Application of a novel modified hybrid algorithm for solving Dynamic Economic Dispatch problem with practical constraints

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Application of a Novel Modified Hybrid Algorithm for Solving Dynamic Economic Dispatch Problem with Practical Constraints Mostefa HAMED1, Belkacem MAHDAD1, Kamel SRAIRI1, Nabil MANCER2 1Department of Electrical Engineering, Faculty of Sciences and Technology, Biskra University, BP 145 RP, 07000 Biskra, Algeria 2Department of Electrical Engineering, Faculty of Science Technology, Constantine University 1, A. Hamani Campus, Ain El Bey Road, 25000 Constantine, Algeria [email protected], [email protected], [email protected], [email protected] DOI: 10.15598/aeee.v16i4.2877 Abstract. Dynamic Economic Dispatch (DED) is a highly complex nonlinear optimization problem with practical constraints. The aim of DED is to optimize dynamically the active power of generating units over operating time considering practical constraints such as valve point effect, prohibited zones, ramp rate limits and total power losses. In order to overcome the drawback of the two standard metaheuristics such as Firefly Algorithm (FA) and Time Varying Acceleration based Particle Swarm Optimization (PSOTVAC), a hybrid method called FAPSOTVAC is proposed to improve the solution of DED. The main idea introduced towards combining FA and PSOTVAC is to create a flexible equilibrium between exploration and exploitation during search process. The robustness of the proposed hybrid method is validated on many practical power systems (10 and 30 units) to minimize the total fuel cost considering all practical constraints. The results found prove the efficiency of the proposed FAPSOTVAC in terms of solution quality and convergence characteristics. Keywords A hybrid algorithm, dynamic economic dispatch, FA-PSOTVAC, prohibited operating zones, ramps rate constraints, valve point effect. 1. Introduction Nowadays electric energy resembles a vital artery to our daily life and the main engine for any economic or commercial activity. Such occupied position renders it irreplaceable due to its credibility when compared with any other natural energy. The electric energy demand in our economic has multiplied by 3.2 in 37 years to reach 19738 TWh in 2010. This colossal number indicates the salient position it occupies in the current world economy. The energy system is composed of power station interconnected with transmission lines transporting the produced energy to consumers after several operation and control stages. The non-stocked aspect of this form of energy obliges us to produce it in the time of consumption. The balance between production and consumption should be respected in real time and within the capacity of power generating units. This problem is generally called the problem of Static Economic Dispatch (SED). The main task of electric power system is to ensure instantaneously the equilibrium between production and demand. The determination of the optimal state of each generator interconnected with the electric network during the twenty-four hours complicates the solution of the faced problem. Rather than being more static, this problem becomes dynamic in time, in other contexts wherein the complexity of nowadays network increases vis a vis its size that holds hundreds of bus-bars and hundreds of thousands of kilometers of transmission lines, in addition to highly complicated structure of the interconnected network.All these factors make the optimization of the total fuel cost complex and vital objective. In this context, new practical constraints, attached mainly to the construction of thermal units, on the one hand, and to the conditions imposed by the strategy of exploitation, on the other hand, should be respected. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 459 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER The opening of fuel’s valves perturb the quadratic form of cost’s objective operation by introducing a highly non-linear form. Furthermore, another new constraint can be added to complicate the (DED) problem. The latter is generators ramp constraint that does not allow the adjustment of the generated active power only by a pre-imposed value. In [1], Ramp-rate limits have been considered in unit commitment and economic dispatch incorporating rotor fatigue effect. This study explains how the violation of such constraints can highly minimize rotor’s life and increments the maintenance cost. Since the constraints are highly non-linear, they add prohibited operating zones, which are attached directly to the equilibrium of the thermal generating units. The latter should operate away from certain intervals called “Prohibited Operating Zones” to avoid some dangerous vibration at the level of machine’s bearing [2]. In