
Citation: Zhi-xin Zheng, Jun-qing Li, Hong-yan Sang. A hybrid invasive weed optimization algorithm for the economic load dispatch problem in power systems[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2775-2794. doi: 10.3934/mbe.2019138
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Economic dispatch (ED) [1] in power systems is an important issue for obtaining the steady-state and economic operations of systems that is a typical constrained optimization problem with multiple variables. The optimization goal of the ED problem is to determine the most economic power outputs of generators while satisfying multiple constraints, such as the generation capacity limits, power demand balance, network transmission losses, ramp rate limits and prohibited operating zones. Considering the valve-point effects (VPE) of multivalve steam turbines for the ED problem, the objective cost function is a nonlinear and nonconvex function, which is hard to solve [2]. Especially in large-scale power systems with multiple generators, the ED problem is a complex optimization problem with several local optimal solutions, and thus the global optimal solution is hard to find.
In recent years, several optimization algorithms, including conventional algorithms and meta-heuristic algorithms, have been proposed to solve the ED problems. Some conventional algorithms, such as linear programming (LP) [3], self-adaptive dynamic programming (SADP) [4], iterative dynamic programming (IDP) [1] and evolutionary programming (EP) [5], have been applied to solve the ED problems. These methods solve the ED problems using the simplified optimization model in which the valve-point effects, ramp rate limits, prohibited operating zones and transmission losses are not considered. Moreover, the optimal results obtained by these methods may be the local optima and have lower computational accuracy. The drawbacks of conventional algorithms prompt researchers to study meta-heuristic algorithms for solving ED problems.
Recently, many meta-heuristic algorithms have been proposed to solve the various optimization problems, such as flow shop scheduling [6,7,8], steelmaking scheduling [9], job shop scheduling [10,11,12,13], flexible task scheduling [14] and chiller loading optimization [15,16,17]. Due to the better optimization performance, many meta-heuristic algorithms have also been applied to solve the complex ED problems, and these algorithms include the genetic algorithm (GA) [18,19,20,21], particle swarm optimization (PSO) and its variants [22,23,24,25,26], firefly algorithm (FA) [27], oppositional real coded chemical reaction optimization (ORCCRO) [28], differential evolution (DE) [29,30], chaotic bat algorithm (CBA) [31], oppositional invasive weed optimization (OIWO) [32], teaching learning based optimization (TLBO) [33], tournament-based harmony search (THS) [34], grey wolf optimization (GWO) [35,36], hybrid artificial algae algorithm (HAAA) [37], orthogonal learning competitive swarm optimizer (OLCSO) [2], backtracking search algorithm (BSA) [38], social spider algorithm (SSA) [39], civilized swarm optimization (CSO) [40], kinetic gas molecule optimization (KGMO) [41] and hybrid methods [42,43,44,45]. Although the above meta-heuristic algorithms have been shown to be efficient in solving ED problems, the optimal results obtained by these algorithms are not the most economical.
By mimicking the colonization behavior of weeds in nature, the invasive weed optimization (IWO) algorithm was proposed by Mehrabian and Lucas [46] to optimize multidimensional functions. The experimental results demonstrated that IWO can obtain superior optimization results compared to other evolutionary-based algorithms. Due to its robustness, convergence, high accuracy and searching ability, the IWO algorithm has been applied to solve many engineering optimization problems. However, when IWO is used to solve the ED problem in large-scale power systems, the optimization power outputs of generators obtained by IWO consumes more generation costs compared to the reported methods in literature. To further improve the optimization performance of IWO in solving ED problems, especially ED problems in the large-scale power systems, inspired by the effective application of hybrid methods in solving ED problems [37,42,43,44,45], a hybrid invasive weed optimization (HIWO) algorithm that hybridizes IWO with GA is developed in this study. The motivation behind choosing GA integrated with IWO is to get a better dispatch solution using the crossover operation between offspring weed and its parent weed to improve the local search ability of IWO, and executing the mutation operation on offspring weeds to increase the diversity of the population. The main contributions of this study are as follows: (1) the economic dispatch problem with various practical constraints is investigated by minimizing the total power generation cost; (2) the crossover and mutation operations of GA are proposed to improve the optimization performance of IWO; and (3) an effective repair method of handing constraints is investigated to repair the infeasible dispatch solutions.
The rest of this paper is organized as follows. Section 2 gives the mathematical formulation of the ED problem. Section 3 introduces a hybrid invasive weed optimization (HIWO) algorithm. Section 4 presents the application method of HIWO on ED problems. Section 5 shows the experimental results and analysis on six power systems with different scales. The conclusion is finally given in Section 6.
The ED problem in power systems is to find the optimal dispatch solution of the power outputs of generators, while the total power generation cost of the system is minimized and all the constraints are satisfied.
The optimization objective of the ED problem is to minimize the power generation cost (SC) consumed by N number of generators in the power system, as shown in Eq 1.
Min.SC=N∑i=1Ci(Pi) | (1) |
where Pi and Ci are the power output and generation cost of the ith generator, respectively.
For the ED problem neglecting valve-point effects, Ci is calculated by Eq 2. For the ED problem considering valve-point effects, Eq 3 is used to calculate Ci [2,32].
Ci(Pi)=ai⋅Pi2+bi⋅Pi+ci | (2) |
Ci(Pi)=ai⋅Pi2+bi⋅Pi+ci+|ei⋅sin(fi⋅(Pmini−Pi))| | (3) |
where ai, bi and ci are the cost coefficients of the ith generator; ei and fi are valve-point coefficients of the ith generator;
The feasible dispatch solutions of the ED problem should satisfy the following constraints.
The power output of each generator must be in the range specified by the minimum (
Pmini≤Pi≤Pmaxi | (4) |
The power outputs of generators should satisfy the system power demand (PD). For the ED problem neglecting network transmission losses (PL), the power demand balance is expressed as Eq 5 [30]. For the ED problem considering PL, the power demand balance is expressed as Eq 6.
N∑i=1Pi=PD | (5) |
N∑i=1Pi=PD+PL | (6) |
PL can be calculated using the power flow analysis method [47] or the B-coefficients method [48]. This study adopts the following B-coefficients method to calculate PL.
L=N∑i=1N∑j=1PiBijPj+N∑i=1B0iPi+B00 | (7) |
where Bij, B0i and B00 represent the loss coefficients.
In the actual operation of the power system, to avoid the excessive stress on the boiler and combustion equipment, the change rate of the power output of each generating unit should be within the ramp rate limit, as shown in Eq 8.
{Pi−P0i≤URiP0i−Pi≤DRi | (8) |
where
When taking into account both the generation capacity limits and ramp rate limits, the value range of Pi can be rewritten as Eq 9.
max{Pmini,P0i−DRi}≤Pi≤min{Pmaxi,P0i+URi} | (9) |
Considering the operation limitations of machine components, the power outputs of some generators cannot lie in the prohibited zones, as shown in Eq 10.
Pi∈{Pmini≤Pi≤Pli,1Pui,k−1≤Pi≤Pli,kPui,npi≤Pi≤Pmaxik=2,3,⋯,npi | (10) |
where
IWO is a novel evolutionary computation algorithm based on weed swarm intelligence. By simulating the propagation and growth behaviors of weeds in nature, IWO searches for the optimal solution of the problem in the solution space. The calculation steps of IWO include initialization, reproduction, spatial dispersal and selection. The initial population with Nwo weed individuals is randomly generated in the feasible solution space, in which each weed consisting of variables represents a feasible solution. Then, each weed Wj in the population reproduces seeds, and the seeds grow into offspring weeds through spatial dispersal. The amount (Nsj) of seeds reproduced by Wj is calculated by using Eq 11.
Nsj=Fitj−FitminFitmax−Fitmin⋅(Nsmax−Nsmin)+Nsmin | (11) |
where Fitj is the fitness value of Wj; Fitmin and Fitmax are the minimum and maximum fitness values in the weed population, respectively; Nsmin and Nsmax are the minimum and maximum of the number of seeds, respectively.
The parent weeds with higher fitness values can reproduce more seeds, and they have more offspring weeds in the population. This reproduction strategy means that IWO can converge rapidly and reliably to the approximate optimal solution. Offspring weeds are randomly distributed around their parent weed according to a normal distribution with a standard deviation (σit). The calculation formula of σit is shown in Eq 12. Along with the increase of the iteration times, σit is gradually reduced from an initial value (σiv) to a final value (σfv), which makes the search range of IWO be gradually reduced. This strategy makes IWO have the whole space search capability in early iterations and high local convergence in later iterations. After all the seeds grow into weeds, the Nwmax weeds with higher fitness values are selected from all the weeds as the parent weeds of the next iteration. Through Itermax times iterations, the weed with the highest fitness value is the optimal solution of the problem.
σit=(Itermax−Iter)mItermaxm⋅(σiv−σfv)+σfv | (12) |
where m is the nonlinear modulation index, and Iter and Itermax are the current number and maximum of iterations, respectively.
In the proposed HIWO algorithm, IWO is used to explore the solution space around parent weeds. After the seeds reproduced by parent weeds have grown into offspring weeds, the crossover and mutation operations of GA are performed on offspring weeds for improving the quality and diversity of solutions, which can improve the convergence speed and avoid the premature convergence of the algorithm.
The execution flow of HIWO is represented by the pseudo code shown in Figure 1.
Each offspring weed (OW(j, q)) (q = 1, 2, …, Nsj ) crosses with its parent weed (Wj) to generate a new weed (
(a) If
(b) If
After the new offspring weed (
For each offspring weed (OW(j, q)) (q = 1, 2, …, Nsj ), randomly select X mutation points from N variables
σm=(Pmaxi−Pmini)⋅rand(0,1) | (13) |
where rand (0, 1) is a random number between 0 and 1.
In the proposed HIWO algorithm, the first task is the encoding to represent each solution considering all of the constraints. Each weed (Wj) is represented as a row vector consisting of power outputs of generators, as shown in Eq 14. The weed population is initialized by randomly generating the power outputs of generators by using Eq 15. Then, infeasible weeds are repaired into feasible solutions by using the repair method in Section 4.2. Weeds in the initial population are used as the parent weeds to reproduce seeds, which grows into offspring weeds through spatial dispersal. The weeds with higher fitness value can reproduce more seeds. The fitness function used in this study is shown in Eq 16. Each offspring weed will perform the crossover and mutation procedures, like in the canonical GA, and thus can increase the diversity of the population. Then, the repair procedure is applied on the infeasible offspring weeds to make them satisfy with all of the constraints. If the total quantity of parent weeds and offspring weeds is larger than the specified population size, select the weeds with higher fitness values as the parent weeds of the next iteration. Otherwise, all the weeds are used as parent weeds. After multiple times iterations, the best weed with the highest fitness value is selected as the optimal dispatch solution of the ED problem.
