The optimization field has grown tremendously, and new optimization techniques are developed based on statistics and evolutionary procedures. Therefore, it is necessary to identify a suitable optimization technique for a particular application. In this work, Black Widow Optimization (BWO) algorithm is introduced to minimize the cost functions in order to optimize the Multi-Area Economic Dispatch (MAED). The BWO is implemented for two different-scale test systems, comprising 16 and 40 units with three and four areas. The performance of BWO is compared with the available optimization techniques in the literature to demonstrate the strategy’s efficacy. Results show that the optimized cost for four areas with 16 units is found to be 7336.76$/h, whereas it is 121,589$/h for four areas with 40 units using BWO. It is also noted that optimization algorithms other than BWO require higher cost value. The best-optimized solution for emission is achieved at 9.2784e+06 tones/h, and it is observed that there is a considerable difference between the worst and the best values. Also, the suggested technique is implemented for large-scale test systems successfully with high precision, and rapid convergence occurs in MAED.
Black widow optimization algorithmmulti-objective multi-area economic dispatchemission optimizationcost optimizationIntroduction
The problem of Economic Dispatch (ED) and Emission Dispatch (EmD) are significant issues in power systems. The Combined Economic and Emissions Dispatch (CEED) challenge is characterized by a collection of solutions or a compromise solution that minimizes fuel costs and environmental emissions. This solution can extend to multiple locations, which operate together on several benefits such as increasing the availability and safe operations, reducing the investments in new power generation units, decreasing the maintenance costs and improving economic resource planning. The challenge of cost-effective load shipments in multi-area systems is presented here.
One of the primary goals of the system operators is to reduce the generation costs since it lowers the customer energy expenditures and the outcomes promote welfare. Demand Side Management (DSM) is a critical component of smart grids that may provide several advantages to power system operators and customers. The DSM has possible implications on electricity generating costs by considering a thorough and practical model of individual household loads. The consumption of various house hold appliances can be changed from the typical schedules to flexible loads using varied allowable delay durations [1]. Then a modern and improved variant, Novel Symbiotic Organisms Search (NSOS) is employed to satisfy the Multi-Area CEED (MA-CEED). To solve the MA-CEED, the NSOS algorithm is modified in [2] to generate better, more stable and more accurate alternatives than the original SOS algorithm. The reduction of fuel cost and total emissions are the major aims for ED multi-area systems. Although renewable energy supplies are increasingly penetrated, a significant percentage of the plant includes thermal energy units that utilize fossil fuel increases the environmental concerns. The MAED can determine the transferred power across the various locations by optimizing the overall costs. It computes each region’s optimum cost and accomplishes the best power streams from one area to another [3].
The purpose of maintaining a balance between exploration and operation phases is discussed in [4] to achieve high efficiency. The Dantzig-Wolfe decomposition approach solves the ED issue with limits of line flows and spinning reserve in a conventional linear optimization. It solves a linear problem independently by decomposing sub-programs that may relate to the physical sections of the power network [5]. A novel Generalized Unified Power Flow Controller (GUPFC) is discussed for the multi-area multi-fuel CEED issue. A systematic total power model for GUPFC with two series converters is designed to consider the converter switching losses. Then the multi-objective issue is solved using non-dominated solutions filtering approach and a Particle Swarm Optimization (PSO) algorithm. PSO is an optimization algorithm that optimizes the challenge by periodically seeking a candidate solution through the relation to a particular quality measure [6].
The finest Artificial Bee Colony Optimization (ABCO) algorithm can solve the MAED problem with tie-line constraints in [7]. ABCO is a swarm-based algorithm developed by honey bee food foraging behavioral patterns. It considers transmission losses, multiple fuels, valve-point loading, and Prohibited Operating Zones (POZs). A unit with POZs converts the regular ED into a non-convex optimization issue where the traditional approach cannot be used. It provides the characteristics of the discontinuous input and output of the power generation. A simple and efficient way is described in [8] for incorporating the area in power transfer limitations using the unit commitment and ED process. Also, a method for dispersing savings among participating firms in a pool on an equal basis is provided.
The black widow spider genus evolutionary theory is discussed in [9] with multiple species often described as actual widows [9]. The spiders are commonly referred to as black widow spiders and brown widow spiders based on the DNA sequences from the mitochondrial gene cytochrome c oxidase subunit I. The two well-supported mutually monophyletic clades within the genus are the geometricus clade (Latrodectusrhodesiensis from Africa and its sister species, the cosmopolitan L. geometricus), the mactans clade (taxa from various localities). Although the hybrid evolutionary programming scheme is used to solve MAED issues with different fuel alternatives by evolutionary programming, the Levenberg Marquardt optimization method is designed as an essential Evolutionary Programming (EP) in [10]. It is an introductory level search that determines the best global region’s direction and finds the best solution for the MAED issues. An excellent adaption of the PSO method is described in [11] to tackle various ED issues in power systems. It includes MAED with tie-line constraints, ED with diverse fuel alternatives, CEED and ED generators with POZ.
