Research article Special Issues

A hybrid invasive weed optimization algorithm for the economic load dispatch problem in power systems

  • In this study, a hybrid invasive weed optimization (HIWO) algorithm that hybridizes the invasive weed optimization (IWO) algorithm and genetic algorithm (GA) has been proposed to solve economic dispatch (ED) problems in power systems. In the proposed algorithm, the IWO algorithm is used as the main optimizer to explore the solution space, whereas the crossover and mutation operations of the GA are developed to significantly improve the optimization ability of IWO. In addition, an effective repair method is embedded in the proposed algorithm to repair infeasible solutions by handing various practical constraints of ED problems. To verify the optimization performance of the proposed algorithm and the effectiveness of the repair method, six ED problems in the different-scale power systems were tested and compared with other algorithms proposed in the literature. The experimental results indicated that the proposed HIWO algorithm can obtain the more economical dispatch solutions, and the proposed repair method can effectively repair each infeasible dispatch solution to a feasible solution. The convergence capability, applicability and effectiveness of HIWO were also demonstrated through the comprehensive comparison results.

    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

    Related Papers:

    [1] Yixin Zhuo, Ling Li, Jian Tang, Wenchuan Meng, Zhanhong Huang, Kui Huang, Jiaqiu Hu, Yiming Qin, Houjian Zhan, Zhencheng Liang . Optimal real-time power dispatch of power grid with wind energy forecasting under extreme weather. Mathematical Biosciences and Engineering, 2023, 20(8): 14353-14376. doi: 10.3934/mbe.2023642
    [2] Ning Zhou, Chen Zhang, Songlin Zhang . A multi-strategy firefly algorithm based on rough data reasoning for power economic dispatch. Mathematical Biosciences and Engineering, 2022, 19(9): 8866-8891. doi: 10.3934/mbe.2022411
    [3] Lihe Liang, Jinying Cui, Juanjuan Zhao, Yan Qiang, Qianqian Yang . Ultra-short-term forecasting model of power load based on fusion of power spectral density and Morlet wavelet. Mathematical Biosciences and Engineering, 2024, 21(2): 3391-3421. doi: 10.3934/mbe.2024150
    [4] Guohao Sun, Sen Yang, Shouming Zhang, Yixing Liu . A hybrid butterfly algorithm in the optimal economic operation of microgrids. Mathematical Biosciences and Engineering, 2024, 21(1): 1738-1764. doi: 10.3934/mbe.2024075
    [5] Lingling Li, Congbo Li, Li Li, Ying Tang, Qingshan Yang . An integrated approach for remanufacturing job shop scheduling with routing alternatives. Mathematical Biosciences and Engineering, 2019, 16(4): 2063-2085. doi: 10.3934/mbe.2019101
    [6] Shanshan Pan, Jinbao Jian, Linfeng Yang . Solution to dynamic economic dispatch with prohibited operating zones via MILP. Mathematical Biosciences and Engineering, 2022, 19(7): 6455-6468. doi: 10.3934/mbe.2022303
    [7] Bowen Ding, Zhaobin Ma, Shuoyan Ren, Yi Gu, Pengjiang Qian, Xin Zhang . A genetic algorithm with two-step rank-based encoding for closed-loop supply chain network design. Mathematical Biosciences and Engineering, 2022, 19(6): 5925-5956. doi: 10.3934/mbe.2022277
    [8] Yanmei Jiang, Mingsheng Liu, Jianhua Li, Jingyi Zhang . Reinforced MCTS for non-intrusive online load identification based on cognitive green computing in smart grid. Mathematical Biosciences and Engineering, 2022, 19(11): 11595-11627. doi: 10.3934/mbe.2022540
    [9] Yejun Hu, Liangcai Dong, Lei Xu . Multi-AGV dispatching and routing problem based on a three-stage decomposition method. Mathematical Biosciences and Engineering, 2020, 17(5): 5150-5172. doi: 10.3934/mbe.2020279
    [10] Mehrdad Ahmadi Kamarposhti, Ilhami Colak, Kei Eguchi . Optimal energy management of distributed generation in micro-grids using artificial bee colony algorithm. Mathematical Biosciences and Engineering, 2021, 18(6): 7402-7418. doi: 10.3934/mbe.2021366
  • In this study, a hybrid invasive weed optimization (HIWO) algorithm that hybridizes the invasive weed optimization (IWO) algorithm and genetic algorithm (GA) has been proposed to solve economic dispatch (ED) problems in power systems. In the proposed algorithm, the IWO algorithm is used as the main optimizer to explore the solution space, whereas the crossover and mutation operations of the GA are developed to significantly improve the optimization ability of IWO. In addition, an effective repair method is embedded in the proposed algorithm to repair infeasible solutions by handing various practical constraints of ED problems. To verify the optimization performance of the proposed algorithm and the effectiveness of the repair method, six ED problems in the different-scale power systems were tested and compared with other algorithms proposed in the literature. The experimental results indicated that the proposed HIWO algorithm can obtain the more economical dispatch solutions, and the proposed repair method can effectively repair each infeasible dispatch solution to a feasible solution. The convergence capability, applicability and effectiveness of HIWO were also demonstrated through the comprehensive comparison results.


