobo optimizer and
its application for solar cell parameter estimation. Sustainabil-
ity. 2021;13(7):3863. doi:10.3390/su13073863
61. Hassan MH, Kamel S, Salih SQ, Khurshaid T, Ebeed M. Devel-
oping chaotic artificial ecosystem-based optimization algorithm
for combined economic emission dispatch. IEEE Access. 2021;9:
51146-51165. doi:10.1109/ACCESS.2021.3066914
62. Zimmerman RD, Murillo-Sanchez CE, Thomas RJ.
MATPOWER: steady-state operations, planning, and analysis
tools for power systems research and education. IEEE Trans
Power Syst. 2011;26(1):12-19. doi:10.1109/TPWRS.2010.2051168
63. Part WT. Design Requirements; IEC 61400-1. Geneva,
Switzerland: International Electrotechnical Commission; 2005.
64. Morales JM, Conejo AJ, Liu K, Zhong J. Pricing electricity in
pools with wind producers. IEEE Trans Power Syst. 2012;27(3):
1366-1376. doi:10.1109/TPWRS.2011.2182622
65. The UK Renewable Energy. Wind speed distribution Weibull.
http://www.reuk.co.uk/wordpress/wind/wind-speed-
distribution-weibull/. Accessed July 18, 2020.
66. Black V. Cost and performance data for power generation tech-
nologies. National Renewable Energy Lab.(NREL), Golden,
CO, USA, Technical Report. 2012.
67. Biswas PP, Suganthan PN, Mallipeddi R, Amaratunga GAJ.
Optimal power flow solutions using differential evolution algo-
rithm integrated with effective constraint handling techniques.
Eng Appl Artif Intel. 2018;68:81-100. doi:10.1016/j.engappai.
2017.10.019
68. Bouchekara HREH, Chaib AE, Abido MA, El-Sehiemy RA.
Optimal power flow using an improved colliding bodies optimi-
zation algorithm. Appl Soft Comput. 2016;42:119-131. doi:10.
1016/j.asoc.2016.01.041
69. Bouchekara HREH, Chaib AE, Abido MA. Multiobjective
optimal power flow using a fuzzy based grenade explosion
method. Energy Syst. 2016;7(4):699-721. doi:10.1007/s12667-
016-0206-8
70. Mahdad B, Srairi K. Security constrained optimal power flow
solution using new adaptive partitioning flower pollination
34 HASSAN ET AL.
algorithm. Appl Soft Comput. 2016;46:501-522. doi:10.1016/j.
asoc.2016.05.027
71. Abaci K, Yamacli V. Differential search algorithm for solving
multi-objective optimal power flow problem. Int J Electr
Power Energy Syst. 2016;79:1-10. doi:10.1016/j.ijepes.2015.
12.021
72. Roberge V, Tarbouchi M, Okou F. Optimal power flow based
on parallel metaheuristics for graphics processing units. Electr
Power Syst Res. 2016;140:344-353. doi:10.1016/j.epsr.2016.
06.006
73. Ramesh Kumar A, Premalatha L. Optimal power flow for a
deregulated power system using adaptive real coded
biogeography-based optimization. Int J Electr Power Energy
Syst. 2015;73:393-399. doi:10.1016/j.ijepes.2015.05.011
74. El-Hana Bouchekara HR, Abido MA, Chaib AE. Optimal
power flow using an improved electromagnetism-like mecha-
nism method. Electr Power Compon Syst. 2016;44(4):434-449.
doi:10.1080/15325008.2015.1115919
75. Rezaei Adaryani M, Karami A. Artificial bee colony algorithm
for solving multi-objective optimal power flow problem. Int J
Electr Power Energy Syst. 2013;53:219-230. doi:10.1016/j.ijepes.
2013.04.021
76. Kumari MS, Maheswarapu S. Enhanced genetic algorithm
based computation technique for multi-objective optimal
power flow solution. Int J Electr Power Energy Syst. 2010;32(6):
736-742. doi:10.1016/j.ijepes.2010.01.010
77. Sulaiman MH, Mustaffa Z. Optimal power flow incorporating
stochastic wind and solar generation by metaheuristic opti-
mizers. Microsyst Technol. 2021;27(9):3263-3277. doi:10.1007/
s00542-020-05046-7
78. Farhat M, Kamel S, Atallah AM, Khan B. Optimal power flow
solution based on jellyfish search optimization considering
uncertainty of renewable energy sources. IEEE Access. 2021;9:
100911-100933. doi:10.1109/ACCESS.2021.3097006
79. Hossain MA, Sallam KM, Elsayed SS, Chakrabortty RK,
Ryan MJ. Optimal power flow considering intermittent
solar and wind generation using multi-operator differential
evolution algorithm. Energy Fuel Technol. 2021. doi:10.
20944/preprints202103.0228.v1
80. Pandya S, Jariwala HR. Single- and multiobjective optimal
power flow with stochastic wind and solar power plants using
moth flame optimization algorithm. Smart Sci. 2021;1-41. doi:
10.1080/23080477.2021.1964692
81. Riaz M, Hanif A, Hussain SJ, Memon MI, Ali MU, Zafar A. An
optimization-based strategy for solving optimal power flow
problems in a power system integrated with stochastic solar
and wind power energy. Appl Sci. 2021;11(15):6883. doi:10.
3390/app11156883
82. Taher MA, Kamel S, Jurado F, Ebeed M. Modified grasshopper
optimization framework for optimal power flow solution. Electr
Eng. 2019;101(1):121-148. doi:10.1007/s00202-019-00762-4
83. Ghasemi M, Ghavidel S, Rahmani S, Roosta A, Falah H. A
novel hybrid algorithm of imperialist competitive algorithm
and teaching learning algorithm for optimal power flow prob-
lem with non-smooth cost functions. Eng Appl Artif Intel. 2014;
29:54-69. doi:10.1016/j.engappai.2013.11.003
84. Ghasemi M, Ghavidel S, Gitizadeh M, Akbari E. An improved
teaching–learning-based optimization algorithm using Lévy
mutation strategy for non-smooth optimal power flow. Int J
Electr Power Energy Syst. 2015;65:375-384. doi:10.1016/j.ijepes.
2014.10.027
85. Vaisakh K, Srinivas LR. Evolving ant direction differential evo-
lution for OPF with non-smooth cost functions. Eng Appl Artif
Intel. 2011;24(3):426-436. doi:10.1016/j.engappai.2010.10.019
How to cite this article: Hassan MH, Elsayed SK,
Kamel S, Rahmann C, Taha IBM. Developing
chaotic Bonobo optimizer for optimal power flow
analysis considering stochastic renewable energy
resources. Int J Energy Res. 2022;1‐35. doi:10.1002/
er.7928
HASSAN ET AL.35
