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Optimal Power Flow Solution with Uncertain RES using Augmented Grey Wolf Optimzation

Authors:

Abstract

This work focuses on implementing the optimal power flow (OPF) problem, considering wind, solar and hy-dropower generation in the system. The stochastic nature of renewable energy sources (RES) is modelled using Weibull, Log-normal and Gumbel probability density functions. The system-wide economic aspect is examined with additional cost functions such as penalty and reserve costs for under and overestimating the imbalance of RES power outputs. Also, a carbon tax is imposed on carbon emissions as a separate objective function to enhance the contribution of green energy. For solving the optimization problem, a simple and efficient augmentation to the basic grey wolf optimization (GWO) algorithm is proposed, in order to enhance the algorithm's exploration capabilities. The performance of the new augmented GWO (AGWO) approach, in terms of robustness and scalability, is confirmed on IEEE-30, 57 and 118 bus systems. The obtained results of the AGWO algorithm are compared with modern heuristic techniques for a case of OPF incorporating RES. Numerical simulations indicate that the proposed method has better exploration and exploitation capabilities to reduce operational costs and carbon emissions.
Optimal Power Flow Solution with Uncertain RES
using Augmented Grey Wolf Optimization
Inam Ullah Khan1,, Nadeem Javaid2, C. James Taylor1, Kelum A. A Gamage3and Xiandong MA1
1Engineering Department, Lancaster University, Bailrigg, Lancaster LA1 4YW, UK
2Department of Computer Science, COMSATS University Islamabad, Islamabad, 44000, Pakistan
3James Watt School of Engineering, University of Glasgow, Glasgow G12 8QQ, UK
*Correspondence: i.u.khan@lancaster.ac.uk
Abstract—This work focuses on implementing the optimal
power flow (OPF) problem, considering wind, solar and hy-
dropower generation in the system. The stochastic nature of
renewable energy sources (RES) is modelled using Weibull, Log-
normal and Gumbel probability density functions. The system-
wide economic aspect is examined with additional cost functions
such as penalty and reserve costs for under and overestimating
the imbalance of RES power outputs. Also, a carbon tax is
imposed on carbon emissions as a separate objective function
to enhance the contribution of green energy. For solving the
optimization problem, a simple and efficient augmentation to the
basic grey wolf optimization (GWO) algorithm is proposed, in
order to enhance the algorithm’s exploration capabilities. The
performance of the new augmented GWO (AGWO) approach,
in terms of robustness and scalability, is confirmed on IEEE-
30, 57 and 118 bus systems. The obtained results of the AGWO
algorithm are compared with modern heuristic techniques for a
case of OPF incorporating RES. Numerical simulations indicate
that the proposed method has better exploration and exploitation
capabilities to reduce operational costs and carbon emissions.
Index Terms—Optimal power flow, Renewable energy sources,
Carbon emission, Meta-heuristic techniques
I. INTRODUCTION
Optimal power flow (OPF) has proved to be an essential tool
for the efficient and secure operation of power networks since
its inception. The main objective of OPF is to find optimal set-
tings of the control variables with certain objective functions
while satisfying system equality and inequality constraints.
The system control variables that need adjustment include
generated active power, the voltage of all generation buses
and tap settings of the transformer. During the optimization
process, system constraints such as transmission line capacity,
power flow balance, voltage profile of all buses and generator
capability constraints need to be maintained.
OPF with only traditional thermal power generators (TPGs)
is widely studied in the literature [1]. However, with increased
penetration of RES, it is necessary to incorporate associated
uncertainty into the power network. Under recent studies, sys-
tems that consider both TPGs and RES are in pursuit of similar
objective functions studied in the past [2]–[6]. The work in
[2] conducts an extensive study on the over/underestimation
of wind power generation (WPG) in the classical economic
dispatch model. In this study, the Weibull probability density
function (PDF) is used to model the uncertainty of WPG
output. For economic dispatch strategies, it provides valuable
insight into the integrated wind system. However, the chal-
lenge of wind speed variation on the optimal dispatch schedule
of power plants remains unaddressed. Also, the reactive power
capability of WPGs, bus voltage constraints and loading effect
of transmission line were not considered in [3].
