# Smart Microgrid Energy Management Using a Novel Artificial Shark Optimization

Authors: Pawan Singh; Baseem Khan
Journal: Complexity
 (2017)

---

## Abstract

At present, renewable energy sources (RESs) integration using microgrid (MG) technology is of great importance for demand side management. Optimization of MG provides enhanced generation from RES at minimum operation cost. The microgrid optimization problem involves a large number of variables and constraints; therefore, it is complex in nature and various existing algorithms are unable to handle them efficiently. This paper proposed an artificial shark optimization (ASO) method to remove the limitation of existing algorithms for solving the economical operation problem of MG. The ASO algorithm is motivated by the sound sensing capability of sharks, which they use for hunting. Further, the intermittent nature of renewable energy sources is managed by utilizing battery energy storage (BES). BES has several benefits. However, all these benefits are limited to a certain fixed area due to the stationary nature of the BES system. The latest technologies, such as electric vehicle technologies (EVTs), provide all benefits of BES along with mobility to support the variable system demands. Therefore, in this work, EVTs incorporated grid connected smart microgrid (SMG) system is introduced. Additionally, a comparative study is provided, which shows that the ASO performs relatively better than the existing techniques.


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## Body

## 1. Introduction

As the world is transforming from the conventional grid system to the smart grid system, renewable energy sources’ integration has become a vital issue in the current situation. The intermittency and climate dependency of RESs make their integration more complex and difficult. A microgrid (MG) offers an efficient way to incorporate distributed RESs in large electric power systems for supplying the continually growing demand. The smart microgrid (SMG) consist of RESs (especially wind turbine (WT) and solar photovoltaic (SPV)), BESs, electric vehicle technologies (EVTs), and electrical demands. BESs and EVTs have a bidirectional battery charging system as well as automatic sensors for detecting over- or undergeneration. SMG coalesced with RESs, BESs, and EVTs is a preferable alternative to manage the increased power demand and is a supplement to the centralized smart power grids [1]. Recently, there are rising issues and concerns regarding the instability and discontinuity of RESs in the MG. Therefore, the MG central operator (MGCO) recommends the incorporation of BES in the MG for accumulating surplus power and feedback to the MG during the peak load. Further, the latest EVTs (battery electric vehicle (BEV) and plug-in hybrid vehicle (PHEV)) along with BES play a vital role to store excess power during high availability. The advantages of EVTs are their mobility and ability to supply the stored power in the energy deficient areas during peak hours. Therefore, the computation of the suitable capacity of BES, BEV, and PHEV is highly essential for an economized operation of SMG. Currently, the attention is shifted towards more efficient fuel cell technologies (FCTs), such as automotive fuel cell electric vehicle (FCEV) and stationary FC power generator (FCPG) [2, 3], due to their numerous advantages (less CO2 emissions, extremely less noise and vibrations). FCEV and FCPG have a lower maintenance cost, as they consist of fewer rotating parts. Additionally, FCTs do not require recharging, unlike BESs. FCTs are expandable and can be grouped for the required power rating [4]. FCTs decrease manufacturing, shipping, and security concerns related to MG [5].The economical operation of SMG is one of the most significant optimization problems for MGCO, wherein the economical power output of BES, FCPG, microturbine (MT), EVTs, and RESs is computed by fulfilling all equality and inequality constraints. Various studies are performed for the economical operation of MG. Mitra [6] proposed a method to estimate the capacity of BES for removing the intermittency problem of DGs. SA is applied by Ekren and Ekren Banu [7] to optimize the capacity of DGs and minimize the total cost. Mohammadi et al. [8] applied a variation of GA to find an optimal power output as well as the cost of MG under the deregulated electricity market. Bahmani-Firouzi and Azizipanah-Abarghooee [9] developed an IBA for the economical operation of MG.Various studies focus on the economical operation problem of MG without taking into account the effect of optimum sizing of BES. Chakraborty et al. [10] applied linear programming for the economical operation of MG and improved the BES’s charge states. Sortomme and El-Sharkawi utilized PSO for the economical operation of MG [11]. Niknam et al. [12] utilized the honeybee mating optimization for the economical operation of MG, which includes PV, WT, and FC. Sharma et al. [13] presented a comparative study of metaheuristic techniques for microgrid optimization.Nowadays, EVTs are the integral part of MG technology. Various optimization studies are performed on EVTs incorporated MG systems. Laureri et al. [14] focused on EVTs and their active participation in grid optimization by adopting power modulation under V2G mode. Bai et al. [15] designed a two-way charging system for bidirectional flow of current to convert it as a mobile energy storage system. Chen et al. [16] proposed a microgrid layout of EVT charging station, which combines with renewable energy sources and battery storage. Gunter et al. [17] presented a framework and an optimization technique to design utility coupled MG with BES, distributed generation, and PHEV chargers.Conventional optimization techniques have a number of limitations together with continuity and derivability of the objective function. Existing methods which are based on metaheuristic search techniques [19, 20] can be regarded as appropriate alternatives for handling optimization problems since these metaheuristic methods provide the best global solution, handle large constraints, and are derivative-free. These methods have some shortcomings like getting stuck in local optima and take a long time to find global optima; consequently, selection of the appropriate evolutionary algorithm has great importance. In previous algorithms, the main limitation is to handle problems with a large number of variables and various constraints.This paper proposed an artificial shark optimization (ASO) method to remove the limitation of existing algorithms for solving the economical operation problem of MG. The ASO algorithm is motivated by the sound sensing capability of sharks which they use for hunting. ASO is capable of providing extremely reasonable results of various standard functions in comparison with other renowned metaheuristic techniques. The global and local searching capabilities of ASO algorithm are far better than the former optimization method. These capabilities of ASO have encouraged the authors to utilize this algorithm to find the economical operation of SMG (EOSMG). The result obtained from ASO is compared with the results of GA, SA, and PSO to show its feasibility. The key contributions of this work are summarized as follows: (1) developing a novel ASO technique for solving EOSMG, (2) introducing V2G, G2V, and V2H technology based on SMG system, and (3) comparative analysis of developed ASO and existing methods on EOSMG and standard benchmark functions.Section2 describes the problem formulation of EOSMG. In Section 3, a novel optimization method called artificial shark optimization is proposed. Section 4 provides the results and discussion followed by the conclusion.


## 2. Problem Formulation

The required objective function and constraints for the formulation of an economical operation problem of SMG (EOSMG) are as follows.Minimize total costs:(1)minCX=∑t=1NTct+OMDG+CTCPD,where (2)CTCPD=TCPDBES+TCPDBEV+TCPDPHEV,ct=Costgrid,t+CostDG,t+CostBES,t+CostBEV,t+CostPHEV,t+SUCFC,t+SUCMT,t+SUCFCEV,t,Costgrid,t=Bidgrid,tPgrid,tifPgrid,t>01-taxBidgrid,tPgrid,tifPgrid,t<00ifPgrid,t=0,CostDG,t=BidMT,tPMT,tuMT,t+BidFC,tPFC,tuFC,t+BidFCEV,tPFCEV,tuFCEV,t+BidPVi,tPPVi,t+BidWTi,tPWTi,t,SUCFC,t=StartFC∗max⁡0,uFC,t-uFC,t-1,SUCFCEV,t=StartFCEV∗max⁡0,uFCEV,t-uFCEV,t-1,SUCMT,t=StartMT∗max⁡0,uMT,t-uMT,t-1,OMDG=OMMT+OMFC+OMFCEV+OMPVi+OMWTi∗NT.The total cost of the SMG is the summation of the following costs:(A) Operation cost of utility, BES, BEV, PHEV, and FCEV(B) Fuel, operation, and maintenance cost of DGs(C) Start-up costs of FC, FCEV, and MT(D) The cumulative total cost per dayCTCPD of batteries used in BES, BEV, and PHEVThe overall cost of batteries depends on the sum of fixed cost (FX) and maintenance cost (MC), where FX and MC are one-time purchasing cost and a variable annual maintenance cost, respectively. Therefore, the total cost, proportional to the size of batteries, isFX+MC∗Cmax, where Cmax is the maximum capacity of a battery. The study is conducted for a day and the cost of operation is computed on an hourly basis (24 values); therefore, TCPD is modeled in €ct/day. Let IR and LT be the rate of interest for funding the installed battery and its lifetime, respectively; then, the TCPDs in €ct/day has been calculated as follows [18]:(3)TCPDBES=CBES,max365IR1+IRLT1+IRLT-1FCBES+MCBES,TCPDBEV=CBEV,max365IR1+IRLT1+IRLT-1FCBEV+MCBEV,TCPDPHEV=CPHEV,max365IR1+IRLT1+IRLT-1FCPHEV+MCPHEV.The proposed operation cost minimization problem handles the constraints as presented in Table 1.Table 1
Constraints.
Constraint
Formulation
Electrical demand balance
Pgrid,t+PMT,tuMT,t+PFC,tuFC,t+PPVi,t+PWTi,t+PBES,tuBES,t+PBEV,tuBEV,t+PPHEV,tuPHEV,t+PFCEV,tuFCEV,t=PDemand,tt=1,…,NT
Dispatchable DGs constraints
P
F
C
,
m
i
n
≤
P
F
C
,
t
≤
P
F
C
,
m
a
x
t
=
1
,
…
,
N
T PMT,min≤PMT,t≤PMT,maxt=1,…,NT PFCEV,min≤PFCEV,t≤PFCEV,maxt=1,…,NT
BES constraints
Discharging mode [18]: CBES,t+1=max⁡CBES,t-ΔtPBES,tηdischarge,CBES,mint=1,…,NT P_BES,t≤PBES,t≤P¯BES,tt=1,…,NT Charging mode: CBES,t+1=min⁡CBES,t-ΔtPBES,tηcharge,CBES,maxt=1,…,NT P_BES,t≤PBES,t≤P¯BES,tt=1,…,NT P¯BES,t=min⁡PBES,maxCBES,t-CBES,minηdischargeΔtt=1,…,NT P_BES,t=max⁡PBES,minCBES,t-CBES,maxηchargeΔtt=1,…,NT
BEV constraints
Discharging mode:CBEV,t+1=max⁡CBEV,t-ΔtPBEV,tηdischarge,CBEV,mint=1,…,NT P_BES,t≤PBES,t≤P¯BES,tt=1,…,NT Charging mode:  CBEV,t+1=min⁡CBEV,t-ΔtPBEV,tηcharge,CBEV,maxt=1,…,NT P_BEV,t≤PBEV,t≤P¯BEV,tt=1,…,NT P¯BEV,t=min⁡PBEV,maxCBEV,t-CBEV,min⁡ηdischargeΔtt=1,…,NT P_BEV,t=max⁡PBEV,minCBEV,t-CBEV,maxηchargeΔtt=1,…,NT
PHEV constraints
Discharging mode:CPHEV,t+1=max⁡CPHEV,t-ΔtPPHEV,tηdischarge,CPHEV,mint=1,…,NT P_PHEV,t≤PPHEV,t≤P¯PHEV,tt=1,…,NT Charging mode:  CPHEV,t+1=min⁡CPHEV,t-ΔtPPHEV,tηcharge,CPHEV,maxt=1,…,NT P_PHEV,t≤PPHEV,t≤P¯PHEV,tt=1,…,NT P¯PHEV,t=min⁡PPHEV,maxCPHEV,t-CPHEV,min⁡ηdischargeΔtt=1,…,NT P_PHEV,t=max⁡PPHEV,minCPHEV,t-CPHEV,max⁡ηchargeΔtt=1,…,NT
Grid  constraints
P
g
r
i
d
,
m
i
n
≤
P
g
r
i
d
,
t
≤
P
g
r
i
d
,
m
a
x
Operating reserve constraints
In SMG systems, reliability is achieved by acquiring the energy storage (e.g., BES, EVTs, and operating reserves). In each time step, operating reserve (OR) is the addition of standby generation capacity of turned-on BES, EVTs, FC, MT, and grid. It can be supplied to the SMG in less than 10 min and is defined byPFC,maxuFC,t+PMT,maxuMT,t+Pgrid,max+P¯BES,tuBES,t+P¯BEV,tuBEV,t+P¯PHEV,tuPHEV,t+P¯FCEV,tuFCEV,t≥ORt+PDemand,tt=1,…,NT, where ORt is the 10 min OR requirement at time t


## 3. Artificial Shark Optimization Algorithm

Metaheuristic optimization algorithms are best suited to solve the complex engineering problems as they depend on simple and easily implantable concepts, bypass local optima, and do not need gradient data.


### 3.1. Motivation

Nature finds solutions in a better manner than humans. Modeling biological, physical, or behavioral phenomena of natural objects provides better capabilities to solve optimization problems. Such optimization methods are categorized as nature inspired metaheuristic algorithms. The great hunting capabilities of sharks have become a motivation to propose this optimization algorithm. Sharks are great swimmers with fantastic sudden and smooth turning capabilities. Sharks swim in the ocean in groups and search for food by identifying the proper location of prey. Sharks have strong sensing potential; they can hear their prey in the water even from approximately 3000 feet away [21]. This makes sharks a deadly hunter that survived for millions of years even before the dinosaurs.The sound of the prey helps sharks to locate the appropriate position of their prey. The shark receives a fraction of these sounds due to different noises, available between the shark and the prey. The movement of the shark depends on the sounds of nearby (Snear) or faraway (Sfar) prey. The motion of a shark is random until the probability of finding prey is less than or equal to the threshold value (Tp). If the probability goes more than Tp, the shark starts moving in a shrinking spiral motion to grab the prey.


