A novel population initialization strategy for accelerating Levy flights based multi-verse optimizer

Abstract

In this paper, we have designed a new optimization technique, which is named as the Improved Multi-verse Algorithm with Levy Flights (ILFMVO) algorithm. The quality of the population is an important factor that can directly or indirectly affect the strength of an algorithm in searching for the given search space for an optimal solution. Also, having an initialization of the initial population with randomly generated candidate solutions is not an effective idea in every case, especially when the search space is large. Hence, we have updated the Levy flights based Multi-verse Optimizer (LFMVO) by dividing initialization into two parts. To investigate the ability of ILFMVO, we have solved a constrained economic dispatch problem with a non-smooth, non-convex cost functions of three, six, and twenty thermal generator systems and two design engineering problems with nonlinear objectives and complex nonlinear constraints. We have compared our results with other standard algorithms. We have presented the sensitivity analysis to check the robustness and stability of our approach. The outcome demonstrated that ILFMVO has better accuracy, stability, and convergence.

Keywords

Antlion optimizer Economic load dispatch Design engineering problems Firefly algorithm Improved Multi-verse optimizer with Levy flights Lambda iteration Particle swarm optimization

1 Introduction

Economic load dispatch problem lies at an important position among different problems in power industry. It is a highly constrained and complex optimization problem. In the last few decades, several researchers have tried to solve ELD problems by incorporating several types of constraints combining them with multiple objectives.

Optimisation techniques like λ-iteration method [1], BPPF (base point and participation factors) method [1], Gradient based techniques [2, 3] required the input domain to be continuous. Resultantly, all previous techniques were unable to tackle problems in discrete domains, for example, problems with prohibited zones. By adding these difficulties it make ELD problems more complex and thus leading the algorithm to consume high computation time. ELD problem consisting of valve-point effects was solved by using Dynamic Programming (DP). Although, DP obtained promising results for problems with small dimensions but due to the lengthy solution procedure of DP, the method was vulnerable to solve higher dimension problem in ELD [4].

Furthermore, evolutionary computation and random search techniques such as Genetic Algorithm (GA) [5], Particle Swarm Optimization (PSO) [6, 7], Multi-verse Optimizer (MVO) [8, 9], Multi-verse Optimizer with Levy Flights (LFMVO) [10], have been implemented to solve the ELD problems at hand. Evolutionary algorithms were successfully implemented to different types of engineering problems in power industry, like Load Flow problems and hydro-electric generator tuning [11]. A co-evolutionary particle swarm optimization was used to solve engineering design problems [12], the temperature profile subject to the optimal design of porous fin model is discussed by using cuckoo search algorithm In addition, hybridized algorithms of GA with Simulated Annealing (SA) [13] and Tabu Search (TS) [14] were also proposed in [15]. With several modifications new evolutionary algorithms are developed to solve ELD problems. Despite of the fruitful results obtained by GA and others, they still face some weaknesses like higher computation time and slow convergence due to lack in balance between exploration and exploitation characteristics. Particularly, solving the problems with objective functions which include interrelated parameters. GA often converge to local optimum [15].

In order to overcome the difficulties faced by its predecessor methods in evolutionary computation, in this paper, we have proposed a novel population initialization strategy for accelerating LFMVO, to solve the ELD with a non-smooth cost functions. Which are constrained, quadratic with generator constraints, power losses and ramp rate limits. The results obtained from our proposed method ILFMVO are compared with λ-iteration, GA, PSO and others. Furthermore, ILFMVO is applied to solve design engineering problems to test its capability of handling complex objective functions and highly nonlinear constraints. The proposed methodology is proved to be a robust optimization technique for solving ELD problem for various search landscapes and power systems.

In the rest of the paper, section II contains mathematical formulation of ELD problems, section III contains MVO and LFMVO algorithms, section IV contains ILFMVO algorithms, section V contains simulation settings, results and discussion. Section VI contains conclusion.

2 Mathematical Formulation of ELD Problem

In this sections, ELD problem is formulated and discussed, where the resulting objective function(s) depend on several equality and inequality constraints. The transmission losses are the main issue to be minimized for the optimum load dispatch.

2.1 Objective function

The objective of ELD problem is to reduce the cost due to power generation which is formulated as an accumulative cost due to all power units deployed in the system. The objective function is quadratic with constant coefficients as in Eq. 1, $Min . C_{t} = \sum_{i = 1}^{n} α_{i} + β_{i} P_{i} + γ_{i} P_{i}^{2},$ (1) where C_t denotes the total cost and n represents the number of units. The coefficients in the quadratic objective functions are given by α_i, β_i and γ_i. P_i are decision variables representing i^th generating units. Another case of the objective function is formulated by considering the valve-point effects which is represented with a sinusoidal term given in Eq. 2,

$\begin{matrix} Min . C_{t} & = & \sum_{i = 1}^{n} α_{i} + β_{i} P_{i} + γ_{i} P_{i}^{2} \\ + | δ_{i} sin (ε_{i} (P_{i}^{\min} - P_{i})) |, \end{matrix}$ (2) where δ_i and ε_i are the values for coefficients of the valve-point for i^th unit respectively. $P_{i}^{\min}$ is the lower bound on power generation capacity of the i^th generator. Different types of fuel may be considered to operate a power generator. Mathematically, k fuel options can be represented as in Eq. 3, $C_{Ti} = {\begin{matrix} α_{i 1} + β_{i 1} P_{i} + γ_{i 1} P_{i}^{2} fuel 1 P_{i}^{\min} \leq P_{i} \leq P_{i 1}, \\ α_{i 2} + β_{i 2} P_{i} + γ_{i 2} P_{i}^{2} fuel 2 P_{i}^{\min} \leq P_{i} \leq P_{i 2}, \\ ⋮ \\ ⋮ \\ α_{iN} + β_{iN} P_{i} + γ_{iN} P_{i}^{2} fuel N P_{i}^{\min} \leq P_{i} \leq P_{iN}, \end{matrix}$ (3) where fuel cost function for the i^th generator is represented by C_Ti. The coefficients α_iN, β_iN and γ_iN are cost coefficients of the i^th unit using fuel type N. When multiple areas are considered for an ED problem, we get a collective cost for all areas as in Eq. 4, $Min . C_{T} = \sum_{i = 1}^{Na} C_{Ti},$ (4) where C_Ti represents the overall cost for an i^th and Na represents the total number of areas.

2.2 Constraints

Objective functions in the Equations (1 –4) are subject to the following types of constraints:

2.2.1 Constraint on generation of power

Generation capacity of each power generator is supposed between maximum and minimum bounds: $P_{i}^{\min} \leq P_{i} \leq P_{i}^{\max} .$ (5)

2.2.2 Power balance constraint

The power balance constraint is given by, $\sum_{i = 1}^{n} P_{i} = P_{D} + P_{L},$ (6) where P_D represents the total requirements of the power and P_L is power lost during transmission. For network losses, the B-coefficients is used as in [15], where the objective function can be written in the form,

$P_{L} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} P_{i} B_{ij} P_{j} + \sum_{i = 1}^{n} B_{0 i} P_{i} + B_{00} .$ (7)

3 Mathematical Formulation of Design Engineering Problems

In this section, two design optimization problems are formulated and discussed, where the resulting objective function(s) depend on several equality and inequality constraints. These problems are solved by using ILFMVO.