Authors in [4] combined advanced variant of differential
evolution with an effective constraint handling technique for
a system that considers both solar and wind power generation
in the OPF problem. The uncertain and intermittent nature of
both RES were modelled with lognormal and Weibull PDFs.
However, the resulting SHADE-SF algorithm sometimes at-
tains premature convergence (i.e., becomes trapped in a local
solution) and the convergence rate can be prolonged. The
scalability and robustness of the proposed algorithm were not
verified since the algorithm was only verified on the IEEE-
30 bus system. This does not guarantee good performance
over medium and higher bus systems (IEEE-57 and IEEE-
118). In general, OPF with the incorporation of RES needs
further attention.
II. MATHEMATICAL MODEL
In this work, the IEEE-30, 57 and 118 bus systems are
used to validate the performance of the proposed AGWO
algorithm in the OPF problem. The essential characteristics
of these bus systems are provided in Table I. Along with
the TPGs, RES such as wind, solar and small hydro (WSH)
generators are selected as power generation sources for the
OPF framework. The power output from RES is variable in
nature and power output instability needs to be minimized and
balanced by the aggregation of the power outputs of all the
generators and spinning reserve. Thus, total power generation
cost is the combination of operating cost of all generators,
reserve and penalty cost (due to the intermittent nature of
power generation from RES). In subsequent subsections, cost
models are discussed in detail.
A. Stochastic Wind Power
The behaviour of the wind speed v(m/s)distribution can
be modelled with the help of Weibull PDF fv(v)by adjusting
scale parameter cand shape parameter kas established by
[3] and [4]. The probability of wind speed during any time
interval is expressed as follows:
fv(v) = k
c(v
c)k1exp h(v
c)ki,0< v < (1)
978-1-7281-6350-5/20/$31.00 ©2020 IEEE 1
In the modified IEEE-30 bus system, TPGs at bus 5 and bus 11
are replaced with the WPGs. The values for scale cand shape
kparameters are given in Table II. The wind speed behavior
for WPG 1 and WPG 2 at buses 5 and 11 follow the Weibull
PDF. For each WPG, the relationship between wind speed and
output power is expressed in Eq. 2 [3]:
PW=
0, v < vci or v > vco
PW r, vr< v vco
PW r(vvci
vrvci ), vci vvr,
(2)
where vis forecasted wind speed in m/s,vci,vco and vrare
cut-in, cut-out and rated wind speeds, PW r is rated output
power for the WPG.
B. Stochastic Solar Power
Similarly, the TPG at bus 13 of the modified IEEE-30 bus
system is replaced with the solar power generator (SPG). The
output power from SPG depends upon the solar irradiance
which follows lognormal PDF. The probability with standard
deviation λand mean σcan be calculated as follows [4]:
fX(X) = 1
Xσ2πexp [ln Xλ]2
2σ2, X > 0(3)
The values for λand σare given in Table II. The relationship
between the solar irradiance X (W/m2)and output power of
SPG is expressed as follows:
PS(X) =
PSr (X2
XstdCI),0< X < CI
PSr (X
Xstd ), X > CI,
(4)
where Xis forecasted solar irradiance, Xstd is standard solar
irradiance value set as 800 W/m2,CIis certain irradiance
point (120W/m2) and PSR is rated SPG power output.
C. Stochastic Hydropower
It is well known that the Gumbel distribution is followed
for river flow rate calculations. The probability calculation of
Gumbel distribution for river flow rate with scale parameter ω
and location parameter γis formulated in Eq. 5 [5]:
fH(Gh) = 1
ωexp (Ghγ)
ω) exp exp (Ghγ)
ω
(5)
In the modified IEEE 30-bus system, the conventional TPG
at bus 13 is replaced together with 45 MW SPG and 5 MW
small HPG. Table II provides PDF values for these fittings and
many of these values are realistically chosen in a study given
by Ref. [5]. The output of HPG as a function of pressure head
and water flow rate is calculated as follows:
Ph(Gh) = αβgGhPh(6)
where αand βrepresent efficiency of the generating unit and
density of water volume, respectively. The numerical values
for calculation of HPG output are assumed: α= 0.85, β1000
kg/m3,Ph= 25m and g= 9.81m/s2.