A decomposition method to solve the issue of multi-area generation scheduling is discussed in [12]. The goal is to keep the system’s operating costs as low as possible while still meeting the network limits. Early methods did not consider the generator nonlinearities, power pool topological linkages, or tie-line restrictions. Two-layer decomposition is employed to tackle the issues since it is a large-scale mixed-integer non-linear system model. An efficient direct search approach is discussed in [13] to address the ED issue while considering transmission capacity limits. It deals with various inequality and equality restrictions and units with fuel cost functions. To improve the performance of the direct search technique, a unique strategy is integrated with multi-level convergence. It reduces the total number of iterations in the searching process.
The CEED dispatch issue is discussed in [14], accounting for cost and emission reduction. It solves the challenging conflicting-objective function problem. The goal programming approaches are best suited to such challenges. The CEED dispatch problem is tackled using linear and non-linear goal programming techniques in this case. The effectiveness of various EP strategies for all types of ED challenges is explored in [15]. The three EP approaches under consideration are Gaussian, Cauchy, and combination Gaussian–Cauchy mutations.
A unified power flow controller has an effective and reliable evolutionary-based method to reduce fuel costs [16]. It operates within limits and is associated with transmission constraints that require less CPU time than other conventional techniques. The Shuffle Frog Leaping Algorithm (SFLA) is a memetic meta-heuristic algorithm that has been intended to solve stochastic optimization issues. The SFLA is a natural memetics-inspired population based cooperative search metaphor. It does a local independent searching in each memeplex simultaneously. The first SFLA frequently converges to local optima. A novel technique that takes advantage of the Simulated Annealing (SA) method to enhance local search around the global optima is discussed in [17]. It is a meta-heuristic algorithm used to estimate global optimum in a vast search process and increases the likelihood of convergence to global optima. The SLFA described in [18] addresses the multi-objective Optimum Power Flow (OPF) problem while considering the economic and pollution. Adding the emission objective to the OPF issue makes the situation more complicated than the previous one and requires an exact algorithm to solve the problems. The power loss and voltage variation are discussed in [19]. The Distribution Feeder Reconfiguration (DFR) is the main issue in the conventional distribution systems. In the modernized distribution systems, the operating problem has virtually ignored the security difficulties caused by the distributed generation, which might threaten power system security.
A new mapping approach for quadratic 0-1 programming problems and linear equality and inequality constraints are discussed in [20]. Hopfield neural networks are performed better while addressing the combinatorial optimization tasks. The modified Gee and Prager’s approach is used to remedy the ED with transmission capacity limits. A Chaotic Global Best Artificial Bee Colony (CGBABC) method is discussed in [21]. During the search for a global optimum, chaotic sequencing from the chaotic map is mapped to provide the design variables rather than the random number generator’s sequence. The CGBABC is used to solve the MAED issue by considering valve-point effects, transmission line losses, multi-fuel sources, POZs, tie-line capacity and power transfer system’s cost across various areas and compared with Network Flow Programming (NFP), Harmonic Search (HS), Hybrid HS (HHS) and Hybrid PSO (HPSO) [21]. A PSO technique is described in [22] for tackling the ED problem in power networks in which smooth cost functions are evaluated in an operational generator. The MAED calculates the amount of electricity generated cheaply in one location and transported into another to reduce the generation in the second area. The network flow model does not compute the actual power transmission between locations. A considerable number of the issue constraints may be phrased as network-type constraints, and as a result, the technique delivers a robust and high-speed solution [23].
Black Widow Optimization (BWO) is a novel meta-heuristic method [4] appropriate for continuous non-linear optimization problems. Cannibalism is an exclusive step in this approach. As a result of this stage, species with insufficient fitness are excluded from the circle, resulting in early convergence. The BWO algorithm is tested on 51 objective functions to ensure its efficacy in generating optimal solutions. A Quasi-Oppositional Group Search Optimization (QOGSO) algorithm is discussed in [24], which addresses the multi-area dynamic ED issue with different fuels and valve-point loading. A biologically realistic approach is a group search optimization, which is inspired by animal searching behavior [25]. QOGSO is utilized in this case to increase the solution’s efficacy and quality. An enhanced Multi-Objective PSO (MO-PSO) method is designed in [26] that provides a collection of Pareto-optimal solutions. Local search is employed in the MO-PSO to improve search efficiency.
The purpose of electric power brokerage networks developed in Florida is to minimize the actual electricity generation by conducting short-term transfers between the utilities (economy energy exchanges). Nevertheless, the present procedures for determining buyers and sellers and the quantities transferred may be improved to maximize the savings [27]. The optimization approaches based on meta-heuristic processes might help the power generation policy with the objective of reducing generation costs. In this respect, this analysis aims to provide a unique strategy for addressing ED issues based on the HS algorithm and to provide a practicable alternative to the existing methods [28]. The use of meta-heuristic algorithms to successfully tackle the problem of CEED with peak load control for a medium-sized power system is discussed in [29]. The cuckoo search and grasshopper optimization algorithms are used to solve the CEED issue through a composite function of four objectives with weight ratios and the cost penalty factors. It controls the peak load condition at the generating units of expensive locations, implemented in the IEEE 30-bus system with six generating units [30]. The MAED can reduce the total generation cost in multi-area power networks by obtaining active electricity from other economic power plants. Hence, in this work, the Meta-heuristic algorithm, BWO is employed to reduce the cost based on the black widow spider’s unique matching behavior.