3.1 Welded beam design

The goal of the given problem to minimize the welded beam production cost. In this problem we have four unknowns parameters weld’s thickness (h), clamped bar’s length (l), bar’s height (t), bar’s thickness (b) and four constraints are beam’s end deflection (δ). shear stress (τ), beam’s bending stress (θ), bar’s buckling load (P_c), problem formulation is as follows: $Consider \vec{x} = [x_{1}, x_{2}, x_{3}, x_{4}] = [h, l, t, b]$ $\begin{matrix} Minimize & f (\vec{x}) = 1.10471 x_{1}^{2} x_{2} \\ + 0.04811 x_{3} x_{4} (14.0 + x_{2}) \\ Subject to & g_{1} (\vec{x}) = τ (\vec{x}) - τ_{max} \leq 0 \\ g_{2} (\vec{x}) = σ (\vec{x}) - σ_{max} \leq 0 \\ g_{3} (\vec{x}) = δ (\vec{x}) - δ_{max} \leq 0 \\ g_{4} (\vec{x}) = δ (\vec{x}) - δ_{max} \leq 0 \\ g_{5} (\vec{x}) = r_{1} - x_{4} \leq 0 \\ g_{6} (\vec{x}) = 0.125 - x_{1} \leq 0 \\ - 5 x_{1} \leq 2 \\ 0.1 \leq x_{2} \leq 10 \\ 0.1 \leq x_{3} \leq 10 \\ 0.1 \leq x_{4} \leq 2 \end{matrix}$ where $τ (\vec{x}) = \sqrt{{(τ^{'})}^{2} + 2 τ^{'} τ^{''} \frac{x_{2}}{2 R} + {(τ^{''})}^{2}}$ $\begin{matrix} τ^{'} = \frac{P}{\sqrt{2} x_{1} x_{2}}, τ^{''} = \frac{MR}{J}, M = P (L + \frac{x_{2}}{2}) \\ R = \sqrt{\frac{x_{2}^{2}}{4} + {(\frac{x_{1} + x_{3}}{2})}^{2}} \\ J = 2 {\sqrt{2} x_{1} x_{2} [\frac{x_{2}^{2}}{2} + {(\frac{x_{1} + x_{3}}{2})}^{2}]} \\ σ (\vec{x}) = \frac{4.013 E \sqrt{\frac{x_{3}^{2}}{36}}}{x_{4} x_{3}^{2}}, δ (\vec{x}) = \frac{6 P L^{3}}{E x_{3}^{2} x_{4}} \\ P_{c} (\vec{x}) = \frac{4.013 E \sqrt{\frac{x_{3}^{2} x_{4}^{6}}{36}}}{L^{2}} (1 - \frac{x_{3}}{2 L} \sqrt{\frac{E}{4 G}}) \\ E = 30 \times 10^{6} psi, G = 12 \times 10^{6} psi \\ τ_{\max} = 13, 600 psi, σ_{max} = 30, 000 psi . \end{matrix}$ Researchers use numerous algorithms to solve Welded beam design problems (for example Deb [16, 17] and Coello [18] worked on GA, HS used Lee and Geem [19], improved HS utilized by Mahdavi et al. [20], Kaveh and Khayatzad [21] worked on RO and Kaveh and Mahdavi [22] used CBO) while Radgsdell and Philips [23] adopted mathematical approaches like Simplex method, Richardson’s random method, Davidon Fletcher-Powell, Griffith and Stewart’s successive linear approximation. The results of ILFMVO are compared with the literature in Table 1. The statistical results of the algorithms are given in Table 2 for 30 independent runs. In which ILFMVO perform best result in minimum number of function evaluation and iteration.

Table 1
Comparison of ILFMVO optimization results with literature for the welded beam design problem

Algorithm Optimum variables Optimum

h l t b cost

ILFMVO 0.20564426 3.472578742 9.03662391 0.2057296 1.7250022

WOA 0.205396 3.484293 9.037426 0.206276 1.730499

GSA 0.182129 3.856979 10 0.202376 1.879952

CBO 0.2057220 3.470410 9.0372760 0.2057350 1.7246630

RO 0.2036870 3.5284670 9.0042330 0.2072410 1.7353440

Improved HS 0.205730 3.470490 9.03662 0.20570 1.72480

GA (Coello) N/A N/A N/A N/A 1.82450

GA (Deb) N/A N/A N/A N/A 2.38

GA (Deb) 0.24890 6.1730 8.17890 0.25330 2.43310

HS (Lee and Geem) 0.24420 6.22310 8.29150 0.24430 2.3807

Random 0.45750 4.73130 5.08530 0.660 4.11850

Simplex 0.27920 5.62560 7.75120 0.27960 2.53070

David 0.24340 6.25520 8.29150 0.24440 2.38410

APPROX 0.24440 6.21890 8.29150 0.24440 2.38150

Algorithm	Optimum variables	Optimum
ILFMVO	0.20564426	3.472578742	9.03662391	0.2057296	1.7250022
WOA	0.205396	3.484293	9.037426	0.206276	1.730499
GSA	0.182129	3.856979	10	0.202376	1.879952
CBO	0.2057220	3.470410	9.0372760	0.2057350	1.7246630
RO	0.2036870	3.5284670	9.0042330	0.2072410	1.7353440
Improved HS	0.205730	3.470490	9.03662	0.20570	1.72480
GA (Coello)	N/A	N/A	N/A	N/A	1.82450
GA (Deb)	N/A	N/A	N/A	N/A	2.38
GA (Deb)	0.24890	6.1730	8.17890	0.25330	2.43310
HS (Lee and Geem)	0.24420	6.22310	8.29150	0.24430	2.3807
Random	0.45750	4.73130	5.08530	0.660	4.11850
Simplex	0.27920	5.62560	7.75120	0.27960	2.53070
David	0.24340	6.25520	8.29150	0.24440	2.38410
APPROX	0.24440	6.21890	8.29150	0.24440	2.38150

Table 2

Comparison of ILFMVO statistical results with literature for the welded beam design problem

Algorithm	Average	Standard deviation	Function evaluation
ILFMVO	1.73190	0.02160	8500
WOA	1.732	0.0226	9900
PSO	1.7422	0.01275	13770
GSA	3.5761	1.2874	10750

3.2 Pressure vessel design

In the given problem our objective is to reduce overall cost (forming, material and welding) for cylindrical pressure vessel. The head of vessel has a hemi-spherical shape, while it’s both end are capped. In the given problem we have four unknowns parameters and four constraints, the unknowns parameters are shell’s thickness (T_s) , Head Thickness (T_h) , Internal radius (R) , length of cylindrical section without considering the head (L). Problem’s Mathematical formulation are as follows: $Consider \vec{x} = [x_{1}, x_{2}, x_{3}, x_{4}] = [T_{s}, T_{h}, R, L]$ $\begin{matrix} Minimize f (\vec{x}) & = 0.6224 x_{1} x_{3} x_{4} + 1.7781 x_{2} x_{3}^{2} \\ + & 3.1661 x_{1}^{2} x_{4} + 19.84 x_{1}^{2} x_{3} \\ Subject to & g_{1} (\vec{x}) = - x_{1} + 0.0193 x_{3} \leq 0 \\ g_{2} (\vec{x}) = - x_{3} + 0.00954 x_{3} \leq 0 \\ g_{3} (\vec{x}) = - π x_{3}^{2} x_{4} - \frac{4}{3} π x_{3}^{3} + \\ 1, 296, 000 \leq 0 \\ g_{4} (\vec{x}) = x_{4} - 240 \leq 0 \\ 0 \leq x_{2} \leq 99 \\ 10 & \leq x_{3} \leq 200 \\ 10 \leq & x_{4} \leq 200 . \end{matrix}$

For solving above test problem numerous researchers use metaheuristics like GA, ACO, ES, PSO, DE and improved HS [12 , 24–29] respectively. Some of researchers used mathematical methods just like lagrangian multiplier and branch-and-bound. Comparison of the obtained result with the literature are given in Table 3. ILFMVO is the proficient optimizer and perform best optimal result for the pressure vessel design problem. Table 4 shows the statistical results of pressure design problem on some of the algorithms, in which ILFMVO perform best result in minimum number of function evaluation and iteration.