such situation, the form of objective function must be modified and adapted to take into consideration the effects of these prohibited zones. Several mathematical methods have been applied for solving such non-linear problem. Most of them have exploited the mathematical characteristics of the cost function function for discovering the continuity and hessian derivation,..etc. In this sense, authors in [3] made a comparison between the solution of the iterative Lamda method and the metaheuristic algorithm “Brent method” for solving the problem of dynamic economic dispatching with losses and ramp constraints. Moreover, authors in [4] have applied the dynamic programming in order to find the solution to the same problem by considering the prohibited operating zones. Whereas, reserve constraints have been considered in [5] by applying Lagrange relaxation method. Besides, authors in [6] have applied the same method to investigate unit commitment problem. All these methods are swift and all what they need is one launch either to find the optimum solution or stay inapt towards the different mentioned constraints. The ordinary methods of optimization cannot cover the entire space designed for research in order to find a low cost for they can be trapped at a local rather than a global optimum following an exaggerated time that can never be applied in real time. The application of artificial intelligence methods present an alternative to the conventional methods, which leads to the development and the application of many techniques such as Genetic Algorithms (GA) [7], Particle Swarm Optimization (PSO) [8] and their modified versions. Authors in [9] used the modified version of PSO which they called Modulated particle swarm optimization to solve the problem of muti-objective dynamic economic dispatch. In addition, a kenetic gas molecule optimization algorithm has been proposed in [2] to solve the static and dynamic economic dispatch problems. Whereas, authors in [10] applied the Modified Real Coded Genetic Algorithm (MRCGA) to solve the problem of multi-objective Dynamic Economic Dispatch (DED). Furthermore, authors in [11] solved the large scale problem DED by using the Crisscross optimization algorithm. Meanwhile, authors in [12] suggested the Alternating Direction Method of Multipliers (ADMM) for solving environmental economic dispatch. In [13] Differential Evolution (DE) algorithm was applied to solve the DED problem considering ramp rates constraints. For being able to cover the whole research area, limited by an important number of constraints as well as the huge non-linearity, on one hand, and to solve the problem of a large size DED on the other hand, many hybrid algorithms have been suggested. These hybrid techniques have been developed to overcome the drawback of the standard metaheuristic methods by creating flexible equilibrium between diversification and intensification during search process. Authors in [14] used Modified Particle Swarm Optimization and Genetic Algorithm (MPSO-GA) for solving the problem of static economic dispatch with prohibited operation zones, ramp constraints and multi fuel. Besides, authors in [15] proposed the hybrid method (MILP-MDSD) to solve the problem of dynamic economic dispatch with valve points effects. Authors in [16] have used the Improved Dynamic Programming (IDP), which is a recursive of a dynamic programming to solve the problem of economic dispatch with prohibited zones and ramp rate constraints. In addition in [17] a Chaotic self-adaptive Differential Harmony Search algorithm (CDHS) applied in order to solve the problem of dynamic economic dispatch wherein, the prohibited operation zones and ramp-rate constraints are taken into consideration simultaneously. Authors in [18] proposed metaheuristic Two Stage Mixed Integer Linear Programming (TSMILP) as a method to solve the problem of dynamic economic dispatch considering the effects of valves and transmission losses. On the other hand authors in [19] used Fast Evolutionary Programming with Swarm Direction for solving DED problem. Whereas, authors in [20] applied the hybrid technique of Cross-Entropy Method and Sequential Quadratic Programming to solve the same problem. This article intends to solve the problem of multi constraints non-linear dynamic economic dispatch to investigate valves point effects, ramps constraints, by introducing prohibited operating zones that have never been treated together before according to review of literature. The huge number of constraints and complication problem obliged us to introduce new hybrid algorithms such as FA-PSOTVAC and BBO-PSOTVAC to achieve the desired low cost by respecting all the practical operation constraints imposed. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 460 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER 2. Nomenclature: BBO: Biogeography-Based Optimization algorithm. Cit: The unit iproduction cost at time t. DED: Dynamic Economic Dispatch. FA: Firfly algorithm. ng: The number of generation units. ni: The number of prohibited operating zones in the ith generating unit. Pd(t): Load demand at time t. Pmin i, P max i: The maximum and the minimum production of unit i. Pit: Power output of unit at time t. Ploss: Transmission losses. PSOTVAC: Particles Swarm Algorithm with a variable acceleration coefficient. T: The total number of hours in the operation period. T C: Total Cost ($). URi, DRi: The ramp up and the ramp down rate limit’s respectively. 3. Mathematical Formulation 3.1. Objective Function The objective function of (DED) problem is to minimize the total production cost over the operation period, which can be written as [21]: min T C = T X t=1 ng X i=1 Cit(Pit),(1) where Cit is the cost of ith generating unit at time t,ng is the number of generation units and Pit is the power output of it unit at time t.Tis the total number of hours in the operation period. The fuel cost function of generating units considering valve-point effect can be expressed using the following equation [15]: F(P gi) = ng X i=1 ai+biP git +ciP g2 it +|eisin(fi(P gmin it −P git))|, (2) where ai, bi, ci, ei, fiare the cost coefficients of ith power generating units. This objective function should be minimized considering the following equality and inequality constraints [22]. 3.2. The Equality Constraints: ng X i=1 Pit =Pd(t) + Ploss, t = 1,2, ..., T. (3) 3.3. Inequality Constraints [23] and [24] Pmin i≤Pit ≤Pmax i, i = 1, ..., ng t = 1, ..., T. (4) Pmin i, P max iare the minimum and the maximum of unit’s production. 1) Ramp Rate Constraints [25] Pit −Pi(t−1) ≤URi,(5) Pi(t−1) −Pit ≤DRi.(6) 2) Prohibited Operation Zone The Prohibited Operation Zones [26] and [27] are mathematically expressed by the following equation: Pi∈     Pmin i≤Pi≤PL i1, Pik−1≤Pi≤PL ik, Pu izi ≤Pi≤Pmax i, (7) where: niis the number of prohibited operating zones in the ith generating unit. kis the index of the prohibited operating zones of the ith generating unit. PL iK , PU iK are the lower and upper bounds of kth prohibited operating zones of unit i. 4. Optimization Algorithms 4.1. Particles Swarm Algorithm with a Variable Acceleration Coefficient PSOTVAC Particle Swarm Optimization with Time Variable Acceleration (PSOTVAC) is a dynamic variant of the standard PSO algorithm. This algorithm presents a modified version of the basic algorithm PSO, though it somewhat differs from the standard algorithm by its cognitive and social coefficients that change during search process. The dynamic behavior of these two coefficients allows to create equilibrium between exploration and exploitation [28] and [29]. The position and the speed of each particle are presented in the following equations:    V(t+ 1) = w∗V(t) + α1rand1∗(Pi−X(t))+ +α1rand2∗(P b −X(t)), X(t+ 1) = X(t)+(t+ 1), (8) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 461 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER (α1= (c1f−c1i)iter itermax +c1i, α1= (c2f−c2i)iter itermax +c2i,(9) w= (wmax −wmin)∗(itermax −itermin) itermax +wmin,(10) where x(t)is the initial position of the particle. v(t) presents the initial speed of the particle. v(t+1) is the new speed of the particle. x(t+1) is the new position of the particle. P i is the best local solution. P b is the best global solution. wis the inertia factor presented by 0.4≤w≤0.9.iter is the iteration number. itermax is the maximum iteration number. α1, α2are respectively the cognitive and the social factors. C1i, C2i, C1f, C2f represents the initial and final values of the cognitive and the social factors which are respectively 2.5, 0.5, 0.5 and 2.5. The flowchart of the PSOTVAC is shown in Fig. 1. Fig. 1: Flowchart of PSOTVAC. Initialize the data power system and parameters of, PSO-TVAC Create a random initial population Evaluate the objective function 1. Update iteration count 2. Update velocity and position Evaluate the objective function by power flow calculation by the new population START Yes Select the best solution END No Convergence? Fig. 1: Flowchart of PSOTVAC. 