Solution of Non-Convex and Dynamic Economic Load Dispatch Problem of Small Scale Power Systems using Dragonfly Algorithm

DOI : 10.17577/IJERTCONV4IS15046

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Solution of Non-Convex and Dynamic Economic Load Dispatch Problem of Small Scale Power Systems using Dragonfly Algorithm

Vikram Kumar Kamboj Department of Electrical Engineering, DAV University, Jalandhar,

Punjab, India

Ashutosh Bhadoria

Department of Electrical Engineering, DAV University, Jalandhar,

Punjab, India

Pawanpreet Singh

    1. ech Research Scholar, Electrical Engineering Department, DAV University,

      Jalandhar

      S. K. Bath

      Department of Electrical Engineering, GZS Campus College of Engineering & Technology,

      Bathinda, Punjab

      AbstractDragonfly algorithm is a novel intelligence optimization technique, which simulates the static and dynamic swarming behaviours of dragonflies in environment. Exploration and exploitation in dragonfly algorithm is achieved by modelling the social interaction of dragonflies in navigating, searching for foods and avoiding enemies when swarming dynamically or statistically. This paper presents the application of dragonfly algorithm for the solution of non-convex and dynamic economic load dispatch problem of electric power system. The performance of dragonfly algorithm is tested for economic load dispatch problem of six IEEE benchmarks of small scale power systems and the results are verified by a comparative study with Lambda Iteration Method, Particle Swarm Optimization (PSO) algorithm, Genetic Algorithm (GA), Simulated Annealing( SA), Artificial Bee Colony (ABC), Evolutionary Programming (EP) and Grey Wolf Optimizer(GWO). Comparative results show that the performance of Dragonfly algorithm is better than recently developed GWO algorithm and other well known heuristics and meta-heuristics search algorithms.

      Keywords Economic Load Dispatch Problem (ELDP), Dragonfly Algorithm (DA), Grey Wolf Optimizer (GWO)

      1. INTRODUCTION

        In modern power system networks, there are various generating resources like thermal, hydro, nuclear etc. Also, the load demand varies during a day and attains different peak values. Thus, it is required to decide which generating unit to turn on and at what time it is needed in the power system network and also the sequence in which the units must be shut down keeping in mind the cost effectiveness of turning on and shutting down of respective units. The entire process of computing and making these decisions is known as unit commitment (UC). The unit which is decided or scheduled to be connected to the power system network, as and when required, is known to be committed unit. Unit commitment in power systems refers to the problem of determining the on/off states of generating units that minimize the operating cost for a given time horizon. Electrical power plays a pivotal role in the modern world to satisfy various needs. It is therefore very

        important that the electrical power generated is transmitted and distributed efficiently in order to satisfy the power requirement. Electrical power is generated in several ways. The most significant crisis in the planning and operation of electric power generation system is the effective scheduling of all generators in a system to meet the required demand. The Economic Load Dispatch (ELD) problem is the most important optimization problem in scheduling the generation among thermal generating units in power system.

        Economic dispatch in electric power system refers to the short-term discernment of the optimal generation output of various electric utilities, to meet the system load demand, at the minimum possible cost, subject to various system and operating constraints viz. operational and transmission constraints. The Economic Load Dispatch Problem (ELDP) means that the electric utilities (i.e. generator's) real and reactive power are allowed to vary within certain limits so as to meet a particular load demand within lowest fuel cost. The ultimate aim of the ELD problem is to minimize the operation cost of the power generation system, while supplying the required power demanded. In addition to this, the various operational constraints of the system should also be satisfied. The problem of ELD is usually multimodal, discontinuous and highly nonlinear. Although the cost curve of thermal generating units are generally modelled as a smooth curve, the input-output characteristics are nonlinear by nature because of valve-point loading effects, Prohibited Operating Zones (POZ), ramp rate limits etc.

        In recent years, various evolutionary, heuristic and meta- heuristics optimization algorithms have been developed simulating natural phenomena such as: Genetic Algorithm(GA) [1], Ant Colony Optimization (ACO) [2], Particle Swarm Optimization[3], Simulating Annealing(SA)[4], Gravitational Local Search (GLSA) [5], Big-Bang Big-Crunch (BBBC) [6], Gravitational Search Algorithm (GSA) [7], Curved Space Optimization (CSO) [8], Charged System Search (CSS) [9], Central Force Optimization (CFO) [10], Artificial Chemical Reaction Optimization Algorithm (ACROA) [11], Black Hole (BH)