Formulation of the Problem
The main objective of this research work is to develop a meta-heuristic algorithm to optimize the cost function. Here the cost functions we have considered are generating cost and emission cost. We had utilized the proposed algorithm to reduce the objective functions. In today’s world, power generation and distribution involve many men and materials. The resources have to be effectively and intelligently used. In this regard, the meta-heuristic-based BWO algorithm is utilized. By reducing the cost, we can effectively use the precious resources. As mentioned, the fuel cost and emission are the objective functions of the underlying problem, which are defined as follow:Mincost=∑r=1R∑s=1NrCrs(Prs)+∑r=1R∑s=r+1RfCrsTrk(Prs)
where Crs(Prs) is represented as cost. An electricity system with NU generating units and an R Area is deemed in various regions. Each area has Nr generating units. Trkis the tie-line power flow to area k from area r. For an area with surplus powers, it is essential to find a tie-line to transmit extra powers to a deficient area of power.cost=ars+brsPrs+crsPrs2where Prsis the powers flow of generating units s from areas r; Crsis the fuel cost function related with unit r from area s, which a quadratic polynomial function can express, and Trsand fcrs are the tie-line power flow to area s from area r and its equivalent generation cost function.cost=ars+brsPrs+crsPrs2+|drssin[ers(Prsmin−Prs)]|
where, ars, brs, crs are the factors of generating unit i in area s. ers and frs are constants of unit r from area s signifying the valve-point effects. Furthermore, the objective function is reduced subject to the subsequent limits [9]. The active powers outputs of the unit should be between their lower and upper limits.Prsmin≤Prs≤Prsmaxr=1,2…….Nr;S=1,2…….R
where, Prsmin and Prsmax are the actual power operational limits for unit r from area s. The tie-line real power flow to area r from area k (Trk) should be amongst the limits of tie-line power transmission capability.Trkmin≤Trk≤Trkmaxr=1,2…….R;k=2,….Rr≠k
where,Trkmax,Trkmin and are the maximum and minimum capabilities of the tie-line for the power transmission to area r from area k. The thermal generation unit ramp rate limits are as follows:max(Prsmin,Prso−Drs)≤Prs≤min(Prsmax,Prso−URrs)
The DRijand URij are the down and up ramp rate limits of unit r from s and Prsoare the real power output of unit r from area s. Inclining support from thermal generation would be a significant characteristic considering the enormous scope of unlimited usage and changing load shape. The central electricity authority’s technical standards for construction of the electrical plant and electric lines, Regulations 2010 endorse +/ - 3% each moment slope rate for coal terminated plants, and the Indian electricity grid code has arrangements requiring just +/ - 1% each moment incline rate as it are. The real incline rate given by the warm machines has been concentrated on dependent on authentic information accessible at Regional Load Dispatch Centres, and the National Load Dispatch Centre’s the report ready in such manner is encased as Annexe−1. On an All-India level, just around 35% of coal-terminated creating units (438 Nos.) have given the slope – Up/Down capacity of at minimum 1%/Min. In this manner, reasonable arrangements regarding execution observing concerning incline rate might be remembered for the central electricity regulatory commission’s terms and conditions of tariff, regulations, 2019.
The real power balance constraint so the system for areas r without considerations of networks loss can be provided.PGr=PDr+∑k,=1,k≠rNrTrkk=1,2,…Nr
where PGr is the overall engendered power in area r, PDr is the load requirement in area r.At the point when an unsettling influence happens in the framework (e.g., cut off, trip, etc.), it might bring about the awkwardness between the power generated and the full load power. On the off chance that the complete generated power is more than the load power (counting the network misfortunes), the frequency recurrence will rise; in any case, assuming the generated power influence is not precisely the load influence, the framework recurrence will fall. In light of the different recurrence variances and the genuine activity status of the framework, the related measures, primarily including managing the generator dynamic result, breaking down the generator, crumbling the heap, etc., will be considered.