Table 3
Comparison of WOA optimization results with literature for the pressure vessel design problem

Algorithm Optimum variables Optimum cost

Ts Th R L

ILFMVO 0.812500 0.437500 42.098446 176.636596 6059.714339

WOA 0.81250 0.43750 42.09826990 176.6389980 6059.7410

Improved HS 1.1250 0.6250 58.290150 43.692680 7197.730

GSA 1.1250 0.6250 55.98865980 84.45420250 8538.83590

PSO (He and Wang) 0.81250 0.43750 42.0912660 176.74650 6061.07770

GA (Coello) 0.81250 0.43450 40.32390 200.0 6288.74450

GA (Coello and Montes) 0.81250 0.43750 42.0973980 176.654050 6059.9463

GA (Deb and Gene) 0.93750 0.50 48.3290 112.6790 6410.38110

ES (Montes and Coello) 0.81250 0.43750 42.0980870 176.6405180 6059.7456

DE (Huang et al.) 0.81250 0.43750 42.0984110 176.637690 6059.7340

ACO (Kaveh and Talataheri) 0.81250 0.43750 42.1036240 176.5726560 6059.08880

(infeasible)

Lagrangian multiplier (Kannan) 1.1250 0.6250 58.2910 43.690 7198.04280

Branch-bound (Sandgren) 1.1250 0.6250 47.70 117.7010 8129.10360

Algorithm	Optimum variables	Optimum cost
ILFMVO	0.812500	0.437500	42.098446	176.636596	6059.714339
WOA	0.81250	0.43750	42.09826990	176.6389980	6059.7410
Improved HS	1.1250	0.6250	58.290150	43.692680	7197.730
GSA	1.1250	0.6250	55.98865980	84.45420250	8538.83590
PSO (He and Wang)	0.81250	0.43750	42.0912660	176.74650	6061.07770
GA (Coello)	0.81250	0.43450	40.32390	200.0	6288.74450
GA (Coello and Montes)	0.81250	0.43750	42.0973980	176.654050	6059.9463
GA (Deb and Gene)	0.93750	0.50	48.3290	112.6790	6410.38110
ES (Montes and Coello)	0.81250	0.43750	42.0980870	176.6405180	6059.7456
DE (Huang et al.)	0.81250	0.43750	42.0984110	176.637690	6059.7340
ACO (Kaveh and Talataheri)	0.81250	0.43750	42.1036240	176.5726560	6059.08880
					(infeasible)
Lagrangian multiplier (Kannan)	1.1250	0.6250	58.2910	43.690	7198.04280
Branch-bound (Sandgren)	1.1250	0.6250	47.70	117.7010	8129.10360

Table 4

Comparison of WOA statistical results with literature for the pressure vessel design problem

Algorithm	Average	Standard deviation	Function evaluation
ILFMVO	6065.21551	60.58393	5933
WOA	6068.050	65.65190	6300
PSO	6531.10	154.371600	14790
GSA	8932.950	683.54750	7110

4 Levy Flights Based Multi-verse Optimizer

Multi-Verse Optimizer (MVO) is a novel optimization technique proposed in [30]. MVO simulates the idea of the theory of multiple verse in physics. This theory is based on three facts: white, black and wormholes. The exploration of the search space is represented by phenomena of white hole and black hole. On the other hand, exploitation is achieved by implementing the concept of wormholes. In basic MVO technique, when the search is focused on the neighborhood of current best by perturbing a number of candidate universes around it. This tends the algorithm to get stuck in local optima.

In case, self-solutions are not improved by the universes in several iterations, then Levy flights are employed to increase diversity in the population. The algorithm is named as Levy flights based Multi-verse Optimizer (LFMVO) [10, 31]. The self-solutions are re-configured with Levy flights. Thus the most preferred universes achieved so far in the process remains unaffected. Levy flights are used in the following manner;

$\begin{matrix} U_{i}^{t + 1} & = & U_{i}^{t} + K \times ({Lb}_{i} + ({Ub}_{i} - {Lb}_{i}) \\ * Levy (x)) \times U_{i}^{t}, \end{matrix}$ (8) in the above equation,

K = The Levy weight factor which scales the creation of present universe based on the previous universe.

Lb = Lowest bound of the most feasible region.

Ub = The upper bound, in the region of maximum feasibility.

Global search and local search are facilitated through heavy Levy weight and smaller Levy weight, respectively. So the K component in the equation is of great importance for LFMVO in connection with its convergence behaviour. In order to balance global exploration with local exploitation and thus get more refined solutions, it is necessary to provide a suitable value of Levy weight. During the initial stages large value of Levy weight is used in order to tune the whole search area. But at later stages, for the sake of fine-tuning the area, small Levy weight is used. This factor K, is known as Adaptive Levy Weight Factor (ALWF) and it can be represented as following,

K = ({Max}_{-} Iter - t) / {Max}_{-} Iter,

(9)Max_-Iter = maximum iterations and t = present iteration.

It is a complicated process to create samples of step size s through use of Levy flights [32]. Mantegna algorithm is one of the most efficient and straight forward approach for the generation of these step sizes [33]. In this algorithm, the step size is given as, $s = \frac{u}{| v |^{\frac{1}{β}}},$ (10)u and v belongs to normal distributions, $u \sim N (0, σ^{2}), v = N (0, σ^{2}),$ (11) $σ = {(\frac{Γ (1 + β) \times \sin (\frac{π β}{2})}{Γ (\frac{1 + β}{2}) \times β \times 2^{\frac{β - 1}{2}}})}^{\frac{1}{β}}, σ_{v} = 1 .$ (12) Hence, the depiction of a simple scheme can be presented as, $Levy (x) = 0.01 \times \frac{u - σ}{| v |^{\frac{1}{β}}},$ (13)β = 1.5 (constant) and σ is calculated in Equation 12.

Existence probability of the wormholes can be represented by WEP in universes. During optimization process, to intensify the exploitation it linearly increases with respect to iterations. $WEP = W_{\min} + l \times (\frac{W_{\max} - W_{\min}}{L}),$ (14) where the minimum is (generally set to 0.2) represented by W_min, the maximum is (frequently set to 1) represented by W_max, the current iteration is l, and the maximum iteration is L.

TDR is defined to calculate the perturbation in the candidate solution, due to wormhole, around the current best solution. The rate of exploitation is directly proportional to the value of TDR over the iterations, $TDR = 1 - \frac{l^{\frac{1}{p}}}{L^{\frac{1}{p}}},$ (15) where p (set to 6 in this paper) indicates the exploitation accuracy over the iterations.