D. Cost Model for Thermal Power Generators
TPGs require fossil fuel for their operation. The relationship
between generated power (MW) and fuel cost ($/h) can be
calculated with the help of following quadratic equation:
CT=
NT
X
i=1
ai+biPT g,i +ciP2
T g,i (7)
Practically, the valve point loading effect needs to be con-
sidered to model accurate cost function. Hence, the overall
thermal power generation cost ($/h) becomes:
CT=
NT
X
i=1
ai+biPT g,i +ciP2
T g,i+
di×sinei×(Pm
T g,i PT g,i)
(8)
where ai,bi,ciare the cost coefficients and di,eiare fuel
cost coefficients for the i-th TPG. PT g,i is the output power,
NTis total number of the TPGs in the system and Pm
T g,i the
minimum power when i-th TPG is in operation. All emission
and cost coefficients pertaining to TPGs are given in Table III.
E. Cost Model for Renewable Energy Sources
The total cost of the RES consists of the direct cost associ-
ated with scheduled power, reserve cost for overestimation and
penalty cost for underestimation. These models are developed
in line with the concept presented in Refs. [3]–[6].
The direct, reserve and penalty costs of WPG as a function
of scheduled power are represented in Eqs. 9–11 as follows:
CDW,j =dw,j PW S,j (9)
CRW,j =rw,j ZPW S,j
0
(PW S,j W)fw(W)dW (10)
CP W,j =pw,j ZPW R,j
PW S,j
(WPW S,j )fw(W)dW (11)
where dw,j ,rw,j and pw,j are direct, reserve and penalty cost
coefficients pertaining to j-th WPG. PW S,j is the scheduled
power and fw(W)is PDF of same WPG.
The total cost of WPG can be calculates as:
CT W,j =CDW,j +CRW,j +CP W,j (12)
Likewise, the SPG also has uncertain power output. The direct,
reserve and penalty costs pertaining to the k-th SPG are
represented as:
CDS,k =ds,k PSS,k (13)
CRS,k =rs,k ·P r(PAS,k < PSS,k )·
[(PSS,k E(PAS,k < PSS,k )] (14)
CP S,k =ps,k ·P r(PAS,k > PS S,k )·
[(E(PAS,k > PSS,k )PSS,k ](15)
In Eqs. 13–15, ds,k,rs,k and ps,k are direct, reserve and
penalty cost coefficients pertaining to k-th SPG. PAS,k and
TABLE I: Characteristics of Bus Systems under Consideration
Items IEEE-30 Bus System IEEE-57 Bus System IEEE-118 Bus System
Quantity Details Quantity Details Quantity Details
Number of buses 30 [4] 57 [5] 118 [1]
Number of Branches 41 [4] 80 [5] 186 [1]
Number of TPGs 3 Connect at bus 1 (Swing),
2 and 8 5Connect at bus 1 (slack),
3, 8 and 12 54 [1]
Number of WPGs 2 Connect at bus 5 and 11 2 Connect at bus 2 and 6 2 Connect at bus 5 and 11
Number of SPGs 1 Connect at bus 13 1 Connect at bus 9 1 Connect at bus 9
Number of HPG 1 Connect at bus 11 1 Connect at bus 11 1 Connect at bus 11
Connected load 283.4 MW, 126.2 MVAr 1250.8 MW, 336.4.2 MVAr 4242 MW, 1439 MVAr
Control variables 24 —– 33 —– 120 —–
Load Bus voltage
range 24 [0.95-1.06] p.u. 50 [0.94-1.06] p.u. 64 [0.94-1.06] p.u.
TABLE II: PDF Parameters for Wind, Solar and Hydropower Generation [4], [5].
Wind power generation plants Solar + Hydropower generation plant (bus 13)
Windfarm # No. of
wind turbines
Total rated
power
Weibull PDF
parameters
Rated power
of SPG
Lognormal PDF
parameters
Rated power
of HPG
Gumbel PDF
parameters
1 at bus 5 25 75 MW c = 9, k = 2 45 MW λ= 6,σ= 0.65 MW ω= 15, γ= 1.2
2 at bus 11 20 60 MW c = 10, k = 2
TABLE III: Thermal Power Generators Cost and Emission Coefficients for the System [4].