Valve-Point Effect (VPE) & Multi-Fuel Operations (MFO) Cost Function
The stated cost objective function description is relevant for solo fuel units. Consequently, it is not applicable for multi-fuel objective generating units. Units can have varied fuels in real life, which can be described as the equation for their respective cost function. The VPE fuel cost feature, which may be expressed as an Equation, is another popular cost function model in MAED problems.MinFcost=∑m=1M∑n=1NmrFmn(Pmn)+∑m=1M∑n=m+1MFmnTmk(Pmn)Fcost=amn+bmnPmn+cmnPmn2
Considering the valve-points effect of the unit, the fuel costing functions can be described as follows Eq. (6). Fmn (Pmn) is represents as FcostFcost=amn+bmnPmn+cmnPmn2+|dmnsin[emn(Pmnmin−Pmn)]|
The power generation limits are as follows:Pmnmin≤Pmn≤Pmnmaxm=1,2,…..Nm,n=1,2…..M
where PmnminPmnmax are the actual power operational limits for unit m from area n using the above equation, the power generation limits can be easily estimated, and thereby proper calculations can be provided for the optimization algorithm. Since these are the main parameters that have to be considered for economic power dispatch issues, a little care has been provided. The thermal generation unit ramp rate limits are as follows:max(Pmnmin,Pmno−Dmn)≤Pmn≤min(Pmnmax,Pmno−URmn)
The real power balance are defined asPGm=PDm+∑k,=1,k≠mNmTmkk=1,2,…Nm
MA-CEED is a problem that optimizes two functions concurrently to minimize: Overall TotalCost [[P][T]] operations (or FuelCost [P] if transmission expenses are ignored) and emissions levels; (Emission[P]). One way of dealing with the MA-CEED issue is to make it a single target optimization problem [11].Ψ=w∙TotalCost[[P],[T]]+(1−w)∙∑i=1N∑j=1Mi(FGij(Pij,max)Eij(Pij,max))∙(Eij)(Pij)
where w is a weighting factor having [0,1] values. The FGij ratio is a scaling factor corresponding to the generator j in field i. FGij(Pij, max)/Eij. The two minimized functions (TotalCost and Emission) set competing targets. Hence, there are maximum emissions when the entire operating costs are minimized and vice versa. The two scenarios contribute to the determination of two extreme points in the space of the objectives (Total Cost, Emission). By resolving the problems of MA-ED and MA-EmD, the final points are separately defined. To acquire more points (items) from the Pareto front, which is defined as a group of non-dominated solutions in each objective is often seen as equally significant. The W factor is gradually adjusted from 0.1 to 0.9. The system operator has to select almost one operational point as a compromise between the two opposing objectives (Total Cost and Emission) [12,13]. Therefore, to help the system operator select what to do, the issue of categorizing solutions from the Pareto Front and determining the Best Compromising Solution (BCS) is recognized. Considering the Pareto-front known non-dominated solutions, this choice of BCS may be achieved via a fuzzy mechanism composed [14]. This is below:
Step 1. A value based on a linear membership function is generated for each goal I and each solution k from the P to front:μi,k={fimax−fi,kfimax−fiminfi,k≤fiminfimin<fi,k<fimaxi=1,2…o,k=1,2….Pfi,k≥fimax
The lowest and maximal magnitude of an ith objective function is represented by fi, min and fi.
Step 2. The normalized linear membership function m*k is computed as for each non-dominated solution k.:μk∗=∑i=1oμi,k∑i=1o∑i=1oμi,kk=1,2….,Pwhere P means the non-dominant number of alternatives, whereas O is the target number of functions.
Step 3.M * k max 1⁄4 max (m*k, k 1⁄4 1, 2,…, P) is the best-compromised option for this. Parameters of the non-dominated and dominated approach, the Pareto front, etc., are available in references.
Each generator, in actuality, has distinct POZs. Shaft bearing or other machine components such as pumps and boilers might harm from generating in these regions. In the operational area spectrum, the POZs are disconnected. From a technical standpoint, it is not possible or advisable to operate in POZs because the multiplication of vibrations in the shaft bearing may lead to volatile operational instability [10]. The POZs idea is created to prevent these instabilities. POZs lead to discontinuities, which are harmful to mathematical formulation such as the POZ equations 1.∀n∈Nm,∀m∈M,Pmn∈{Pmnmin≤Pmn≤Pmin1LPmnzU≤Pmn≤PminzLPmnzU≤Pmn≤Pmnmaxz=2,3….Z
Proposed BWO Algorithm
The suggested method begins with previous evolutionary algorithms with an original population of spiders to make each spider possible. These first spiders attempt to replicate the new generation in pairs [15]. The man is eaten during or after the match by a female Black Widow. Then it carries in her sperm thecae stored sperms and frees them into egg sackcloth. Spreads arise from the egg sacks as soon as 11 days when they are deposited [4]. For many days to a week, they live together on the maternal orca, during which cannibalism occurs. Afterward, they go by the wind.
4.1 BWO Algorithm
This technique for optimal BWO relies on three parameters: Reproductive Rate (RP), Cannibal Rate (CP), and Mutation Rate (MR). RP- By managing the generation of diverse offspring, this parameter provides additional variety and allows for a more detailed exploration of the search space. CP- It is a cannibalism operator’s regulating parameter, which excludes the incorrect individuals from the population. By agents of shifting search from global stage local and vice-versa, the appropriate parameter’s value can assure exploitation stage performance. MR- This parameter can govern the transition of search agents from the global stage to the local stage and urge them toward the optimal solution.
The breeding rate monitors the young spiders and helps to grow the search space to find a better answer. The cannibalism rate limits the lowest fit population in the iteration, while the following generation is only permitted with better fit populations [16]. The transformation rate regulates the diversity in the next generation. A pseudo-code is a simple form of describing programming that does not require any specific syntax of programming language. It can be used to develop such programming framework, which is a simple form of describing programming that does not require any specific syntax of programming language. It can be used to develop such programming framework, which is presented below in section 4.2; the principal phases of BWO have been summed up. The workflow of the BWO algorithm is shown in Fig. 1. The BWO algorithm is as follows:
Flowchart of the BWO algorithm
•Step 1:Initial population:
Selecting parents randomly for the procreation stage guarantees that the search domain is explored. The examination of the search field is also emphasized while producing many children in procreation.
•Step 2:Procreate:
The step to circumvent the local optima flap is by the BWO algorithm. The BWO algorithm draws attention from local optima, as it employs many search agents to calculate global optima.
•Step 3:Cannibalism:
Eliminating unsuitable options help BWO progress exceptionally quickly to the optimal. The cannibalism step assures excellent operating performance, ensuring rapid closure of the BWO process.
•Step 4:Mutation:
The transformation process confirms the equilibrium between the phases of exploitation and research. The mutation rate of the Mute group is computed.