5 Our Proposed Initialization Strategy

According to the trend in stochastic techniques, randomization in each generation is used repeatedly to initialize algorithms without knowing any information about the problem at hand. The idea of a random initialization often prompts the algorithm to explore a well-defined domain while exploiting around the best solutions is not done with this type of initialization. We use an initialization approach in which LFMVO is balanced in terms of exploration and exploitation. We have developed a two-step startup strategy: in the first step, the algorithm is randomly initialized for a certain number of solutions scattered throughout the search domain.In the second step, when algorithm completing a number specified genration and accumulat the best points form the search space, we combine LFMVO with the population of these optimal solutions to focus the algorithm on exploiting the best solution in space. Therefore, in this article we have updated the LFMVO by dividing the algorithm’s capabilities into two parts. In the first part, the algorithm is initialized to evaluate a specific number of functions evaluations with a random population Equation 16

$U_{i} = {Lb}_{i} + ({Ub}_{i} - {Lb}_{i}) * rand (0, 1) .$ (16) In part second, algorithm focuses on the best universes that have been found so far during previous evaluations. This approach has proven to be extremely effective in getting the best results with fewer function evaluations and less time taken for solving test problems. ILFMVO algorithm’s graphical overview and pseudocode is given in Figure 1, 2.

Fig. 1

Pseudocode of ILFMVO algorithm.

Fig. 2

Graphical overview of ILFMVO algorithm.

6 Results and Comparison

To investigate the capability and competency of the proposed ILFMVO algorithm, standard case studies of ELD problems with valve-point, smooth and non-smooth quadratic cost functions, line flow constraints, loading effects, prohibited operating areas and ramp-rate limits are solved. These cases are individually explained in the following sections.

6.1 Case Study 1

Firstly, to examine the proposed method, it is applied to a standard ED problem. In order to do so, a power generation system of 3-unit is taken into consideration. The details of which are given in [34]. The input data and B_mn i.e. the loss coefficients are taken from [35] which is presented in Table 5.

$B_{mn} = 10^{- 5} [\begin{matrix} 7.1 & 3.0 & 2.5 \\ 3.0 & 6.9 & 3.2 \\ 2.5 & 3.2 & 8.0 \end{matrix}]$ For this case study, the objective function can be written as, $C_{1} = 0.0350 + 38.3050 P_{1} + 1243.5310 P_{1}^{2},$ $C_{2} = 0.0210 + 36.3270 P_{2} + 1658.5760 P_{2}^{2},$ $C_{3} = 0.0170 + 38.2700 P_{3} + 1356.6590 P_{3}^{2} .$ All constraints on the decision variables, active power constraints, are taken into account. Their lower and upper bounds of power generation ranges are given as under: $35.0 \leq P_{1} \leq 210.0,$ $130.0 \leq P_{2} \leq 325.0,$ $125.0 \leq P_{3} \leq 315.0 .$ For the generation cost, the performance data is presented in Table 6. The results obtained from our algorithm are compared against λ-iteration method [35], Firefly Algorithm (FFA) [35] and Antlion Optimizer (ALO) [36] in Table 7, Figure 4 and the rate of convergence is given in Figure 5.

Table 5
Generating unit data for case study 1

Unit α _i β_i γ _i Max Min

1 0.035 38.305 1243.531 35 210

2 0.021 36.327 1658.576 130 325

3 0.017 38.27 1356.659 125 315

Unit	α _i	β_i	γ _i	Max	Min
1	0.035	38.305	1243.531	35	210
2	0.021	36.327	1658.576	130	325
3	0.017	38.27	1356.659	125	315

Table 6

Case study 1 results with 600 MW total demand and loss

Unit generation output (MW)	FFA	ALO	LFMVO	ILFMVO
P1	130.02	130.02	150	161.56
P2	250.84	250.85	253.86	263.43
P3	236.43	236.44	213.55	191.83
Total	617.29	617.31	617.41	616.82
Ploss	17.3	17.2	17.41	16.83

Table 7

Performances of various algorithms (case study 1)

Sr.No	Method	Generation Cost ($)	CPU Time (s)
1	λ-iteration	30359.3	9.7
2	FFA	30334	10.2
3	ALO	30333.98	9.23
4	LFMVO	29865.63	9.87

Fig. 3

Sensitivity analysis of ILFMVO towards parameters p (used in TDR), universes (size of population), random probability, and WEP values.

Fig. 4

Case 1 results, comparison with different methods.

Fig. 5

Case Study 1, Fitness plot for ILFMVO, LFMVO and MVO.

Resultantly, ILFMVO algorithm shows solutions of high quality as compared to classical algorithms.

6.2 Case Study 2

In this case study, for load demand of 800 MW, we have considered a 6-generation unit system. The input data and B_mn i.e the loss coefficients are taken from [35] which is presented in Table 8. The objective function can be written as, $C_{1} = 0.152 + 38.539 P_{1} + 756.7988 P_{1}^{2},$ $C_{2} = 0.105 + 46.159 P_{2} + 451.3251 P_{2}^{2},$ $C_{3} = 0.028 + 40.396 P_{3} + 1049.997 P_{3}^{2},$ $C_{4} = 0.035 + 38.305 P_{4} + 1243.531 P_{4}^{2},$ $C_{5} = 0.021 + 36.328 P_{5} + 1658.559 P_{5}^{2},$ $C_{6} = 0.018 + 38.27 P_{6} + 1356.659 P_{6}^{2} .$ The unit operating ranges are: $10.0 \leq P 1 \leq 125.0,$ $10.0 \leq P 2 \leq 150.0,$ $35.0 \leq P 3 \leq 225.0,$ $35.0 \leq P 4 \leq 210.0,$ $130.0 \leq P 5 \leq 325.0,$ $125.0 \leq P 6 \leq 315.0 .$ For six generator system, the input data and B_mn i.e. the loss coefficients are taken from [35] and is given below. $B_{mn} = 10^{- 5} [\begin{matrix} 1.40 & 1.70 & 1.50 & 1.90 & 2.60 & 2.20 \\ 1.70 & 6.00 & 1.30 & 1.60 & 1.50 & 2.00 \\ 1.50 & 1.30 & 6.50 & 1.70 & 2.40 & 1.90 \\ 1.90 & 1.60 & 1.70 & 7.20 & 3.00 & 2.50 \\ 2.60 & 1.50 & 2.40 & 3.00 & 6.90 & 3.20 \\ 2.20 & 2.00 & 1.90 & 2.50 & 3.20 & 8.50 \end{matrix}]$

Table 8
Generating unit data for case study 2

Unit α _i β_i γ _i Max Min

1 0.152 38.539 756.7988 10 125

2 0.105 46.159 451.3251 10 150

3 0.028 40.396 1049.997 35 225

4 0.035 38.305 1243.531 35 210

5 0.021 36.328 1658.559 130 325

6 0.018 38.27 1356.659 125 315

Unit	α _i	β_i	γ _i	Max	Min
1	0.152	38.539	756.7988	10	125
2	0.105	46.159	451.3251	10	150
3	0.028	40.396	1049.997	35	225
4	0.035	38.305	1243.531	35	210
5	0.021	36.328	1658.559	130	325
6	0.018	38.27	1356.659	125	315

The results obtained from our proposed method are compared with conventional methods [36], Particle Swarm Optimization (PSO) [37], Lambda iteration [35], Firefly Algorithm (FFA) [35] and Antlion Optimizer (ALO) [36] in Table 9. The comparison for the generation cost is presented in Table 10 and Figure 6 and the convergence is given in Figure 7.

Table 9

Case study 2 results with 800 MW total demand and loss

Sr.No	Method	Generation Cost ($)	CPU Time (s)
1	PSO	41896.66	14.87
2	λ-iteration	41959	13.9
3	FFA	41896.9	14.5
4	ALO	41896.62	13.3
5	LFMVO	41765.86	13
6	ILFMVO	41572.23	12.8

Table 10

Performances of various algorithms (case study 2)

Sr.No	Method	Generation Cost ($)	CPU Time (s)
1	PSO	41896.66	14.87
2	λ-iteration	41959	13.9
3	FFA	41896.9	14.5
4	ALO	41896.62	13.3
5	LFMVO	41765.86	13
6	ILFMVO	41572.23	12.8

Fig. 6

Case 2 results, comparison with different methods.