Wj=(P1,P2,⋯,PN) | (14) |
Pi=(Pmaxi−Pmini)⋅rand(0,1)+Pminii=1,2,⋯,N | (15) |
Fitj=1SCj | (16) |
where SCj and Fitj represent the power generation cost and fitness value of the jth weed, respectively.
An effective repair method of handing constraints is proposed in this study to repair infeasible weeds into feasible solutions. The detail repair steps are stated in the following.
Step 1: Modify the
Pi={max{Pmini,P0i−DRi}if Pi<max{Pmini,P0i−DRi}min{Pmaxi,P0i+URi}if Pi>min{Pmaxi,P0i+URi} | (17) |
Step 2: Calculate the constraint violation (V) of the power demand balance. For the ED problem considering transmission losses, V is calculated by using Eq 18. For the ED problem neglecting transmission losses, V is calculated by using Eq 19. If
V=|N∑i=1Pi−PD−N∑i=1N∑j=1PiBijPj−N∑i=1B0iPi−B00| | (18) |
V=|N∑i=1Pi−PD| | (19) |
Step 3: Determine the modification sequence of N generators. For each generator i (i = 1, 2, …, N), calculate the modification value
P′i=PD−∑r∈RPr | (20) |
Bii(P′i)2+(2∑r∈RPrBir+B0i−1)P′i+(PD+∑r∈R∑t∈RPrBrtPt+∑r∈RB0rPr−∑r∈RPr+B00)=0 | (21) |
P′i=−(2∑r∈RPrBir+B0i−1)−√(2∑r∈RPrBir+B0i−1)2−4Bii(PD+∑r∈R∑t∈RPrBrtPt+∑r∈RB0rPr−∑r∈RPr+B00)2Bii | (22) |
For each generator i (i = 1, 2, …, N), assume that the ith generator is selected as the revised generator, and Pi is replaced by
CXi=Ci(P′i)−Ci(Pi) | (23) |
PCVi=CXi−min(CX)max(CX)−min(CX)+PVi−min(PV)max(PV)−min(PV) | (24) |
Step 4: Modify the power output of each generator in turn according to the modification sequence stored in S until the power demand balance constraint is satisfied. When the ith (
Step 5: Output the modified weed (Wj).
To validate the optimization ability of HIWO on ED problems with various practical constraints, six classical ED problems in the small, medium, large and very large-scale power systems were selected as the studied test cases. For each test case, the optimal dispatch results obtained by HIWO in 50 independent runs, including the minimum cost (SCmin), average cost (SCavg), maximum cost (SCmax) and standard deviation of the costs (SCstd), are compared to those of algorithms reported in the literature. The best optimization performance among these algorithms is shown in boldface. The parameters of HIWO on six test systems are set as follows: the initial population size Nwo = 30, maximum population size Nwmax = 50, minimum number of seeds Nsmin = 1, maximum number of seeds Nsmax = 5, nonlinear modulation index m = 5, initial standard deviation
The 15-generator power system [2,24] considering transmission losses, ramp rate limits and prohibited operating zones is selected as the small-scale test system. The power load demand of the system is 2630 MW. In this test system study, the optimal power outputs of generators obtained by HIWO are shown in Table 1.The optimal dispatch results of HIWO are compared to those of OLCSO [2], WCA [49], ICS [50], FA [27], RTO [51], EMA [52] and IWO, as shown in Tables 2. Compared to other algorithms in terms of minimum, average, maximum and standard deviation of costs in 50 runs, the dispatch solution obtained by HIWO consumes the least cost.
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 455.0000 | 4 | 130.0000 | 7 | 430.0000 | 10 | 159.7871 | 13 | 25.0000 |
2 | 380.0000 | 5 | 170.0000 | 8 | 71.2594 | 11 | 80.0000 | 14 | 15.0000 |
3 | 130.0000 | 6 | 460.0000 | 9 | 58.4944 | 12 | 80.0000 | 15 | 15.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
EMA [52] | 32704.4503 | 32704.4504 | 32704.4506 | NA |
FA [27] | 32704.5000 | 32856.1000 | 33175.0000 | 147.17022 |
ICS [50] | 32706.7358 | 32714.4669 | 32752.5183 | NA |
WCA [49] | 32704.4492 | 32704.5096 | 32704.5196 | 4.513e-05 |
RTO [51] | 32701.8145 | 32704.5300 | 32715.1800 | 5.07 |
OLCSO [2] | 32692.3961 | 32692.3981 | 32692.4033 | 0.0022 |
IWO | 32691.8615 | 32691.9392 | 32692.1421 | 0.0927 |
HIWO | 32691.5614 | 32691.8615 | 32691.8616 | 0.0001 |
The 40-generator power system [32] considering valve-point effects and transmission losses is selected as the medium-scale test system. The power load demand of the system is 10500 MW. The optimal power outputs obtained by HIWO are shown in Table 3. The optimal dispatch results of HIWO are compared to those of ORCCRO [28], BBO [28], DE/BBO [28], SDE [29], OIWO [32], HAAA [37] and IWO, as shown in Tables 4. Compared to other algorithms in the literature, the proposed HIWO algorithm can obtain the cheapest dispatch solution in terms of minimum, average and maximum of costs in 50 runs.
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 113.9993 | 9 | 289.4281 | 17 | 489.2798 | 25 | 523.2794 | 33 | 190.0000 |
2 | 113.9993 | 10 | 279.5996 | 18 | 489.2793 | 26 | 523.2794 | 34 | 200.0000 |
3 | 120.0000 | 11 | 243.5995 | 19 | 511.2795 | 27 | 10.0000 | 35 | 199.9999 |
4 | 179.7330 | 12 | 94.0000 | 20 | 511.2793 | 28 | 10.0000 | 36 | 164.7999 |
5 | 87.7999 | 13 | 484.0391 | 21 | 523.2794 | 29 | 10.0000 | 37 | 109.9998 |
6 | 139.9998 | 14 | 484.0390 | 22 | 523.2794 | 30 | 87.7999 | 38 | 110.0000 |
7 | 300.0000 | 15 | 484.0393 | 23 | 523.2794 | 31 | 190.0000 | 39 | 109.9999 |
8 | 299.9997 | 16 | 484.0391 | 24 | 523.2794 | 32 | 190.0000 | 40 | 549.9999 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 136855.19 | 136855.19 | 136855.19 | NA |
BBO [28] | 137026.82 | 137116.58 | 137587.82 | NA |
DE/BBO [28] | 136950.77 | 136966.77 | 137150.77 | NA |
SDE [29] | 138157.46 | NA | NA | NA |
OIWO [32] | 136452.68 | 136452.68 | 136452.68 | NA |
HAAA [37] | 136433.5 | 136436.6 | NA | 3.341896 |
IWO | 136543.8580 | 137009.5641 | 137679.1073 | 292.9686 |
HIWO | 136430.9504 | 136435.2127 | 136441.1059 | 4.3238 |
To verify the dispatch performance of HIWO on large-scale power systems with multiple local optimal solutions, two cases studies are performed to compare the optimization results of HIWO and other algorithms. The detail information of these two cases is shown as follows.
Case Ⅰ: The 80-generator power system [37] considering valve-point effects. The power load demand is 21000 MW.
Case Ⅱ: The 110-generator power system [20,32] neglecting valve-point effects and transmission losses. The power load demand is 15000 MW.
In the case Ⅰ study, the optimal dispatch solution obtained by HIWO is shown in Table 5. The comparison results of generation costs generated by HIWO, THS [34], CSO [40], HAAA [37], GWO [35] and IWO are summarized in Table 6. It can be found from Table 6 that HIWO can obtain the cheapest dispatch solution compared to other algorithms.
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 110.8335 | 17 | 489.3362 | 33 | 189.9994 | 49 | 284.6071 | 65 | 523.2794 |
2 | 111.5439 | 18 | 489.2794 | 34 | 165.1983 | 50 | 130.0000 | 66 | 523.2835 |
3 | 97.3834 | 19 | 511.2731 | 35 | 199.9997 | 51 | 94.0040 | 67 | 10.0000 |
4 | 179.7603 | 20 | 511.2666 | 36 | 199.9998 | 52 | 94.0000 | 68 | 10.0000 |
5 | 87.9806 | 21 | 523.2525 | 37 | 109.9999 | 53 | 214.7298 | 69 | 10.0000 |
6 | 139.9997 | 22 | 523.2805 | 38 | 110.0000 | 54 | 394.2675 | 70 | 87.8052 |
7 | 259.5584 | 23 | 523.2794 | 39 | 109.9987 | 55 | 394.2967 | 71 | 190.0000 |
8 | 284.7677 | 24 | 523.2794 | 40 | 511.2603 | 56 | 304.4839 | 72 | 189.9997 |
9 | 284.6331 | 25 | 523.2794 | 41 | 110.9296 | 57 | 489.3082 | 73 | 189.9991 |
10 | 130.0000 | 26 | 523.2958 | 42 | 110.8195 | 58 | 489.2773 | 74 | 164.7786 |
11 | 169.0220 | 27 | 10.0000 | 43 | 97.3706 | 59 | 511.2121 | 75 | 199.9994 |
12 | 94.0000 | 28 | 10.0000 | 44 | 179.7187 | 60 | 511.2992 | 76 | 200.0000 |
13 | 214.7422 | 29 | 10.0000 | 45 | 87.8560 | 61 | 523.2830 | 77 | 109.9990 |
14 | 394.1929 | 30 | 89.6856 | 46 | 139.9995 | 62 | 523.3201 | 78 | 110.0000 |
15 | 394.2794 | 31 | 189.9993 | 47 | 259.6320 | 63 | 523.2794 | 79 | 109.9996 |
16 | 394.3050 | 32 | 189.9992 | 48 | 284.6702 | 64 | 523.2794 | 80 | 511.2482 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
THS [34] | 243192.6899 | 243457.36 | NA | 120.9889 |
CSO [40] | 243195.3781 | 243546.6283 | 244038.7352 | NA |
HAAA [37] | 242815.9 | 242883 | 242944.5 | 29.2849 |
GWO [35] | 242825.4799 | 242829.8192 | 242837.1303 | 0.093 |
IWO | 246386.4038 | 248088.2077 | 249888.0623 | 844.0919 |
HIWO | 242815.2096 | 242836.1110 | 242872.4662 | 10.3458 |
In the case Ⅱ study, the optimal dispatch solution obtained by HIWO is shown in Table 7. The generation cost generated by HIWO are compared to those of ORCCRO [28], BBO [28], DE/BBO [28], OIWO [32], OLCSO [2] and IWO, which are summarized in Table 8. Compared to other algorithms in terms of minimum, average, maximum and standard deviation of costs in 50 runs, the optimal dispatch solution obtained by HIWO generates the least generation cost.