        1. algorithm, Ray Optimization algorithm(ROA) [13], Small-World Optimization Algorithm (SWOA) [14], Galaxy- based Search Algorithm (GbSA) [15], Shuffled Frog Leaping Algorithm(SFLA)[16], Snake Algorithm[17], Biogeography Based Optimization[18], Marriage in Honey Bees Optimization Algorithm (MBO) [19] ,Artificial Fish-Swarm Algorithm (AFSA) [20] , Termite Algorithm (TA)[21] , Wasp Swarm Algorithm(WSA) [22] , Monkey Search Algorithm(MSA) [23] , Bee Collecting Pollen Algorithm (BCPA) [24] , Cuckoo Search Algorithm (CSA) [25], Dolphin Partner Optimization (DPO)[26] , Firefly Algorithm[27], Krill Herd (KH) algorithm [28] , Fruit fly Optimization Algorithm (FOA) [29], Distributed BBO[30]. Out of these heuristics evolutionary search algorithm, some of these are used to solve Economic Load Dispatch Problem(ELDP), Combined Economic Load Dispatch Problem(CELDP), Dynamic Economic Dispatch Problem(DEDP) and Combined Economic Emission Dispatch (CEED) and are reported in numerous literatures as: Evolutionary Programming [31], Particle Swarm Optimization[32], Genetic Algorithm[32,33], Improved Genetic Algorithm[34], Adaptive PSO and Chaotic PSO[35], cardinal Priority ranking based Decision making[36], Gravitational Search Algorithm[37, 42, 45], Biogeography Based Optimization[38, 39, 44], Intelligent Water Drop Algorithm[40], Hybrid Harmony Search Algorithm[41], Firefly Algorithm[43], Cuckoo Search Algorithm[46, 54], Biogeography Based Optimization[44], Differential harmony Search[47], Hybrid Particle Swarm Optimization and Gravitational Search Algorithm[48], Differential Evolution[49], Modified Ant Colony Optimization[50], Modified Harmony Search[51], Hybrid GA-MGA[52], Artificial Bee Colony[53]. Although no optimization algorithm can perform general enough to solve all optimizations problems, each optimization algorithm have their own advantages and disadvantages. The limitations of some of these well known optimization algorithms are listed below:

          The major limitations of the numerical techniques and dynamic programming method are the size or dimensions of the problem, large computational time and complexity in programming. The mixed integer programming methods for solving the economic load dispatch problem fails when the participation of number of units increases because they require a large memory and suffer from great computational delay. Gradient Descent method is distracted for Non-Differentiable search spaces. The Lagrangian Relaxation (LR) approach fails to obtain solution feasibility and solution quality of problems and becomes complex if the number of units are more. The Branch and Bound (BB) method employs a linear function o represent fuel cost, start-up cost and obtains a lower and upper bounds. The difficulty of this method is the exponential growth in the execution time for systems of a large practical size. An Expert System (ES) algorithm rectifies the complexity in calculations and saving in computation time. But it faces the problem if the new schedule is differing from schedule in database. The fuzzy theory method using fuzzy set solves the forecasted load schedules error but it suffers from complexity. The Hopfield neural network technique considers more constraints but it may suffer from numerical convergence due to its training process. The Simulated

          Annealing (SA) and Tabu Search (TS) are powerful, general- purpose stochastic optimization technique, which can theoretically converge asymptotically to a global optimum solution with probability one. But it takes much time to reach the near-global minimum. Particle swarm optimization (PSO) has simple concept, easy implementation, relative robustness to control parameters and computational efficiency[55], although it has numerous advantages, it get trapped in a local minimum, when handling heavily constrained problems due to the limited local/global searching capabilities [56, 57]. Differential Evolution (DE) algorithm has the ability to find the true global minimum regardless of the initial parameters values and requires few control parameters. It has parallel processing nature and fast convergence as compared to conventional optimization algorithm. Although, it does not always give an exact global optimum due to premature convergence and may require tremendously high computation time because of a large number of fitness evaluations. The Biogeography Based Optimization (BBO) is an efficient algorithm for Power System optimization, which does not take unnecessary computational time and is good for exploiting the solutions. The solutions obtained by BBO algorithm does not die at the end of each generation like the other optimization algorithm, but the convergence becomes slow for medium and large scale systems. Gravitational Search algorithm has the advantages to explore better optimized results, but due to the cumulative effect of the fitness function on mass, masses get heavier and heavier over the course of iteration. This causes masses to remain in close proximity and neutralise the gravitational forces of each other in later iterations, preventing them from rapidly exploiting the optimum [55]. Therefore, increasing effect of the cost function on mass, masses get greater over the course of iteration and search process and convergence becomes slow. To overcome the limitation of GSA, Seyedali Mirjalili [55] proposed an Adaptive gbest- Guided Gravitational Search algorithm (AgGGSA), in which the best mass is archived and utilised to accelerate the exploitation phase, enriching the weakness of GSA. Grey wolf Optimizer (GWO) is a recently developed powerful evolutionary algorithm proposed by Seyedali Mirjalili [57] and has the ability to converge to a better quality near-optimal solution and possesses better convergence characteristics than other prevailing techniques reported in the recent literatures. Also, GWO has a good balance between exploration and exploitation that result in high local optima avoidance, but the computation of GWO algorithm becomes slow, when applied to economic dispatch problem of medium and large scale power system. To overcome the drawbacks of GWO algorithm, recently developed intelligence Dragonfly Algorithm (DA), developed by Seyedali Mirjalili [59], is tested for the solution of non-convex and dynamic economic load dispatch problem of electric power system.