### 3.2. Mathematical Model

In the initial phase, the shark searches for potential prey in a random motion. If the probability of getting the prey is high, then it circulates in a shrinking spiral motion to grab the prey.


#### 3.2.1. Probability

The shark is a hungry animal; it searches for prey until potential prey is not located. The probability of finding preytp is an increasing real number and it ranges from 0 to 1. The threshold value of probability to find prey is Tp. If the shark estimates the probability to be more than the threshold value, it starts moving in a shrinking spiral motion to grab the prey.


#### 3.2.2. Random Motion

The minimum and maximum audible sounds from nearby (faraway) prey areSnearmin and Snearmax (Sfarmin and Sfarmax), respectively. A function rand· returns a random value in the range [0,1]. Figure 1 shows the random motion of a shark.Figure 1
Random motion.The sound of nearby prey for iteration iter is formulated as(4)Sneariter=Snearmin+rand·Snearmax-Snearmin.The sound of faraway prey for iteration iter is formulated as(5)Sfariter=Sfarmin+rand·Sfarmax-Sfarmin.The speed that defines the amount of change applied to the shark is modeled as(6)Sditer=Sditer-1+Snearmin+rand·Snearmax-Snearmin∗Pbestiter-Piter-1+Sfarmin+rand·Sfarmax-Sfarmin∗Gbest-Piter-1.The position of a shark is evaluated as(7)Piter=Piter-1+Sditer.


#### 3.2.3. Shrinking Spiral Motion

Let the position of the prey (the best solution achieved so far) beGbest and the previous location (position) of the shark be Piter-1; then, the distance between them Dt is modeled as(8)Dtiter=absoluteGbest-Piter-1.Then, the position of a shark is evaluated as(9)Piter=Dtiter∗cos⁡θiter∗exiter+Gbest,where xiter=a1-rand·∗2+iter/max⁡_iter, θiter=2πxiter/a, and  rand· is a function that returns a random value in the range [0,1]. Figure 2 presents the spiral motion of the shark.Figure 2
Spiral motion.


### 3.3. Pseudocode of the Artificial Shark Optimization Algorithm

See Algorithm1.Algorithm 1
Initialize the whole Shark populationShi(i=1ton)
Random initialization of the positions of each Shark
while (iter < maximum number of iterations max_iter)
for each Shark Shi(i=1ton)
Calculate the objective function value of every Sharkf(Shi)
Select the local best position & objective function value of every SharkShi
Select the universal best position and objective function value of Sharks
Update the value oftp
Iftp <= threshold probability of finding pray Tp
Speed of a sharkShi is updated by using equation (6)
Position of a SharkShi is updated by using equation (7)
else  Position of a Shark Shi is updated by using equation (9)
end if
Check the new position of SharkShi is within its limit
Evaluate the direction of the speed of a SharkShi
end for
iter = iter + 1
end while
return universal best Shark.


### 3.4. Performance Assessment

To observe the efficiency of the proposed ASO technique, fifteen renowned unimodal, multimodal, and composite test functions are utilized. The parametric values ofSnearmax (Sfarmax) and Snearmin (Sfarmin) are assumed to be 2 and 0 for this study, respectively. The formulation of these test functions is described in Tables 2(a), 2(b), and 2(c) [22].Table 2
(a)
Unimodal benchmark functions
Functions
Dim
Range
fmin
F
1
x
=
∑
i
=
1
n
x
1
2
30, 200
[
-
100,100
]
0
F
2
x
=
∑
i
=
1
n
x
i
+
∏
i
=
1
n
x
i
30, 200
[
-
10,10
]
0
F
3
x
=
∑
i
=
1
n
x
i
+
0.5
2
30, 200
[
-
100,100
]
0
F
4
x
=
∑
i
=
1
n
i
x
1
4
+
random
0
,
1
30, 200
[
-
1.28,1.28
]
0
(b)
Multimodal benchmark functions
Functions
Dim
Range
fmin
F
5
x
=
∑
i
=
1
n
-
x
i
sin
⁡
x
i
30, 200
-
500,500
-
418.9829
∗
D
i
m
a
F
6
x
=
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
2
π
x
i
+
20
+
e
30, 200
-
32,32
0
F
7
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
30, 200
-
600,600
0
F
8
x
=
π
n
10
sin
⁡
π
y
1
+
∑
i
=
1
n
-
1
y
i
-
1
2
1
+
10
sin
2
⁡
π
y
i
+
1
+
y
n
-
1
2
+
∑
i
=
1
n
u
x
i
,
10,100,4 yi=1+xi+14 uxi,a,k,m=kxi-amxi>a0-a<xi<ak(-xi-a)mxi<-a
30, 200
-
50,50
0
F
9
x
=
0.1
sin
2
⁡
3
π
x
1
+
∑
i
=
1
n
x
i
-
1
2
1
+
s
i
n
2
3
π
x
i
+
1
+
x
n
-
1
2
1
+
s
i
n
2
2
π
x
n
+
∑
i
=
1
n
u
(
x
i
,
5,100,4
)
30, 200
-
50,50
0
aDim in F5 indicates the count of variables.
(c)
Composite benchmark functions
Functions
Dim
Range
fmin
F
10
C
F
1: f1,f2,f3,…,f10=SphereFunction [σ1,σ2,σ3,…,σ10]=[1,1,1,…,1] λ1,λ2,λ3,…,λ10=5100,5100,5100,…,5100
10
-
5,5
0
F
11
C
F
2: f1,f2,f3,…,f10=  Griewank’s Function  σ1,σ2,σ3,…,σ10=1,1,1,…,1 λ1,λ2,λ3,…,λ10=5100,5100,5100,…,5100
10
-
5,5
0
F
12
C
F
3:  f1,f2,f3,…,f10= Griewank’s Function  σ1,σ2,σ3,…,σ10=1,1,1,…,1 λ1,λ2,λ3,…,λ10=1,1,1,…,1
10
-
5,5
0
F
13
C
F
4:  f1,f2 = Ackley’s Function  f3,f4 = Rastrigin’s Function  f5,f6=WeierstrassFunction  f7,f8 = Griewank’s Function  f9,f10=SphereFunction  σ1,σ2,σ3,…,σ10=1,1,1,…,1 λ1,λ2,λ3,…,λ10=532,532,1,1,50.5,50.5,5100,5100,5100,5100
10
[
-
5,5
]
0
F
14
C
F
5:  f1,f2 = Rastrigin’s Function  f3,f4=WeierstrassFunction  f5,f6 = Griewank’s Function  f7,f8= Ackley’s Function  f9,f10=SphereFunction  σ1,σ2,σ3,…,σ10=1,1,1,…,1  λ1,λ2,λ3,…,λ10=15,15,50.5,50.5,5100,5100,532,532,5100,5100
10
-
5,5
0
F
15
C
F
6:  f1,f2 = Rastrigin’s Function  f3,f4=WeierstrassFunction  f5,f6= Griewank’s Function  f7,f8= Ackley’s Function  f9,f10= Sphere Function  σ1,σ2,σ3,…,σ10=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1 λ1,λ2,λ3,…,λ10=0.1∗15,0.2∗15,0.3∗50.5,0.4∗50.5,0.5∗5100,0.6∗5100,0.7∗532,0.8∗532,0.9∗5100,1∗5100
10
[
-
5,5
]
0Initially, three benchmark functions (1 unimodal and 2 multimodal) are selected to observe the change in performance with respect to the increase in population. The above-mentioned test functions are applied on the ASO technique along with SA, GA, and PSO; 30 independent trials are made to observe the results. SA, GA, and PSO depend on the population number and the maximum number of function evaluations; therefore, they are best suited to compare the performance of ASO. Every standard function is evaluated for 50, 100, 250, and 500 dimensions. Four criteria, that is, mean, best, worst, and std. values, are selected to estimate the efficiency of optimization techniques [22]. The size of the population and the number of iterations for all the applied algorithms are 50 and 500, respectively. Table 3 presents the above-mentioned criteria for different dimensions of all benchmark functions obtained by ASO and other techniques. From the results, it is observed that, for the dimensions 100, 250, and 500, the proposed ASO technique calculates the best values for all four criteria. In case of 50 dimensions, the proposed algorithm ASO stands second.Table 3
Comparison of the results obtained by different algorithms.
Function
D
Criteria
SA
GA
PSO
ASO
Sphere
50
Mean
2.80
E
+
00
1.08
E
+
00
3.69
E
-
01
3.25
E
-
02
Best
5.90
E
-
03
3.77
E
-
01
8.66
E
-
02
6.69
E
-
03
Worst
1.07
E
+
01
2.31
E
+
00
5.70
E
-
01
7.00
E
-
02
Std.
3.99
E
+
00
6.54
E
-
01
2.02
E
-
01
2.38
E
-
02
100
Mean
1.96
E
+
02
1.37
E
+
01
6.12
E
+
00
5.25
E
-
01
Best
1.57
E
+
02
6.77
E
+
00
3.00
E
+
00
6.59
E
-
02
Worst
2.43
E
+
02
1.88
E
+
01
1.07
E
+
01
8.90
E
-
01
Std.
3.90
E
+
01
4.56
E
+
00
2.60
E
+
00
2.64
E
-
01
250
Mean
8.48
E
+
02
1.55
E
+
02
8.17
E
+
01
1.96
E
+
00
Best
8.19
E
+
02
1.38
E
+
02
6.23
E
+
01
3.41
E
-
01
Worst
8.67
E
+
02
1.92
E
+
02
1.43
E
+
02
3.67
E
+
00
Std.
1.87
E
+
01
1.93
E
+
01
3.09
E
+
01
1.29
E
+
00
500
Mean
3.54
E
+
25
6.07
E
+
02
8.36
E
+
02
1.46
E
+
01
Best
4.87
E
+
16
5.46
E
+
02
3.02
E
+
02
1.02
E
+
00
Worst
1.77
E
+
26
6.51
E
+
02
9.93
E
+
02
2.70
E
+
01
Std.
7.08
E
+
25
3.76
E
+
01
2.68
E
+
02
9.66
E
+
00
Rastrigin
50
Mean
9.29
E
+
01
1.19
E
+
02
9.37
E
+
01
2.63
E
-
03
Best
8.56
E
+
01
8.54
E
+
01
7.26
E
+
01
3.49
E
-
04
Worst
1.02
E
+
02
1.42
E
+
02
1.46
E
+
02
7.37
E
-
03
Std.
5.72
E
+
00
2.13
E
+
01
2.67
E
+
01
2.57
E
-
03
100
Mean
2.78
E
+
02
2.69
E
+
02
2.03
E
+
02
2.11
E
+
00
Best
2.72
E
+
02
2.40
E
+
02
1.65
E
+
02
1.46
E
+
00
Worst
2.90
E
+
02
2.99
E
+
02
2.25
E
+
02
2.89
E
+
00
Std.
7.18
E
+
00
2.39
E
+
01
2.26
E
+
01
5.13
E
-
01
250
Mean
1.60
E
+
03
1.25
E
+
03
7.07
E
+
02
1.45
E
+
00
Best
1.53
E
+
03
9.89
E
+
02
6.07
E
+
02
1.52
E
-
01
Worst
1.73
E
+
03
1.52
E
+
03
7.86
E
+
02
3.57
E
+
00
Std.
7.21
E
+
01
1.73
E
+
02
6.18
E
+
01
1.49
E
+
00
500
Mean
4.76
E
+
03
2.89
E
+
03
2.16
E
+
03
3.56
E
+
00
Best
4.53
E
+
03
2.58
E
+
03
2.04
E
+
03
1.44
E
+
00
Worst
4.97
E
+
03
3.04
E
+
03
2.33
E
+
03
9.72
E
+
00
Std.
1.55
E
+
02
1.67
E
+
02
1.25
E
+
02
3.13
E
+
00
Griewank
50
Mean
1.72
E
-
04
1.54
E
+
00
9.73
E
-
02
2.85
E
-
03
Best
1.46
E
-
04
1.32
E
+
00
5.37
E
-
02
3.69
E
-
04
Worst
1.99
E
-
04
1.75
E
+
00
1.58
E
-
01
3.70
E
-
03
Std.
1.91
E
-
05
1.51
E
-
01
4.00
E
-
02
1.26
E
-
03
100
Mean
6.33
E
+
00
1.84
E
+
01
1.35
E
+
00
1.00
E
-
01
Best
5.16
E
+
00
1.38
E
+
01
1.07
E
+
00
3.81
E
-
03
Worst
7.07
E
+
00
2.30
E
+
01
1.76
E
+
00
2.91
E
-
01
Std.
6.54
E
-
01
2.91
E
+
00
2.58
E
-
01
1.10
E
-
01
250
Mean
1.07
E
+
03
2.35
E
+
02
3.02
E
+
01
1.23
E
+
00
Best
1.03
E
+
03
2.03
E
+
02
2.64
E
+
01
7.11
E
-
01
Worst
1.15
E
+
03
2.64
E
+
02
3.90
E
+
01
1.88
E
+
00
Std.
4.04
E
+
01
2.00
E
+
01
4.71
E
+
00
3.92
E
-
01
500
Mean
5.12
E
+
03
1.12
E
+
03
2.14
E
+
02
7.52
E
-
01
Best
5.03
E
+
03
1.02
E
+
03
2.08
E
+
02
1.29
E
-
03
Worst
5.21
E
+
03
1.29
E
+
03
2.28
E
+
02
1.37
E
+
00
Std.
6.39
E
+
01
1.02
E
+
02
7.10
E
+
00
5.49
E
-
01Figure3 shows the variation in fitness value with respect to the population for SA, GA, PSO, and ASO. From Figure 3, it is clear that the developed ASO algorithm shows less volatility as compared to other methods.Figure 3
Fitness value variations with respect to population.
(a)
(b)
(c)
(d)Five agents are deputed over 250 iterations to compute all standard functions and computed values are shown in Figure4. The first row in Figure 4 shows the 2D representation of the functions. In the second row, the exploration and exploitation space of the best shark’s position during optimization is presented. The ASO method is inclined to investigate the potential areas of the search space comprehensively on the standard functions. This shows that the shark’s agents are able to investigate the search space efficiently. Directional search history is presented in the third row in Figure 4, which shows in which direction sharks search the space efficiently.Figure 4
Search space, directional search history, trajectory in the first dimension, average function value of all sharks, and ASO convergence curve.The trajectory of the first variable of the first shark over 250 iterations is illustrated in the fourth row of Figure4. There are rapid changes in the early iterations which decrease progressively over the course of iterations. As per Van den Bergh and Engelbrecht [23], this conduct ensures that a technique in due course converges to a point and investigates locally the search space. The average (mean) value of all sharks and convergence curves are presented in the last two rows of Figure 4, respectively. The mean value of sharks has a descending tendency on all of the standard functions. The same tendency is analyzed in convergence curves. This proves that the approximate value of the best search agent becomes more precise with the increase in iterations. An additional reality observed is the accelerated style of convergence curves as a consequence of the emphasis on exploration and exploitation. Tables 4(a), 4(b), and 4(c) present the comparison of the results obtained by the developed ASO algorithm with five different techniques (GA, PSO, states of matter search (SMS), bat algorithm (BA), and firefly (FF)) on fifteen standard benchmark functions under three different categories: unimodal, multimodal, and composite. It is observed from Table 4(a) that, in unimodal benchmark function F3, the performance of ASO is the best, while in F1 and F2 it stands second and in F4 third. It is clear from Table 4(b) that, in multimodal benchmark functions F5, F7, and F8, the performance of ASO is the best, while in F9 it stands second and in F6 third.Table 4
(a)
Comparison of results on unimodal benchmark functions
Function
PSO
SMS
BA
FF
GA
ASO
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
F1
2.7000
E
-
09
1.0000
E
-
09
5.6987
E
-
02
1.4689
E
-
02
7.7362
E
-
01
5.2813
E
-
01
3.9615
E
-
02
1.4490
E
-
02
1.1884
E
-
01
1.2561
E
-
01
8.6073
E
-
07
5.0293
E
-
07
F2
7.1500
E
-
05
2.2600
E
-
05
6.8480
E
-
03
1.5770
E
-
03
3.3458
E
-
01
3.8160
E
+
00
5.0346
E
-
02
1.2348
E
-
02
1.4522
E
-
01
5.3227
E
-
02
1.9506
E
-
04
1.1703
E
-
04
F3
5.7300
E
-
07
2.7400
E
-
06
1.2532
E
-
01
8.4998
E
-
02
7.7885
E
-
01
6.7392
E
-
01
5.8730
E
-
02
1.4477
E
-
02
1.6792
E
-
01
8.6864
E
-
01
5.7109
E
-
07
3.7268
E
-
07
F4
1.3980
E
-
03
1.2600
E
-
03
3.0400
E
-
04
2.5800
E
-
04
1.3748
E
-
01
1.1267
E
-
01
8.5300
E
-
04
5.0400
E
-
04
1.0073
E
-
02
3.2630
E
-
03
3.0397
E
-
02
8.2079
E
-
03
(b)
Comparison of results on multimodal benchmark functions
Function
PSO
SMS
BA
FF
GA
ASO
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
F5
-
5.5670
E
+
03
1.4641
E
+
02
-
4.2074
E
+
00
9.3600
E
-
16
-
1.0659
E
+
03
8.5850
E
+
02
-
9.5835
E
+
03
3.5327
E
+
02
-
1.1078
E
+
04
2.4724
E
+
00
-
1.2042
E
+
04
4.5619
E
+
02
F6
1.1100
E
-
09
2.3900
E
-
11
8.8800
E
-
06
8.5600
E
-
09
1.2936
E
-
01
4.3251
E
-
02
1.6831
E
-
01
5.0796
E
-
02
9.5611
E
-
01
8.0770
E
-
01
3.9519
E
-
03
2.0783
E
-
03
F7
2.7367
E
-
01
2.0435
E
-
01
7.0609
E
-
01
9.0795
E
-
01
1.4516
E
+
00
5.7031
E
-
01
9.9815
E
-
02
2.4466
E
-
02
4.8781
E
-
01
2.1778
E
-
01
1.1974
E
-
05
1.7799
E
-
05
F8
9.4200
E
-
09
2.3100
E
-
10
1.2334
E
-
01
4.0898
E
-
02
3.9598
E
-
01
9.9333
E
-
01
1.2608
E
-
01
2.6320
E
-
01
1.1077
E
-
01
2.1520
E
-
03
7.4432
E
-
09
1.2944
E
-
08
F9
1.3500
E
-
07
2.8800
E
-
08
1.3500
E
-
02
2.8800
E
-
04
3.8663
E
-
01
1.2199
E
-
01
2.1300
E
-
03
1.2380
E
-
03
1.2900
E
-
01
6.8851
E
-
02
2.4142
E
-
06
3.0311
E
-
06
(c)
Comparisons of results on the composite benchmark functions
Function
PSO
SMS
BA
FF
GA
ASO
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
F10
1.2000
E
+
02
1.3166
E
+
02
7.7648
E
+
02
5.2100
E
-
12
1.8248
E
+
02
1.1702
E
+
02
1.5017
E
+
02
9.7159
E
+
01
1.1461
E
+
02
2.6962
E
+
01
8.0259
E
+
01
4.0952
E
+
01
F11
1.6291
E
+
02
1.1924
E
+
02
8.7375
E
+
02
9.7162
E
+
00
4.8720
E
+
02
1.6141
E
+
02
3.1447
E
+
02
9.2934
E
+
01
1.3146
E
+
02
7.1634
E
+
00
3.9530
E
-
04
4.8323
E
-
04
F12
3.6324
E
+
02
1.5131
E
+
02
9.6128
E
+
02
6.7216
E
+
01
5.8819
E
+
02
1.3779
E
+
02
7.3454
E
+
02
2.0397
E
+
02
3.2544
E
+
02
5.1668
E
+
01
8.5887
E
-
04
1.5236
E
-
03
F13
4.5007
E
+
02
1.5785
E
+
02
8.9986
E
+
02
1.9900
E
-
05
7.5698
E
+
02
1.6010
E
+
02
8.1857
E
+
02
1.0997
E
+
02
5.3631
E
+
02
2.9568
E
+
01
2.9632
E
+
02
8.0027
E
+
01
F14
1.7534
E
+
02
1.7554
E
+
02
7.4097
E
+
02
7.8580
E
-
01
5.4220
E
+
02
2.2020
E
+
02
1.3352
E
+
02
2.1560
E
+
02
9.0369
E
+
01
1.3730
E
+
01
4.3268
E
-
06
6.8172
E
-
06
F15
9.0159
E
+
02
8.3859
E
-
01
9.0048
E
+
02
8.4420
E
-
01
8.1850
E
+
02
1.5250
E
+
02
8.6222
E
+
02
1.2596
E
+
02
5.2119
E
+
02
2.7985
E
+
01
3.3478
E
-
02
3.1866
E
-
02The best performance of the proposed algorithm ASO can be observed from Table4(c) on complex composite benchmark functions, which are the reflection of the complex practical problems. Hence, it can be concluded that the developed ASO algorithm is best suited for the complex constrained practical problems.Figure5 provides the comparison between different techniques on the basis of their convergence characteristics using different benchmark functions. Figure 5 demonstrates that the ASO illustrates the best convergence for composite and multimodal benchmark functions.Figure 5
Comparison based on convergence characteristic.