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 2.4000 | 23 | 68.9000 | 45 | 659.9999 | 67 | 70.0000 | 89 | 82.4977 |
2 | 2.4000 | 24 | 350.0000 | 46 | 616.2499 | 68 | 70.0000 | 90 | 89.2333 |
3 | 2.4000 | 25 | 400.0000 | 47 | 5.4000 | 69 | 70.0000 | 91 | 57.5687 |
4 | 2.4000 | 26 | 400.0000 | 48 | 5.4000 | 70 | 359.9999 | 92 | 99.9986 |
5 | 2.4000 | 27 | 499.9992 | 49 | 8.4000 | 71 | 399.9999 | 93 | 439.9998 |
6 | 4.0000 | 28 | 500.0000 | 50 | 8.4000 | 72 | 399.9998 | 94 | 499.9999 |
7 | 4.0000 | 29 | 199.9997 | 51 | 8.4000 | 73 | 105.2864 | 95 | 600.0000 |
8 | 4.0000 | 30 | 99.9998 | 52 | 12.0000 | 74 | 191.4091 | 96 | 471.5717 |
9 | 4.0000 | 31 | 10.0000 | 53 | 12.0000 | 75 | 89.9996 | 97 | 3.6000 |
10 | 64.5432 | 32 | 19.9993 | 54 | 12.0000 | 76 | 49.9999 | 98 | 3.6000 |
11 | 62.2465 | 33 | 79.9950 | 55 | 12.0000 | 77 | 160.0000 | 99 | 4.4000 |
12 | 36.2739 | 34 | 249.9998 | 56 | 25.2000 | 78 | 295.4962 | 100 | 4.4000 |
13 | 56.6406 | 35 | 359.9999 | 57 | 25.2000 | 79 | 175.0102 | 101 | 10.0000 |
14 | 25.0000 | 36 | 399.9997 | 58 | 35.0000 | 80 | 98.2829 | 102 | 10.0000 |
15 | 25.0000 | 37 | 39.9998 | 59 | 35.0000 | 81 | 10.0000 | 103 | 20.0000 |
16 | 25.0000 | 38 | 69.9996 | 60 | 45.0000 | 82 | 12.0000 | 104 | 20.0000 |
17 | 154.9999 | 39 | 99.9998 | 61 | 45.0000 | 83 | 20.0000 | 105 | 40.0000 |
18 | 154.9993 | 40 | 119.9984 | 62 | 45.0000 | 84 | 199.9999 | 106 | 40.0000 |
19 | 155.0000 | 41 | 157.4299 | 63 | 184.9996 | 85 | 324.9972 | 107 | 50.0000 |
20 | 155.0000 | 42 | 219.9999 | 64 | 184.9996 | 86 | 440.0000 | 108 | 30.0000 |
21 | 68.9000 | 43 | 439.9999 | 65 | 184.9984 | 87 | 14.0886 | 109 | 40.0000 |
22 | 68.9000 | 44 | 559.9998 | 66 | 184.9997 | 88 | 24.0910 | 110 | 20.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 198016.29 | 198016.32 | 198016.89 | NA |
BBO [28] | 198241.166 | 198413.45 | 199102.59 | NA |
DE/BBO [28] | 198231.06 | 198326.66 | 198828.57 | NA |
OIWO [32] | 197989.14 | 197989.41 | 197989.93 | NA |
OLCSO [2] | 197988.8576 | 197989.5832 | 197990.4551 | 0.3699 |
IWO | 198252.3594 | 198621.3233 | 198902.7697 | 138.4714 |
HIWO | 197988.1927 | 197988.1969 | 197988.2045 | 0.0025 |
To investigate the dispatch performance of HIWO on very large-scale power systems, the following two cases studies are performed for comparing the optimization results of HIWO and other algorithms.
Case Ⅰ: The 140-generator Korea power system [23,32] neglecting transmission losses. The 12 generators consider the valve point effects. The power load demand is 49342 MW.
Case Ⅱ: The 160-generator power system [32] considering valve-point effects. The power load demand is 43200 MW.
In the case Ⅰ study, the optimal dispatch solution obtained by HIWO is shown in Table 9. The optimal results of HIWO are compared to those of SDE [29], OIWO [32], HAAA [37], GWO [35], KGMO [41] and IWO, as shown in Table 10. The corrected optimal result of OIWO is shown in italics. Compared to other algorithms in terms of minimum, average, maximum and standard deviation of costs in 50 runs, HIWO can obtain the cheapest dispatch solution.
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 115.2442 | 29 | 500.9998 | 57 | 103.0000 | 85 | 115.0000 | 113 | 94.0000 |
2 | 189.0000 | 30 | 500.9994 | 58 | 198.0000 | 86 | 207.0000 | 114 | 94.0000 |
3 | 190.0000 | 31 | 505.9993 | 59 | 311.9941 | 87 | 207.0000 | 115 | 244.0000 |
4 | 190.0000 | 32 | 505.9997 | 60 | 281.1604 | 88 | 175.0000 | 116 | 244.0000 |
5 | 168.5393 | 33 | 506.0000 | 61 | 163.0000 | 89 | 175.0000 | 117 | 244.0000 |
6 | 189.9932 | 34 | 505.9998 | 62 | 95.0000 | 90 | 175.0000 | 118 | 95.0000 |
7 | 489.9992 | 35 | 499.9996 | 63 | 160.0000 | 91 | 175.0000 | 119 | 95.0000 |
8 | 489.9996 | 36 | 500.0000 | 64 | 160.0000 | 92 | 579.9998 | 120 | 116.0000 |
9 | 495.9997 | 37 | 240.9993 | 65 | 489.9465 | 93 | 645.0000 | 121 | 175.0000 |
10 | 495.9994 | 38 | 240.9999 | 66 | 196.0000 | 94 | 983.9998 | 122 | 2.0000 |
11 | 495.9997 | 39 | 773.9996 | 67 | 489.9717 | 95 | 977.9993 | 123 | 4.0000 |
12 | 496.0000 | 40 | 769.0000 | 68 | 489.9908 | 96 | 681.9997 | 124 | 15.0000 |
13 | 506.0000 | 41 | 3.0000 | 69 | 130.0000 | 97 | 719.9998 | 125 | 9.0000 |
14 | 509.0000 | 42 | 3.0000 | 70 | 234.7202 | 98 | 717.9993 | 126 | 12.0000 |
15 | 506.0000 | 43 | 249.2474 | 71 | 137.0000 | 99 | 719.9997 | 127 | 10.0000 |
16 | 504.9997 | 44 | 246.0287 | 72 | 325.4950 | 100 | 963.9998 | 128 | 112.0000 |
17 | 505.9997 | 45 | 249.9973 | 73 | 195.0000 | 101 | 958.0000 | 129 | 4.0000 |
18 | 505.9997 | 46 | 249.9863 | 74 | 175.0000 | 102 | 1006.9992 | 130 | 5.0000 |
19 | 504.9994 | 47 | 241.0622 | 75 | 175.0000 | 103 | 1006.0000 | 131 | 5.0000 |
20 | 505.0000 | 48 | 249.9950 | 76 | 175.0000 | 104 | 1012.9999 | 132 | 50.0000 |
21 | 504.9998 | 49 | 249.9916 | 77 | 175.0000 | 105 | 1019.9996 | 133 | 5.0000 |
22 | 505.0000 | 50 | 249.9995 | 78 | 330.0000 | 106 | 953.9999 | 134 | 42.0000 |
23 | 504.9998 | 51 | 165.0000 | 79 | 531.0000 | 107 | 951.9998 | 135 | 42.0000 |
24 | 504.9996 | 52 | 165.0000 | 80 | 530.9995 | 108 | 1005.9996 | 136 | 41.0000 |
25 | 536.9997 | 53 | 165.0000 | 81 | 398.6524 | 109 | 1013.0000 | 137 | 17.0000 |
26 | 536.9995 | 54 | 165.0000 | 82 | 56.0000 | 110 | 1020.9998 | 138 | 7.0000 |
27 | 548.9998 | 55 | 180.0000 | 83 | 115.0000 | 111 | 1014.9996 | 139 | 7.0000 |
28 | 548.9993 | 56 | 180.0000 | 84 | 115.0000 | 112 | 94.0000 | 140 | 26.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
SDE [29] | 1560236.85 | NA | NA | NA |
OIWO [32] | 1559712.2604 | NA | NA | NA |
HAAA [37] | 1559710.00 | 1559712.87 | 1559731.00 | 4.1371 |
GWO [35] | 1559953.18 | 1560132.93 | 1560228.40 | 1.024 |
KGMO [41] | 1583944.60 | 1583952.14 | 1583963.52 | NA |
IWO | 1564050.0027 | 1567185.2227 | 1571056.6280 | 1678.8488 |
HIWO | 1559709.5266 | 1559709.6956 | 1559709.8959 | 0.0856 |
In the case Ⅱ study, the optimal dispatch solution obtained by HIWO is shown in Table 11. The optimal results of HIWO are compared to those of ORCCRO [28], BBO [28], DE/BBO [28], CBA [31], OIWO [32] and IWO, as shown in Table 12. Compared to other algorithms, HIWO can also obtain the cheapest dispatch solution in terms of minimum, average, maximum and standard deviation of costs.
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 218.6095 | 33 | 280.6560 | 65 | 279.6118 | 97 | 287.7203 | 129 | 431.0758 |
2 | 209.2361 | 34 | 238.9676 | 66 | 238.5645 | 98 | 238.6988 | 130 | 275.8790 |
3 | 279.6486 | 35 | 279.9554 | 67 | 287.7296 | 99 | 426.2750 | 131 | 219.6189 |
4 | 240.3113 | 36 | 240.9831 | 68 | 241.2519 | 100 | 272.6741 | 132 | 210.4739 |
5 | 280.0206 | 37 | 290.1069 | 69 | 427.7708 | 101 | 217.5647 | 133 | 281.6640 |
6 | 238.4301 | 38 | 240.0425 | 70 | 272.9907 | 102 | 211.9593 | 134 | 238.9676 |
7 | 288.2326 | 39 | 426.3102 | 71 | 218.5918 | 103 | 280.6578 | 135 | 276.5752 |
8 | 239.5051 | 40 | 275.6392 | 72 | 212.7020 | 104 | 239.2363 | 136 | 239.3707 |
9 | 425.6549 | 41 | 219.6195 | 73 | 281.6629 | 105 | 276.3263 | 137 | 287.7806 |
10 | 275.6903 | 42 | 210.9690 | 74 | 238.9676 | 106 | 240.7144 | 138 | 238.5645 |
11 | 217.5646 | 43 | 282.6711 | 75 | 279.3688 | 107 | 290.0715 | 139 | 430.7874 |
12 | 212.4544 | 44 | 240.3113 | 76 | 237.6239 | 108 | 238.8332 | 140 | 275.8606 |
13 | 280.6558 | 45 | 279.7868 | 77 | 289.9995 | 109 | 425.7918 | 141 | 218.6539 |
14 | 238.6988 | 46 | 237.4895 | 78 | 239.9082 | 110 | 275.2705 | 142 | 210.7215 |
15 | 279.9370 | 47 | 287.7274 | 79 | 425.2406 | 111 | 217.5671 | 143 | 281.6640 |
16 | 240.7144 | 48 | 240.0425 | 80 | 276.0112 | 112 | 212.2069 | 144 | 239.3707 |
17 | 287.6968 | 49 | 427.4497 | 81 | 218.5923 | 113 | 281.6664 | 145 | 276.3578 |
18 | 239.7738 | 50 | 275.6817 | 82 | 212.2069 | 114 | 239.6394 | 146 | 239.6394 |
19 | 427.4049 | 51 | 219.6197 | 83 | 282.7049 | 115 | 276.0940 | 147 | 287.7565 |
20 | 275.6990 | 52 | 213.4447 | 84 | 237.7582 | 116 | 240.3113 | 148 | 239.3707 |
21 | 217.5665 | 53 | 282.6717 | 85 | 279.7940 | 117 | 290.0972 | 149 | 426.3023 |
22 | 212.2069 | 54 | 237.8926 | 86 | 239.3707 | 118 | 239.5051 | 150 | 275.6371 |
23 | 283.6805 | 55 | 276.2856 | 87 | 290.0916 | 119 | 429.4367 | 151 | 217.5647 |
24 | 239.7738 | 56 | 239.5051 | 88 | 239.2363 | 120 | 275.6690 | 152 | 212.2069 |
25 | 279.9011 | 57 | 287.6883 | 89 | 427.0504 | 121 | 217.5656 | 153 | 279.6493 |
26 | 240.9831 | 58 | 238.5645 | 90 | 275.7937 | 122 | 210.2264 | 154 | 238.4301 |
27 | 290.0737 | 59 | 429.9489 | 91 | 217.5643 | 123 | 280.6617 | 155 | 279.9078 |
28 | 240.8488 | 60 | 275.5096 | 92 | 212.9496 | 124 | 239.7738 | 156 | 240.4457 |
29 | 427.1007 | 61 | 218.5915 | 93 | 282.6732 | 125 | 275.9409 | 157 | 287.7385 |
30 | 276.2995 | 62 | 212.9496 | 94 | 240.4457 | 126 | 240.1769 | 158 | 238.5645 |
31 | 219.6189 | 63 | 282.6705 | 95 | 279.4854 | 127 | 287.6965 | 159 | 426.9110 |
32 | 211.7117 | 64 | 239.9082 | 96 | 240.1769 | 128 | 238.4301 | 160 | 272.7775 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 10004.20 | 10004.21 | 10004.45 | NA |
OIWO [32] | 9981.9834 | 9982.991 | 9983.998 | NA |
BBO [28] | 10008.71 | 10009.16 | 10010.59 | NA |
DE/BBO [28] | 10007.05 | 10007.56 | 10010.26 | NA |
CBA [31] | 10002.8596 | 10006.3251 | 10045.2265 | 9.5811 |
IWO | 9984.8409 | 9985.5127 | 9986.1947 | 0.3252 |
HIWO | 9981.7867 | 9982.0010 | 9982.1922 | 0.0934 |
To illustrate the convergence ability of HIWO for solving different-scale ED problems with various constraints, the convergence curves of HIWO and IWO on six test systems are drawn, as shown in Figure 2. It can be found from Figure 2 that HIWO can converge to the optimal areas in the six test systems, and the convergence speed of HIWO on the 15, 40, 80, 110 and 140-generator power systems, is faster than that of IWO. Although the convergence speed of HIWO on the 160-generator power system is slower than that of IWO in the early evolutionary stage, it is faster than that of IWO in the later evolutionary stage. The reason is that the crossover and mutation decrease the fitness value of offspring weeds in 160-generator power system having lots of constraints, and then reduce the convergence speed in the early evolutionary stage, but increase the diversity of the population to jump out local optimization in the later stage.
In this paper, a hybrid HIWO algorithm combining IWO with GA is proposed to solve ED problems in power systems. The HIWO adopts IWO to explore the various regions in the solution space, while the crossover and mutation operations of GA are applied to improve the quality and diversity of solutions, thereby preventing the optimization from prematurity and enhancing the search capability. Moreover, an effective repair method is proposed to repair infeasible solutions to feasible solutions. The experimental results of the six test systems studies show that HIWO can obtain the cheapest dispatch solutions compared to other algorithms in the literature, and have a better optimization ability and faster convergence speed compared to the classical IWO. In summary, the proposed HIWO algorithm is an effective and promising approach for solving ED problems in different-scale power systems.
This research is partially supported by the National Science Foundation of China under grant numbers 61773192 and 61773246, the Key Laboratory of Computer Network and Information Integration (Southeast University), the Ministry of Education (K93-9-2017-02), and the State Key Laboratory of Synthetical Automation for Process Industries (PAL-N201602).
The authors declare no conflict of interest.
[1] | Z. X. Liang and J. D. Glover, A zoom feature for a dynamic programming solution to economic dispatch including transmission losses, IEEE T. Power Syst., 7 (1992), 544–550. |
[2] | G. Xiong and D. Shi, Orthogonal learning competitive swarm optimizer for economic dispatch problems, Appl. Soft. Comput. J., 66 (2018), 134–148. |
[3] | R. A. Jabr, A. H. Coonick and B. J. Cory, A homogeneous linear programming algorithm for the security constrained economic dispatch problem, IEEE T. Power Syst., 15 (2000), 930–936. |
[4] | S. Muralidharan, K. Srikrishna and S. Subramanian, Self-adaptive dynamic programming technique for economic power dispatch, Int. J. Power Energy Syst., 27 (2007), 340–345. |
[5] | N. Sinha, R. Chakrabarti and P. K. Chattopadhyay, evolutionary programming techniques for economic load dispatch, IEEE T. Evol. Comput., 7 (2003), 83–94. |
[6] | J. Q. Li, H. Y. Sang, Y. Y. Han, et al., Efficient multi-objective optimization algorithm for hybrid flow shop scheduling problems with setup energy consumptions, J. Clean. Prod., 181 (2018), 584–598. |
[7] | J. Q. Li, S. C. Bai, P. Y. Duan, et al., An improved artificial bee colony algorithm for addressing distributed flow shop with distance coefficient in a prefabricated system, Int. J. Prod. Res., (2019). |
[8] | J. Q. Li, Q. K. Pan and K. Mao, A discrete teaching-learning-based optimisation algorithm for realistic flowshop rescheduling problems, Eng. Appl. Artif. Intell., 37 (2015), 279–292. |
[9] | J. Q. Li, P. Y. Duan, H. Y. Sang, et al., An efficient optimization algorithm for resource-constrained steelmaking scheduling problems, IEEE Access, 6 (2018), 33883–33894. |
[10] | J. Q. Li, P. Y. Duan, J. D. Cao, et al., A hybrid Pareto-based tabu search for the distributed flexible job shop scheduling problem with E/T criteria, IEEE Access, 6 (2018), 58883–58897. |
[11] | J. Q. Li, Q. K. Pan and M. F. Tasgetiren, A discrete artificial bee colony algorithm for the multi-objective flexible job-shop scheduling problem with maintenance activities, Appl. Math. Model., 38 (2014), 1111–1132. |
[12] | J. Q. Li, Q. K. Pan and S. X. Xie, An effective shuffled frog-leaping algorithm for multi-objective flexible job shop scheduling problems, Appl. Math. Comput., 218 (2012), 9353–9371. |
[13] | J. Q. Li, Q. K. Pan and K. Z. Gao, Pareto-based discrete artificial bee colony algorithm for multi-objective flexible job shop scheduling problems, Int. J. Adv. Manuf. Technol., 55 (2011), 1159–1169. |
[14] | J. Q. Li, A hybrid multi-objective artificial bee colony algorithm for flexible task scheduling problems in Cloud computing system, Cluster Comput., (2019), 1–24. |
[15] | P. Y. Duan, J. Q. Li, Y. Wang, et al., Solving chiller loading optimization problems using an improved teaching‐learning‐based optimization algorithm, Optim, Control Appl. Met., 39 (2018), 65–77. |
[16] | Z. X. Zheng, J. Q. Li and P.Y. Duan, Optimal chiller loading by improved artificial fish swarm algorithm for energy saving, Math. Comput. Simul., 155 (2019), 227–243. |
[17] | Z. X. Zheng and J. Q. Li, Optimal chiller loading by improved invasive weed optimization algorithm for reducing energy consumption, Energy Build, 161(2018), 80–88. |
[18] | S. Kumar and R. Naresh, Nonconvex economic load dispatch using an efficient real-coded genetic algorithm, Appl. Soft Comput. J., 9 (2009), 321–329. |
[19] | P. Subbaraj, R. Rengaraj and S. Salivahanan, Enhancement of Self-adaptive real-coded genetic algorithm using Taguchi method for Economic dispatch problem, Appl. Soft Comput. J., 11 (2011), 83–92. |
[20] | S. O. Orero and M. R. Irving, Large scale unit commitment using a hybrid genetic algorithm, Int. J. Electr. Power Energy Syst., 19 (1997), 45–55. |
[21] | P. Chen and H. Chang, Large-scale economic dispatch by genetic algorithm, IEEE T. Power Syst., 10 (1995), 1919–1926. |
[22] | A. I. Selvakumar, A new particle swarm optimization solution to nonconvex economic dispatch problems, IEEE T. Power Syst., 22 (2007), 42–51. |
[23] | J. B. Park, Y. W. Jeong, J. R. Shin, et al., An improved particle swarm optimization for nonconvex economic dispatch problems, IEEE T. Power Syst., 25 (2010), 156–166. |
[24] | Z. Gaing, Particle swarm optimization to solving the economic dispatch considering the generator constraints, IEEE T. Power Syst., 18 (2003), 1187–1195. |
[25] | K. T. Chaturvedi, M. Pandit and L. Srivastava, Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch, IEEE T. Power Syst., 23 (2008), 1079–1087. |
[26] | Q. Qin, S. Cheng, X. Chu, et al., Solving non-convex/non-smooth economic load dispatch problems via an enhanced particle swarm optimization, Appl. Soft Comput. J., 59 (2017), 229–242. |
[27] | X. S. Yang, S. S. S. Hosseini and A. H. Gandomi, Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect, Appl. Soft Comput. J., 12 (2012), 1180–1186. |
[28] | K. Bhattacharjee, A. Bhattacharya and S. H. N. Dey, Oppositional real coded chemical reaction optimization for different economic dispatch problems, Int. J. Electr. Power Energy Syst., 55 (2014), 378–391. |
[29] | A. S. Reddy and K. Vaisakh, Shuffled differential evolution for large scale economic dispatch, Electr. Power Syst. Res., 96 (2013), 237–245. |
[30] | D. Zou, S. Li, G. G. Wang, et al., An improved differential evolution algorithm for the economic load dispatch problems with or without valve-point effects, Appl. Energy, 181 (2016), 375–390. |
[31] | B. R. Adarsh, T. Raghunathan, T. Jayabarathi, et al., Economic dispatch using chaotic bat algorithm, Energy, 96 (2016), 666–675. |
[32] | A. K. Barisal and R. C. Prusty, Large scale economic dispatch of power systems using oppositional invasive weed optimization, Appl. Soft Comput. J., 29 (2015), 122–137. |
[33] | S. Banerjee, D. Maity and C. K. Chanda, Teaching learning based optimization for economic load dispatch problem considering valve point loading effect, Int. J. Electr. Power Energy Syst., 73 (2015), 456–464. |
[34] | M. A. Al-Betar, M. A. Awadallah, A. T. Khader, et al., Tournament-based harmony search algorithm for non-convex economic load dispatch problem, Appl. Soft Comput. J., 47 (2016), 449–459. |
[35] | M. Pradhan, P. K. Roy and T. Pal, Grey wolf optimization applied to economic load dispatch problems, Int. J. Electr. Power Energy Syst., 83 (2016), 325–334. |
[36] | T. Jayabarathi, T. Raghunathan, B. R. Adarsh, et al., Economic dispatch using hybrid grey wolf optimizer, Energy, 111 (2016), 630–641. |
[37] | M. Kumar and J. S. Dhillon, Hybrid artificial algae algorithm for economic load dispatch, Appl. Soft Comput., 71 (2018), 89–109. |
[38] | M. Modiri-Delshad and N. A. Rahim, Solving non-convex economic dispatch problem via backtracking search algorithm, Energy, 77 (2014), 372–381. |
[39] | J. J. Q. Yu and V. O. K. Li, A social spider algorithm for solving the non-convex economic load dispatch problem, Neurocomputing, 171 (2016), 955–965. |
[40] | A. I. Selvakumar and K. Thanushkodi, Optimization using civilized swarm: Solution to economic dispatch with multiple minima, Electr. Power Syst. Res., 79 (2009), 8–16. |
[41] | M. Basu, Kinetic gas molecule optimization for nonconvex economic dispatch problem, Int. J. Electr. Power Energy Syst., 80 (2016), 325–332. |
[42] | J. Cai, Q. Li, L. Li, et al., A hybrid FCASO-SQP method for solving the economic dispatch problems with valve-point effects, Energy, 38 (2012), 346–353. |
[43] | J. S. Alsumait, J. K. Sykulski and A. K. Al-Othman, A hybrid GA-PS-SQP method to solve power system valve-point economic dispatch problems, Appl. Energy, 87 (2010), 1773–1781. |
[44] | J. Cai, Q. Li, L. Li, et al., A hybrid CPSO-SQP method for economic dispatch considering the valve-point effects, Energy Convers. Manag., 53 (2012), 175–181. |
[45] | S. Sayah and A. Hamouda, A hybrid differential evolution algorithm based on particle swarm optimization for nonconvex economic dispatch problems, Appl. Soft Comput. J., 13 (2013), 1608–1619. |
[46] | A. R. Mehrabian and C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecol. Inform., 1 (2006), 355–366. |
[47] | T. A. A. Victoire and A. E. Jeyakumar, Reserve constrained dynamic dispatch of units with valve-point effects, IEEE T. Power Syst., 20 (2005), 1273–1282. |
[48] | T. Niknam and F. Golestaneh, Enhanced adaptive particle swarm optimisation algorithm for dynamic economic dispatch of units considering valve-point effects and ramp rates, IET Gener. Transm. Distrib., 6 (2012), 424–435. |
[49] | M. A. Elhameed and A. A. El-Fergany, Water cycle algorithm-based economic dispatcher for sequential and simultaneous objectives including practical constraints, Appl. Soft Comput. J., 58 (2017), 145–154. |
[50] | E. Afzalan and M. Joorabian, An improved cuckoo search algorithm for power economic load dispatch, Int. Trans. Electr. Energy Syst., 25 (2015), 958–975. |
[51] | Y. Labbi, D. B. Attous, H. A. Gabbar, et al., A new rooted tree optimization algorithm for economic dispatch with valve-point effect, Int. J. Electr. Power Energy Syst., 79 (2016), 298–311. |
[52] | N. Ghorbani and E. Babaei, Exchange market algorithm for economic load dispatch, Int. J. Electr. Power Energy Syst., 75 (2016), 19–27. |
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Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 455.0000 | 4 | 130.0000 | 7 | 430.0000 | 10 | 159.7871 | 13 | 25.0000 |
2 | 380.0000 | 5 | 170.0000 | 8 | 71.2594 | 11 | 80.0000 | 14 | 15.0000 |
3 | 130.0000 | 6 | 460.0000 | 9 | 58.4944 | 12 | 80.0000 | 15 | 15.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
EMA [52] | 32704.4503 | 32704.4504 | 32704.4506 | NA |
FA [27] | 32704.5000 | 32856.1000 | 33175.0000 | 147.17022 |
ICS [50] | 32706.7358 | 32714.4669 | 32752.5183 | NA |
WCA [49] | 32704.4492 | 32704.5096 | 32704.5196 | 4.513e-05 |