      2. ECONOMIC LOAD DISPATCH PROBLEM FORMULATION

        The scheduling of electric utilities along with the distribution of the generation power which must be planned to meet the load demand for a specific time period represents the Unit Commitment Problem (UCP). Economic Load

        Dispatch Problem (ELDP) refers the optimal generation schedule for the generation system to deliver the

        The equation (5) represent the unconstrained economic

        U U

        U U

        B P P

        required load demand plus transmission loss with the optimal

        load dispatch problem including penalty factor of n1 m1

        nm n m

        generation fuel cost. Noteworthy economical benefits can be achieved by searching a better solution to the Economic Load Dispatch Problem (ELDP). The economic dispatch problem is defined so as to optimize the total operational cost of an electric power system while meeting the total load demand plus transmission losses within utilities generating limits [56]. The overall objective of Economic Load Dispatch Problem (ELDP) of electric power system is to plan the devoted (Committed) electric utilities outputs so as to congregate the load demand at optimal operating cost while satisfying all generating utilities constraints and various operational constraints of the electric utilities. The economic load dispatch problem (ELDP) is a constrained optimization problem and it can be mathematically expressed as follows [56]:

        U

        . The complete unconstrained economic load dispatch problem having (U-1) variables can be represented as:

        U U U U

        U U U U

        n n n n n n n Demand nm n m

        n n n n n n n Demand nm n m

        min[FC(P )] ( P 2 P ) 1000 * abs(P P B P P )

        n1 n1 n1 m1

        (6)

        The complete unconstrained economic load dispatch problem with valve point effect having (U-1) variables can be represented as:

        U U U U

        U U U U

        n n n n n n n n n n n Demand nm n m

        n n n n n n n n n n n Demand nm n m

        min[FC(P )] ( P 2 P ( sin( (P min P ) 1000*abs(P P B P P )

        n1 n1 n1 m1

        (7)

      3. DRAGONFLY ALGORITHM AND MATHEMATICAL FORMULATION

        n n n n n n

        n n n n n n

        min[FC(P )] ( P 2 P )

        n1

        subject to:

          1. The energy balance equation:

            U

            U

            Pn PDemand PLoss .

            n1

          2. The inequality constraints:

        $/Hour (1)

        (2)

        Dragonfly Algorithm (DA) is a novel intelligence optimization technique proposed by Seyedali Mirjalili [59], which simulates the behaviours of dragonflies stationary and energetic swarming in environment .Exploration and exploitation in dragonfly algorithm is obtained by imitating the social communication of dragonflies in navigating, searching for foods and avoiding enemy when swarming

        n n n

        n n n

        P min P P max (n 1, 2,3,…, U).

        (3)

        where, n , n and n are cost coefficients.

        PDemand is Load Demand.

        PLoss is power transmission Loss.

        U is the number of generating units.

        Pn is real power generation and will act as decision variable.

        The most simple and approximate method of expressing

        statistically or energetically. The exploration and exploitation in dragonfly algorithm is achieved by following steps:

        • Separation: This refers to the static smash avoidance of the individuals from other individuals in the Neighbourhood.

        • Alignment: which indicates velocity similar of individuals to that of other individuals in neighbourhood?

        • Cohesion: which refers to the inclination of individuals towards the centre of the mass of the neighbourhood? The main function of any swarm is endurance, so all of the individuals should be attracted towards food sources and distracted outward enemies. Considers these two behaviours, there are five main factors in position

        power transmission loss,

        PLoss

        as a function of generator

        updating of individuals in swarms. The behaviours of

        U U

        U U

        powers is through George's Formula using B-coefficients and mathematically can be expressed as [56]:

        PLoss Pg Bnm Pg

        each is mathematically modelled as follows: The separation process in dragonfly algorithm can be updated as ollows:

        N

        n1 m1

        P

        n m

        MW (4)

        P

        Si X XJ J 1

        (8)

        where, gn and gm are the real power generations at the nth and mth buses respectively.