Thermal generator Bus number a b c d e f g h k l
TPG1 1 0 2 0.00375 18 0.037 4.091 -5.554 6.49 0.0002 6.667
TPG2 2 0 1.75 0.0175 16 0.038 2.543 -6.047 5.638 0.0005 3.333
TPG3 8 0 3.25 0.00834 12 0.045 5.326 -3.55 3.38 0.002 2
PSS,k represent available and scheduled power from SPG.
Finally, the total cost of SPG can be calculated as:
CT S,k =CDS,k +CRS,k +CP S,k (16)
As a third RES, we consider a small hydropower generator
(HPG) in this study. The output of HPG is very less (10–20
%of total install capacity) [5]. It is therefore combined with
SPG and assumed to be owned by a single private operator.
Following Eqs 13–15, the direct, reserve cost for overestima-
tion and penalty cost for underestimation of combined solar
hydropower generation system is:
CSH =dsPSSH,s +dhPS SH,h (17)
CRSH =rsh,m ·P r(PAS H < PSS H )·
[(PSSH E(PASH < PSS H )] (18)
CP SH =psh,m ·P r(PAS H > PSS H )·
[(E(PASH > PS SH )PSS H ](19)
where PSSH,s and PSS H,h represent scheduled power from
SPG and HPG, respectively. dh,m,rsh,m and psh,m are direct,
reserve and penalty cost coefficients pertaining to m-th HPG.
PASH and PSSH represent available and scheduled output
power from combined solar hydropower generator. Finally, the
total cost of HPG is calculated as follows:
CT SH =CDS H +CRSH +CP S H (20)
F. Carbon Tax based Emission Model
Unlike RES, producing power from TPGs emits the harmful
gases into the environment. The emission E (ton/h) is calcu-
lated as follows:
FC=
NT
X
i=1
[(ai+biPT i +ciP2
T i)×0.01 + dieliPT i ](21)
The combustion fossil fuels on which TPGs run is the main
source of greenhouse gases (GHGs) emission. To control
GHGs and make clean energy economy, the carbon emission
tax (emission cost) is modelled as follows:
CE=E·Ctax (22)
where CEis the emission cost and Ctax represents the carbon
tax per unit of carbon emission.
III. PROB L EM FORMULATION
The main objective of the OPF problem is formulated by
incorporating all the cost functions described in the above
sections. The first objective F1of the optimization problem is
to achieve a minimum total generation cost. However, emission
cost is not included in its formulation. To analyze the impact of
the carbon tax on generation scheduling, the second objective
F2is modelled by adding the carbon emission cost within the
first objective function.
The objective is as follows: Minimize
F1=
NT
X
i=1
CT G +
NW
X
j=1
CT W +
NS
X
k=1
CT S +
NSH
X
m=1
CT SH (23)
where NW g,NS g and NSH g are the numbers of WSH
generators in the system. The second objective F2of the
optimization is: Minimize
F2=F1+CE(24)
where CEis the emission cost, calculated in Eq. 22.
Both OPF objective functions in Eqs. 23 and 24 are based
on system equality and inequality constraints.
IV. THE GREY WOLF OPTIMIZATION ALGO RI THM
In GWO [7], wolves are categorised into four different
levels: alpha (α), beta (β), delta (δ)and omega (ω)wolves.
The accurate determination of prey location is treated as the
optimization problem (fittest solution) whilst, the position of
the wolves relative to the prey determines the best solution.
The position of the αwolves is said to be the best solution
found so far in the search space, because they are expected
to be closer to the prey than other wolves in the pack. To
allocate their position in the search space, these wolves are
represented as Xα,Xβand Xδ. Fourth level ωwolves update
their position Xωfollowing the relative position of the α,β
and δwolves. Finally, hunting for prey is achieved by adopting
four main steps, namely encircling, hunting, attacking and
searching again.
The prey encircling behaviour of the grey wolves is:
X(t+1) =
Xp(t)
A×
Dwhere,
D=|
C×
Xp(t)
X(t)|
(25)
where tindicates current iteration,
X(t)and
Xp(t)are
position vectors representing the current location of the grey
wolf and prey in the search space, respectively. The coefficient
vectors
Aand
Care determined as follows:
A= 2
a×
r1
aand
C= 2 ×
r2(26)
To control exploration and exploitation proceses, the compo-
nents of
aare linearly decreased from 2 to 0 over the course
of an iteration. Note that
r1and
r2are random vectors
whose values are chosen between [0, 1]. To reach prey position
(Xp, Yp), the current position of a grey wolf (X, Y )is updated
with Eqs. 25–26. The value of
ais assumed the same for all
the wolves in a population. A wolf can update its position
according to the best agent in different places by setting the
values of
Cand
A.
After finding the prey location, the grey wolves encircle
it. The αwolves guide the pack for prey hunting, while β
and δwolves also contribute. Initially, the α,βand δwolves
location are saved as the ‘locations, representing their better
knowledge to recognize prey location. The remaining search
agents, mainly ωwolves, update their location following the
position of the best search agents. For α,βand δwolves,
position location is determined as follows:
Dα=|
C1×
Xα(t)
X(t)|,
Dβ=|
C2×
Xβ(t)
X(t)|
(27)
Dδ=|
C3×
Xδ(t)
X(t)|,
X1=|
XαA1×
Dα|(28)
X2=|
XβA2×
Dβ|,
X3=|
XδA3×
Dδ|(29)
X(t+ 1) =
X1+
X2+
X3
3(30)
At iteration t, the distance between
X(t)and the three best
hunt agents (
Xα),(
Xβ)are (
Xδ)are determined using Eqs.
27–29, in which A1,A2and A3are random vectors as defined
in Eq. 26. Finally, wolves movement towards prey is updated
by Eq. 30.
V. AUGMENTED GREY WOLF OPTIMIZATION
In this work, we propose a new modification to augment
the exploration capabilities of the GWO algorithm without
affecting its flexibility, simplicity and global optimization
characteristics. In the GWO algorithm, parameter Ais the most
important parameter responsible for controlling the exploration
and exploitation abilities in the search space stated in Eq. 26.
The value of Adepends on a, which changes linearly from
2 to 0 in the GWO algorithm. In the proposed augmentation
(AGWO) algorithm, the value of parameter achanges ran-
domly and non-linearly from 2 to 1 to avoid stagnation given
in Eq. 31. Due to this modification, chances of exploration
gets higher than exploitation [8].
a= 2 cos(rand)×t/Max_iter (31)
In the original GWO algorithm, α,βand δwolves are involved
for hunting and decision making process of the algorithm
as in Eqs. 27 and 28. However, in the proposed AGWO
algorithm, these processes are controlled only by αand β
wolves expressed as:
Dα=|
C1×
Xα(t)
X(t)|,
Dβ=|
C2×
Xβ(t)
X(t)|
(32)
X1=|
XαA1×
Dα|,
X2=|
XβA2×
Dβ|(33)
X(t+ 1) =
X1+
X2
2(34)
Due to the proposed augmentation, the AGWO gains many
advantages over the basic GWO algorithm. Some of these
are better convergence to find global optima, computational
efficiency and better exploration and exploitation capabilities.
VI. CAS E STU D IE S FOR IEEE-30 BU S SYSTEM
A. Case 1: Optimization of Total Generation Cost
The objective of Case-1 is to optimize the power gener-
ation schedule of all RES and TPGs to reduce total power
generation cost using Eq. 23. For illustrative purposes, the
values of direct, reserve and penalty cost coefficients for WSH
generation system are d= 1.6, r= 3 and p= 1.5, respectively.
The total generation cost achieved by AGWO is 781.13 $/h
and that of GWO is 781.77 $/h shown in Table IV. These
results are compared with the results obtained from ABC and
SHADE-SF algorithms, i.e., 784.24 $/h and 782.82 $/h. More
details about these algorithms can be found in Refs. [2] and
[4]. Fig. 1a shows that AGWO has faster convergence and less
computational time than the other three algorithms.
0 50 100 150 200 250 300 350 400 450 500
Number of Iterations
780
785
790
795
800
805
810
815
820
Total Generation Cost ($/hr)
Objective Space
ABC
SHADE-SF
GWO
AGWO
(a) IEEE-30 Bus System Convergence
Characteristics for Case-1
0 50 100 150 200 250 300 350 400 450 500
Number of Iterations
810
815
820
825
Total Generation Cost ($/hr)
Objective Space
ABC
SHADE-SF
GWO
AGWO
(b) IEEE-30 Bus System Convergence
Characteristics for Case-2
0 50 100 150 200 250 300 350 400 450 500
Number of Iterations
2.125
2.13
2.135
2.14
2.145
2.15
Total generation cost ($/h)
104Objective space
ABC
SHADE-SF
GWO
AGWO
(c) IEEE-57 Bus System Convergence
Characteristics for Case-3
0 50 100 150 200 250 300 350 400 450 500
Number of Iterations
2.14
2.16
2.18
2.2
2.22
2.24
2.26
2.28
2.3
Total Generation Cost ($/h)
104Objective space
ABC
SHADE-SF
GWO
AGWO
2.14483
2.14484
2.14485
104
(d) IEEE-57 Bus System Convergence
Characteristics for Case-4
0 50 100 150 200 250 300 350 400 450 500
Number of Iterations
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
Total Generation Cost ($/h)
104Objective space
ABC
SHADE-SF
GWO
AGWO
(e) IEEE-118 Bus System Convergence
Characteristics for Case-5
0 50 100 150 200 250 300 350 400 450 500
Number of Iterations
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Total Generation Cost ($/h)
105Objective space
ABC
SHADE-SF
GWO
AGWO
(f) IEEE-118 Bus System Convergence
Characteristics for Case-6
Fig. 1: Convergence Characteristics of AGWO and Recent Techniques for Case-1–Case-6.
TABLE IV: Comparison Between AGWO and other Algorithms for IEEE-30 bus System using Case-1 and Case-2.
Case-I Case-II
Min Max (ABC) (SHADE-SF) [4] (GWO) (AGWO) (ABC) (SHADE-SF) [4] (GWO) (AGWO)
PT g,1(MW) 50 140 131.4 130.6 129.6 130.1 108.4 109.6 109.8 108.1
PT g,2(MW) 20 80 38.5 37.6 38.1 36.2 43.7 44.7 44.7 41.3
PW g,1(MW) 0 75 37.5 43.8 48.9 39.5 42.8 43.5 42.4 41.7
PT g,3(MW) 10 35 10.4 10 10 10 12.1 10.5 11.05 16.3
PW g,2(MW) 0 60 39.8 40.0 37.8 40.1 44.0 43.9 43.9 43.7
PSg (MW) 0 50 31.2 31.9 31.9 32.8 36.9 35.7 36 36.3
Total cost ($/hr) 784.24 782.82 781.77 781.13 813.81 811.43 810.17 810.15
Elapsed time (Seconds) 367 272 279 230 395 272 286 246
Carbon emission (ton/h) 1.42 1.35 1.28 1.48 0.7 0.58 0.42 0.39
B. Case 2: Optimizing Fuel Cost and Carbon Emission
The main objective of Cas-2 is to minimize total generation
costs while imposing a carbon tax on the amount of carbon
emission from TPGs. Total generation cost, including the
carbon tax, is calculated with the help of Eq. 24. Carbon tax
(Ctax) is considered at the rate of $20/ton [4]. The optimized
generation schedule of all generators, total power generation
cost, values of carbon emissions and other parameters for all
algorithms are provided in Table IV. It is clearly depicted that
RES contribution gets higher when the carbon tax is imposed
in Case-2, compared to Case-1 (when there is no tax on carbon
emission). The obtained result of emission gases by AGWO
is 0.39259 ton/h, which is the lowest value compared with
0.42503 ton/h, 0.58487 ton/h and 0.7049 ton/h obtained by
GWO, ABC and SHADE-SF, respectively, as given in Table
IV. The convergence properties of AGWO, basic GWO and
other approaches is shown in Fig. 1b.
VII. CASE STUDIES FOR IEEE-57 BUS SY STE M
A. Case 3: Optimization of Total Generation Cost
The objective of Case-3 is to optimize the power generation
schedule of three RES and TPGs to reduce total power
generation costs in the IEEE-57 bus system. It is similar to
Case-1 in the IEEE-30 bus system and the objective function
of the quadratic fuel cost is given in Eq. 23. The total cost
obtained by the AGWO algorithm is 21215 $/h, which hits the
best minima in search space compared to the ABC, SHADE-
SF and GWO. The fuel cost value by ABC is 21262 $/h, by
SHADE-SF is 21260 $/h and by the GWO is 21247 $/h, as
given in Table V. The convergence properties of AGWO and
other optimization methods are shown in Fig. 1c.
TABLE V: Simulation Results for IEEE-57 Bus system using Case-3 and Case-4.
Bus System IEEE-57
Objective function ABC SHADE-SF [4] GWO AGWO
Case-3 Case-4 Case-3 Case-4 Case-3 Case-4 Case-3 Case-4
Cost (MW/h) 21262 21450 21260 22693 21247 21448 21215 21448
Carbon emission (ton/h) 33 16 39 23 36 10 31 9.42
Computational time (Sec) 870 448 330 298 247 255 220 243
TABLE VI: Simulation results for IEEE-118 Bus System using Case-5 and Case-6.
Bus System IEEE-118
Objective function ABC SHADE-SF [4] GWO AGWO
Case-5 Case-6 Case-5 Case-6 Case-5 Case-6 Case-5 Case-6
Cost (MW/h) 69934 92963 113523 125189 77606 98231 70014 98231
Carbon emission (ton/h) 128 119 133 99 144 97 113 95
Computational time (Sec) 6319 7700 1223 1772 2200 3679 2377 3517
B. Case 4: Optimizing Fuel Cost and Carbon Emission
This Case study is conducted to optimize the OPF solution
for quadratic fuel cost and carbon emission control for the
objective function given in Eq. 24. It is evident from Table V
that AGWO obtains the lowest values for this Case study with
fuel cost and carbon emission values of 21448 $/h and 9.42
ton/h, respectively. The variation of total fuel cost between
AGWO and other algorithms are shown in Fig. 1d.
VIII. CASE STUDIES FOR IEEE-118 BUS SY STE M
A. Case 5: Optimization of Total Generation Cost
In this Case study, the generation system total fuel cost
minimization is taken as an objective function given by Eq.
23. The cost computed by AGWO for this Case is 70014 $/h,
which is better than SHADE-SF and the original GWO [7],
which are respectively 77606 $/h and 129509 $/h. The ABC
algorithm achieves the minimum cost for this case study with
an obtained value of 69934 $/h. Table VI provides obtained
values comparison for generation costs, carbon emissions
and computational time for all algorithms. The convergence
graph in Fig. 1e reveals that AGWO has better convergence
characteristics than GWO and other approaches reported in the
literature.
B. Case 6: Optimizing Fuel Cost and Carbon Emission
Both quadratic fuel cost and emission gases minimization is
the aim of this Case study. The objective function calculation
is based on Eq. 24. With carbon tax imposition, the value
of emission is significantly reduced from 113 ton/h in Case
5 to 95 ton/h. The ABC algorithm obtained lower costs for
Case-5 and Case-6 but at the cost of computational time.
The AGWO algorithm requires the least computation time,
suggesting that it is a highly promising algorithm for industrial
applications. Fig. 1f compares the convergence characteristics
of all algorithms for 500 trial run.
IX. CONCLUSION
This paper presents a solution strategy for OPF study
considering traditional TPGs and the intermittent nature of
renewable energy sources (RES). Different PDFs were used to
model WPG, SPG and HPG uncertainty, and their integration
methods were discussed. Several case studies were investigated
to evaluate the performance of the proposed algorithm and the
results were compared with other well recognized evolutionary
algorithms. Hence, novel contributions include the proposed
objective functions that consider RES; the use of an AGWO
approach to address the non-convex OPF problem, and its ap-
plication both in small, medium and higher-scale bus systems
with evaluation via simulation.
The new results in this article show the AGWO proves
to be very useful and reliable with a fast convergence rate
to find a global solution for considered objective functions.
It outperforms other algorithms in terms of total cost
and convergence time minimization, whilst simultaneously
addressing the necessary system constraints.
Acknowledgements The authors acknowledge funding
support from COMSATS University Islamabad, Lahore
campus and Lancaster University UK to support this project.
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