RTO [51] | 32701.8145 | 32704.5300 | 32715.1800 | 5.07 |
OLCSO [2] | 32692.3961 | 32692.3981 | 32692.4033 | 0.0022 |
IWO | 32691.8615 | 32691.9392 | 32692.1421 | 0.0927 |
HIWO | 32691.5614 | 32691.8615 | 32691.8616 | 0.0001 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 113.9993 | 9 | 289.4281 | 17 | 489.2798 | 25 | 523.2794 | 33 | 190.0000 |
2 | 113.9993 | 10 | 279.5996 | 18 | 489.2793 | 26 | 523.2794 | 34 | 200.0000 |
3 | 120.0000 | 11 | 243.5995 | 19 | 511.2795 | 27 | 10.0000 | 35 | 199.9999 |
4 | 179.7330 | 12 | 94.0000 | 20 | 511.2793 | 28 | 10.0000 | 36 | 164.7999 |
5 | 87.7999 | 13 | 484.0391 | 21 | 523.2794 | 29 | 10.0000 | 37 | 109.9998 |
6 | 139.9998 | 14 | 484.0390 | 22 | 523.2794 | 30 | 87.7999 | 38 | 110.0000 |
7 | 300.0000 | 15 | 484.0393 | 23 | 523.2794 | 31 | 190.0000 | 39 | 109.9999 |
8 | 299.9997 | 16 | 484.0391 | 24 | 523.2794 | 32 | 190.0000 | 40 | 549.9999 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 136855.19 | 136855.19 | 136855.19 | NA |
BBO [28] | 137026.82 | 137116.58 | 137587.82 | NA |
DE/BBO [28] | 136950.77 | 136966.77 | 137150.77 | NA |
SDE [29] | 138157.46 | NA | NA | NA |
OIWO [32] | 136452.68 | 136452.68 | 136452.68 | NA |
HAAA [37] | 136433.5 | 136436.6 | NA | 3.341896 |
IWO | 136543.8580 | 137009.5641 | 137679.1073 | 292.9686 |
HIWO | 136430.9504 | 136435.2127 | 136441.1059 | 4.3238 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 110.8335 | 17 | 489.3362 | 33 | 189.9994 | 49 | 284.6071 | 65 | 523.2794 |
2 | 111.5439 | 18 | 489.2794 | 34 | 165.1983 | 50 | 130.0000 | 66 | 523.2835 |
3 | 97.3834 | 19 | 511.2731 | 35 | 199.9997 | 51 | 94.0040 | 67 | 10.0000 |
4 | 179.7603 | 20 | 511.2666 | 36 | 199.9998 | 52 | 94.0000 | 68 | 10.0000 |
5 | 87.9806 | 21 | 523.2525 | 37 | 109.9999 | 53 | 214.7298 | 69 | 10.0000 |
6 | 139.9997 | 22 | 523.2805 | 38 | 110.0000 | 54 | 394.2675 | 70 | 87.8052 |
7 | 259.5584 | 23 | 523.2794 | 39 | 109.9987 | 55 | 394.2967 | 71 | 190.0000 |
8 | 284.7677 | 24 | 523.2794 | 40 | 511.2603 | 56 | 304.4839 | 72 | 189.9997 |
9 | 284.6331 | 25 | 523.2794 | 41 | 110.9296 | 57 | 489.3082 | 73 | 189.9991 |
10 | 130.0000 | 26 | 523.2958 | 42 | 110.8195 | 58 | 489.2773 | 74 | 164.7786 |
11 | 169.0220 | 27 | 10.0000 | 43 | 97.3706 | 59 | 511.2121 | 75 | 199.9994 |
12 | 94.0000 | 28 | 10.0000 | 44 | 179.7187 | 60 | 511.2992 | 76 | 200.0000 |
13 | 214.7422 | 29 | 10.0000 | 45 | 87.8560 | 61 | 523.2830 | 77 | 109.9990 |
14 | 394.1929 | 30 | 89.6856 | 46 | 139.9995 | 62 | 523.3201 | 78 | 110.0000 |
15 | 394.2794 | 31 | 189.9993 | 47 | 259.6320 | 63 | 523.2794 | 79 | 109.9996 |
16 | 394.3050 | 32 | 189.9992 | 48 | 284.6702 | 64 | 523.2794 | 80 | 511.2482 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
THS [34] | 243192.6899 | 243457.36 | NA | 120.9889 |
CSO [40] | 243195.3781 | 243546.6283 | 244038.7352 | NA |
HAAA [37] | 242815.9 | 242883 | 242944.5 | 29.2849 |
GWO [35] | 242825.4799 | 242829.8192 | 242837.1303 | 0.093 |
IWO | 246386.4038 | 248088.2077 | 249888.0623 | 844.0919 |
HIWO | 242815.2096 | 242836.1110 | 242872.4662 | 10.3458 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 2.4000 | 23 | 68.9000 | 45 | 659.9999 | 67 | 70.0000 | 89 | 82.4977 |
2 | 2.4000 | 24 | 350.0000 | 46 | 616.2499 | 68 | 70.0000 | 90 | 89.2333 |
3 | 2.4000 | 25 | 400.0000 | 47 | 5.4000 | 69 | 70.0000 | 91 | 57.5687 |
4 | 2.4000 | 26 | 400.0000 | 48 | 5.4000 | 70 | 359.9999 | 92 | 99.9986 |
5 | 2.4000 | 27 | 499.9992 | 49 | 8.4000 | 71 | 399.9999 | 93 | 439.9998 |
6 | 4.0000 | 28 | 500.0000 | 50 | 8.4000 | 72 | 399.9998 | 94 | 499.9999 |
7 | 4.0000 | 29 | 199.9997 | 51 | 8.4000 | 73 | 105.2864 | 95 | 600.0000 |
8 | 4.0000 | 30 | 99.9998 | 52 | 12.0000 | 74 | 191.4091 | 96 | 471.5717 |
9 | 4.0000 | 31 | 10.0000 | 53 | 12.0000 | 75 | 89.9996 | 97 | 3.6000 |
10 | 64.5432 | 32 | 19.9993 | 54 | 12.0000 | 76 | 49.9999 | 98 | 3.6000 |
11 | 62.2465 | 33 | 79.9950 | 55 | 12.0000 | 77 | 160.0000 | 99 | 4.4000 |
12 | 36.2739 | 34 | 249.9998 | 56 | 25.2000 | 78 | 295.4962 | 100 | 4.4000 |
13 | 56.6406 | 35 | 359.9999 | 57 | 25.2000 | 79 | 175.0102 | 101 | 10.0000 |
14 | 25.0000 | 36 | 399.9997 | 58 | 35.0000 | 80 | 98.2829 | 102 | 10.0000 |
15 | 25.0000 | 37 | 39.9998 | 59 | 35.0000 | 81 | 10.0000 | 103 | 20.0000 |
16 | 25.0000 | 38 | 69.9996 | 60 | 45.0000 | 82 | 12.0000 | 104 | 20.0000 |
17 | 154.9999 | 39 | 99.9998 | 61 | 45.0000 | 83 | 20.0000 | 105 | 40.0000 |
18 | 154.9993 | 40 | 119.9984 | 62 | 45.0000 | 84 | 199.9999 | 106 | 40.0000 |
19 | 155.0000 | 41 | 157.4299 | 63 | 184.9996 | 85 | 324.9972 | 107 | 50.0000 |
20 | 155.0000 | 42 | 219.9999 | 64 | 184.9996 | 86 | 440.0000 | 108 | 30.0000 |
21 | 68.9000 | 43 | 439.9999 | 65 | 184.9984 | 87 | 14.0886 | 109 | 40.0000 |
22 | 68.9000 | 44 | 559.9998 | 66 | 184.9997 | 88 | 24.0910 | 110 | 20.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 198016.29 | 198016.32 | 198016.89 | NA |
BBO [28] | 198241.166 | 198413.45 | 199102.59 | NA |
DE/BBO [28] | 198231.06 | 198326.66 | 198828.57 | NA |
OIWO [32] | 197989.14 | 197989.41 | 197989.93 | NA |
OLCSO [2] | 197988.8576 | 197989.5832 | 197990.4551 | 0.3699 |
IWO | 198252.3594 | 198621.3233 | 198902.7697 | 138.4714 |
HIWO | 197988.1927 | 197988.1969 | 197988.2045 | 0.0025 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 115.2442 | 29 | 500.9998 | 57 | 103.0000 | 85 | 115.0000 | 113 | 94.0000 |
2 | 189.0000 | 30 | 500.9994 | 58 | 198.0000 | 86 | 207.0000 | 114 | 94.0000 |
3 | 190.0000 | 31 | 505.9993 | 59 | 311.9941 | 87 | 207.0000 | 115 | 244.0000 |
4 | 190.0000 | 32 | 505.9997 | 60 | 281.1604 | 88 | 175.0000 | 116 | 244.0000 |
5 | 168.5393 | 33 | 506.0000 | 61 | 163.0000 | 89 | 175.0000 | 117 | 244.0000 |
6 | 189.9932 | 34 | 505.9998 | 62 | 95.0000 | 90 | 175.0000 | 118 | 95.0000 |
7 | 489.9992 | 35 | 499.9996 | 63 | 160.0000 | 91 | 175.0000 | 119 | 95.0000 |
8 | 489.9996 | 36 | 500.0000 | 64 | 160.0000 | 92 | 579.9998 | 120 | 116.0000 |
9 | 495.9997 | 37 | 240.9993 | 65 | 489.9465 | 93 | 645.0000 | 121 | 175.0000 |
10 | 495.9994 | 38 | 240.9999 | 66 | 196.0000 | 94 | 983.9998 | 122 | 2.0000 |
11 | 495.9997 | 39 | 773.9996 | 67 | 489.9717 | 95 | 977.9993 | 123 | 4.0000 |
12 | 496.0000 | 40 | 769.0000 | 68 | 489.9908 | 96 | 681.9997 | 124 | 15.0000 |
13 | 506.0000 | 41 | 3.0000 | 69 | 130.0000 | 97 | 719.9998 | 125 | 9.0000 |
14 | 509.0000 | 42 | 3.0000 | 70 | 234.7202 | 98 | 717.9993 | 126 | 12.0000 |
15 | 506.0000 | 43 | 249.2474 | 71 | 137.0000 | 99 | 719.9997 | 127 | 10.0000 |
16 | 504.9997 | 44 | 246.0287 | 72 | 325.4950 | 100 | 963.9998 | 128 | 112.0000 |
17 | 505.9997 | 45 | 249.9973 | 73 | 195.0000 | 101 | 958.0000 | 129 | 4.0000 |
18 | 505.9997 | 46 | 249.9863 | 74 | 175.0000 | 102 | 1006.9992 | 130 | 5.0000 |
19 | 504.9994 | 47 | 241.0622 | 75 | 175.0000 | 103 | 1006.0000 | 131 | 5.0000 |
20 | 505.0000 | 48 | 249.9950 | 76 | 175.0000 | 104 | 1012.9999 | 132 | 50.0000 |
21 | 504.9998 | 49 | 249.9916 | 77 | 175.0000 | 105 | 1019.9996 | 133 | 5.0000 |
22 | 505.0000 | 50 | 249.9995 | 78 | 330.0000 | 106 | 953.9999 | 134 | 42.0000 |
23 | 504.9998 | 51 | 165.0000 | 79 | 531.0000 | 107 | 951.9998 | 135 | 42.0000 |
24 | 504.9996 | 52 | 165.0000 | 80 | 530.9995 | 108 | 1005.9996 | 136 | 41.0000 |
25 | 536.9997 | 53 | 165.0000 | 81 | 398.6524 | 109 | 1013.0000 | 137 | 17.0000 |
26 | 536.9995 | 54 | 165.0000 | 82 | 56.0000 | 110 | 1020.9998 | 138 | 7.0000 |
27 | 548.9998 | 55 | 180.0000 | 83 | 115.0000 | 111 | 1014.9996 | 139 | 7.0000 |
28 | 548.9993 | 56 | 180.0000 | 84 | 115.0000 | 112 | 94.0000 | 140 | 26.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
SDE [29] | 1560236.85 | NA | NA | NA |
OIWO [32] | 1559712.2604 | NA | NA | NA |
HAAA [37] | 1559710.00 | 1559712.87 | 1559731.00 | 4.1371 |
GWO [35] | 1559953.18 | 1560132.93 | 1560228.40 | 1.024 |
KGMO [41] | 1583944.60 | 1583952.14 | 1583963.52 | NA |
IWO | 1564050.0027 | 1567185.2227 | 1571056.6280 | 1678.8488 |
HIWO | 1559709.5266 | 1559709.6956 | 1559709.8959 | 0.0856 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 218.6095 | 33 | 280.6560 | 65 | 279.6118 | 97 | 287.7203 | 129 | 431.0758 |
2 | 209.2361 | 34 | 238.9676 | 66 | 238.5645 | 98 | 238.6988 | 130 | 275.8790 |
3 | 279.6486 | 35 | 279.9554 | 67 | 287.7296 | 99 | 426.2750 | 131 | 219.6189 |
4 | 240.3113 | 36 | 240.9831 | 68 | 241.2519 | 100 | 272.6741 | 132 | 210.4739 |
5 | 280.0206 | 37 | 290.1069 | 69 | 427.7708 | 101 | 217.5647 | 133 | 281.6640 |
6 | 238.4301 | 38 | 240.0425 | 70 | 272.9907 | 102 | 211.9593 | 134 | 238.9676 |
7 | 288.2326 | 39 | 426.3102 | 71 | 218.5918 | 103 | 280.6578 | 135 | 276.5752 |
8 | 239.5051 | 40 | 275.6392 | 72 | 212.7020 | 104 | 239.2363 | 136 | 239.3707 |
9 | 425.6549 | 41 | 219.6195 | 73 | 281.6629 | 105 | 276.3263 | 137 | 287.7806 |
10 | 275.6903 | 42 | 210.9690 | 74 | 238.9676 | 106 | 240.7144 | 138 | 238.5645 |
11 | 217.5646 | 43 | 282.6711 | 75 | 279.3688 | 107 | 290.0715 | 139 | 430.7874 |
12 | 212.4544 | 44 | 240.3113 | 76 | 237.6239 | 108 | 238.8332 | 140 | 275.8606 |
13 | 280.6558 | 45 | 279.7868 | 77 | 289.9995 | 109 | 425.7918 | 141 | 218.6539 |
14 | 238.6988 | 46 | 237.4895 | 78 | 239.9082 | 110 | 275.2705 | 142 | 210.7215 |
15 | 279.9370 | 47 | 287.7274 | 79 | 425.2406 | 111 | 217.5671 | 143 | 281.6640 |
16 | 240.7144 | 48 | 240.0425 | 80 | 276.0112 | 112 | 212.2069 | 144 | 239.3707 |
17 | 287.6968 | 49 | 427.4497 | 81 | 218.5923 | 113 | 281.6664 | 145 | 276.3578 |
18 | 239.7738 | 50 | 275.6817 | 82 | 212.2069 | 114 | 239.6394 | 146 | 239.6394 |
19 | 427.4049 | 51 | 219.6197 | 83 | 282.7049 | 115 | 276.0940 | 147 | 287.7565 |
20 | 275.6990 | 52 | 213.4447 | 84 | 237.7582 | 116 | 240.3113 | 148 | 239.3707 |
21 | 217.5665 | 53 | 282.6717 | 85 | 279.7940 | 117 | 290.0972 | 149 | 426.3023 |
22 | 212.2069 | 54 | 237.8926 | 86 | 239.3707 | 118 | 239.5051 | 150 | 275.6371 |
23 | 283.6805 | 55 | 276.2856 | 87 | 290.0916 | 119 | 429.4367 | 151 | 217.5647 |
24 | 239.7738 | 56 | 239.5051 | 88 | 239.2363 | 120 | 275.6690 | 152 | 212.2069 |
25 | 279.9011 | 57 | 287.6883 | 89 | 427.0504 | 121 | 217.5656 | 153 | 279.6493 |
26 | 240.9831 | 58 | 238.5645 | 90 | 275.7937 | 122 | 210.2264 | 154 | 238.4301 |
27 | 290.0737 | 59 | 429.9489 | 91 | 217.5643 | 123 | 280.6617 | 155 | 279.9078 |
28 | 240.8488 | 60 | 275.5096 | 92 | 212.9496 | 124 | 239.7738 | 156 | 240.4457 |
29 | 427.1007 | 61 | 218.5915 | 93 | 282.6732 | 125 | 275.9409 | 157 | 287.7385 |
30 | 276.2995 | 62 | 212.9496 | 94 | 240.4457 | 126 | 240.1769 | 158 | 238.5645 |
31 | 219.6189 | 63 | 282.6705 | 95 | 279.4854 | 127 | 287.6965 | 159 | 426.9110 |
32 | 211.7117 | 64 | 239.9082 | 96 | 240.1769 | 128 | 238.4301 | 160 | 272.7775 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 10004.20 | 10004.21 | 10004.45 | NA |
OIWO [32] | 9981.9834 | 9982.991 | 9983.998 | NA |
BBO [28] | 10008.71 | 10009.16 | 10010.59 | NA |
DE/BBO [28] | 10007.05 | 10007.56 | 10010.26 | NA |
CBA [31] | 10002.8596 | 10006.3251 | 10045.2265 | 9.5811 |
IWO | 9984.8409 | 9985.5127 | 9986.1947 | 0.3252 |
HIWO | 9981.7867 | 9982.0010 | 9982.1922 | 0.0934 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 455.0000 | 4 | 130.0000 | 7 | 430.0000 | 10 | 159.7871 | 13 | 25.0000 |
2 | 380.0000 | 5 | 170.0000 | 8 | 71.2594 | 11 | 80.0000 | 14 | 15.0000 |
3 | 130.0000 | 6 | 460.0000 | 9 | 58.4944 | 12 | 80.0000 | 15 | 15.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
EMA [52] | 32704.4503 | 32704.4504 | 32704.4506 | NA |
FA [27] | 32704.5000 | 32856.1000 | 33175.0000 | 147.17022 |
ICS [50] | 32706.7358 | 32714.4669 | 32752.5183 | NA |
WCA [49] | 32704.4492 | 32704.5096 | 32704.5196 | 4.513e-05 |
RTO [51] | 32701.8145 | 32704.5300 | 32715.1800 | 5.07 |
OLCSO [2] | 32692.3961 | 32692.3981 | 32692.4033 | 0.0022 |
IWO | 32691.8615 | 32691.9392 | 32692.1421 | 0.0927 |
HIWO | 32691.5614 | 32691.8615 | 32691.8616 | 0.0001 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 113.9993 | 9 | 289.4281 | 17 | 489.2798 | 25 | 523.2794 | 33 | 190.0000 |
2 | 113.9993 | 10 | 279.5996 | 18 | 489.2793 | 26 | 523.2794 | 34 | 200.0000 |
3 | 120.0000 | 11 | 243.5995 | 19 | 511.2795 | 27 | 10.0000 | 35 | 199.9999 |
4 | 179.7330 | 12 | 94.0000 | 20 | 511.2793 | 28 | 10.0000 | 36 | 164.7999 |
5 | 87.7999 | 13 | 484.0391 | 21 | 523.2794 | 29 | 10.0000 | 37 | 109.9998 |
6 | 139.9998 | 14 | 484.0390 | 22 | 523.2794 | 30 | 87.7999 | 38 | 110.0000 |
7 | 300.0000 | 15 | 484.0393 | 23 | 523.2794 | 31 | 190.0000 | 39 | 109.9999 |
8 | 299.9997 | 16 | 484.0391 | 24 | 523.2794 | 32 | 190.0000 | 40 | 549.9999 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 136855.19 | 136855.19 | 136855.19 | NA |
BBO [28] | 137026.82 | 137116.58 | 137587.82 | NA |
DE/BBO [28] | 136950.77 | 136966.77 | 137150.77 | NA |
SDE [29] | 138157.46 | NA | NA | NA |
OIWO [32] | 136452.68 | 136452.68 | 136452.68 | NA |
HAAA [37] | 136433.5 | 136436.6 | NA | 3.341896 |
IWO | 136543.8580 | 137009.5641 | 137679.1073 | 292.9686 |
HIWO | 136430.9504 | 136435.2127 | 136441.1059 | 4.3238 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 110.8335 | 17 | 489.3362 | 33 | 189.9994 | 49 | 284.6071 | 65 | 523.2794 |
2 | 111.5439 | 18 | 489.2794 | 34 | 165.1983 | 50 | 130.0000 | 66 | 523.2835 |
3 | 97.3834 | 19 | 511.2731 | 35 | 199.9997 | 51 | 94.0040 | 67 | 10.0000 |
4 | 179.7603 | 20 | 511.2666 | 36 | 199.9998 | 52 | 94.0000 | 68 | 10.0000 |
5 | 87.9806 | 21 | 523.2525 | 37 | 109.9999 | 53 | 214.7298 | 69 | 10.0000 |
6 | 139.9997 | 22 | 523.2805 | 38 | 110.0000 | 54 | 394.2675 | 70 | 87.8052 |
7 | 259.5584 | 23 | 523.2794 | 39 | 109.9987 | 55 | 394.2967 | 71 | 190.0000 |
8 | 284.7677 | 24 | 523.2794 | 40 | 511.2603 | 56 | 304.4839 | 72 | 189.9997 |
9 | 284.6331 | 25 | 523.2794 | 41 | 110.9296 | 57 | 489.3082 | 73 | 189.9991 |
10 | 130.0000 | 26 | 523.2958 | 42 | 110.8195 | 58 | 489.2773 | 74 | 164.7786 |
11 | 169.0220 | 27 | 10.0000 | 43 | 97.3706 | 59 | 511.2121 | 75 | 199.9994 |
12 | 94.0000 | 28 | 10.0000 | 44 | 179.7187 | 60 | 511.2992 | 76 | 200.0000 |
13 | 214.7422 | 29 | 10.0000 | 45 | 87.8560 | 61 | 523.2830 | 77 | 109.9990 |
14 | 394.1929 | 30 | 89.6856 | 46 | 139.9995 | 62 | 523.3201 | 78 | 110.0000 |
15 | 394.2794 | 31 | 189.9993 | 47 | 259.6320 | 63 | 523.2794 | 79 | 109.9996 |
16 | 394.3050 | 32 | 189.9992 | 48 | 284.6702 | 64 | 523.2794 | 80 | 511.2482 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
THS [34] | 243192.6899 | 243457.36 | NA | 120.9889 |
CSO [40] | 243195.3781 | 243546.6283 | 244038.7352 | NA |
HAAA [37] | 242815.9 | 242883 | 242944.5 | 29.2849 |
GWO [35] | 242825.4799 | 242829.8192 | 242837.1303 | 0.093 |
IWO | 246386.4038 | 248088.2077 | 249888.0623 | 844.0919 |
HIWO | 242815.2096 | 242836.1110 | 242872.4662 | 10.3458 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 2.4000 | 23 | 68.9000 | 45 | 659.9999 | 67 | 70.0000 | 89 | 82.4977 |
2 | 2.4000 | 24 | 350.0000 | 46 | 616.2499 | 68 | 70.0000 | 90 | 89.2333 |
3 | 2.4000 | 25 | 400.0000 | 47 | 5.4000 | 69 | 70.0000 | 91 | 57.5687 |
4 | 2.4000 | 26 | 400.0000 | 48 | 5.4000 | 70 | 359.9999 | 92 | 99.9986 |
5 | 2.4000 | 27 | 499.9992 | 49 | 8.4000 | 71 | 399.9999 | 93 | 439.9998 |
6 | 4.0000 | 28 | 500.0000 | 50 | 8.4000 | 72 | 399.9998 | 94 | 499.9999 |
7 | 4.0000 | 29 | 199.9997 | 51 | 8.4000 | 73 | 105.2864 | 95 | 600.0000 |
8 | 4.0000 | 30 | 99.9998 | 52 | 12.0000 | 74 | 191.4091 | 96 | 471.5717 |
9 | 4.0000 | 31 | 10.0000 | 53 | 12.0000 | 75 | 89.9996 | 97 | 3.6000 |
10 | 64.5432 | 32 | 19.9993 | 54 | 12.0000 | 76 | 49.9999 | 98 | 3.6000 |
11 | 62.2465 | 33 | 79.9950 | 55 | 12.0000 | 77 | 160.0000 | 99 | 4.4000 |
12 | 36.2739 | 34 | 249.9998 | 56 | 25.2000 | 78 | 295.4962 | 100 | 4.4000 |
13 | 56.6406 | 35 | 359.9999 | 57 | 25.2000 | 79 | 175.0102 | 101 | 10.0000 |
14 | 25.0000 | 36 | 399.9997 | 58 | 35.0000 | 80 | 98.2829 | 102 | 10.0000 |
15 | 25.0000 | 37 | 39.9998 | 59 | 35.0000 | 81 | 10.0000 | 103 | 20.0000 |
16 | 25.0000 | 38 | 69.9996 | 60 | 45.0000 | 82 | 12.0000 | 104 | 20.0000 |
17 | 154.9999 | 39 | 99.9998 | 61 | 45.0000 | 83 | 20.0000 | 105 | 40.0000 |
18 | 154.9993 | 40 | 119.9984 | 62 | 45.0000 | 84 | 199.9999 | 106 | 40.0000 |
19 | 155.0000 | 41 | 157.4299 | 63 | 184.9996 | 85 | 324.9972 | 107 | 50.0000 |
20 | 155.0000 | 42 | 219.9999 | 64 | 184.9996 | 86 | 440.0000 | 108 | 30.0000 |
21 | 68.9000 | 43 | 439.9999 | 65 | 184.9984 | 87 | 14.0886 | 109 | 40.0000 |
22 | 68.9000 | 44 | 559.9998 | 66 | 184.9997 | 88 | 24.0910 | 110 | 20.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 198016.29 | 198016.32 | 198016.89 | NA |
BBO [28] | 198241.166 | 198413.45 | 199102.59 | NA |
DE/BBO [28] | 198231.06 | 198326.66 | 198828.57 | NA |
OIWO [32] | 197989.14 | 197989.41 | 197989.93 | NA |
OLCSO [2] | 197988.8576 | 197989.5832 | 197990.4551 | 0.3699 |
IWO | 198252.3594 | 198621.3233 | 198902.7697 | 138.4714 |
HIWO | 197988.1927 | 197988.1969 | 197988.2045 | 0.0025 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 115.2442 | 29 | 500.9998 | 57 | 103.0000 | 85 | 115.0000 | 113 | 94.0000 |
2 | 189.0000 | 30 | 500.9994 | 58 | 198.0000 | 86 | 207.0000 | 114 | 94.0000 |
3 | 190.0000 | 31 | 505.9993 | 59 | 311.9941 | 87 | 207.0000 | 115 | 244.0000 |
4 | 190.0000 | 32 | 505.9997 | 60 | 281.1604 | 88 | 175.0000 | 116 | 244.0000 |
5 | 168.5393 | 33 | 506.0000 | 61 | 163.0000 | 89 | 175.0000 | 117 | 244.0000 |
6 | 189.9932 | 34 | 505.9998 | 62 | 95.0000 | 90 | 175.0000 | 118 | 95.0000 |
7 | 489.9992 | 35 | 499.9996 | 63 | 160.0000 | 91 | 175.0000 | 119 | 95.0000 |
8 | 489.9996 | 36 | 500.0000 | 64 | 160.0000 | 92 | 579.9998 | 120 | 116.0000 |
9 | 495.9997 | 37 | 240.9993 | 65 | 489.9465 | 93 | 645.0000 | 121 | 175.0000 |
10 | 495.9994 | 38 | 240.9999 | 66 | 196.0000 | 94 | 983.9998 | 122 | 2.0000 |
11 | 495.9997 | 39 | 773.9996 | 67 | 489.9717 | 95 | 977.9993 | 123 | 4.0000 |
12 | 496.0000 | 40 | 769.0000 | 68 | 489.9908 | 96 | 681.9997 | 124 | 15.0000 |
13 | 506.0000 | 41 | 3.0000 | 69 | 130.0000 | 97 | 719.9998 | 125 | 9.0000 |
14 | 509.0000 | 42 | 3.0000 | 70 | 234.7202 | 98 | 717.9993 | 126 | 12.0000 |
15 | 506.0000 | 43 | 249.2474 | 71 | 137.0000 | 99 | 719.9997 | 127 | 10.0000 |
16 | 504.9997 | 44 | 246.0287 | 72 | 325.4950 | 100 | 963.9998 | 128 | 112.0000 |
17 | 505.9997 | 45 | 249.9973 | 73 | 195.0000 | 101 | 958.0000 | 129 | 4.0000 |
18 | 505.9997 | 46 | 249.9863 | 74 | 175.0000 | 102 | 1006.9992 | 130 | 5.0000 |
19 | 504.9994 | 47 | 241.0622 | 75 | 175.0000 | 103 | 1006.0000 | 131 | 5.0000 |
20 | 505.0000 | 48 | 249.9950 | 76 | 175.0000 | 104 | 1012.9999 | 132 | 50.0000 |
21 | 504.9998 | 49 | 249.9916 | 77 | 175.0000 | 105 | 1019.9996 | 133 | 5.0000 |
22 | 505.0000 | 50 | 249.9995 | 78 | 330.0000 | 106 | 953.9999 | 134 | 42.0000 |
23 | 504.9998 | 51 | 165.0000 | 79 | 531.0000 | 107 | 951.9998 | 135 | 42.0000 |
24 | 504.9996 | 52 | 165.0000 | 80 | 530.9995 | 108 | 1005.9996 | 136 | 41.0000 |
25 | 536.9997 | 53 | 165.0000 | 81 | 398.6524 | 109 | 1013.0000 | 137 | 17.0000 |
26 | 536.9995 | 54 | 165.0000 | 82 | 56.0000 | 110 | 1020.9998 | 138 | 7.0000 |
27 | 548.9998 | 55 | 180.0000 | 83 | 115.0000 | 111 | 1014.9996 | 139 | 7.0000 |
28 | 548.9993 | 56 | 180.0000 | 84 | 115.0000 | 112 | 94.0000 | 140 | 26.0000 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
SDE [29] | 1560236.85 | NA | NA | NA |
OIWO [32] | 1559712.2604 | NA | NA | NA |
HAAA [37] | 1559710.00 | 1559712.87 | 1559731.00 | 4.1371 |
GWO [35] | 1559953.18 | 1560132.93 | 1560228.40 | 1.024 |
KGMO [41] | 1583944.60 | 1583952.14 | 1583963.52 | NA |
IWO | 1564050.0027 | 1567185.2227 | 1571056.6280 | 1678.8488 |
HIWO | 1559709.5266 | 1559709.6956 | 1559709.8959 | 0.0856 |
Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi | Generators | Pi |
1 | 218.6095 | 33 | 280.6560 | 65 | 279.6118 | 97 | 287.7203 | 129 | 431.0758 |
2 | 209.2361 | 34 | 238.9676 | 66 | 238.5645 | 98 | 238.6988 | 130 | 275.8790 |
3 | 279.6486 | 35 | 279.9554 | 67 | 287.7296 | 99 | 426.2750 | 131 | 219.6189 |
4 | 240.3113 | 36 | 240.9831 | 68 | 241.2519 | 100 | 272.6741 | 132 | 210.4739 |
5 | 280.0206 | 37 | 290.1069 | 69 | 427.7708 | 101 | 217.5647 | 133 | 281.6640 |
6 | 238.4301 | 38 | 240.0425 | 70 | 272.9907 | 102 | 211.9593 | 134 | 238.9676 |
7 | 288.2326 | 39 | 426.3102 | 71 | 218.5918 | 103 | 280.6578 | 135 | 276.5752 |
8 | 239.5051 | 40 | 275.6392 | 72 | 212.7020 | 104 | 239.2363 | 136 | 239.3707 |
9 | 425.6549 | 41 | 219.6195 | 73 | 281.6629 | 105 | 276.3263 | 137 | 287.7806 |
10 | 275.6903 | 42 | 210.9690 | 74 | 238.9676 | 106 | 240.7144 | 138 | 238.5645 |
11 | 217.5646 | 43 | 282.6711 | 75 | 279.3688 | 107 | 290.0715 | 139 | 430.7874 |
12 | 212.4544 | 44 | 240.3113 | 76 | 237.6239 | 108 | 238.8332 | 140 | 275.8606 |
13 | 280.6558 | 45 | 279.7868 | 77 | 289.9995 | 109 | 425.7918 | 141 | 218.6539 |
14 | 238.6988 | 46 | 237.4895 | 78 | 239.9082 | 110 | 275.2705 | 142 | 210.7215 |
15 | 279.9370 | 47 | 287.7274 | 79 | 425.2406 | 111 | 217.5671 | 143 | 281.6640 |
16 | 240.7144 | 48 | 240.0425 | 80 | 276.0112 | 112 | 212.2069 | 144 | 239.3707 |
17 | 287.6968 | 49 | 427.4497 | 81 | 218.5923 | 113 | 281.6664 | 145 | 276.3578 |
18 | 239.7738 | 50 | 275.6817 | 82 | 212.2069 | 114 | 239.6394 | 146 | 239.6394 |
19 | 427.4049 | 51 | 219.6197 | 83 | 282.7049 | 115 | 276.0940 | 147 | 287.7565 |
20 | 275.6990 | 52 | 213.4447 | 84 | 237.7582 | 116 | 240.3113 | 148 | 239.3707 |
21 | 217.5665 | 53 | 282.6717 | 85 | 279.7940 | 117 | 290.0972 | 149 | 426.3023 |
22 | 212.2069 | 54 | 237.8926 | 86 | 239.3707 | 118 | 239.5051 | 150 | 275.6371 |
23 | 283.6805 | 55 | 276.2856 | 87 | 290.0916 | 119 | 429.4367 | 151 | 217.5647 |
24 | 239.7738 | 56 | 239.5051 | 88 | 239.2363 | 120 | 275.6690 | 152 | 212.2069 |
25 | 279.9011 | 57 | 287.6883 | 89 | 427.0504 | 121 | 217.5656 | 153 | 279.6493 |
26 | 240.9831 | 58 | 238.5645 | 90 | 275.7937 | 122 | 210.2264 | 154 | 238.4301 |
27 | 290.0737 | 59 | 429.9489 | 91 | 217.5643 | 123 | 280.6617 | 155 | 279.9078 |
28 | 240.8488 | 60 | 275.5096 | 92 | 212.9496 | 124 | 239.7738 | 156 | 240.4457 |
29 | 427.1007 | 61 | 218.5915 | 93 | 282.6732 | 125 | 275.9409 | 157 | 287.7385 |
30 | 276.2995 | 62 | 212.9496 | 94 | 240.4457 | 126 | 240.1769 | 158 | 238.5645 |
31 | 219.6189 | 63 | 282.6705 | 95 | 279.4854 | 127 | 287.6965 | 159 | 426.9110 |
32 | 211.7117 | 64 | 239.9082 | 96 | 240.1769 | 128 | 238.4301 | 160 | 272.7775 |
Algorithms | SCmin ($) | SCavg ($) | SCmax ($) | SCstd |
ORCCRO [28] | 10004.20 | 10004.21 | 10004.45 | NA |
OIWO [32] | 9981.9834 | 9982.991 | 9983.998 | NA |
BBO [28] | 10008.71 | 10009.16 | 10010.59 | NA |
DE/BBO [28] | 10007.05 | 10007.56 | 10010.26 | NA |
CBA [31] | 10002.8596 | 10006.3251 | 10045.2265 | 9.5811 |
IWO | 9984.8409 | 9985.5127 | 9986.1947 | 0.3252 |
HIWO | 9981.7867 | 9982.0010 | 9982.1922 | 0.0934 |