    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=Ni=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)=aiPi2+biPi+ci (2)
    Ci(Pi)=aiPi2+biPi+ci+|eisin(fi(PminiPi))| (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; Pmini is the lower limit of Pi.

    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) and maximum (Pmaxi) of the power output of the ith generator, as shown in Eq 4.

    PminiPiPmaxi (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.

    Ni=1Pi=PD (5)
    Ni=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=Ni=1Nj=1PiBijPj+Ni=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.

    {PiP0iURiP0iPiDRi (8)

    where P0i is the power output of the ith generator at the previous time interval. URi and DRi represent the upper limits of ramp up and ramp down rate of the ith generator, respectively.

    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,P0iDRi}Pimin{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{PminiPiPli,1Pui,k1PiPli,kPui,npiPiPmaxik=2,3,,npi (10)

    where Pui,k and Pli,k represent the upper and lower limits of the kth prohibited zone, respectively. npi is the number of the prohibited zones of Pi.

    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=FitjFitminFitmaxFitmin(NsmaxNsmin)+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=(ItermaxIter)mItermaxm(σivσfv)+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.

    Figure 1.  Pseudo code of the hybrid HIWO algorithm.

    Each offspring weed (OW(j, q)) (q = 1, 2, …, Nsj ) crosses with its parent weed (Wj) to generate a new weed (OW(j,q)). For each variable Pi (i = 1, 2, …, N), calculate the generation cost (COWi(j,q)) consumed by the Pi of OW(j, q) and the generation cost (CWij) consumed by the Pi of Wj, , and then determine the value of the Pi of OW(j,q) according to the following two cases.

    (a) If COWi(j,q)CWij, the Pi of OW(j,q) is equal to that of OW(j, q).

    (b) If COWi(j,q)>CWij, the Pi of OW(j,q) is equal to that of Wj.

    After the new offspring weed (OW(j,q)) is generated by the crossover operation, set OW(j,q)=OW(j,q).

    For each offspring weed (OW(j, q)) (q = 1, 2, …, Nsj ), randomly select X mutation points from N variables (P1,P2,,PN), and then modify the Pi of each mutation point using a random number that is distributed around Pi according to a normal distribution with a standard deviation (σm). The calculation formula of σm is expressed as Eq 13.

    σm=(PmaxiPmini)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=(PmaxiPmini)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(i=1,2,,N) of Wj to satisfy the generation capacity limits and ramp rate limits by using Eq 17. If Pi violates the constraint of prohibited operating zone, update Pi using a upper or lower limit of the prohibited operating zones that is closest to Pi.

    Pi={max{Pmini,P0iDRi}if Pi<max{Pmini,P0iDRi}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 V0, go to Step 3. Otherwise, go to Step 5.

    V=|Ni=1PiPDNi=1Nj=1PiBijPjNi=1B0iPiB00| (18)
    V=|Ni=1PiPD| (19)

    Step 3: Determine the modification sequence of N generators. For each generator i (i = 1, 2, …, N), calculate the modification value Pi of Pi using the power outputs of the other N-1 generators. Create the set R = R={1,2,,N}{i} to store the indexes of the other N1 generators excluding the ith generator. For the ED problem neglecting transmission losses, Eq 5 is modified to Eq 20 to calculate Pi. For the ED problem considering transmission losses, Eq 6 can be expressed as a second-order equation in Pi as shown in Eq 21. One root of the second-order equation is chosen as Pi, as shown in Eq 22. Then, modify Pi using Eq 17 to satisfy the generation capacity limits.

    Pi=PDrRPr (20)
    Bii(Pi)2+(2rRPrBir+B0i1)Pi+(PD+rRtRPrBrtPt+rRB0rPrrRPr+B00)=0 (21)
    Pi=(2rRPrBir+B0i1)(2rRPrBir+B0i1)24Bii(PD+rRtRPrBrtPt+rRB0rPrrRPr+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 Pi. Calculate the cost change (CXi) using Eq 23 and the constraint violation (PVi) of the power demand balance using Eq 18 or 19. Then, calculate the percentage (PCVi) according to CXi and PVi, as shown in Eq 24. Finally, create a set (S) to store the modification sequence of N variables, which is determined by the value of PCVi sorted in ascending order.