Fig. 7

Case Study 2, Fitness plot for ILFMVO, LFMVO and MVO.

6.3 Case Study 3

This case study consists of 20 generation-unit power system for load demand of 2500 MW [38]. The cost function can be written as, $\begin{matrix} C_{1} & = 0.0007 + 18.19 P_{1} + 1000 P_{1}^{2}, \\ C_{2} & = 0.0007 + 19.26 P_{2} + 970 P_{2}^{2}, \\ C_{3} & = 0.0065 + 19.8 P_{3} + 600 P_{3}^{2}, \\ C_{4} & = 0.005 + 19.1 P_{4} + 700 P_{4}^{2}, \\ C_{5} & = 0.0074 + 18.1 P_{5} + 420 P_{5}^{2}, \\ C_{6} & = 0.0061 + 19.26 P_{6} + 360 P_{6}^{2}, \\ C_{7} & = 0.0079 + 17.14 P_{7} + 490 P_{7}^{2}, \\ C_{8} & = 0.0081 + 18.92 P_{8} + 660 P_{8}^{2}, \\ C_{9} & = 0.0052 + 18.27 P_{9} + 765 P_{9}^{2}, \\ C_{10} & = 0.0057 + 18.92 P_{10} + 770 P_{10}^{2}, \\ C_{11} & = 0.0048 + 16.69 P_{11} + 800 P_{11}^{2}, \\ C_{12} & = 0.0031 + 16.76 P_{12} + 970 P_{12}^{2}, \\ C_{13} & = 0.0085 + 17.36 P_{13} + 900 P_{13}^{2}, \\ C_{14} & = 0.0051 + 18.7 P_{14} + 700 P_{14}^{2}, \\ C_{15} & = 0.004 + 18.7 P_{15} + 450 P_{15}^{2}, \\ C_{16} & = 0.0712 + 14.26 P_{16} + 370 P_{16}^{2}, \\ C_{17} & = 0.0089 + 19.14 P_{17} + 480 P_{17}^{2}, \\ C_{18} & = 0.0071 + 18.92 P_{18} + 680 P_{18}^{2}, \\ C_{19} & = 0.0062 + 18.47 P_{19} + 700 P_{19}^{2}, \\ C_{20} & = 0.0077 + 19.79 P_{20} + 850 P_{20}^{2} . \end{matrix}$

The unit operating ranges are: $150 \leq P_{1} \leq 600,$ $50 \leq P_{2} \leq 200,$ $50 \leq P_{3} \leq 200,$ $50 \leq P_{4} \leq 200,$ $50 \leq P_{5} \leq 160,$ $20 \leq P_{6} \leq 100,$ $25 \leq P_{7} \leq 125,$ $50 \leq P_{8} \leq 150,$ $50 \leq P_{9} \leq 200,$ $30 \leq P_{10} \leq 150,$ $100 \leq P_{11} \leq 300,$ $150 \leq P_{12} \leq 500,$ $40 \leq P_{13} \leq 160,$ $20 \leq P_{14} \leq 130,$ $25 \leq P_{15} \leq 185,$ $20 \leq P_{16} \leq 80,$ $30 \leq P_{17} \leq 85,$ $30 \leq P_{18} \leq 120,$ $40 \leq P_{19} \leq 120,$ $30 \leq P_{20} \leq 100 .$ For twenty generator system, the input data and B_mn i.e. the loss coefficients are taken from [38] and is given in Table 11.

Table 11
Generating unit data for case study 3

Unit α _i β_i γ _i P _min P _max

1 0.0007 18.19 1000 150 600

2 0.0007 19.26 970 50 200

3 0.0065 19.8 600 50 200

4 0.005 19.1 700 50 200

5 0.0074 18.1 420 50 160

6 0.0061 19.26 360 20 100

7 0.0079 17.14 490 25 125

8 0.0081 18.92 660 50 150

9 0.0052 18.27 765 50 200

10 0.0057 18.92 770 30 150

11 0.0048 16.69 800 100 300

12 0.0031 16.76 970 150 500

13 0.0085 17.36 900 40 160

14 0.0051 18.7 700 20 130

15 0.004 18.7 450 25 185

16 0.0712 14.26 370 20 80

17 0.0089 19.14 480 30 85

18 0.0071 18.92 680 30 120

19 0.0062 18.47 700 40 120

20 0.0077 19.79 850 30 100

Unit	α _i	β_i	γ _i	P _min	P _max
1	0.0007	18.19	1000	150	600
2	0.0007	19.26	970	50	200
3	0.0065	19.8	600	50	200
4	0.005	19.1	700	50	200
5	0.0074	18.1	420	50	160
6	0.0061	19.26	360	20	100
7	0.0079	17.14	490	25	125
8	0.0081	18.92	660	50	150
9	0.0052	18.27	765	50	200
10	0.0057	18.92	770	30	150
11	0.0048	16.69	800	100	300
12	0.0031	16.76	970	150	500
13	0.0085	17.36	900	40	160
14	0.0051	18.7	700	20	130
15	0.004	18.7	450	25	185
16	0.0712	14.26	370	20	80
17	0.0089	19.14	480	30	85
18	0.0071	18.92	680	30	120
19	0.0062	18.47	700	40	120
20	0.0077	19.79	850	30	100

The results obtained from our proposed methods are compared and presented in Table 12 and Figure 8 with BBO [36], LI [38], Hidden Markov Model (HM) [38] and ALO method [36]. The rate of convergence is given in Figure 9.

Table 12

Results of case study 3 with 2500 MW total demand

Unit generation	BBO [21]	LI [23]	HM [23]	ALO [21]	LFMVO	ILFMVO
(MW)
P1	513.08920	512.78050	512.78040	512.780	528.450	600
P2	173.35330	169.10330	169.10350	169.110	176.740	200
P3	126.92310	126.88980	126.88970	126.890	146	200
P4	103.32920	102.86570	102.86560	102.870	55	50
P5	113.77410	113.63860	113.68360	113.680	136.50	160
P6	73.066940	73.5710	73.57090	73.5680	89.980	100
P7	114.98430	115.28780	115.28760	115.290	78.650	25
P8	116.42380	116.39940	116.39940	116.40	105.20	50
P9	100.69480	100.40620	100.40630	100.410	116.430	200
P10	99.999790	106.02670	106.02670	106.020	88.50	30
P11	148.9770	150.23940	150.23950	150.240	123.10	100
P12	294.02070	292.76480	292.76470	292.770	253.410	150
P13	119.57540	119.11540	119.11550	119.120	50.34	40
P14	30.547860	30.8340	30.83420	30.8310	132.430	130
P15	116.45460	115.80570	115.80560	115.81	143.670	185
P16	36.227870	36.25450	36.25450	36.2540	24.3	20
P17	66.859430	66.8590	66.8590	66.8570	69.70	85
P18	88.547010	87.9720	87.9720	87.9750	87.570	30
P19	100.98020	100.80330	100.80330	100.80	103.90	120
P20	54.27250	54.3050	54.3050	54.305	78.65	100
Total generation	2592.101	2591.9670	2591.9670	2591.9670	2588.550	2575
Transmission loss	92.1011	91.967	91.9669	91.9662	88.55	75

Fig. 8

Case 3 results, comparison with different methods.