        Bnm is the loss coefficients which are constant under certain assumed conditions and U is the number of generating units.

        The constrained Economic Load Dispatch Problem can be converted to unconstrained ELD Problem using Penalty of

        Where, N is the number of neighbouring individuals, X is the current individual position, X J is the position J-th neighbouring individual.

        Alignment process in dragonfly algorithm can be updated using following recursive relation:

        N

        N

        VJ

        Ai J 1

        definite value, which can be mathematically expressed as:

        U U U U

        U U U U

        min[FC(Pn )] Fn (Pn ) 1000 * (Pn PDemand BnmPnPm )

        N

        where, VJ

        individual.

        (9)

        shows the velocity of J-th neighbouring

        n1 n1 n1 m1

        (5)

        The cohesion in dragonfly algorithm is calculated as follows:

        N

        N

        XJ

        Ci J 1 X

        N (10)

        Where, X is current individual position, N is the number

        of neighbourhoods and X J

        neighbouring individual.

        is the position of J-th

        Attraction towards a food source is calculated as follows:

        F X X

        i (11)

        Where, X is the current individual position and X

        position shows the food source.

        Interruption outwards an enemies is calculated as follows

        Fig.1: PSEUDO code for Dragonfly algorithm

      4. TEST SYSTEMS, RESULTS AND DISCUSSION

i

i

E X X

(12)

Where, X is the current individual position and X shows the position of the enemy.

For updating the position of imitation dragonflies in search space and imitate their activities, two vectors are considered: step ( X ) and position (X). The step vector is similar to velocity vector of PSO algorithm shows the direction of the movement of the dragonflies and mathematically defined as follows:

In order to show the effectiveness of the dragonfly algorithm for economic load dispatch Problem, four benchmark test system of small scale power systems having standard IEEE bus systems have been taken into consideration. The performance of the proposed dragonfly algorithm is tested in MATLAB 2013a (8.1.0.604) software on Intel® core i-5-3470S CPU@ 2.90 GHz, 4.00 GB RAM system. The PSEUDO code for Dragonfly algorithm is

Xt 1

(sSi aAi cCi fFi eEi ) wXt

S

(13)

mentioned in Fig.1

  1. Test System-I: 3-Generating Unit System considering

    Where, s shows the separation weight, i indicates of the separation of i-th individual, a is the alignment weight, Ai is alignment of the i-th individual, c is indicates the cohesion weight, Ci is the cohesion of the i-th individual, f is the food ffacacttoorr,, FFii is the i-th individual food source , e is indicate the enemy factor, Ei is the position of enemy of the i-th individual, w is indicate the inertia weight, and t is indicate the iteration counter.

    After calculating the step vector, the position vectors are

    calculated as follows

    transmission losses

    The first test system consists of 3-Generating units with a load demand of 150 MW [60]. Test data of 3-Generating Unit System are taken from [60], Loss Coefficients Matrices are used to calculate the corresponding Transmission losses. The algorithm is tested for 250 iterations and The corresponding results are compared with lambda iteration method [60] and Particle Swarm Optimization (PSO) [60] and Grey Wolf Optimizer (GWO)[59]. Table-I shows that optimal fuel cost for 3-unit generating model for 150MW load demand using GWO and DA algorithm is 1597.4815 Rs./hour, power loss

    Xt 1

    Xt Xt 1

    (14)

    using DA is 2.3420 MW and Iteration time for DA algorithm is 4.322344 seconds, which shows the superiority of DA

    To improve the uncertainty, stochastic behaviour and exploration of the synthetic dragonflies, they are essential to fly around the search space using a unsystematic walk (Levy flight) when there is no neighbouring solutions obtain. In this condition, the position updating dragonflies is using the following equation:

    algorithm over GWO and population based PSO algorithm. For 3-generating units system, DA completely converges in 58 iterations and takes Iteration time of 3.463332 seconds while GWO algorithm takes 92 iterations for convergence and converges times of 4.761541 seconds.

  2. Test System-II: 3-Generating Unit System without

    Xt 1 Xt Levy(d )Xt

    (15)

    transmission losses

    Where, t is indicating the current iteration, and d is indicating the dimension of the position vectors.

    The Levy flight is calculated as follows:

    Levy(x) 0.001 r1

    r 1/

    2 (16)

    Where, r1 and r2 are two random numbers in [0,1], b is a constant and is calculated as follows:

    (1 ) sin( / 2)

    The second test system also consisting of 3-Generating Unit System [58] is tested for two different load demands of 850 MW and 1050 MW including transmission losses. The corresponding results are compared with lambda iteration method [58], Genetic Algorithm (GA)[58], Particle Swarm Optimization(PSO)[58,60], Artificial Bee Colony(ABC)[58] and Grey Wolf Optimizer(GWO) [61]. Table-II shows the comparison of results with different methodologies and it is found that optimal value of fuel cost obtained by DA is much less that lambda iteration, GA, PSO, ABC and GWO. The

    convergence curve of test case-II is shown in Fig.2 (b)-(c).

    1 ( )

    1 ( )

    1

    ( ) 2 2

    2

    (17)

  3. Test System-III: 5-Generating unit system considering valve point effect

    The third test system consists of 5-Generating Unit System

    [58] is tested for load demand of 730 MW. Valve point effect is taken into consideration, but transmission losses are neglected while calculating optimal fuel cost. The results obtained by ALO algorithm are compared with lambda iteration method [58], Genetic Algorithm (GA)[58], Particle Swarm Optimization(PSO)[58], APSO[58], Artificial Bee Colony(ABC)[58], Evolutionary Programming(EP)[58] and Grey Wolf Optimizer(GWO) [61]. Table-IV shows the comparison of results with different methodologies and it is found that optimal value of fuel cost obtained by DA is much less that lambda iteration, GA, PSO, APSO, ABC, EP and GWO. The convergence curve of test case-III is shown in Fig.3 (a).

  4. Test System-IV: 6-Generating Unit System without valve point effect.

    The fourth test case consists of 6-Generating unit System without valve point loading [60]. The results of 6-generating units systems are tested for load demands of 600 MW, 700 MW, 800 MW, 900 MW and 1000MW and are shown in Table-V and effectiveness of ALO for 6-generating unit system is compared with lambda iteration method [60], particle swarm optimization (PSO)[60] and Grey Wolf Optimizer(GWO)[61]. Corresponding analysis of results (Table-V) shows that DA algorithm yields better fuel cost and power loss as compared to Lambda-Iteration Method, Particle Swarm Optimization Algorithm and Grey Wolf Optimizer. Also, the convergence of proposed algorithm is much better than these algorithms. The convergence curve of test case-IV

    is shown in Fig.3 (b). Another test benchmark of 6-generating units is tested for load demand of 1263 MW and experimentally it is found that the results obtained by DA are much better than FA[65], BBO[66], ABC[66], SOH-PSO[67], NMPSO[68], PSO-LRS[70], NPSO-LRS[70], DE[65], GA[69] and SA[65].

  5. Test System-V: 13-Generating unit system considering valve point effect

    The fifth test system consists of 13-Generating Unit System [64] is tested for load demand of 2520 MW. alve point effect is taken into consideration, but transmission losses are neglected while calculating optimal fuel cost. The results obtained by Dragonfly algorithm are compared with Simulated Annealing [64] and Genetic Algorithm (GA) [64]. Table-VI shows the comparison of results with GA, SA and it is found that optimal value of fuel cost obtained by DA is much less than Simulated Annealing (SA) and Genetic Algorithm (GA). The convergence curve of test case-IV is shown in Fig. 2(a).

  6. Test System-VI: 20-Generating unit system considering valve point effect

    The sixth test system consists of 20-Generating Unit System [71] is tested for load demand of 2500 MW considering transmission losses. The results obtained by Dragonfly algorithm are compared with ABC [72], ABCNN [71], BBO [73], LI [74], HM [75], QP [76] and GAMS [76].

    Table-VIII shows the comparison of results with ABCNN, BBO, LI, HM, QP, GAMS and it is found that optimal value of fuel cost obtained by DA is much less than these well known heuristics algorithms.

    Table-I: Economic Load Dispatch for 3-Generating Units System (Load Demand=150MW)

    Method

    Load Demand

    P1 (MW)

    P2(MW)

    P3(MW)

    Fuel Cost (Rs./h)

    Ploss (MW)

    No. of Iteration

    Elapsed Time(Seconds)

    Lambda Iteration [60]

    150 MW

    33.4401

    64.0974

    55.1011

    1599.9

    2.66

    250

    NA

    PSO [60]

    150 MW

    33.0858

    64.4545

    54.8325

    1598.79

    2.37

    250

    NA

    GWO

    150 MW

    30.4998

    64.6208

    54.8994

    1597.4815

    2.3444

    250

    4.761541

    DA[Proposed Method]

    150 MW

    32.8101

    64.595

    54.9369

    1597.4815

    2.3420

    250

    4.322344

    Table-II: Economic Load Dispatch for 3-Generating Units System (Load Demand=850MW)

    Method

    Load Demand

    Generation Scheduling

    Fuel Cost (Rs./h)

    Best Cost

    Average Cost

    Worst Cost

    Iteration Time(sec.)