•Step 5:Convergence:
Three stop circumstances can be considered, like previous evolutionary algorithms: (a) The number of iterations that have been preset. (b) No observance for multiple iterations of the fitness value of the best widow. (c) The level of precision required.
The BWO algorithm relies on three important metrics: Generation Rate RP, CP, and MR. The multiplication rate, which tells the sexual reproduction effect of the female spider controls the age, growth rate of youthful bugs and gives freedoms to investigate the quest space for tracking down a superior arrangement. The sibling flesh consumption rate controls the more vulnerable fittest populace in the age, and just the fittest populaces are considered the future. The change rate controls the variety in the current reproduction to the future generations. In this research work, we thought the above three main parameters to evaluate the fittest population and thereby decide the optimization level.
Pseudo Code for the Proposed Work
Input: Maximize the number of iterations, cannibalism, Mutation rate, etc.
Output: Optimal solution for the selected objective function
Initialization:
Select a random value as the initial population of Black Widow spiders
Loop until satisfactory optimization is achieved
2a. calculate the number of off-springs ‘nr’
2b. Select the best nr and its associate population as pop1
For i = 1 to nr, do
Randomly select two parents from pop1
Generate the off-spring D based on Eq. (1)
Remove the father
Randomly remove some off-spring to represent off-spring cannibalism
Save the result and name as pop2
End
Based on the removal (mutation), calculate the mutation rate ‘mr’
For 1to mr, Do
Select a solution from pop1
Apply mutation operation
Save the results as pop3
End
Update population by adding pop2 and pop3
Return the best solution
Simulations and Comparisons
The performances of the BWO proposed are evaluated by the four areas, along with a 4-area system with 16 generator units and a 4-area system with 40 generating units, to demonstrate the performance and capabilities of the proposed algorithm of hybrid PSO (HPSO) that combines the traditional PSO framework with the crossover operation of a genetic algorithm using the crossover operation in PSO, it not only inhibits early convergence to the local optimum, but it also efficiently explores and uses exciting locations in the search space. To validate the efficiency of the suggested strategy, numerical experiments on a large-scale 40 units test system with valve-point effects are carried out [28].
The Teaching Learning Based Optimization (TLBO) algorithm considers the impact of a teacher on students. The optimization technique is based on phase angles rather than design variables, which allows the non-linear aspects of the issue to be treated more effectively. A new learning strategy is presented to prevent being trapped in local optima [29].
The same values are utilized in comparative algorithms for a fair comparison with similar parameters. The MATLAB 7.1 on a core i5−4110U processor with 2, 40 GHz, or 4 GB RAM is used to build comparison methods and reproduce them for all the test systems for 30 separate runs.
Case I: The total cost of generating is individually reduced as an objective function.
Case II: Emission targets are reduced as a function
Case III: Total costs and emission targets are concurrently reduced.
Test System 1: 4 Areas with 16-Unit
To demonstrate the performance of the BWO method in addressing the MAED (Case I), MA-EmD (Case II), and MA-CEED (Case III) issues, it is applied to a medium size thermal system with 16 units grouping into four zones, each with four generating units. Six tie lines link the four areas together.
Tab. 1 displays data for the cost coefficients (a, b, c), power operating limits (Prs,min, Prs,max), transmission cost coefficient (Cip), and emission coefficients (α,β, γ, δ, λ).
The emission coefficients for the 16-unit system
Units
α(t/MW2h)
β(t/MWh)
γ(t/h)
δ(t/h)
λ(1/MW)
1
0.085
−3.08
80
1.31
0.0569
2
0.095
−1.98
100
1.42
0.0677
3
0.048
−2.22
60
1.28
0.0561
4
0.082
−1.89
50
0.99
0.0406
5
0.06
−2.67
121
1.23
0.0552
6
0.072
−2.13
97
1.38
0.0467
7
0.043
−2.29
65
1.24
0.0489
8
0.065
−1.72
45
1.12
0.0456
9
0.046
−2.58
75
1.25
0.0502
10
0.069
−2.25
98
1.52
0.0622
11
0.028
−2.59
70
1.38
0.0511
12
0.059
−1.63
80
0.87
0.0423
13
0.066
−1.98
63
1.39
0.0431
14
0.088
−2.35
70
1.46
0.0631
15
0.055
−2.78
100
1.51
0.0588
16
0.081
−2.43
40
1.27
0.0378
Tab. 2 provides the best answers by comparing algorithms and then the BWO receives the lowest cost of 7336. 76$/h, despite possible solutions identified by all the algorithms.