## 3.1. Motivation

Nature finds solutions in a better manner than humans. Modeling biological, physical, or behavioral phenomena of natural objects provides better capabilities to solve optimization problems. Such optimization methods are categorized as nature inspired metaheuristic algorithms. The great hunting capabilities of sharks have become a motivation to propose this optimization algorithm. Sharks are great swimmers with fantastic sudden and smooth turning capabilities. Sharks swim in the ocean in groups and search for food by identifying the proper location of prey. Sharks have strong sensing potential; they can hear their prey in the water even from approximately 3000 feet away [21]. This makes sharks a deadly hunter that survived for millions of years even before the dinosaurs.The sound of the prey helps sharks to locate the appropriate position of their prey. The shark receives a fraction of these sounds due to different noises, available between the shark and the prey. The movement of the shark depends on the sounds of nearby (Snear) or faraway (Sfar) prey. The motion of a shark is random until the probability of finding prey is less than or equal to the threshold value (Tp). If the probability goes more than Tp, the shark starts moving in a shrinking spiral motion to grab the prey.


## 3.2. Mathematical Model

In the initial phase, the shark searches for potential prey in a random motion. If the probability of getting the prey is high, then it circulates in a shrinking spiral motion to grab the prey.


### 3.2.1. Probability

The shark is a hungry animal; it searches for prey until potential prey is not located. The probability of finding preytp is an increasing real number and it ranges from 0 to 1. The threshold value of probability to find prey is Tp. If the shark estimates the probability to be more than the threshold value, it starts moving in a shrinking spiral motion to grab the prey.


### 3.2.2. Random Motion

The minimum and maximum audible sounds from nearby (faraway) prey areSnearmin and Snearmax (Sfarmin and Sfarmax), respectively. A function rand· returns a random value in the range [0,1]. Figure 1 shows the random motion of a shark.Figure 1
Random motion.The sound of nearby prey for iteration iter is formulated as(4)Sneariter=Snearmin+rand·Snearmax-Snearmin.The sound of faraway prey for iteration iter is formulated as(5)Sfariter=Sfarmin+rand·Sfarmax-Sfarmin.The speed that defines the amount of change applied to the shark is modeled as(6)Sditer=Sditer-1+Snearmin+rand·Snearmax-Snearmin∗Pbestiter-Piter-1+Sfarmin+rand·Sfarmax-Sfarmin∗Gbest-Piter-1.The position of a shark is evaluated as(7)Piter=Piter-1+Sditer.


### 3.2.3. Shrinking Spiral Motion

Let the position of the prey (the best solution achieved so far) beGbest and the previous location (position) of the shark be Piter-1; then, the distance between them Dt is modeled as(8)Dtiter=absoluteGbest-Piter-1.Then, the position of a shark is evaluated as(9)Piter=Dtiter∗cos⁡θiter∗exiter+Gbest,where xiter=a1-rand·∗2+iter/max⁡_iter, θiter=2πxiter/a, and  rand· is a function that returns a random value in the range [0,1]. Figure 2 presents the spiral motion of the shark.Figure 2
Spiral motion.


## 3.2.1. Probability

The shark is a hungry animal; it searches for prey until potential prey is not located. The probability of finding preytp is an increasing real number and it ranges from 0 to 1. The threshold value of probability to find prey is Tp. If the shark estimates the probability to be more than the threshold value, it starts moving in a shrinking spiral motion to grab the prey.


## 3.2.2. Random Motion

The minimum and maximum audible sounds from nearby (faraway) prey areSnearmin and Snearmax (Sfarmin and Sfarmax), respectively. A function rand· returns a random value in the range [0,1]. Figure 1 shows the random motion of a shark.Figure 1
Random motion.The sound of nearby prey for iteration iter is formulated as(4)Sneariter=Snearmin+rand·Snearmax-Snearmin.The sound of faraway prey for iteration iter is formulated as(5)Sfariter=Sfarmin+rand·Sfarmax-Sfarmin.The speed that defines the amount of change applied to the shark is modeled as(6)Sditer=Sditer-1+Snearmin+rand·Snearmax-Snearmin∗Pbestiter-Piter-1+Sfarmin+rand·Sfarmax-Sfarmin∗Gbest-Piter-1.The position of a shark is evaluated as(7)Piter=Piter-1+Sditer.