    U1

    U2

    U3

    Lambda Iteration

    850 MW

    382.258

    127.419

    340.323

    8575.68

    GA

    850 MW

    382.2552

    127.4184

    340.3202

    8575.64

    PSO

    850 MW

    394.5243

    200

    255.4756

    8280.81

    ABC

    850 MW

    300.266

    149.733

    400

    8253.1

    DA[Proposed Method]

    850MW

    300.266

    149.733

    400

    8253.1052

    8253.1052

    8253.1052

    8253.1052

    9.3128

    Table-III: Economic Load Dispatch for 3-Generating Units System (Load Demand=1050MW)

    Method

    Load Demand

    Generation Scheduling

    Cost(Rs./Hour)

    Best Cost

    Average Cost

    Worst Cost

    Iteration Time(sec.)

    U1

    U2

    U3

    Lambda Iteration

    1050 MW

    487.5

    162.5

    400

    10212.459

    GA

    1050 MW

    487.498

    162.499

    400

    10212.44

    PSO

    1050 MW

    492.699

    157.3

    400

    10123.73

    ABC

    1050 MW

    492.6991

    157.301

    400

    10123.73

    DA[Proposed Method]

    1050MW

    492.69

    157.3

    400

    10123.7347

    9.3281

    Table-IV: Economic Load Dispatch for 5-Generating Units (Load Demand=730 MW)

    Method

    Load Demand

    Units Generation Scheduling

    Cost(Rs./Hour)

    Best

    Average

    Worst

    U1

    U2

    U3

    U4

    U5

    Lambda Iteration

    730 MW

    218.028

    109.014

    147.535

    28.38

    272.042

    2412.709

    GA

    730 MW

    218.0184

    109.0092

    147.5229

    28.37844

    227.0275

    2412.538

    PSO

    730 MW

    229.5195

    125

    175

    75

    125.4804

    2252.572

    APSO

    730 MW

    225.3845

    113.02

    109.4146

    73.11176

    209.0692

    2140.97

    EP

    730 MW

    229.803

    101.5736

    113.7999

    75

    209.8235

    2030.673

    ABC

    730 MW

    229.5247

    102.0669

    113.4005

    75

    210.0079

    2030.259

    DA[Proposed Method]

    730MW

    229.5196

    102.91

    112.72

    75

    209.83

    2029.823

    2029.823

    2076.946

    2124.07

    Table-V: Economic Load Dispatch for 6-Generating Units

    Comparison of Results for 6-Generating Units System

    Load Demand

    Methods

    P1(MW)

    P2(MW)

    P3(MW)

    P4(MW)

    P5(MW)

    P6(MW)

    Fuel Cost(Rs./h)

    Ploss

    Iteration Time(Sec.)

    600 MW

    Lambda Iteration

    23.7909

    10.22

    95.25

    10.12309

    202.967

    181.34

    32132.29

    14.7988

    PSO

    23.8602

    10

    95.6394

    100.7081

    202.8315

    181.1978

    32094.72

    14.2373

    DA

    23.8705

    10

    95.6365

    100.7078

    202.8302

    181.1922

    32094.6783

    4.23721

    11.818428

    700 MW

    Lambda Iteration

    28.29

    10.0901

    118.9873

    118

    230.2372

    213.9068

    36912.32

    19.5114

    PSO

    28.29

    10

    118.9583

    118.6747

    230.763

    212.7449

    36912.22

    19.43

    DA

    28.2991

    10

    119.0333

    118.6142

    230.7032

    212.7813

    36912.1448

    19.431

    11.863085

    800 MW

    Lambda Iteration

    32.9521

    14.7126

    141.5988

    136.0345

    258.1009

    243.8011

    41897.25

    27.5

    PSO

    32.586

    14.4839

    141.5475

    136.0435

    257.6624

    243.0073

    41896.7

    25.33

    DA

    32.6006

    14.4782

    141.5441

    136.0404

    257.6578

    243.0098

    41896.6286

    25.3309

    11.937735

    900 MW

    Lambda Iteration

    36.9889

    22.1821

    163.01

    153.2168

    284.1482

    273.0581

    47045.32

    32.6131

    NA

    PSO

    36.848

    21.0774

    163.9304

    153.263

    284.1696

    272.7301

    47045.25

    31.98

    NA

    DA

    36.8638

    21.0785

    163.9289

    153.2192

    284.243

    272.6538

    47045.1565

    31.9873

    11.89715

    1000 MW

    Lambda Iteration

    40.3969

    28.1002

    187

    171.2136

    310.721

    303.1006

    52362.07

    40.5323

    NA

    PSO

    41.1657

    27.7786

    186.5604

    170.5795

    310.8297

    302.568

    52361.65

    39.4821

    NA

    DA

    41.1849

    27.8074

    186.061

    170.7025

    311.2873

    302.4481

    52361.1604

    39.4912

    11.81442

    Table-VII: Economic Load Dispatch for 6-Generating Units (Load Demand=1263 MW)

    Unit Power Output

    DA

    FA[65]

    BBO[66]

    AB[65]C

    SOH- PSO[67]

    New MPSO[68]

    PSO[69]

    PSO- LRS[70]

    NPSO[70]

    NPSO- LRS[70]

    DE[65]

    GA[69]

    SA[65]

    P1(MW)

    500

    445.08

    447.3997

    438.65

    438.21

    446.71

    447.5

    447.444

    447.4734

    446.96

    400

    474.81

    447.08

    P2(MW)

    154.1458

    173.08

    173.2392

    167.9

    172.58

    173.01

    173.32

    173.343

    173.1012

    173.3944

    186.55

    178.64

    173.18

    P3(MW)

    236.4782

    264.42

    263.3163

    262.82

    257.42

    265

    263.47

    263.3646

    262.6804

    262.3436

    289

    262.21

    263.92

    P4(MW)

    135.1084

    139.59

    138.0006

    136.77

    141.09

    139

    139.06

    139.1279

    139.4156

    139.512

    150

    134.28

    139.06

    P5(MW)

    151.2559

    166.02

    165.4104

    171.76

    179.37

    165.23

    165.48

    165.5076

    165.3002

    164.7089

    200

    151.9

    165.58

    P6(MW)

    98.4635

    87.21

    87.07979

    97.67

    86.88

    86.78

    87.13

    87.1698

    87.9761

    89.0162

    50

    74.81

    86.63

    Total Power Output

    1275.5419

    1275.4

    1275.446

    1275.57

    1275.55

    1275.7

    1276.1

    1275.95

    1275.95

    1275.94

    1275.55

    1276.03

    1275.47

    Total Transmission loss(MW)

    12.4519

    12.4

    12.446

    12.57

    12.55

    12.958

    12.9571

    12.9471

    12.947

    12.9361

    12.55

    13.022

    12.47

    Total Generation Cost($/Hour)

    15406.5198

    15443

    15443.0963

    15445.4

    15446.02

    15447

    15450

    15450

    15450

    15450

    15452

    15459

    15466

    Iteration Time

    11.9101

    11.52

    0.0325

    2.82

    0.0633

    0.0379

    0.06

    NA

    NA

    NA

    6.2

    0.22

    62.02

    Table-VIII: Economic Load Dispatch for 20-Generating Units (Load Demand=2500 MW)

    Unit

    DA

    ABCNN[71]

    ABC[72]

    BBO[73]

    LI[74]

    HM[75]

    QP[76]

    GAMS[76]

    P1

    600

    599.9972

    599.882

    513.0892

    512.7805

    512.7804

    600

    512.782

    P2

    133.7124

    172.4309

    172.866

    173.3533

    169.1033

    169.1035

    200

    169.102

    P3

    50

    50

    106.993

    126.9231

    126.8898

    126.8897

    50

    126.891

    P4

    50

    50

    63.1275

    103.3292

    1028657

    102.8656

    56.92

    102.891

    P5

    92.724

    115.8288

    70.9701

    113.7174

    113.6836

    113.6836

    94.28

    113.683

    P6

    31.986

    39.5509

    52.1022

    73.06694

    73.571

    73.5709

    33.72

    73.572

    P7

    125

    120.0216

    119.142

    114.9843

    115.2878

    115.2876

    125

    115.29

    P8

    50

    71.7034

    50

    116.4238

    116.3994

    116.3994

    60.24

    116.4

    P9

    106.8898

    129.4382

    76.3559

    100.6948

    100.4062

    100.4063

    103.28

    100.405

    P10

    49.941

    30

    102.403

    99.99979

    106.0267

    106.0267

    79.49

    106.027

    P11

    263.5682

    2304784

    263.905

    148.977

    150.2395

    150.2395

    221.14

    150.239

    P12

    407.4554

    469.0286

    362.23

    294.0207

    292.7648

    292.7647

    347.05

    292.766

    P13

    160

    104.1452

    123.52

    1195754

    119.1154

    119.1155

    127.38

    119.114

    P14

    72.7019

    80.0902

    47.7657

    30.54786

    30.834

    30.8342

    60.29

    30.832

    P15

    90.3428

    59.3637

    56.4597

    116.4546

    115.8057

    115.8056

    116.7

    115.805

    P16

    35.0882

    34.0204

    34.0936

    36.22787

    36.2545

    36.2545

    36.25

    36.254

    P17

    33.1827

    41.623

    31.4734

    66.87943

    66.859

    66.859

    30

    66.859

    P18

    46.9723

    30

    30

    88.54701

    87.972

    87.972

    58.21

    87.967

    P19

    83.53

    55.3963

    118.464

    1,009,802

    100.8033

    100.8033

    85.52

    100.8033

    P20

    30

    30

    30

    54.2725

    54.305

    54.305

    30

    54.305

    Total Power Output

    2513.0945

    2513.1164

    2511.8

    2592.1011

    2591.967

    2591.967

    2515.48

    2591.976

    Total Transmission loss(MW)

    13.0945

    13.1163

    11.7527

    92.1011

    91.967

    91.9669

    15.48

    91.967

    Total Generation Cost($/Hour)

    60427.444

    60446.377

    60540

    62456.779

    62456.639

    62456.634

    62456.63

    62456.63

    Table-VI: Economic Load Dispatch for 13- Generating Units (Load Demand=2520 MW) ELD for 13-units test system using DA

    Unit

    Generated Power(MW)

    Unit

    Generated Power(MW)

    1

    1166.877271

    8

    60.03842743

    2

    303.8276937

    9

    109.8665501

    3

    299.7904073

    10

    40

    4

    60

    11

    40

    5

    109.8665501

    12

    55

    6

    60

    13

    55

    7

    159.7331001

    Comparison of Results

    Method

    Cost(Rs./Hour)

    SA[64]

    24970.91

    GA[64]

    24398.23

    DA[Proposed Method]

    24386.86

    1. CONCLUSIONS

In this research paper, application of Dragonfly algorithm is presented for the solution of non-convex and dynamic economic load dispatch problem of electric power system. Performance of ALO algorithm is tested for small scale power plants. The effectiveness of proposed Dragonfly algorithm is tested with the standard IEEE bus system consisting of 3, 5 and 6 generating units model considering transmission losses (Power Loss) and valve point effect.

The results obtained show that Dragonfly algorithm have been successfully implemented to solve different ELD problems moreover, Dragonfly algorithm is able to provide

algorithm has the ability to converge to a better quality near- optimal solution and possesses better convergence characteristics than other widespread techniques reported in the recent literatures. It is also clear from the results obtained by different trials show that the Dragonfly algorithm shows a good balance between exploration and exploitation that result in high local optima avoidance.

Thus, this algorithm may become very promising for solving some more complex power system optimizations problems such as: Economic Load Dispatch for quadratic and cubical cost function, Single and Multi-objective Economic Load Dispatch including valve point effect, Economic Load Dispatch incorporating wind Power , Economic Load Dispatch incorporating Solar Power, Hydro-Thermal and Wind-Thermal Scheduling of electric power system. Thermal Scheduling incorporating Smart Grids, Hydro-Thermal Scheduling incorporating Smart Grids, Single and Multi Objective Unit Commitment Problem formulation, Multi- Objective and Multi-Area Unit Commitment Problem

Convergence of DA for 13-Generating Units System[ Load Demand=2520 MW]

DA

DA

4.396

10

4.395

10

4.394

10

4.393

Fuel Cost——>

Fuel Cost——>

10

4.392

10

4.391

10

4.39

very spirited results in terms of minimizing total fuel cost and lower transmission loss. Also, convergence of Dragonfly algorithm is very fast as compared to Lambda Iteration Method, Particle Swarm Optimization (PSO) algorithm, Genetic algorithm (GA), APSO, Artificial Bee Colony (ABC), and Grey Wolf Optimizer (GWO) for small scale power systems. Also, It has been observed that the Dragonfly

10

4.389

10

4.388

10

0 100 200 300 400 500 600 700

Iteration——>

Fig.2(a): Convergence of DA for 13-Generaing unit System

Fig.2(b): Convergence of DA Algorithm for 3-Generating Units test system [load demand = 850MW]

Fig.2(c): Convergence of DA Algorithm for 3-Generating Units test system [load demand = 1050MW]

Fig.3(a): The convergence curve of test case-III

Fig.3(c): The convergence curve of test case-IV

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