Results by proposed method with alternative resolution techniques on case I, II, III (16-unitssystem, 4 area, PD =1250 (32/16/28/24) (400, 200, 350, 300)
Methods
HPSO
NFP
HS
HHS
BWO cost
CGBABCCost
BWOEmi
CGBABCEmi
BWO BCS
CGBABCBCS
P1,1
150
150
150
150
150
150
66.5105
66.5207
126.845
126.8754
P1,2
100
100
100
100
100
100
52.8989
52.8691
86.5117
86.4816
P1,3
67.366
66.97
67.29
66.86
66.85
67.0142
79.2084
79.2482
67.5639
67.5640
P1,4
100
100
100
100
100
99.9999
76.3091
76.2889
92.9262
92.9259
P2,1
56.613
56.97
56.13
57.04
57
57.0015
77.5055
77.4657
90.6991
90.7287
P2,2
95.474
96.25
95.35
96.22
96
96.2596
77.7704
77.8002
81.4799
81.4698
P2,3
41.617
41.87
41.51
41.74
41.96
41.8803
93.194
93.2139
51.1958
51.2062
P2,4
72.356
72.52
71.74
72.5
72.55
72.5023
83.8338
83.8241
68.5503
68.5204
P3,1
50
50
50
50
50
50
90.1573
90.1575
61.7767
61.7564
P3,2
35.973
36.27
36.25
36.24
36
36.2553
63.7019
63.7014
48.9469
48.9570
P3,3
38.21
38.49
38.41
38.39
38.83
38.5029
94.5990
94.4779
54.6724
54.6623
P3,4
37.162
37.32
37.32
37.2
37.1
37.3107
94.4600
94.4319
41.7872
41.8075
P4,1
150
150
150
150
150
150
86.0000
86.1495
130.127
130.1275
P4,2
100
100
100
100
100
100
59.000
59.0007
86.6531
86.6829
P4,3
57.83
57.05
57.65
56.9
57.7
57.0077
73.1061
73.2077
69.2998
69.2595
P4,4
97.349
96.27
97.82
96.2
96
96.2650
81.7441
81.6419
90.9638
90.9740
T12
0
0
0.28
0
0
7.35−7
−100
−82.933
−26.15
−28.6319
T13
22.588
18.18
22.9
16.86
16.85
19.6978
−25.07
−52.139
0
2.4789
T14
−5.176
−1.21
−5.29
0
0
−2.6836
0
9.9999
0
0.000076
T23
66.064
69.73
65.09
70.61
71.31
68.233
32.15
−10.629
65.78
63.28892
T24
−0.004
−2.11
−0.08
−3.11
−3.4
−0.5891
0.15
60
0
0.00450
T43
100
100
100
100
100
100
0
70
77.04
77.04867
Cost
7336.93
7337.00
7333.26
7329.85
7336.76
7337.01
9702.59
9828.91
7724.69
7727.11
Emission
Pd = 1249.47
Pd = 1249.29
17178.35
17184.75
5696.65
5697.70
9566.19
9569.40
The results of the BWO are much better than those of other algorithms; compared to those of HPSO, NFP, HS, HHS and CGBABC [21], the results for reducing the emission-objective-functions. Whereas, the algorithm is superior CGBABC, which is nearer to the BWO also best compromising solution (BCS) between CGBABC and BWO are shown in Tab. 2.
Tab. 3 represents the analysis of the test system’s best, average and worst emission objective values, which provide the best, average, worst costs and BWO’s standard deviation of 30 separate runs, compared to the HPSO, NFP, HS, HHS and CGBABC. The comparative analysis shows that the BWO algorithm-based test system performs better with far smaller variations. Furthermore, in this test system, VPE, MFO and active transmission losses are considered. The capacity of the suggested method to handle various limitations of the issue is another essential aspect seen when one looks at the energy flow in ties and generator outputs in various locations. Then the existing algorithm for Case III resolution on a 16-unit test system is obtained through multi-area dispatch is comparatively shown in Tab. 4.
Analysis of the best, average and worst emission objective values on the 16-units (30 runs)
Algorithms
Best solution (tones/h)
Mean value (tones/h)
Worst solution (tones/h)
Standard deviation
BWO
1.3408 e+06
1.3440 e+06
1.3468 e+06
2.2218 e+03
HPSO-NFP
1.3418 e+06
1.3448 e+06
1.3474 e+06
2.2258 e+03
HS
1.3469 e+06
1.3479 e+06
1.3489 e+06
2.2997 e+03
HHS
1.3484 e+06
1.3495 e+06
1.3511 e+06
3.4578 e+03
CGBABC
1.3498 e+06
1.3530 e+06
1.3577 e+06
3.1168 e+03
Comparison among a suggested and existing algorithm for Case II resolution (16-units)
Method
Output active power of units (kW)
Power flows of Tie-lines (kW)
P1, 1
P1, 2
P1, 3
P1, 4
P2, 1
P2, 2
P2, 3
P3, 1
P3, 2
P3, 3
T21
T23
T23
BWO
250
230
420
265
403.27
265
218.55
265
210.56
200
100
99.90
97.25
HPSO-NFP
250
230
420.02
265
406.34
265
218.44
265
217.61
200.27
100
99.88
96.94
HS
250
230
420.17
265
358.67
265
210.47
265
269.64
201.59
100
99.73
44.43
HHS
250
230
420.05
265
344.42
252.27
236.85
265
258.01
213.34
100
99.85
44.51
CGBABC
250
230
420.20
265
325.62
264.82
227.24
265
276.36
210.02
100
99.71
29.62
Tab. 4 illustrates the recommended technique is 30 times autonomously of each other. This table displays the best results for the presented HPSO based NFP, HS, HHS [30] and CGBABC the value of the standard emission deviation. In addition, the poorest solution of the algorithms is presented. The conversions of the test system 1 algorithm are also quicker and have better search results than other techniques, which show that the suggested methodology. Among that the two techniques has (BWO and CGBABC) superior best solution. Moreover, the generation cost of various case studies produced by the suggested approach is shown in Fig. 2.
Analysis of test system 1
Test System 2: 4 Areas with 40-Units
There are four regions and 40 production units in the second test system. The data concerning the cost coefficients and power operating limitations are shown in Tab. 5. In addition, an ABCO, TLBO, GBABC are used in comparison with BWO to check further the effectiveness of the suggested method. Tab. 5 represents the overall Power Demand (PD) for the system is 10,500 MW, then the PD1, PD2, PD3, and PD4 are 1575, 4200, 3150 and 1575 respectively. Furthermore, the maximum flow limit of the region is 200 MW. The boundary uses 100 MW lines between regions. Whereas, the CPU time (sec) is mentioned in Tab. 6, which is the amount of time that the CPU spends processing data for a certain application or operation and it is also called as processing time. The comparative algorithms for case 2 solution methods for 40-unit test systems are shown in Tab. 6.
Comparisons between the suggested and existing case I, case II, algorithms on (40-unitsystem, 4 area PD =10, 500 (15/40/30/15)) test equipment are obtained from multi-area dispatching results
Objective
Cost minimization
Emission minimization
Best compromise solution
Methods
TLBO
ABCO
GBABC
BWO
GBABC
BWO
GBABC
BWO
P1, 1
110.879
111.102
110.80027
110.79899
114
114
110.79991
110.800121
P1, 2
112.955
109.977
110.80129
110.79999
114
114
110.80107
110.79924
P1, 3
97.4151
100.923
97.40028
97.39939
120
120
97.400527
97.39924
P1, 4
179.946
190
179.73309
179.743014
169.36759
169.35994
175.72731
175.73024
P1, 5
89.4955
96.939
93.27311
93.281254
97
97
87.901305
87.89924
P1, 6
139.893
96.9675
140
139.99831
124.25792
124.24883
105.40064
105.39964
P1, 7
259.733
259.695
259.59984
259.60245
299.71207
299.69201
259.60082
259.59941
P1, 8
284.638
276.872
284.59962
284.60014
297.91579
297.93809
284.58921
284.60014
P1, 9
284.741
300
284.60018
284.59931
297.26087
297.27323
284.59351
284.60013
P1, 10
130.115
130.697
130
130
130
130
130.00005
130
P2, 1
168.831
245.100
168.79977
168.80141
298.40939
298.39021
318.39508
318.40341
P2, 2
168.821
94
168.79982
168.80142
298.02460
298.01549
318.31958
318.29914
P2, 3
125.062
125
214.75990
214.75271
433.55916
433.54566
394.27940
394.26421
P2, 4
394.279
434.806
394.27939
394.27924
421.72623
421.74700
394.27934
394.26419
P2, 5
394.252
390.674
394.27947
394.27931
422.77879
422.77895
394.2807
394.27921
P2, 6
484.042
395.004
394.27934
394.27891
422.78048
422.77946
394.28060
394.30412
P2, 7
489.284
500
489.28041
489.28047
439.41285
439.3930
399.52272
399.49211
P2, 8
489.270
500
489.28014
489.27995
439.3999
439.42216
399.52129
399.53142
P2, 9
511.334
530.788
511.27940
511.28001
439.41489
439.40230
508.32639
508.29414
P2, 10
511.454
514.409
511.28095
511.27999
439.41204
439.42162
508.20962
508.19954
P3, 1
523.281
527.198
523.27969
523.28004
439.44981
439.44648
517.22598
517.24511
P3, 2
523.432
502.079
523.27931
523.28000
439.44526
439.44558
517.52154
517.53412
P3, 3
523.377
530.365
523.27940
523.28005
439.77281
439.78082
517.96121
517.95421
P3, 4
523.597
542.342
523.27937
523.27945
439.77060
439.77360
517.72197
517.72197
P3, 5
523.549
520.244
523.27949
523.27947
440.11122
440.10133
433.53137
433.52998
P3, 6
523.277
533.638
523.27936
523.27970
440.11129
440.11128
433.52085
433.51995
P3, 7
10.1442
10
10
10
28.99148
28.99348
10
10
P3, 8
10.0248
10
10
10
28.99102
28.99315
10.00014
10.00012
P3, 9
10.0862
10
10
10
28.99410
28.99344
10
10
P3, 10
88.2354
96.7699
87.82548
87.82444
97
97
87.81343
87.79984
P4, 1
189.919
190
162.32646
162.31524
172.33144
172.33157
159.80640
159.79958
P4, 2
189.971
168.684
189.99999
189.99999
172.33198
172.33249
159.87119
159.88484
P4, 3
190
173.616
190
190
172.33644
172.33117
159.88096
159.87354
P4, 4
164.892
186.374
164.79985
164.80041
200
200
200
200
P4, 5
165.134
200
164.80235
164.79998
200
200
200
200
P4, 6
165.232
164.957
164.80048
164.80241
200
200
200
200
P4, 7
90.2758
92.5627
89.11794
89.11801
100.83870
100.83821
89.14191
89.13472
P4, 8
109.981
96.9911
89.12062
89.11924
100.83834
100.83799
89.12031
89.11123
P4, 9
90.2019
109.815
89.12437
89.12521
100.84003
100.83776
89.11481
89.13547
P4, 10
458.937
431.401
511.27938
511.28000
439.41275
439.44341
421.53870
421.5999
T12
185.586
191.707
198.62448
198.63318
188.52190
199.96684
188.70936
188.69987
T31
23.6686
6.674
17.18298
17.17994
99.992352
88.55537
49.34195
49.34175
T32
183.086
183.185
167.4386
167.42175
143.44028
138.82542
108.99600
108.98345
T41
47.1037
86.859
99.99975
99.99985
100
100
67.55302
67.54357
T42
94.6933
95.3237
97.61822
97.62074
100
83.89262
90.87187
90.88354
T43
97.7497
57.2192
42.75351
43.00715
83.929721
100
35.04941
35.05071
PG1
-
-
1690.8077
1689.79145
1763.5142
1763.4922
1646.81438
1646.8254
PG2
-
-
3736.3186
3736.31450
4054.9183
4054.8959
4029.41476
4029.39499
PG3
-
-
3257.5021
3257.49754
2822.6376
2822.6691
3055.29653
3055.30145
PG4
-
-
1815.3714
1816.3951
1858.9297
1858.9426
1768.47431
1768.46451
Fuel Cost ($/h)
121760.
124009.
121595.833
121589.825
176682.264
176682.263
124576.576
124576.498
Emission (ton/h)
-
-
350449.698
350441.248
129995.189
129995.172
254489.640
254489.587
Comparative analysis of Case II: 40-unit test system results for multi-area dispatching
P1, 1
P1, 2
P1, 3
P1, 4
P1, 5
P1, 6
P1, 7
P1, 8
Area/Unit (kW)
103.92
111.13
117.14
157.66
84.63
137.12
213.48
225.22
P1, 9
P1, 10
P2, 1
P2, 2
P2, 3
P2, 4
P2, 5
P2, 6
217.90
239.82
374.25
374.25
381.49
222.89
291.92
237.84
P2, 7
P2, 8
P2, 9
P2, 10
P3, 1
P3, 2
P3, 3
P3, 4
499.00
499.00
516.30
548.91
311.79
547.64
359.01
446.76
P3, 5
P3, 6
P3, 7
P3, 8
P3, 9
P3, 10
P4, 1
P4, 2
494.58
458.94
149.70
142.30
147.17
88.71
190.00
190.00
P4, 3
P4, 4
P4, 5
P4, 6
P4, 7
P4, 8
P4, 9
P4, 10
188.85
175.50
200.00
171.06
110.00
69.68
90.80
413.62
Tie-line (kW)
T21
T41
T32
T31
T42
T43
Emission (tones/h)
CPU time (s)
102.16
200.00
200.00
130.86
48.02
72.53
9.2784EP06
78.19
Tab. 6 illustrates the Case II 40-unit test system emission from the technique suggested and the TLBO, ABCO and GBABC methodologies. A comparison between the best, medium and worst solutions found by the proposed method, with BWO being able to converge in a better solution and being rugged than three other algorithms of optimization, is shown in Tab. 7.
Emission objective values for 30 separate runs between the best, medium and worst
Algorithms
Best solution (tones/h)
Mean value (tones/h)
Worst solution (tones/h)
Standard deviation
BWO
9.2784e+06
9.6734e+06
9.7784e+06
1.0584e+06
TLBO
9.7738e+06
1.0142e+07
1.0609e+07
3.8297e+05
ABCO
1.0596e+07
1.0698e+07
1.0823e+07
1.1505e+05
GBABC
1.0639e+07
1.0839e+07
1.1076e+07
1.5538e+05
The minimal cost for fuel obtained by BWO, as seen in Tab. 5, is 121,589.8254 $/h, which is far less than the other large-scale MAED, MA-EmD solution. In particular, BWO is the least standard deviation, which indicates BWO is the most resilient of the algorithm. The BWO differs considerably from other algorithms. In addition, BWO shows higher search performance, as shown in Fig. 3.
The cost comparison of unit 40 test system
Fig. 3 demonstrates that the proposed method may converge to a superior solution that displays the potential to resolve the suggested MAED problem globally or virtually worldwide. While BWO may achieve pretty good test system solutions, the effectiveness of the BWO has yet been considerably improved and also compared to other algorithms. It is clear that BWO converges rapidly and steadily towards optimal alternatives, showing the durability and resilience of the BWO, where the BWO has the lowest cost of 121,589 $/h among the other algorithms.
Conclusions
In this paper, the BWO technique is implemented for tackling the MAED issue considering the total cost for the generation and emission as multi-objective issues. It also considers the actual power system restrictions such as tie-line capacity and unit limits. In a 40-unit test system, the total cost and emission of the generation are reduced, further enhancing the optimization outcomes. In contrast, the analysis of the best, average and worst emission objective values on the 16-unit system for 30 separate runs are also provided. A comparison among the suggested and existing algorithms for a 16-unit test and 40-unit test system is also provided. The results show that the BWO algorithm outperforms conventional algorithms. The optimized cost for four areas 16 units is found to be 7336.76$/h, whereas for four areas 40 units it is found to be 121,589$/h. In the future, the MA-CEED can be solved by other techniques like genetic algorithms and cuckoo search. A new meta-heuristic optimization technique will be interesting for future studies to tackle additional complex energy-related improvements such as the demand response parameter for smart grids and production for hydroelectric plants.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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