Fig. 9

Case Study 3, Fitness plot for ILFMVO, LFMVO and MVO.

The outcome of our experiments suggests that ILFMVO is better than other state-of-the-art algorithms.

6.4 Case Study 4

For the load demand of 1263 MW, we have considered a 6 thermal generation units system. The cost function can be written as, $C_{1} = 240 + 7 P_{1} + 0.007 P_{1}^{2},$ $C_{2} = 200 + 10 P_{2} + 0.009 P_{2}^{2},$ $C_{3} = 220 + 8.5 P_{3} + 0.009 P_{3}^{2},$ $C_{4} = 200 + 11 P_{4} + 0.009 P_{4}^{2},$ $C_{5} = 220 + 10.5 P_{5} + 0.008 P_{5}^{2},$ $C_{6} = 190 + 12 P_{6} + 0.008 P_{6}^{2} .$ The unit operating ranges are: $500 \leq P_{1} \leq 100,$ $200 \leq P_{2} \leq 50,$ $300 \leq P_{3} \leq 80,$ $150 \leq P_{4} \leq 50,$ $200 \leq P_{5} \leq 50,$ $120 \leq P_{6} \leq 50 .$ All units have prohibited ramp-rate limits and operating zones. The values of the generation units are given in Table 14.

Table 13
Performances of various algorithms (case study 3)

S.No Method Generation Cost ($) CPU Time (s)

1 BBO 62456.77926 47.87

2 LI 62456.6391 40.43

3 HM 62456.63441 53.12

4 ALO 62456.63309 43.67

5 LFMVO 62404.45 42.34

6 ILFMVO 61410.5145 38.55

S.No	Method	Generation Cost ($)	CPU Time (s)
1	BBO	62456.77926	47.87
2	LI	62456.6391	40.43
3	HM	62456.63441	53.12
4	ALO	62456.63309	43.67
5	LFMVO	62404.45	42.34
6	ILFMVO	61410.5145	38.55

Table 14

Variable coefficients, upper and lower limits on variables for case study 4

Unit	α _i	β_i	γ _i	P _max	P _min	P _o	UR _i	DR _i	Prohibit zones
1	240	7	0.007	500	100	440	80	120	[210 240] [350 380]
2	200	10	0.009	200	50	170	50	90	[90 110] [140 160]
3	220	8.5	0.009	300	80	200	65	100	[150 170] [210 240]
4	200	11	0.009	150	50	150	50	90	[80 90] 110 120]
5	220	10.5	0.008	200	50	190	50	90	[90 110] [140 150]
6	190	12	0.008	120	50	110	50	90	[75 85] [100 105]

The B coefficients matrix with base capacity 100 MVA with network losses are given as follows: $B_{ij} = 10^{- 3} [\begin{matrix} 1.7 & 1.2 & 0.7 & - 0.1 & - 0.5 & - 0.2 \\ 1.2 & 1.4 & 0.9 & 0.1 & - 0.6 & - 0.1 \\ 0.7 & 0.9 & 3.1 & 0.0 & - 1 & - 0.6 \\ - 0.1 & 0.1 & 0.0 & 2.4 & - 0.6 & - 0.8 \\ - 0.5 & - 0.6 & - 0.1 & - 0.6 & 12.9 & - 0.2 \\ - 0.2 & - 0.2 & - 0.6 & - 0.8 & - 0.2 & 15.0 \end{matrix}]$

$\begin{matrix} B_{0 i} & = 0.001 \times \\ [- 0.3908 - 0.1297 0.7047 0.0591 0.2161 - 0.6635], \end{matrix}$ $B_{00} = 0.0056 .$ We have compared our results with GA [39], PSO [36], ESO [40], DE [41], HS [42] and HHS [43]. The results are presented in Table 15. Table 16 and Figure 10 shows the performance of our proposed method, Figure 11 shows the convergence for this case study.

Fig. 10

Case Study 4 results, comparison with different methods.

Fig. 11

Case Study 4: Fitness plot for ILFMVO, LFMVO and MVO.

7 Analysis and discussion

In the following sections, we have presented a detailed discussion on our results in terms of solution quality, efficiency of our proposed algorithm, its robustness and comparison of lowest cost obtained by ILFMVO with different algorithms.

7.1 Solution quality

From the experimental outcome it can be seen in Tables 7, 10, 13 and 16 that our proposed technique ILFMVO can beat other algorithms by producing lowest generation cost among other as given in literature [10 , 43]. Keeping in view the quality of results obtained by ILFMVO, it is evident that our algorithm can consistently solve large non-convex type of optimization problems. Also the results indicate that our algorithm can descend to global minima in a consistent way.

Moreover, the convergence characteristics of our algorithm, as indicated in Figures 5, 7, 9 and 11, is superior as compared to other optimization techniques.

7.2 Computational efficiency

The results given in Tables 6, 9, 12 and 15 dictates that ILFMVO has achieved the best costs for all four test case studies. As in Table 7, ILFMVO takes 8.92 sec as compared to λ-iteration, FFA, ALO and LFMVO with time taken as 9.7 sec, 10.2 sec, 9.23 sec and 9.87 sec respectively.

Table 15
Case Study 4 Results with 1263 MW total demand

Unit generation GA PSO ESO DE HS HHS LFMVO ILFMVO

P1 474.81 447.5 451.56 447.74 449.381 447.496 456 500

P2 178.64 173.32 173.44 173.41 173.53 173.314 173.5 200

P3 262.21 263.48 263.99 263.41 263.524 263.445 269.65 271.1

P4 134.28 139.06 147.46 139.08 132.049 139.055 133.76 118.52

P5 151.9 165.48 164.68 165.36 167.262 165.475 162.54 118.12

P6 74.18 87.13 71.32 86.94 90.262 87.125 77.65 62.29

Total 1276.03 1275.96 1272.46 1275.95 1276.01 1275.91 1273.1 1270.03

Ploss 13.02 12.96 12.82 12.96 13.08 12.95 10.1 7.035

Unit generation	GA	PSO	ESO	DE	HS	HHS	LFMVO	ILFMVO
P1	474.81	447.5	451.56	447.74	449.381	447.496	456	500
P2	178.64	173.32	173.44	173.41	173.53	173.314	173.5	200
P3	262.21	263.48	263.99	263.41	263.524	263.445	269.65	271.1
P4	134.28	139.06	147.46	139.08	132.049	139.055	133.76	118.52
P5	151.9	165.48	164.68	165.36	167.262	165.475	162.54	118.12
P6	74.18	87.13	71.32	86.94	90.262	87.125	77.65	62.29
Total	1276.03	1275.96	1272.46	1275.95	1276.01	1275.91	1273.1	1270.03
Ploss	13.02	12.96	12.82	12.96	13.08	12.95	10.1	7.035

Furthermore, ILFMVO took 12.8 sec as in Table 10 while PSO, λ-iteration, FFA, ALO and LFMVO are slower in solving the problem by taking 14.87 sec, 13.9 sec, 14.5 sec, 13.3 sec and 13.0 sec respectively.

In Table 13, again ILFMVO is superior in terms of time required to solve case study 3 by taking 38.55 sec as compared with times taken by BBO, LI, HM, ALO and LFMVO as 47.87 sec, 40.43 sec, 53.12 sec, 43.67 sec and 42.34 sec respectively.

Finally, case study 4 is solved with comparatively less time as 10.4 sec taken by ILFMVO and PSO, ESO, DE, HS, HHS, LFMVO with 10.4 sec, 11.49 sec, 17.8 sec, 11.49 sec, 11.58 sec, 10.4 sec, 10.8 sec respectively. It is important to note that GA was better in terms of time taken to solve this case study.

We have found that ILFMVO solved the test problems faster than other techniques in less time. The time taken by ILFMVO is either less or comparable to others mentioned techniques Tables 7, 10, 13 and 16. Thus we conclude that ILFMVO is efficient in terms of computational time it takes to solve the test case studies.

Table 16

Performances of various algorithms (case study 4)

Method	Max cost ($).	Min cost($).	Average cost ($)	CPU time (s)
GA	15224	15459	15469	8.16
PSO	15492	15450	15454	11.49
ESO	15470	15408	15430	17.8
DE	15450	15450	15450	11.49
HS	15449	15449	15449	11.58
HHS	15453	15449	15450	10.4
LFMVO	15440	15420	15430	10.8
ILFMVO	15424.47	15424.47	15424.47	10.4

7.3 Robustness

Random initialization of population or initial solutions is an inherent property of stochastic optimization algorithms. We have balanced the randomization and local search in ILFMVO by initializing the algorithm in two stages:

In the first stage, the algorithm is initialized randomly for certain number of function evaluations. After a limit of generations are completed and the algorithm produces sufficient number of best so far solutions in each iteration then the algorithm move to the next stage. In second phase, the ILFMVO is initialized with a collection of best so far solutions obtained during previous iterations. This strategy prevents the algorithm to stuck in local optima as well as randomization improves the global search capability of the algorithm. This strategy gets the best solution in lower number of trials and reduces the computational time. We have fixed the maximum number of trials as taken by other techniques in the literature to check the consistency of ILFMVO algorithm.

Tables 7, 10, 13 and 16 lists the best costs for all case studies along with the time taken by each algorithm. In all these results ILFMVO wins by producing consistently lower cost as shown in Figures 5, 7, 9, 11.

7.4 Comparison of best generation cost

The best solutions calculated by ILFMVO for case study 1 are compared with available results, taken from [35, 36], in Table 7. It can be seen that ILFMVO produced the minimum generation cost as 29584.14 $/hour by comparing it to λ-iteration, FFA, ALO, LFMVO as 30359.3 $/hour, 30334.0 $/hour, 30333.98 $/hour, 29865.63 $/hour respectively.

Similarly the result obtained by ILFMVO for case study 2 is shown in Table 10 and it is 41572 $/hour which is minimum cost as compared with PSO, λ-iteration, FFA, ALO, LFMVO with 41896.66 $/hour, 41959.0 $/hour, 41896.9 $/hour, 41896.62 $/hour, 41765.86 $/hour respectively.

For case study 3, the outcome of our experiment is presented in Table 13 which show that ILFMVO have lowest cost 61410.5145 $/hour as compared to BBO, λ-iteration, HM, ALO, LFMVO with 62456.77926 $/hour, 62456.6391 $/hour, 62456.63441 $/hour, 62456.63309 $/hour, 62404.45 $/hour respectively.

At last, for case study 4 the result of our proposed technique in terms of generating minimum cost is shown in Table 16. ILFMVO obtained the generating cost as 15424.47 $/hour and it is compared with GA, PSO, ESO, DE, HS, HHS and LFMVO as 15469 $/hour, 15454 $/hour, 15430 $/hour, 15450 $/hour, 15449 $/hour and 15450 $/hour respectively.

Hence the above discussion dictates that ILFMVO found superior results in terms of performance, robustness, and time complexity as compared to other methods taken from literature.

8 Conclusion

In current studies, we have improved the LFMVO by initializing it with an adaptive population strategy. The performance of our algorithm is significantly improved, and the exploration and exploitation capabilities of the algorithm are balanced.

Moreover, the quality of the population is an important factor that can directly or indirectly affect the strength of an algorithm in searching the given domain for an optimal solution. Also, having an initialization process with a random generation of candidate solutions will not be an effective idea in every case, especially when the search spaces are large. Hence we have updated the LFMVO by dividing the capabilities of the algorithm into two parts.

The algorithm initializes with a fixed random population for a certain number of function evaluations.

The algorithm is focused on a population of best so far spots found during the earlier evaluations. This strategy is shown to be very efficient in getting the best results with a less number of function evaluations and time.

We have selected an economic dispatch problem to check the performance of the proposed algorithm. Which consists of four different case studies. One with three units of generating stations, two with six units of generating stations, and one with twenty units of generating stations.

Our results show that the ILFMVO method provides solutions for high quality with less cost and less computational time. In this case, the performance of the ILFMVO is comparatively better than the other algorithms.

Conflict of interest

All the authors of this manuscript declare that they have no conflict of interest.

Funding

This project was supported by KMUTT Fixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand.

References

Wood

A.J.

and Wollenberg

B.F.

, Power generation, operation, and control, John Wiley & Sons, 2012.

Snyman

J.A.

and Wilke

D.N.

, Newgradient-based trajectory and approximation methods, in: Practical Mathematical Optimization, Springer, 2018, pp. 197–250.

Pahnehkolaei

S.M.A.

, Alfi

, Sadollah

and Kim

J.H.

, Gradient-based water cycle algorithm with evaporation rate applied to chaos suppression, Applied Soft Computing 53 (2017), 420–440.

Noorbin

S.F.

and Alfi

, Adaptive parameter control of search group algorithm using fuzzy logic applied to networked control systems, Soft Comput 22(23) (2018), 7939–7960. doi:10.1007/s00500-017-2742-0. URL https://doi.org/10.1007/s00500-017-2742-0

Kadri

R.L.

and Boctor

F.F.

, An efficient genetic algorithm to solve the resource-constrained project scheduling problem with transfer times: The single mode case, European Journal of Operational Research 265(2) (2018), 454–462.

Zhou

, Yang

, Lin

, Zhu

and Ji

, Local best particle swarm optimization using crown jewel defense strategy, in: Critical Developments and Applications of Swarm Intelligence, IGI Global, (2018), pp. 27–52.

Shahri

E.S.A.

, Alfi

and Machado

J.T.

, Fractional fixedstructureh? controller design using augmented lagrangian particle swarm optimization with fractional order velocity, Applied Soft Computing 77 (2019), 688–695.

Pei

, Zhao

, Yang

, Cao

and Gong

, Design optimization of a srm motor by a nature-inspired algorithm: Multi-verse optimizer, in: 2018 13th IEEE Conference on Industrial Electronics and Applications (ICIEA), IEEE, (2018)), pp. 1870–1875.

Mousavi

and Alfi

, Fractional calculus-based firefly algorithm applied to parameter estimation of chaotic systems, Chaos, Solitons & Fractals 114 (2018), 202–215.

10.

, Li

, Zhou

, Zhu

and Xu

, A multi-verse optimizer with levy flights for numerical optimization and its application in test scheduling for network-on-chip, PloS one 11(12) (2016), e0167341.

11.

Gomez-Exposito

, Conejo

A.J.

and Canizares

, Electric energy systems: analysis and operation, CRC press, 2018.

12.

and Wang

, An effective co-evolutionary particle swarm optimization for constrained engineering design problems, Engineering Applications of Artificial Intelligence 20(1) (2007), 89–99.

13.

Leno

I.J.

, Sankar

S.S.

and Ponnambalam

, Mip model and elitist strategy hybrid ga–sa algorithm for layout design, Journal of Intelligent Manufacturing 29(2) (2018), 369–387.

14.

Qiu

, Fu

, Eglese

and Tang

, A tabu search algorithm for the vehicle routing problem with discrete split deliveries and pickups, Computers & Operations Research 100 (2018), 102–116.

15.

Chen

P.-H.

and Chang

H.-C.

, Large-scale economic dispatch by genetic algorithm, IEEE Transactions on Power Systems 10(4) (1995), 1919–1926.

16.

Deb

, Optimal design of a welded beam via genetic algorithms, AIAA Journal 29(11) (1991), 2013–2015.

17.

Deb

, An efficient constraint handling method for genetic algorithms, Computer Methods in Applied Mechanics and Engineering 186(2-4) (2000), 311–338.

18.

Coello Coello

C.A.

, Constraint-handling using an evolutionary multiobjective optimization technique, Civil Engineering Systems 17(4) (2000), 319–346.

19.

Lee

K.S.

and Geem

Z.W.

, A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice, Computer Methods in Applied Mechanics and Engineering 194(36-38) (2005), 3902–3933.

20.

Mahdavi

, Fesanghary

and Damangir

, An improved harmony search algorithm for solving optimization problems, Applied Mathematics and Computation 188(2) (2007), 1567–1579.

21.

Kaveh

and Khayatazad

, A new meta-heuristic method: ray optimization, Computers & Structures 112 (2012), 283–294.

22.

Kaveh

and Mahdavi

V.R.

, Colliding bodies optimization: a novel meta-heuristic method, Computers & Structures 139 (2014), 18–27.

23.

Ragsdell

and Phillips

, Optimal design of a class of welded structures using geometric programming (1976).

24.

Coello

C.A.C.

and Montes

E.M.

, Constraint-handling in genetic algorithms through the use of dominance-based tournament selection, Advanced Engineering Informatics 16(3) (2002), 193–203.

25.

Coello

C.A.C.

, Use of a self-adaptive penalty approach for engineering optimization problems, Computers in Industry 41(2) (2000), 113–127.

26.

Deb

, Geneas: A robust optimal design technique for mechanical component design, in: Evolutionary algorithms in engineering applications, Springer, 1997, pp. 497–514.

27.

Kaveh

and Talatahari

, An improved ant colony optimization for constrained engineering design problems, Engineering Computations 27(1) (2010), 155–182.

28.

Mezura-Montes

and Coello

C.A.C.

, An empirical study about the usefulness of evolution strategies to solve constrained optimization problems, International Journal of General Systems 37(4) (2008), 443–473.

29.

, Huang

, Liu

and Wu

, A heuristic particle swarm optimizer for optimization of pin connected structures, Computers & Structures 85(7-8) (2007), 340–349.

30.

Mirjalili

, Mirjalili

S.M.

and Hatamlou

, Multi-verse optimizer: a nature-inspired algorithm for global optimization, Neural Computing and Applications 27(2) (2016), 495–513.

31.

Osman

and Baki

M.F.

, A cuckoo search algorithm to solve transfer line balancing problems with different cutting conditions, IEEE Transactions on Engineering Management (2018).

32.

Yang

X.-S.

and Deb

, Multiobjective cuckoo search for design optimization, Computers & Operations Research 40(6) (2013), 1616–1624.

33.

Mantegna

R.N.

, Fast, accurate algorithm for numerical simulation of levy stable stochastic processes, Physical Review E 49(5) (1994), 4677.

34.

Sydulu

, A very fast and effective noniterative“/spl lambda/-logic based” algorithm for economic dispatch of thermal units, in: TENCON 99. Proceedings of the IEEE Region 10 Conference, Vol. 2, IEEE, (1999), pp. 1434–1437.

35.

Reddy

K.S.

and Reddy

M.D.

, Economic load dispatch using firefly algorithm, International Journal of Engineering Research and Applications 2(4) (2012), 2325–2330.

36.

Nischal

M.M.

and Mehta

, Optimal load dispatch using ant lion optimization, Int J Eng Res Appl 5(8) (2015), 10–19.

37.

Abbas

, Gu

, Farooq

, Asad

M.U.

and El-Hawary

, Solution of an economic dispatch problem through particle swarm optimization: A detailed survey-part i, IEEE Access 5 (2017), 15105–15141.

38.

C.-T.

and Lin

C.-T.

, New approach with a hopfield modeling framework to economic dispatch, IEEE Transactions on Power Systems 15(2) (2000), 541–545.

39.

Sivanandam

and Deepa

, Genetic algorithm optimization problems, in: Introduction to Genetic Algorithms, Springer, (2008), pp. 165–209.

40.

Xie

Y.M.

and Steven

G.P.

, Basic evolutionary structural optimization, in: Evolutionary Structural Optimization, Springer, 1997, pp. 12–29.

41.

Rocca

, Oliveri

and Massa

, Differential evolution as applied to electromagnetics, IEEE Antennas and Propagation Magazine 53(1) (2011), 38–49.

42.

Kim

J.H.

, Geem

Z.W.

and Kim

E.S.

, Parameter estimation of the nonlinear muskingum model using harmony search 1, JAWRA Journal of the American Water Resources Association 37(5) (2001), 1131–1138.

43.

Wang

, Pan

Q.-K.

and Tasgetiren

M.F.

, A hybrid harmony search algorithm for the blocking permutation flow shop scheduling problem, Computers & Industrial Engineering 61(1) (2011), 76–83.

A novel population initialization strategy for accelerating Levy flights based multi-verse optimizer

Abstract

Keywords

1 Introduction

2 Mathematical Formulation of ELD Problem

2.1 Objective function

2.2.1 Constraint on generation of power

3.1 Welded beam design

6.1 Case Study 1

Table 5 Generating unit data for case study 1 Unit α i β i γ i Max Min 1 0.035 38.305 1243.531 35 210 2 0.021 36.327 1658.576 130 325 3 0.017 38.27 1356.659 125 315

Table 8 Generating unit data for case study 2 Unit α i β i γ i Max Min 1 0.152 38.539 756.7988 10 125 2 0.105 46.159 451.3251 10 150 3 0.028 40.396 1049.997 35 225 4 0.035 38.305 1243.531 35 210 5 0.021 36.328 1658.559 130 325 6 0.018 38.27 1356.659 125 315

Table 13 Performances of various algorithms (case study 3) S.No Method Generation Cost ($) CPU Time (s) 1 BBO 62456.77926 47.87 2 LI 62456.6391 40.43 3 HM 62456.63441 53.12 4 ALO 62456.63309 43.67 5 LFMVO 62404.45 42.34 6 ILFMVO 61410.5145 38.55

7.1 Solution quality

7.2 Computational efficiency

7.4 Comparison of best generation cost

8 Conclusion

Conflict of interest

Funding

References

Table 5
Generating unit data for case study 1

Unit α _i β_i γ _i Max Min

1 0.035 38.305 1243.531 35 210

2 0.021 36.327 1658.576 130 325

3 0.017 38.27 1356.659 125 315

Table 8
Generating unit data for case study 2

Unit α _i β_i γ _i Max Min

1 0.152 38.539 756.7988 10 125

2 0.105 46.159 451.3251 10 150

3 0.028 40.396 1049.997 35 225

4 0.035 38.305 1243.531 35 210

5 0.021 36.328 1658.559 130 325

6 0.018 38.27 1356.659 125 315

Table 13
Performances of various algorithms (case study 3)

S.No Method Generation Cost ($) CPU Time (s)

1 BBO 62456.77926 47.87

2 LI 62456.6391 40.43

3 HM 62456.63441 53.12

4 ALO 62456.63309 43.67

5 LFMVO 62404.45 42.34

6 ILFMVO 61410.5145 38.55