## 3.2.3. Shrinking Spiral Motion

Let the position of the prey (the best solution achieved so far) beGbest and the previous location (position) of the shark be Piter-1; then, the distance between them Dt is modeled as(8)Dtiter=absoluteGbest-Piter-1.Then, the position of a shark is evaluated as(9)Piter=Dtiter∗cos⁡θiter∗exiter+Gbest,where xiter=a1-rand·∗2+iter/max⁡_iter, θiter=2πxiter/a, and  rand· is a function that returns a random value in the range [0,1]. Figure 2 presents the spiral motion of the shark.Figure 2
Spiral motion.


## 3.3. Pseudocode of the Artificial Shark Optimization Algorithm

See Algorithm1.Algorithm 1
Initialize the whole Shark populationShi(i=1ton)
Random initialization of the positions of each Shark
while (iter < maximum number of iterations max_iter)
for each Shark Shi(i=1ton)
Calculate the objective function value of every Sharkf(Shi)
Select the local best position & objective function value of every SharkShi
Select the universal best position and objective function value of Sharks
Update the value oftp
Iftp <= threshold probability of finding pray Tp
Speed of a sharkShi is updated by using equation (6)
Position of a SharkShi is updated by using equation (7)
else  Position of a Shark Shi is updated by using equation (9)
end if
Check the new position of SharkShi is within its limit
Evaluate the direction of the speed of a SharkShi
end for
iter = iter + 1
end while
return universal best Shark.


## 3.4. Performance Assessment

To observe the efficiency of the proposed ASO technique, fifteen renowned unimodal, multimodal, and composite test functions are utilized. The parametric values ofSnearmax (Sfarmax) and Snearmin (Sfarmin) are assumed to be 2 and 0 for this study, respectively. The formulation of these test functions is described in Tables 2(a), 2(b), and 2(c) [22].Table 2
(a)
Unimodal benchmark functions
Functions
Dim
Range
fmin
F
1
x
=
∑
i
=
1
n
x
1
2
30, 200
[
-
100,100
]
0
F
2
x
=
∑
i
=
1
n
x
i
+
∏
i
=
1
n
x
i
30, 200
[
-
10,10
]
0
F
3
x
=
∑
i
=
1
n
x
i
+
0.5
2
30, 200
[
-
100,100
]
0
F
4
x
=
∑
i
=
1
n
i
x
1
4
+
random
0
,
1
30, 200
[
-
1.28,1.28
]
0
(b)
Multimodal benchmark functions
Functions
Dim
Range
fmin
F
5
x
=
∑
i
=
1
n
-
x
i
sin
⁡
x
i
30, 200
-
500,500
-
418.9829
∗
D
i
m
a
F
6
x
=
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
2
π
x
i
+
20
+
e
30, 200
-
32,32
0
F
7
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
30, 200
-
600,600
0
F
8
x
=
π
n
10
sin
⁡
π
y
1
+
∑
i
=
1
n
-
1
y
i
-
1
2
1
+
10
sin
2
⁡
π
y
i
+
1
+
y
n
-
1
2
+
∑
i
=
1
n
u
x
i
,
10,100,4 yi=1+xi+14 uxi,a,k,m=kxi-amxi>a0-a<xi<ak(-xi-a)mxi<-a
30, 200
-
50,50
0
F
9
x
=
0.1
sin
2
⁡
3
π
x
1
+
∑
i
=
1
n
x
i
-
1
2
1
+
s
i
n
2
3
π
x
i
+
1
+
x
n
-
1
2
1
+
s
i
n
2
2
π
x
n
+
∑
i
=
1
n
u
(
x
i
,
5,100,4
)
30, 200
-
50,50
0
aDim in F5 indicates the count of variables.
(c)
Composite benchmark functions
Functions
Dim
Range
fmin
F
10
C
F
1: f1,f2,f3,…,f10=SphereFunction [σ1,σ2,σ3,…,σ10]=[1,1,1,…,1] λ1,λ2,λ3,…,λ10=5100,5100,5100,…,5100
10
-
5,5
0
F
11
C
F
2: f1,f2,f3,…,f10=  Griewank’s Function  σ1,σ2,σ3,…,σ10=1,1,1,…,1 λ1,λ2,λ3,…,λ10=5100,5100,5100,…,5100
10
-
5,5
0
F
12
C
F
3:  f1,f2,f3,…,f10= Griewank’s Function  σ1,σ2,σ3,…,σ10=1,1,1,…,1 λ1,λ2,λ3,…,λ10=1,1,1,…,1
10
-
5,5
0
F
13
C
F
4:  f1,f2 = Ackley’s Function  f3,f4 = Rastrigin’s Function  f5,f6=WeierstrassFunction  f7,f8 = Griewank’s Function  f9,f10=SphereFunction  σ1,σ2,σ3,…,σ10=1,1,1,…,1 λ1,λ2,λ3,…,λ10=532,532,1,1,50.5,50.5,5100,5100,5100,5100
10
[
-
5,5
]
0
F
14
C
F
5:  f1,f2 = Rastrigin’s Function  f3,f4=WeierstrassFunction  f5,f6 = Griewank’s Function  f7,f8= Ackley’s Function  f9,f10=SphereFunction  σ1,σ2,σ3,…,σ10=1,1,1,…,1  λ1,λ2,λ3,…,λ10=15,15,50.5,50.5,5100,5100,532,532,5100,5100
10
-
5,5
0
F
15
C
F
6:  f1,f2 = Rastrigin’s Function  f3,f4=WeierstrassFunction  f5,f6= Griewank’s Function  f7,f8= Ackley’s Function  f9,f10= Sphere Function  σ1,σ2,σ3,…,σ10=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1 λ1,λ2,λ3,…,λ10=0.1∗15,0.2∗15,0.3∗50.5,0.4∗50.5,0.5∗5100,0.6∗5100,0.7∗532,0.8∗532,0.9∗5100,1∗5100
10
[
-
5,5
]
0Initially, three benchmark functions (1 unimodal and 2 multimodal) are selected to observe the change in performance with respect to the increase in population. The above-mentioned test functions are applied on the ASO technique along with SA, GA, and PSO; 30 independent trials are made to observe the results. SA, GA, and PSO depend on the population number and the maximum number of function evaluations; therefore, they are best suited to compare the performance of ASO. Every standard function is evaluated for 50, 100, 250, and 500 dimensions. Four criteria, that is, mean, best, worst, and std. values, are selected to estimate the efficiency of optimization techniques [22]. The size of the population and the number of iterations for all the applied algorithms are 50 and 500, respectively. Table 3 presents the above-mentioned criteria for different dimensions of all benchmark functions obtained by ASO and other techniques. From the results, it is observed that, for the dimensions 100, 250, and 500, the proposed ASO technique calculates the best values for all four criteria. In case of 50 dimensions, the proposed algorithm ASO stands second.Table 3
Comparison of the results obtained by different algorithms.
Function
D
Criteria
SA
GA
PSO
ASO
Sphere
50
Mean
2.80
E
+
00
1.08
E
+
00
3.69
E
-
01
3.25
E
-
02
Best
5.90
E
-
03
3.77
E
-
01
8.66
E
-
02
6.69
E
-
03
Worst
1.07
E
+
01
2.31
E
+
00
5.70
E
-
01
7.00
E
-
02
Std.
3.99
E
+
00
6.54
E
-
01
2.02
E
-
01
2.38
E
-
02
100
Mean
1.96
E
+
02
1.37
E
+
01
6.12
E
+
00
5.25
E
-
01
Best
1.57
E
+
02
6.77
E
+
00
3.00
E
+
00
6.59
E
-
02
Worst
2.43
E
+
02
1.88
E
+
01
1.07
E
+
01
8.90
E
-
01
Std.
3.90
E
+
01
4.56
E
+
00
2.60
E
+
00
2.64
E
-
01
250
Mean
8.48
E
+
02
1.55
E
+
02
8.17
E
+
01
1.96
E
+
00
Best
8.19
E
+
02
1.38
E
+
02
6.23
E
+
01
3.41
E
-
01
Worst
8.67
E
+
02
1.92
E
+
02
1.43
E
+
02
3.67
E
+
00
Std.
1.87
E
+
01
1.93
E
+
01
3.09
E
+
01
1.29
E
+
00
500
Mean
3.54
E
+
25
6.07
E
+
02
8.36
E
+
02
1.46
E
+
01
Best
4.87
E
+
16
5.46
E
+
02
3.02
E
+
02
1.02
E
+
00
Worst
1.77
E
+
26
6.51
E
+
02
9.93
E
+
02
2.70
E
+
01
Std.
7.08
E
+
25
3.76
E
+
01
2.68
E
+
02
9.66
E
+
00
Rastrigin
50
Mean
9.29
E
+
01
1.19
E
+
02
9.37
E
+
01
2.63
E
-
03
Best
8.56
E
+
01
8.54
E
+
01
7.26
E
+
01
3.49
E
-
04
Worst
1.02
E
+
02
1.42
E
+
02
1.46
E
+
02
7.37
E
-
03
Std.
5.72
E
+
00
2.13
E
+
01
2.67
E
+
01
2.57
E
-
03
100
Mean
2.78
E
+
02
2.69
E
+
02
2.03
E
+
02
2.11
E
+
00
Best
2.72
E
+
02
2.40
E
+
02
1.65
E
+
02
1.46
E
+
00
Worst
2.90
E
+
02
2.99
E
+
02
2.25
E
+
02
2.89
E
+
00
Std.
7.18
E
+
00
2.39
E
+
01
2.26
E
+
01
5.13
E
-
01
250
Mean
1.60
E
+
03
1.25
E
+
03
7.07
E
+
02
1.45
E
+
00
Best
1.53
E
+
03
9.89
E
+
02
6.07
E
+
02
1.52
E
-
01
Worst
1.73
E
+
03
1.52
E
+
03
7.86
E
+
02
3.57
E
+
00
Std.
7.21
E
+
01
1.73
E
+
02
6.18
E
+
01
1.49
E
+
00
500
Mean
4.76
E
+
03
2.89
E
+
03
2.16
E
+
03
3.56
E
+
00
Best
4.53
E
+
03
2.58
E
+
03
2.04
E
+
03
1.44
E
+
00
Worst
4.97
E
+
03
3.04
E
+
03
2.33
E
+
03
9.72
E
+
00
Std.
1.55
E
+
02
1.67
E
+
02
1.25
E
+
02
3.13
E
+
00
Griewank
50
Mean
1.72
E
-
04
1.54
E
+
00
9.73
E
-
02
2.85
E
-
03
Best
1.46
E
-
04
1.32
E
+
00
5.37
E
-
02
3.69
E
-
04
Worst
1.99
E
-
04
1.75
E
+
00
1.58
E
-
01
3.70
E
-
03
Std.
1.91
E
-
05
1.51
E
-
01
4.00
E
-
02
1.26
E
-
03
100
Mean
6.33
E
+
00
1.84
E
+
01
1.35
E
+
00
1.00
E
-
01
Best
5.16
E
+
00
1.38
E
+
01
1.07
E
+
00
3.81
E
-
03
Worst
7.07
E
+
00
2.30
E
+
01
1.76
E
+
00
2.91
E
-
01
Std.
6.54
E
-
01
2.91
E
+
00
2.58
E
-
01
1.10
E
-
01
250
Mean
1.07
E
+
03
2.35
E
+
02
3.02
E
+
01
1.23
E
+
00
Best
1.03
E
+
03
2.03
E
+
02
2.64
E
+
01
7.11
E
-
01
Worst
1.15
E
+
03
2.64
E
+
02
3.90
E
+
01
1.88
E
+
00
Std.
4.04
E
+
01
2.00
E
+
01
4.71
E
+
00
3.92
E
-
01
500
Mean
5.12
E
+
03
1.12
E
+
03
2.14
E
+
02
7.52
E
-
01
Best
5.03
E
+
03
1.02
E
+
03
2.08
E
+
02
1.29
E
-
03
Worst
5.21
E
+
03
1.29
E
+
03
2.28
E
+
02
1.37
E
+
00
Std.
6.39
E
+
01
1.02
E
+
02
7.10
E
+
00
5.49
E
-
01Figure3 shows the variation in fitness value with respect to the population for SA, GA, PSO, and ASO. From Figure 3, it is clear that the developed ASO algorithm shows less volatility as compared to other methods.Figure 3
Fitness value variations with respect to population.
(a)
(b)
(c)
(d)Five agents are deputed over 250 iterations to compute all standard functions and computed values are shown in Figure4. The first row in Figure 4 shows the 2D representation of the functions. In the second row, the exploration and exploitation space of the best shark’s position during optimization is presented. The ASO method is inclined to investigate the potential areas of the search space comprehensively on the standard functions. This shows that the shark’s agents are able to investigate the search space efficiently. Directional search history is presented in the third row in Figure 4, which shows in which direction sharks search the space efficiently.Figure 4
Search space, directional search history, trajectory in the first dimension, average function value of all sharks, and ASO convergence curve.The trajectory of the first variable of the first shark over 250 iterations is illustrated in the fourth row of Figure4. There are rapid changes in the early iterations which decrease progressively over the course of iterations. As per Van den Bergh and Engelbrecht [23], this conduct ensures that a technique in due course converges to a point and investigates locally the search space. The average (mean) value of all sharks and convergence curves are presented in the last two rows of Figure 4, respectively. The mean value of sharks has a descending tendency on all of the standard functions. The same tendency is analyzed in convergence curves. This proves that the approximate value of the best search agent becomes more precise with the increase in iterations. An additional reality observed is the accelerated style of convergence curves as a consequence of the emphasis on exploration and exploitation. Tables 4(a), 4(b), and 4(c) present the comparison of the results obtained by the developed ASO algorithm with five different techniques (GA, PSO, states of matter search (SMS), bat algorithm (BA), and firefly (FF)) on fifteen standard benchmark functions under three different categories: unimodal, multimodal, and composite. It is observed from Table 4(a) that, in unimodal benchmark function F3, the performance of ASO is the best, while in F1 and F2 it stands second and in F4 third. It is clear from Table 4(b) that, in multimodal benchmark functions F5, F7, and F8, the performance of ASO is the best, while in F9 it stands second and in F6 third.Table 4
(a)
Comparison of results on unimodal benchmark functions
Function
PSO
SMS
BA
FF
GA
ASO
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
F1
2.7000
E
-
09
1.0000
E
-
09
5.6987
E
-
02
1.4689
E
-
02
7.7362
E
-
01
5.2813
E
-
01
3.9615
E
-
02
1.4490
E
-
02
1.1884
E
-
01
1.2561
E
-
01
8.6073
E
-
07
5.0293
E
-
07
F2
7.1500
E
-
05
2.2600
E
-
05
6.8480
E
-
03
1.5770
E
-
03
3.3458
E
-
01
3.8160
E
+
00
5.0346
E
-
02
1.2348
E
-
02
1.4522
E
-
01
5.3227
E
-
02
1.9506
E
-
04
1.1703
E
-
04
F3
5.7300
E
-
07
2.7400
E
-
06
1.2532
E
-
01
8.4998
E
-
02
7.7885
E
-
01
6.7392
E
-
01
5.8730
E
-
02
1.4477
E
-
02
1.6792
E
-
01
8.6864
E
-
01
5.7109
E
-
07
3.7268
E
-
07
F4
1.3980
E
-
03
1.2600
E
-
03
3.0400
E
-
04
2.5800
E
-
04
1.3748
E
-
01
1.1267
E
-
01
8.5300
E
-
04
5.0400
E
-
04
1.0073
E
-
02
3.2630
E
-
03
3.0397
E
-
02
8.2079
E
-
03
(b)
Comparison of results on multimodal benchmark functions
Function
PSO
SMS
BA
FF
GA
ASO
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
F5
-
5.5670
E
+
03
1.4641
E
+
02
-
4.2074
E
+
00
9.3600
E
-
16
-
1.0659
E
+
03
8.5850
E
+
02
-
9.5835
E
+
03
3.5327
E
+
02
-
1.1078
E
+
04
2.4724
E
+
00
-
1.2042
E
+
04
4.5619
E
+
02
F6
1.1100
E
-
09
2.3900
E
-
11
8.8800
E
-
06
8.5600
E
-
09
1.2936
E
-
01
4.3251
E
-
02
1.6831
E
-
01
5.0796
E
-
02
9.5611
E
-
01
8.0770
E
-
01
3.9519
E
-
03
2.0783
E
-
03
F7
2.7367
E
-
01
2.0435
E
-
01
7.0609
E
-
01
9.0795
E
-
01
1.4516
E
+
00
5.7031
E
-
01
9.9815
E
-
02
2.4466
E
-
02
4.8781
E
-
01
2.1778
E
-
01
1.1974
E
-
05
1.7799
E
-
05
F8
9.4200
E
-
09
2.3100
E
-
10
1.2334
E
-
01
4.0898
E
-
02
3.9598
E
-
01
9.9333
E
-
01
1.2608
E
-
01
2.6320
E
-
01
1.1077
E
-
01
2.1520
E
-
03
7.4432
E
-
09
1.2944
E
-
08
F9
1.3500
E
-
07
2.8800
E
-
08
1.3500
E
-
02
2.8800
E
-
04
3.8663
E
-
01
1.2199
E
-
01
2.1300
E
-
03
1.2380
E
-
03
1.2900
E
-
01
6.8851
E
-
02
2.4142
E
-
06
3.0311
E
-
06
(c)
Comparisons of results on the composite benchmark functions
Function
PSO
SMS
BA
FF
GA
ASO
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
Ave.
Std.
F10
1.2000
E
+
02
1.3166
E
+
02
7.7648
E
+
02
5.2100
E
-
12
1.8248
E
+
02
1.1702
E
+
02
1.5017
E
+
02
9.7159
E
+
01
1.1461
E
+
02
2.6962
E
+
01
8.0259
E
+
01
4.0952
E
+
01
F11
1.6291
E
+
02
1.1924
E
+
02
8.7375
E
+
02
9.7162
E
+
00
4.8720
E
+
02
1.6141
E
+
02
3.1447
E
+
02
9.2934
E
+
01
1.3146
E
+
02
7.1634
E
+
00
3.9530
E
-
04
4.8323
E
-
04
F12
3.6324
E
+
02
1.5131
E
+
02
9.6128
E
+
02
6.7216
E
+
01
5.8819
E
+
02
1.3779
E
+
02
7.3454
E
+
02
2.0397
E
+
02
3.2544
E
+
02
5.1668
E
+
01
8.5887
E
-
04
1.5236
E
-
03
F13
4.5007
E
+
02
1.5785
E
+
02
8.9986
E
+
02
1.9900
E
-
05
7.5698
E
+
02
1.6010
E
+
02
8.1857
E
+
02
1.0997
E
+
02
5.3631
E
+
02
2.9568
E
+
01
2.9632
E
+
02
8.0027
E
+
01
F14
1.7534
E
+
02
1.7554
E
+
02
7.4097
E
+
02
7.8580
E
-
01
5.4220
E
+
02
2.2020
E
+
02
1.3352
E
+
02
2.1560
E
+
02
9.0369
E
+
01
1.3730
E
+
01
4.3268
E
-
06
6.8172
E
-
06
F15
9.0159
E
+
02
8.3859
E
-
01
9.0048
E
+
02
8.4420
E
-
01
8.1850
E
+
02
1.5250
E
+
02
8.6222
E
+
02
1.2596
E
+
02
5.2119
E
+
02
2.7985
E
+
01
3.3478
E
-
02
3.1866
E
-
02The best performance of the proposed algorithm ASO can be observed from Table4(c) on complex composite benchmark functions, which are the reflection of the complex practical problems. Hence, it can be concluded that the developed ASO algorithm is best suited for the complex constrained practical problems.Figure5 provides the comparison between different techniques on the basis of their convergence characteristics using different benchmark functions. Figure 5 demonstrates that the ASO illustrates the best convergence for composite and multimodal benchmark functions.Figure 5
Comparison based on convergence characteristic.