    CXi=Ci(Pi)Ci(Pi) (23)
    PCVi=CXimin(CX)max(CX)min(CX)+PVimin(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 (iS generator is selected as the modified generator, the modified value (Pi) of its power output is calculated by using Eq 20 or 22. If all the generators are modified and the power demand balance constraint is still not satisfied, go to Step 3. Otherwise, go to Step 5.

    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 σiv=2, final standard deviation σfv=0.0001, maximum number of iterations Itermax = 2000, and number of mutation points X = 1 in the 160-generator system and X = 3 in the other systems. HIWO is implemented by using the MATLAB (R2016a) environment on an Intel core i7-4790 CPU with 8.00 GB RAM personal computer.

    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.

    Table 1.  Optimal power output of HIWO for the 15-generator system.
    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

     | Show Table
    DownLoad: CSV
    Table 2.  Comparison of the optimal dispatch results for the 15-generator system.
    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

     | Show Table
    DownLoad: CSV

    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.

    Table 3.  Optimal power output of HIWO for the 40-generator system.
    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

     | Show Table
    DownLoad: CSV
    Table 4.  Comparison of the optimal results for the 40-generator system.
    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

     | Show Table
    DownLoad: CSV

    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.

    Table 5.  Optimal power output of HIWO for the 80-generator system (Case Ⅰ).
    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

     | Show Table
    DownLoad: CSV
    Table 6.  Comparison of the optimal results for the 80-generator system (Case Ⅰ).
    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

     | Show Table
    DownLoad: CSV

    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.

    Table 7.  Optimal power output of HIWO for the 110-generator system (Case Ⅱ).
    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

     | Show Table
    DownLoad: CSV
    Table 8.  Comparison of the optimal results for the 110-generator system (Case Ⅱ).
    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

     | Show Table
    DownLoad: CSV

    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.

    Table 9.  Optimal power output of HIWO for the 140-generator system (Case Ⅰ).
    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

     | Show Table
    DownLoad: CSV
    Table 10.  Comparison of the optimal results of HIWO for the 140-generator system (Case Ⅰ).
    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

     | Show Table
    DownLoad: CSV

    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.

    Table 11.  Optimal power output of HIWO for the 160-generator system (Case Ⅱ).
    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

     | Show Table
    DownLoad: CSV
    Table 12.  Comparison of the optimal results for the 160-generator system (Case Ⅱ).
    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

     | Show Table
    DownLoad: CSV

    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.

    Figure 2.  Convergence curves of HIWO and IWO on the six test systems.

    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.
  • This article has been cited by:

    1. Qi Wang, Yufeng Guo, Dongrui Zhang, Yingwei Wang, Ying Xu, Jilai Yu, Research on wind farm participating in AGC based on wind power variogram characteristics, 2022, 19, 1551-0018, 8288, 10.3934/mbe.2022386
    2. Swati Jain, Dr Krishna Teeth Chaturvedi, Review on Economic Load Dispatch and Associated Artificial Intelligence Algorithms, 2021, 7, 2582-4600, 34, 10.24113/ijoscience.v7i3.370
    3. Mahdi Azizi, Milad Baghalzadeh Shishehgarkhaneh, Mahla Basiri, Optimum design of truss structures by Material Generation Algorithm with discrete variables, 2022, 3, 27726622, 100043, 10.1016/j.dajour.2022.100043
    4. Dinu Calin Secui, Monica Liana Secui, Social small group optimization algorithm for large-scale economic dispatch problem with valve-point effects and multi-fuel sources, 2024, 54, 0924-669X, 8296, 10.1007/s10489-024-05517-8
    5. Rahul Gupta, Ashish Khanna, Bal Virdee, AOBL-IPACO: A novel and optimized algorithm to mitigate losses in electrical grid systems, 2024, 2511-2104, 10.1007/s41870-024-02211-3
    6. Thammarsat Visutarrom, Tsung-Che Chiang, Economic dispatch using metaheuristics: Algorithms, problems, and solutions, 2024, 150, 15684946, 110891, 10.1016/j.asoc.2023.110891
    7. Xiaobao Yu, Zhenyu Dong, Dandan Zheng, Siwei Deng, Analysis of critical peak electricity price optimization model considering coal consumption rate of power generation side, 2023, 31, 1614-7499, 41514, 10.1007/s11356-023-29754-5
    8. Balasim M. Hussein, Ahmed I. Jaber, Mohammed W. Abdulwahhab, Hayder J. Mohammed, Nikolay V. Korovkin, 2024, Application of Intelligent Optimization Algorithms on Economic Dispatch Problem, 979-8-3503-6370-8, 453, 10.1109/SCM62608.2024.10554111
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5524) PDF downloads(1086) Cited by(8)

Figures and Tables

Figures(2)  /  Tables(12)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog