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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)k−1exp 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(v−vci

vr−vci ), vci ≤v≤vr,

(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

Xσ√2πexp −[ln X−λ]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 −(Gh−γ)

ω) exp −exp (Gh−γ)

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

(W−PW 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=|−→

Xα−A1×−→

Dα|(28)

−→

X2=|−→

Xβ−A2×−→

Dβ|,−→

X3=|−→

Xδ−A3×−→

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=|−→

Xα−A1×−→

Dα|,−→

X2=|−→

Xβ−A2×−→

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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