## 4. Result and Discussion

The proposed optimization model is utilized to solve the “economically controlled problem” of SMG. The legitimacy and efficacy of ASO are assessed and it is applied for the upgraded SMG system.


### 4.1. Description of a SMG Test System

To evaluate the legitimacy and efficacy of the proposed framework, ASO is applied on an advanced low voltage SMG system as shown in Figure6. The SMG consists of various DGs such as PV, WT, MT, FC, and Li-ion BES. In addition, the proposed SMG system consists of EVTs (BEV, PHEV, and FCEV) technology. In this work, the FCEV is considered as the DG source. Table 5 presents the coefficients and generation limits which are used in the projected approach. If the SMG under study has MT, FC, FCEV, PV, WT, BES, BEV, and PHEV, then the position of mth search agent Xm can be defined as(10)Xm=xm,1xm,2……xm,DCBESmax,1,CBESmax,2,…,CBESmax,s1,CBEVmax,1,CBEVmax,2,…,CBEVmax,s2CPHEVmax,1,CPHEVmax,2,…,CPHEVmax,s3,PMT,1m,PMT,2m,…,PMT,TmPFC,1m,PFC,2m,…,PFC,Tm,PFCEV,1m,PFCEV,2m,…,PFCEV,TmPPV,1m,PPV,2m,…,PPV,Tm,PWT,1m,PWT,2m,…,PWT,TmPBES,1m,PBES,2m,…,PBES,Tm,PBEV,1m,PBEV,2m,…,PBEV,TmPPHEV,1m,PPHEV,2m,…,PPHEV,Tm,PGrid,1m,PGrid,2m,…,PGrid,TmuMT,1m,uMT,2m,…,uMT,Tm,uFC,1m,uFC,2m,…,uFC,TmuFCEV,1m,uFCEV,2m,…,uFCEV,Tm,uPV,1m,uPV,2m,…,uPV,TmuWT,1m,uWT,2m,…,uWT,Tm,uBES,1m,uBES,2m,…,uBES,TmuBEV,1m,uBEV,2m,…,uBEV,Tm,uPHEV,1m,uPHEV,2m,…,uPHEV,TmuGrid,1m,uGrid,2m,…,uGrid,Tm.More information regarding the implementation of SMG can be found in [18, 24, 25]. It is supposed that all of the DGs generate active power at the unity power factor. In this system, it is assumed that 465 (€ct/kW h) and 15 (€ct/kWh) are the fixed and maintenance cost of installation and operation of batteries [14]. The IR and LT for funding the installed batteries are 0.06 and 3, respectively. Ten-percent tax is adopted in this work. Batteries are charged and discharged as the same rate (i.e., 90%). The MATLAB software on a PC with 2.4 GHz Intel i5-4210U CPU and 4 GB RAM is used to implement different algorithms.Table 5
The constraints limit and bids of the DGs, utility, BES, and EVTs.
Type
Min. power (kW)
Max. power (kW)
Bid (€ct/kWh)
OM (€ct/kWh)
Start-up(€ct)
MT
6
30
0.457
0.0446
0.96
FC
3
30
0.294
0.08618
1.65
FCEV
3
30
0.294
0.08618
1.65
PV
0
25
2.584
0.2082
0
WT
0
15
1.073
0.5250
0
BES
−30
30
0.380
—
0
BEV
−30
30
0.380
—
0
PHEV
−30
30
0.380
—
0
Grid
−30
30
—
—
—Figure 6
SMG test system.The algorithms are solved 30 times to validate the performance of ASO. The population size and the maximum number of iterations used in ASO are 50 and 250, respectively. For comparison, SA, GA, and PSO are used. For SA, 50, 0.99, and 0.5 are considered as the population size, the temperature reduction rate, and mutation rate, respectively. For GA, 50, 0.7, and 0.1 are considered as population size, the crossover rate, and mutation rate, respectively. For PSO, 50, 1.5, and 2 are considered as population size, local learning coefficients, and global learning coefficients, respectively. The inertia weight damping ratio is selected as 0.99. The operation reserve and forecasted values of load demand are given in Figure7.Figure 7
(a) Forecasted load demand and (b) operating reserve capacity.
(a)
(b)In current work, two dissimilar scenarios are analyzed to describe the supremacy of the proposed framework.Case 1.
SMG with batteries (BES, BEV, and PHEV) in charging mode.Case 2.
SMG with batteries (BES, BEV, and PHEV) in discharging mode.


#### 4.1.1. Case1

For this scenario, the Li-ion batteries (an elementary component of the SMG) are appended in the form of BES, BEV, and PHEV in the SMG test system. The key advantage of the batteries in SMG is to sustain stability, make the RESs’ integration possible, and improve the quality of power [26–28]. The Li-ion batteries start during the time period when there is no charge; therefore, the discharging is limited to the charging in earlier hours. To validate the effectiveness of selected batteries with optimum capacity, maximum sizes of batteries (CBESmax,CBEVmax,CPHEVmax) are considered as the control parameter. The minimum and maximum capacity are set to 50 and 500 kWh, respectively. In [18], only CBESmax is considered but not as a variable. However, in this work, CBESmax,CBEVmax, and CPHEVmax are considered as control variables. From the above discussion, it is clear that the EOSMG has been solved for minimizing the total cost of operation. Additionally, the economical sizes of batteries as well as optimal output of MT, FC, PV, WT, FCEV, BEV, PHEV, BES, and grid are computed. Table 6 presents the optimal output of DGs, BES, EVTs, and utility calculated by ASO. The economical sizes of BES, BEV, and PHEV in this work are 50 kWh each. The total operation cost of EOSMG in Case 2 is 1673.1215 (€ct/day). A comparison of the results obtained by different algorithms, depicted in Table 7, proves the superiority of ASO. The comparison between convergence curves of ASO, SA, GA, and PSO is given in Figure 8. It is evident from Figure 8 that the proposed method provides a better solution with a high convergence rate which is useful to solve the difficult optimization problems.Table 6
Economical power output and corresponding status of DGs, BES, utility, and EVTS obtained by ASO for Case1.
Hours
Economic power output (kW)
Status
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
(
1
)
6.0000
3.0000
3.0000
0.0000
5.0974
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
2
)
6.0000
3.0000
3.0000
25.0000
0.0000
−30.0000
−13.5018
30.0000
8.9026
1
1
1
1
1
1
1
1
1
(
3
)
6.0000
3.0000
3.0000
0.0000
12.3713
30.0000
27.5859
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
4
)
6.0000
3.0000
30.0000
21.5209
0.0000
−30.0000
21.0944
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
5
)
6.0000
5.4715
3.0000
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
6
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
−30.0000
24.1224
−30.0000
1
1
1
1
1
1
1
1
1
(
7
)
19.5568
3.0000
3.0000
25.0000
0.0000
−30.0000
−30.0000
−30.0000
11.8776
1
1
1
1
1
1
1
1
1
(
8
)
6.0000
3.0000
30.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
9
)
6.0000
22.4326
3.0000
0.0000
0.0000
29.3998
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
10
)
30.0000
3.0000
10.9606
25.0000
0.0000
−20.6393
−2.4427
−13.6555
−20.1118
1
1
1
1
1
1
1
1
1
(
11
)
18.0594
3.1419
16.3532
0.0000
15.0000
30.0000
6.6595
−21.3787
−30.0000
1
1
1
1
1
1
1
1
1
(
12
)
10.2048
11.8372
9.4045
0.0000
0.0000
−30.0000
−11.8259
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
13
)
6.0000
3.0000
13.4757
0.0000
0.0000
−30.0000
−18.5644
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
14
)
6.0000
5.0709
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
15
)
14.3681
7.5256
3.0000
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
16
)
29.2283
4.5969
3.0000
15.7186
3.4992
30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
17
)
6.0000
30.0000
3.0000
0.0000
0.0000
30.0000
18.7101
−5.4274
−30.0000
1
1
1
1
1
1
1
1
1
(
18
)
6.0000
3.0000
16.9083
0.0000
15.0000
30.0000
−18.0426
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
19
)
6.0000
3.0000
7.4556
0.4552
4.2327
−25.8994
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
20
)
6.0000
20.3556
7.3230
0.0000
15.0000
−30.0000
30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
21
)
25.5465
3.0000
5.5935
3.3822
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
22
)
30.0000
3.0000
11.0158
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
23
)
6.0000
3.0000
3.0000
14.1539
0.0000
−30.0000
−10.3277
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
24
)
6.0000
4.5500
3.0000
0.0000
0.0000
30.0000
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1Table 7
Operation cost comparison for Case1 (30 trials), €ct/day.
Methodology
Bestsolution
Worstsolution
Averagesolution
Number of hits tooptimal solution
Standarddeviation
Number oftrials
Populationsize
Iterations
ASO
1673.1215
1764.3676
1690.75
26
15.603299
30
50
250
SA
1709.0489
1799.2431
1760.19
20
23.13305
30
50
250
GA
1693.493
1831.4027
1758.26
15
47.61386
30
50
250
PSO
1684.3069
1843.4294
1717.13
11
52.96354
30
50
250Figure 8
Convergence graph of ASO, SA, GA, and PSO for Case1.


#### 4.1.2. Case2

For this scenario, all the batteries are fully charged; the power outputs of MT, FC, BES, BEV, PHEV, FCEV, PV, WT, and power grid in the SMG are shown in Table8. Due to the economical power supplied by batteries, it is beneficial for EOSMG to purchase power from BES and EVTs. In this case, the system considers batteries of optimum size, 50 kWh, and the total operation costs will be 1882.362 (€ct/day). A comparison of the results computed by various techniques, depicted in Table 9, proves the superiority of ASO. For demonstrating efficiency of the ASO algorithm against SA, GA, and PSO, the convergence graph of these methods is shown in Figure 9. It is evident that the proposed method provides a better solution with a high convergence rate which is useful to solve difficult problems of optimization.Table 8
Economical power output and corresponding status of DGs, BES, utility, and EVTS obtained by ASO for Case2.
Hours
Economic power output (kW)
Status
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
(
1
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
−30.0000
−30.0000
0
0
1
1
1
1
1
1
1
(
2
)
6.0000
3.0000
3.0000
0.0000
11.9570
30.0000
−30.0000
−19.7768
−30.0000
0
0
1
1
1
1
1
1
1
(
3
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
10.6228
−11.0000
−30.0000
0
0
1
1
1
1
1
1
1
(
4
)
6.0000
3.0000
3.0000
0.0000
3.7847
19.0098
−1.0000
30.0000
30.0000
0
0
1
1
1
1
1
1
1
(
5
)
6.0000
3.0000
3.0000
0.0000
0.0000
25.0086
30.0000
0.2333
7.2153
0
0
1
1
1
1
1
1
1
(
6
)
6.0000
9.2127
3.0000
0.0000
0.0000
30.0000
12.2727
30.0000
30.0000
0
1
1
1
1
1
1
1
1
(
7
)
6.0000
7.5602
3.0000
0.0000
0.0000
−30.0000
30.0000
−30.0000
−30.0000
0
1
1
1
1
1
1
1
1
(
8
)
6.0000
3.0000
3.0000
0.0000
1.3623
−29.9667
−13.4169
9.9708
−30.0000
1
1
1
1
1
1
1
1
1
(
9
)
6.0000
3.0000
3.0000
0.0000
0.0000
−16.9160
26.4102
−28.6342
−30.0000
1
1
1
1
1
1
1
1
1
(
10
)
6.0000
3.0000
13.9159
0.0000
0.0000
−0.7986
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
11
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
12
)
6.0000
3.0000
4.8958
0.0000
0.0000
30.0000
−30.0000
11.9779
−24.0000
1
1
1
1
1
1
1
1
1
(
13
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
−18.7099
−30.0000
1
1
1
1
1
1
1
1
1
(
14
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
13.6822
−30.0000
1
1
1
1
1
1
1
1
1
(
15
)
6.0000
3.0000
3.0000
18.5151
0.0000
−0.4999
−30.0000
−9.5256
−30.0000
1
1
1
1
1
1
1
1
1
(
16
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
17
)
6.0000
3.0000
3.0000
0.0000
0.0000
−17.2863
−30.0000
30.0000
−20.0000
1
1
1
1
1
1
1
1
1
(
18
)
6.0000
30.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
10.0578
−30.0000
1
1
1
1
1
1
1
1
1
(
19
)
6.0000
4.3205
8.7349
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
20
)
6.0000
3.0000
3.3085
0.0000
6.2201
24.6153
30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
21
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
5.3846
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
22
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
25.3135
26.6865
−30.0000
1
1
1
1
1
1
1
1
1
(
23
)
6.0000
3.0000
14.9764
0.0000
0.0000
30.0000
30.0000
10.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
24
)
6.0000
3.0000
3.0000
0.0000
0.0000
22.0000
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1Table 9
Comparison of operation cost for Case2 (30 trials), €ct/day.
Methodology
Bestsolution
Worstsolution
Averagesolution
Number of hits tooptimal solution
Standarddeviation
Number oftrials
Populationsize
Iterations
ASO
1882.362
1983.628
1940.603
24
23.289825
30
50
250
SA
1958.546
2140.723
2015.669
10
49.667268
30
50
250
GA
1976.354
2042.508
2003.087
19
27.00037
30
50
250
PSO
1970.121
2041.74
1994.255
18
31.20696
30
50
250Figure 9
Convergence graph of ASO, SA, GA, and PSO for Case2.It can be concluded from the above discussion that the projected MG system, as an EOSMG, is widespread and sufficiently standard to be applied for an illustration of the inconsistency of RESs in the existence of optimal batteries sizing consideration.The PV and WT output generated by our approach can be used as input to stochastic programming algorithms such as the point estimation technique or Monte Carlo simulation [29–31]. The proposed algorithm can be executed with certain adjustments and achieve Pareto optimality [32]. The proposed method can also be used for incorporating EVTs for microgrid optimization.The above-discussed features of the proposed framework can be stated as one of its significant benefits with respect to other methods.The capacities of batteries affect both the fixed and the operational cost of SMG system. To verify the performance and illustrate the advantage of ASO over other methods, the minimum capacities of batteries are lowered to 5 kWh and the experiments are conducted for 30 trials with 250 iterations and 50 search agents in Case1 as well as Case 2. The results computed are stated in Tables 10 and 11.Table 10
Optimized capacities of batteries in Case1.
Methodology
Best
Mean
Worst
BES
BEV
PHEV
BES
BEV
PHEV
BES
BEV
PHEV
ASO
5
5
5
5
5
5
5
5
5
PSO
5.01
5.0447
5.0377
5.1638
5.1343
5.2627
5.4759
5.2772
5.6269
GA
5.0119
5.0232
5.0295
5.0748
5.1717
5.311
5.105
5.5354
5.6876
SA
5
5
5
5.0286
5.0921
5.079
5.0763
5.201
5.201Table 11
Optimized capacities of batteries in Case2.
Methodology
Best
Mean
Worst
BES
BEV
PHEV
BES
BEV
PHEV
BES
BEV
PHEV
ASO
5
5
5
5
5
5
5
5
5
PSO
5.0118
5.0109
5.008
5.0582
5.5696
5.0402
5.181
6.2371
5.0747
GA
5.0363
5.0736
5.0135
5.253
5.4516
5.3123
5.5914
5.9732
5.6427
RCSA
5
5
5
5.1389
5.0601
5.0432
5.472
5.2969
5.1448It is clear from Tables10 and 11 that the ASO algorithm computes consistently minimum batteries’ capacities for Cases 1 and 2, whereas PSO, GA, and RCSA are computing batteries’ capacities greater than the minimum capacity. As per the observation of Tables 10 and 11, it is concluded that ASO is capable of delivering optimal results with the least batteries’ capacities; this is one of the advantages of the proposed ASO.


## 4.1. Description of a SMG Test System

To evaluate the legitimacy and efficacy of the proposed framework, ASO is applied on an advanced low voltage SMG system as shown in Figure6. The SMG consists of various DGs such as PV, WT, MT, FC, and Li-ion BES. In addition, the proposed SMG system consists of EVTs (BEV, PHEV, and FCEV) technology. In this work, the FCEV is considered as the DG source. Table 5 presents the coefficients and generation limits which are used in the projected approach. If the SMG under study has MT, FC, FCEV, PV, WT, BES, BEV, and PHEV, then the position of mth search agent Xm can be defined as(10)Xm=xm,1xm,2……xm,DCBESmax,1,CBESmax,2,…,CBESmax,s1,CBEVmax,1,CBEVmax,2,…,CBEVmax,s2CPHEVmax,1,CPHEVmax,2,…,CPHEVmax,s3,PMT,1m,PMT,2m,…,PMT,TmPFC,1m,PFC,2m,…,PFC,Tm,PFCEV,1m,PFCEV,2m,…,PFCEV,TmPPV,1m,PPV,2m,…,PPV,Tm,PWT,1m,PWT,2m,…,PWT,TmPBES,1m,PBES,2m,…,PBES,Tm,PBEV,1m,PBEV,2m,…,PBEV,TmPPHEV,1m,PPHEV,2m,…,PPHEV,Tm,PGrid,1m,PGrid,2m,…,PGrid,TmuMT,1m,uMT,2m,…,uMT,Tm,uFC,1m,uFC,2m,…,uFC,TmuFCEV,1m,uFCEV,2m,…,uFCEV,Tm,uPV,1m,uPV,2m,…,uPV,TmuWT,1m,uWT,2m,…,uWT,Tm,uBES,1m,uBES,2m,…,uBES,TmuBEV,1m,uBEV,2m,…,uBEV,Tm,uPHEV,1m,uPHEV,2m,…,uPHEV,TmuGrid,1m,uGrid,2m,…,uGrid,Tm.More information regarding the implementation of SMG can be found in [18, 24, 25]. It is supposed that all of the DGs generate active power at the unity power factor. In this system, it is assumed that 465 (€ct/kW h) and 15 (€ct/kWh) are the fixed and maintenance cost of installation and operation of batteries [14]. The IR and LT for funding the installed batteries are 0.06 and 3, respectively. Ten-percent tax is adopted in this work. Batteries are charged and discharged as the same rate (i.e., 90%). The MATLAB software on a PC with 2.4 GHz Intel i5-4210U CPU and 4 GB RAM is used to implement different algorithms.Table 5
The constraints limit and bids of the DGs, utility, BES, and EVTs.
Type
Min. power (kW)
Max. power (kW)
Bid (€ct/kWh)
OM (€ct/kWh)
Start-up(€ct)
MT
6
30
0.457
0.0446
0.96
FC
3
30
0.294
0.08618
1.65
FCEV
3
30
0.294
0.08618
1.65
PV
0
25
2.584
0.2082
0
WT
0
15
1.073
0.5250
0
BES
−30
30
0.380
—
0
BEV
−30
30
0.380
—
0
PHEV
−30
30
0.380
—
0
Grid
−30
30
—
—
—Figure 6
SMG test system.The algorithms are solved 30 times to validate the performance of ASO. The population size and the maximum number of iterations used in ASO are 50 and 250, respectively. For comparison, SA, GA, and PSO are used. For SA, 50, 0.99, and 0.5 are considered as the population size, the temperature reduction rate, and mutation rate, respectively. For GA, 50, 0.7, and 0.1 are considered as population size, the crossover rate, and mutation rate, respectively. For PSO, 50, 1.5, and 2 are considered as population size, local learning coefficients, and global learning coefficients, respectively. The inertia weight damping ratio is selected as 0.99. The operation reserve and forecasted values of load demand are given in Figure7.Figure 7
(a) Forecasted load demand and (b) operating reserve capacity.
(a)
(b)In current work, two dissimilar scenarios are analyzed to describe the supremacy of the proposed framework.Case 1.
SMG with batteries (BES, BEV, and PHEV) in charging mode.Case 2.
SMG with batteries (BES, BEV, and PHEV) in discharging mode.


### 4.1.1. Case1

For this scenario, the Li-ion batteries (an elementary component of the SMG) are appended in the form of BES, BEV, and PHEV in the SMG test system. The key advantage of the batteries in SMG is to sustain stability, make the RESs’ integration possible, and improve the quality of power [26–28]. The Li-ion batteries start during the time period when there is no charge; therefore, the discharging is limited to the charging in earlier hours. To validate the effectiveness of selected batteries with optimum capacity, maximum sizes of batteries (CBESmax,CBEVmax,CPHEVmax) are considered as the control parameter. The minimum and maximum capacity are set to 50 and 500 kWh, respectively. In [18], only CBESmax is considered but not as a variable. However, in this work, CBESmax,CBEVmax, and CPHEVmax are considered as control variables. From the above discussion, it is clear that the EOSMG has been solved for minimizing the total cost of operation. Additionally, the economical sizes of batteries as well as optimal output of MT, FC, PV, WT, FCEV, BEV, PHEV, BES, and grid are computed. Table 6 presents the optimal output of DGs, BES, EVTs, and utility calculated by ASO. The economical sizes of BES, BEV, and PHEV in this work are 50 kWh each. The total operation cost of EOSMG in Case 2 is 1673.1215 (€ct/day). A comparison of the results obtained by different algorithms, depicted in Table 7, proves the superiority of ASO. The comparison between convergence curves of ASO, SA, GA, and PSO is given in Figure 8. It is evident from Figure 8 that the proposed method provides a better solution with a high convergence rate which is useful to solve the difficult optimization problems.Table 6
Economical power output and corresponding status of DGs, BES, utility, and EVTS obtained by ASO for Case1.
Hours
Economic power output (kW)
Status
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
(
1
)
6.0000
3.0000
3.0000
0.0000
5.0974
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
2
)
6.0000
3.0000
3.0000
25.0000
0.0000
−30.0000
−13.5018
30.0000
8.9026
1
1
1
1
1
1
1
1
1
(
3
)
6.0000
3.0000
3.0000
0.0000
12.3713
30.0000
27.5859
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
4
)
6.0000
3.0000
30.0000
21.5209
0.0000
−30.0000
21.0944
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
5
)
6.0000
5.4715
3.0000
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
6
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
−30.0000
24.1224
−30.0000
1
1
1
1
1
1
1
1
1
(
7
)
19.5568
3.0000
3.0000
25.0000
0.0000
−30.0000
−30.0000
−30.0000
11.8776
1
1
1
1
1
1
1
1
1
(
8
)
6.0000
3.0000
30.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
9
)
6.0000
22.4326
3.0000
0.0000
0.0000
29.3998
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
10
)
30.0000
3.0000
10.9606
25.0000
0.0000
−20.6393
−2.4427
−13.6555
−20.1118
1
1
1
1
1
1
1
1
1
(
11
)
18.0594
3.1419
16.3532
0.0000
15.0000
30.0000
6.6595
−21.3787
−30.0000
1
1
1
1
1
1
1
1
1
(
12
)
10.2048
11.8372
9.4045
0.0000
0.0000
−30.0000
−11.8259
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
13
)
6.0000
3.0000
13.4757
0.0000
0.0000
−30.0000
−18.5644
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
14
)
6.0000
5.0709
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
15
)
14.3681
7.5256
3.0000
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
16
)
29.2283
4.5969
3.0000
15.7186
3.4992
30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
17
)
6.0000
30.0000
3.0000
0.0000
0.0000
30.0000
18.7101
−5.4274
−30.0000
1
1
1
1
1
1
1
1
1
(
18
)
6.0000
3.0000
16.9083
0.0000
15.0000
30.0000
−18.0426
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
19
)
6.0000
3.0000
7.4556
0.4552
4.2327
−25.8994
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
20
)
6.0000
20.3556
7.3230
0.0000
15.0000
−30.0000
30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
21
)
25.5465
3.0000
5.5935
3.3822
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
22
)
30.0000
3.0000
11.0158
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
23
)
6.0000
3.0000
3.0000
14.1539
0.0000
−30.0000
−10.3277
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
24
)
6.0000
4.5500
3.0000
0.0000
0.0000
30.0000
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1Table 7
Operation cost comparison for Case1 (30 trials), €ct/day.
Methodology
Bestsolution
Worstsolution
Averagesolution
Number of hits tooptimal solution
Standarddeviation
Number oftrials
Populationsize
Iterations
ASO
1673.1215
1764.3676
1690.75
26
15.603299
30
50
250
SA
1709.0489
1799.2431
1760.19
20
23.13305
30
50
250
GA
1693.493
1831.4027
1758.26
15
47.61386
30
50
250
PSO
1684.3069
1843.4294
1717.13
11
52.96354
30
50
250Figure 8
Convergence graph of ASO, SA, GA, and PSO for Case1.


### 4.1.2. Case2

For this scenario, all the batteries are fully charged; the power outputs of MT, FC, BES, BEV, PHEV, FCEV, PV, WT, and power grid in the SMG are shown in Table8. Due to the economical power supplied by batteries, it is beneficial for EOSMG to purchase power from BES and EVTs. In this case, the system considers batteries of optimum size, 50 kWh, and the total operation costs will be 1882.362 (€ct/day). A comparison of the results computed by various techniques, depicted in Table 9, proves the superiority of ASO. For demonstrating efficiency of the ASO algorithm against SA, GA, and PSO, the convergence graph of these methods is shown in Figure 9. It is evident that the proposed method provides a better solution with a high convergence rate which is useful to solve difficult problems of optimization.Table 8
Economical power output and corresponding status of DGs, BES, utility, and EVTS obtained by ASO for Case2.
Hours
Economic power output (kW)
Status
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
(
1
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
−30.0000
−30.0000
0
0
1
1
1
1
1
1
1
(
2
)
6.0000
3.0000
3.0000
0.0000
11.9570
30.0000
−30.0000
−19.7768
−30.0000
0
0
1
1
1
1
1
1
1
(
3
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
10.6228
−11.0000
−30.0000
0
0
1
1
1
1
1
1
1
(
4
)
6.0000
3.0000
3.0000
0.0000
3.7847
19.0098
−1.0000
30.0000
30.0000
0
0
1
1
1
1
1
1
1
(
5
)
6.0000
3.0000
3.0000
0.0000
0.0000
25.0086
30.0000
0.2333
7.2153
0
0
1
1
1
1
1
1
1
(
6
)
6.0000
9.2127
3.0000
0.0000
0.0000
30.0000
12.2727
30.0000
30.0000
0
1
1
1
1
1
1
1
1
(
7
)
6.0000
7.5602
3.0000
0.0000
0.0000
−30.0000
30.0000
−30.0000
−30.0000
0
1
1
1
1
1
1
1
1
(
8
)
6.0000
3.0000
3.0000
0.0000
1.3623
−29.9667
−13.4169
9.9708
−30.0000
1
1
1
1
1
1
1
1
1
(
9
)
6.0000
3.0000
3.0000
0.0000
0.0000
−16.9160
26.4102
−28.6342
−30.0000
1
1
1
1
1
1
1
1
1
(
10
)
6.0000
3.0000
13.9159
0.0000
0.0000
−0.7986
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
11
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
12
)
6.0000
3.0000
4.8958
0.0000
0.0000
30.0000
−30.0000
11.9779
−24.0000
1
1
1
1
1
1
1
1
1
(
13
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
−18.7099
−30.0000
1
1
1
1
1
1
1
1
1
(
14
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
13.6822
−30.0000
1
1
1
1
1
1
1
1
1
(
15
)
6.0000
3.0000
3.0000
18.5151
0.0000
−0.4999
−30.0000
−9.5256
−30.0000
1
1
1
1
1
1
1
1
1
(
16
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
17
)
6.0000
3.0000
3.0000
0.0000
0.0000
−17.2863
−30.0000
30.0000
−20.0000
1
1
1
1
1
1
1
1
1
(
18
)
6.0000
30.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
10.0578
−30.0000
1
1
1
1
1
1
1
1
1
(
19
)
6.0000
4.3205
8.7349
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
20
)
6.0000
3.0000
3.3085
0.0000
6.2201
24.6153
30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
21
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
5.3846
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
22
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
25.3135
26.6865
−30.0000
1
1
1
1
1
1
1
1
1
(
23
)
6.0000
3.0000
14.9764
0.0000
0.0000
30.0000
30.0000
10.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
24
)
6.0000
3.0000
3.0000
0.0000
0.0000
22.0000
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1Table 9
Comparison of operation cost for Case2 (30 trials), €ct/day.
Methodology
Bestsolution
Worstsolution
Averagesolution
Number of hits tooptimal solution
Standarddeviation
Number oftrials
Populationsize
Iterations
ASO
1882.362
1983.628
1940.603
24
23.289825
30
50
250
SA
1958.546
2140.723
2015.669
10
49.667268
30
50
250
GA
1976.354
2042.508
2003.087
19
27.00037
30
50
250
PSO
1970.121
2041.74
1994.255
18
31.20696
30
50
250Figure 9
Convergence graph of ASO, SA, GA, and PSO for Case2.It can be concluded from the above discussion that the projected MG system, as an EOSMG, is widespread and sufficiently standard to be applied for an illustration of the inconsistency of RESs in the existence of optimal batteries sizing consideration.The PV and WT output generated by our approach can be used as input to stochastic programming algorithms such as the point estimation technique or Monte Carlo simulation [29–31]. The proposed algorithm can be executed with certain adjustments and achieve Pareto optimality [32]. The proposed method can also be used for incorporating EVTs for microgrid optimization.The above-discussed features of the proposed framework can be stated as one of its significant benefits with respect to other methods.The capacities of batteries affect both the fixed and the operational cost of SMG system. To verify the performance and illustrate the advantage of ASO over other methods, the minimum capacities of batteries are lowered to 5 kWh and the experiments are conducted for 30 trials with 250 iterations and 50 search agents in Case1 as well as Case 2. The results computed are stated in Tables 10 and 11.Table 10
Optimized capacities of batteries in Case1.
Methodology
Best
Mean
Worst
BES
BEV
PHEV
BES
BEV
PHEV
BES
BEV
PHEV
ASO
5
5
5
5
5
5
5
5
5
PSO
5.01
5.0447
5.0377
5.1638
5.1343
5.2627
5.4759
5.2772
5.6269
GA
5.0119
5.0232
5.0295
5.0748
5.1717
5.311
5.105
5.5354
5.6876
SA
5
5
5
5.0286
5.0921
5.079
5.0763
5.201
5.201Table 11
Optimized capacities of batteries in Case2.
Methodology
Best
Mean
Worst
BES
BEV
PHEV
BES
BEV
PHEV
BES
BEV
PHEV
ASO
5
5
5
5
5
5
5
5
5
PSO
5.0118
5.0109
5.008
5.0582
5.5696
5.0402
5.181
6.2371
5.0747
GA
5.0363
5.0736
5.0135
5.253
5.4516
5.3123
5.5914
5.9732
5.6427
RCSA
5
5
5
5.1389
5.0601
5.0432
5.472
5.2969
5.1448It is clear from Tables10 and 11 that the ASO algorithm computes consistently minimum batteries’ capacities for Cases 1 and 2, whereas PSO, GA, and RCSA are computing batteries’ capacities greater than the minimum capacity. As per the observation of Tables 10 and 11, it is concluded that ASO is capable of delivering optimal results with the least batteries’ capacities; this is one of the advantages of the proposed ASO.


## 4.1.1. Case1

For this scenario, the Li-ion batteries (an elementary component of the SMG) are appended in the form of BES, BEV, and PHEV in the SMG test system. The key advantage of the batteries in SMG is to sustain stability, make the RESs’ integration possible, and improve the quality of power [26–28]. The Li-ion batteries start during the time period when there is no charge; therefore, the discharging is limited to the charging in earlier hours. To validate the effectiveness of selected batteries with optimum capacity, maximum sizes of batteries (CBESmax,CBEVmax,CPHEVmax) are considered as the control parameter. The minimum and maximum capacity are set to 50 and 500 kWh, respectively. In [18], only CBESmax is considered but not as a variable. However, in this work, CBESmax,CBEVmax, and CPHEVmax are considered as control variables. From the above discussion, it is clear that the EOSMG has been solved for minimizing the total cost of operation. Additionally, the economical sizes of batteries as well as optimal output of MT, FC, PV, WT, FCEV, BEV, PHEV, BES, and grid are computed. Table 6 presents the optimal output of DGs, BES, EVTs, and utility calculated by ASO. The economical sizes of BES, BEV, and PHEV in this work are 50 kWh each. The total operation cost of EOSMG in Case 2 is 1673.1215 (€ct/day). A comparison of the results obtained by different algorithms, depicted in Table 7, proves the superiority of ASO. The comparison between convergence curves of ASO, SA, GA, and PSO is given in Figure 8. It is evident from Figure 8 that the proposed method provides a better solution with a high convergence rate which is useful to solve the difficult optimization problems.Table 6
Economical power output and corresponding status of DGs, BES, utility, and EVTS obtained by ASO for Case1.
Hours
Economic power output (kW)
Status
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
(
1
)
6.0000
3.0000
3.0000
0.0000
5.0974
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
2
)
6.0000
3.0000
3.0000
25.0000
0.0000
−30.0000
−13.5018
30.0000
8.9026
1
1
1
1
1
1
1
1
1
(
3
)
6.0000
3.0000
3.0000
0.0000
12.3713
30.0000
27.5859
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
4
)
6.0000
3.0000
30.0000
21.5209
0.0000
−30.0000
21.0944
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
5
)
6.0000
5.4715
3.0000
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
6
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
−30.0000
24.1224
−30.0000
1
1
1
1
1
1
1
1
1
(
7
)
19.5568
3.0000
3.0000
25.0000
0.0000
−30.0000
−30.0000
−30.0000
11.8776
1
1
1
1
1
1
1
1
1
(
8
)
6.0000
3.0000
30.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
9
)
6.0000
22.4326
3.0000
0.0000
0.0000
29.3998
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
10
)
30.0000
3.0000
10.9606
25.0000
0.0000
−20.6393
−2.4427
−13.6555
−20.1118
1
1
1
1
1
1
1
1
1
(
11
)
18.0594
3.1419
16.3532
0.0000
15.0000
30.0000
6.6595
−21.3787
−30.0000
1
1
1
1
1
1
1
1
1
(
12
)
10.2048
11.8372
9.4045
0.0000
0.0000
−30.0000
−11.8259
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
13
)
6.0000
3.0000
13.4757
0.0000
0.0000
−30.0000
−18.5644
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
14
)
6.0000
5.0709
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
15
)
14.3681
7.5256
3.0000
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
16
)
29.2283
4.5969
3.0000
15.7186
3.4992
30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
17
)
6.0000
30.0000
3.0000
0.0000
0.0000
30.0000
18.7101
−5.4274
−30.0000
1
1
1
1
1
1
1
1
1
(
18
)
6.0000
3.0000
16.9083
0.0000
15.0000
30.0000
−18.0426
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
19
)
6.0000
3.0000
7.4556
0.4552
4.2327
−25.8994
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
20
)
6.0000
20.3556
7.3230
0.0000
15.0000
−30.0000
30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
21
)
25.5465
3.0000
5.5935
3.3822
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
22
)
30.0000
3.0000
11.0158
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
23
)
6.0000
3.0000
3.0000
14.1539
0.0000
−30.0000
−10.3277
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
24
)
6.0000
4.5500
3.0000
0.0000
0.0000
30.0000
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1Table 7
Operation cost comparison for Case1 (30 trials), €ct/day.
Methodology
Bestsolution
Worstsolution
Averagesolution
Number of hits tooptimal solution
Standarddeviation
Number oftrials
Populationsize
Iterations
ASO
1673.1215
1764.3676
1690.75
26
15.603299
30
50
250
SA
1709.0489
1799.2431
1760.19
20
23.13305
30
50
250
GA
1693.493
1831.4027
1758.26
15
47.61386
30
50
250
PSO
1684.3069
1843.4294
1717.13
11
52.96354
30
50
250Figure 8
Convergence graph of ASO, SA, GA, and PSO for Case1.


## 4.1.2. Case2

For this scenario, all the batteries are fully charged; the power outputs of MT, FC, BES, BEV, PHEV, FCEV, PV, WT, and power grid in the SMG are shown in Table8. Due to the economical power supplied by batteries, it is beneficial for EOSMG to purchase power from BES and EVTs. In this case, the system considers batteries of optimum size, 50 kWh, and the total operation costs will be 1882.362 (€ct/day). A comparison of the results computed by various techniques, depicted in Table 9, proves the superiority of ASO. For demonstrating efficiency of the ASO algorithm against SA, GA, and PSO, the convergence graph of these methods is shown in Figure 9. It is evident that the proposed method provides a better solution with a high convergence rate which is useful to solve difficult problems of optimization.Table 8
Economical power output and corresponding status of DGs, BES, utility, and EVTS obtained by ASO for Case2.
Hours
Economic power output (kW)
Status
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
MT
FC
FCEV
SPV
WT
BES
EBEV
PHEV
Grid
(
1
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
−30.0000
−30.0000
0
0
1
1
1
1
1
1
1
(
2
)
6.0000
3.0000
3.0000
0.0000
11.9570
30.0000
−30.0000
−19.7768
−30.0000
0
0
1
1
1
1
1
1
1
(
3
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
10.6228
−11.0000
−30.0000
0
0
1
1
1
1
1
1
1
(
4
)
6.0000
3.0000
3.0000
0.0000
3.7847
19.0098
−1.0000
30.0000
30.0000
0
0
1
1
1
1
1
1
1
(
5
)
6.0000
3.0000
3.0000
0.0000
0.0000
25.0086
30.0000
0.2333
7.2153
0
0
1
1
1
1
1
1
1
(
6
)
6.0000
9.2127
3.0000
0.0000
0.0000
30.0000
12.2727
30.0000
30.0000
0
1
1
1
1
1
1
1
1
(
7
)
6.0000
7.5602
3.0000
0.0000
0.0000
−30.0000
30.0000
−30.0000
−30.0000
0
1
1
1
1
1
1
1
1
(
8
)
6.0000
3.0000
3.0000
0.0000
1.3623
−29.9667
−13.4169
9.9708
−30.0000
1
1
1
1
1
1
1
1
1
(
9
)
6.0000
3.0000
3.0000
0.0000
0.0000
−16.9160
26.4102
−28.6342
−30.0000
1
1
1
1
1
1
1
1
1
(
10
)
6.0000
3.0000
13.9159
0.0000
0.0000
−0.7986
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
11
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
12
)
6.0000
3.0000
4.8958
0.0000
0.0000
30.0000
−30.0000
11.9779
−24.0000
1
1
1
1
1
1
1
1
1
(
13
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
−18.7099
−30.0000
1
1
1
1
1
1
1
1
1
(
14
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
13.6822
−30.0000
1
1
1
1
1
1
1
1
1
(
15
)
6.0000
3.0000
3.0000
18.5151
0.0000
−0.4999
−30.0000
−9.5256
−30.0000
1
1
1
1
1
1
1
1
1
(
16
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
17
)
6.0000
3.0000
3.0000
0.0000
0.0000
−17.2863
−30.0000
30.0000
−20.0000
1
1
1
1
1
1
1
1
1
(
18
)
6.0000
30.0000
3.0000
0.0000
0.0000
−30.0000
−30.0000
10.0578
−30.0000
1
1
1
1
1
1
1
1
1
(
19
)
6.0000
4.3205
8.7349
0.0000
0.0000
−30.0000
−30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
20
)
6.0000
3.0000
3.3085
0.0000
6.2201
24.6153
30.0000
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
21
)
6.0000
3.0000
3.0000
0.0000
0.0000
−30.0000
5.3846
−30.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
22
)
6.0000
3.0000
3.0000
0.0000
0.0000
30.0000
25.3135
26.6865
−30.0000
1
1
1
1
1
1
1
1
1
(
23
)
6.0000
3.0000
14.9764
0.0000
0.0000
30.0000
30.0000
10.0000
−30.0000
1
1
1
1
1
1
1
1
1
(
24
)
6.0000
3.0000
3.0000
0.0000
0.0000
22.0000
−30.0000
30.0000
−30.0000
1
1
1
1
1
1
1
1
1Table 9
Comparison of operation cost for Case2 (30 trials), €ct/day.
Methodology
Bestsolution
Worstsolution
Averagesolution
Number of hits tooptimal solution
Standarddeviation
Number oftrials
Populationsize
Iterations
ASO
1882.362
1983.628
1940.603
24
23.289825
30
50
250
SA
1958.546
2140.723
2015.669
10
49.667268
30
50
250
GA
1976.354
2042.508
2003.087
19
27.00037
30
50
250
PSO
1970.121
2041.74
1994.255
18
31.20696
30
50
250Figure 9
Convergence graph of ASO, SA, GA, and PSO for Case2.It can be concluded from the above discussion that the projected MG system, as an EOSMG, is widespread and sufficiently standard to be applied for an illustration of the inconsistency of RESs in the existence of optimal batteries sizing consideration.The PV and WT output generated by our approach can be used as input to stochastic programming algorithms such as the point estimation technique or Monte Carlo simulation [29–31]. The proposed algorithm can be executed with certain adjustments and achieve Pareto optimality [32]. The proposed method can also be used for incorporating EVTs for microgrid optimization.The above-discussed features of the proposed framework can be stated as one of its significant benefits with respect to other methods.The capacities of batteries affect both the fixed and the operational cost of SMG system. To verify the performance and illustrate the advantage of ASO over other methods, the minimum capacities of batteries are lowered to 5 kWh and the experiments are conducted for 30 trials with 250 iterations and 50 search agents in Case1 as well as Case 2. The results computed are stated in Tables 10 and 11.Table 10
Optimized capacities of batteries in Case1.
Methodology
Best
Mean
Worst
BES
BEV
PHEV
BES
BEV
PHEV
BES
BEV
PHEV
ASO
5
5
5
5
5
5
5
5
5
PSO
5.01
5.0447
5.0377
5.1638
5.1343
5.2627
5.4759
5.2772
5.6269
GA
5.0119
5.0232
5.0295
5.0748
5.1717
5.311
5.105
5.5354
5.6876
SA
5
5
5
5.0286
5.0921
5.079
5.0763
5.201
5.201Table 11
Optimized capacities of batteries in Case2.
Methodology
Best
Mean
Worst
BES
BEV
PHEV
BES
BEV
PHEV
BES
BEV
PHEV
ASO
5
5
5
5
5
5
5
5
5
PSO
5.0118
5.0109
5.008
5.0582
5.5696
5.0402
5.181
6.2371
5.0747
GA
5.0363
5.0736
5.0135
5.253
5.4516
5.3123
5.5914
5.9732
5.6427
RCSA
5
5
5
5.1389
5.0601
5.0432
5.472
5.2969
5.1448It is clear from Tables10 and 11 that the ASO algorithm computes consistently minimum batteries’ capacities for Cases 1 and 2, whereas PSO, GA, and RCSA are computing batteries’ capacities greater than the minimum capacity. As per the observation of Tables 10 and 11, it is concluded that ASO is capable of delivering optimal results with the least batteries’ capacities; this is one of the advantages of the proposed ASO.


## 5. Conclusion

In this work, a new metaheuristic technique ASO is developed and applied as an effective method for solving the SMG economical optimization problem. An optimal model of SMG economical operation is developed with consideration of fuel, operation, and maintenance cost.Numerous observations made are as follows.The developed ASO algorithm shows less volatility as compared to other methods. The results obtained from unimodal and multimodal benchmark functions prove that the developed method stands either first or second with respect to GA, PSO, SMS, BA, FF, and SA in terms of convergence and average optimal value.The best performance of the proposed algorithm ASO can be observed on complex composite benchmark functions, which represent the complex practical problems. Hence, it can be concluded that the developed ASO algorithm is best suited for the complex constrained practical problems.The results obtained from the two case studies show that BES and EVTs can be incorporated within the SMG to decrease the cost. ASO algorithm computes consistently minimum batteries’ capacities for Cases1 and 2, whereas PSO, GA, and RCSA are computing batteries’ capacities greater than the minimum capacity. The reduction in the overall cost is because BES and EVTs are used to store the additional power of RESs and return it appropriately as feedback whenever needed. Also, BES and EVTs make the RESs operation more stable.The advantages of EVT are its mobility and ability to supply the stored power to the energy deficient areas during peak hours. The benefit of batteries is to handle the instability and intermittency of RESs. Therefore, the latest EVTs (BEV, PHEV, and FCEV) along with BES are used to store excess power during high availability in SMG.One of the open problems is to propose a new metaheuristic algorithm which can provide results better than those obtained in this work. The next futuristic enhancement may be to use other latest storage devices to improve the performance of SMG.



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*Source: 2158926-2017-10-08.xml*