4.2. Firefly Algorithm This algorithm is inspired by and based on the principle of attraction between fireflies in nature, which gives many similarities with other metaheuristic methods based on group collective intelligence such as PSO algorithm. Based on the pseudo code of the FFA shown in Alg. 1, the FA algorithm is governed by the three following rules: •All the fireflies are unisex; they will move towards more attractive and brighter ones regardless their sex. •The degree of attractiveness of a firefly is proportional to its brightness which decreases as the distance from the other firefly increases. •Fireflies luminosity is determined by an objective function (an optimized one). Algorithm 1 Firefly Algorithm. Ensure: : Initialize population of m fireflies, xi, i= 1,2,3, . . . m. Ensure: : Compute Light intensity f(xi), for i= 1,2, . . . m. while stopping criteria is not met do for i= 1 to mdo for j= 1 to mdo if (f(xi)> f(xj)) then return Move firefly itowards j(eq 13) end if end for end for Update Light intensity f(xi)for i= 1,2, . . . m. Rank the fireflies and find the current best end while 1) Attractiveness The attractiveness function between fireflies is expressed by the following equation: β(r) = B0exp(γrm),with m≥1,(11) where ris the distance between any two fireflies, B0is the initial attractiveness at r= 0, and γis an absorption coefficient which controls the decrease of the light intensity. 2) Distance During the search process, the distance between two fireflies iand jat location xiand xjcan be defined by the following expression: rij =kxi−xjk=v u u t d X k=1 (xi,k −xj,k)2,(12) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 462 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER where rij is the distance between two fireflies and dis the dimension of the problem. 3) Movement The movement of a firefly iwhich is attracted by a more attractive firefly jis given by the following equation: xt+1 i=xt i+B0exp(−γr2 ij)∗(xi−xj) +α(rand −0.5),(13) where the first term is the current position of a firefly, the second term is used for considering a firefly’s attractiveness to light intensity seen by adjacent fireflies, and the third term is for the random movement of a firefly in case there are not any brighter ones. i and jare two variables which reflect the light intensity that is associated with a specified fitness function of particles to be evaluated [30]. 4.3. BBO Algorithm BBO is relatively a new metaheuristic method introduced by (Simon, 2008) [31] and [32]. This method is inspired by migration of species among islands. The fitness of a geographical area is assessed by a Habitat Suitability Index (HSI). Habitats which are more suitable for species to reside are said to have a high HSI. Similarly, habitats which are less suitable for species to reside are said to have low a HIS (Bansal et al., 2016) [33]. In BBO, a solution is represented by an island consisting of solution features named Suitability Index Variables (SIV), which are represented by real numbers. It is represented for a problem with nd decision variables as: island = [SIV1, SIV2, SIV3, ..., SIVnd].(14) The suitability of sustaining larger number of species of an island can be modeled as a fitness measure referred to Suitability Index (SI) in BBO as: SI =f(island) = f(SIV1, SIV2, SIV3, ..., SIVnd).(15) High SI represents a better quality solution and low SI denotes an inferior solution. The aim is to find optimal solution in terms of SIV that maximizes the SI. Each island, representing a solution point, is characterized by its own immigration rate λand emigration rate µ. A good solution enjoys a higher µand lower λ and vice-versa. The immigration and emigration rates are the functions of the number of species in the island as well shown in Fig. 2, and defined for the kth island as [34]. µk=fk n,(16) Fig.3. Species model of an island. Emigration μ Species count Rate I E Immigration λ Smax Fig. 2: Species model of an island. λk=I1−k n,(17) when E=I, the immigration and emigration rates can be related as: λk+µk=E. (18) 4.4. Proposed Hybrid FA-PSOTVAC In order to exploit the best proprieties of the two well known algorithms, the FA and PSOTVAC, a hybrid method is proposed to improve the solution of DED. The mechanism search of the standard FA is characterized by its possibility to locate the best solution but at high number of iteration. The PSOTVAC algorithm is characterized by its fast convergence, however the solution achieved is not competitive in particular when considering large DED problems. The proposed FAPSOTVAC is adapted and applied to solve the DED of large test system considering simultaneously the prohibited zones, the valve point effect and ramp-rate limits. The flowchart of the proposed hybrid algorithm is shown in Fig. 3. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 463 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Fig. 4. Flow chart of the proposed hybrid algorithm based FA-PSOTVAC. YES Start Read Data base, itermax, Pop-size, n, K Initialization of PSO-TVAC , FA Algorithms parameters Tableau (2) Solution de l’algorithme hybride FA-PSOTVAC . Initialisation des paramètres FA. Start Firfly Algorithm Evaluation Of FA Mechanism search Iter<= itermax/k Start Of PSO-TVAC Generation of the first population related to the same solution of FA. Evaluation of PSO-TVAC Mechanism Iter<=Itermaxitermax/k NO NO YES Store the Best Solution of Cost and Pg Vector End Fig. 3: Flow chart of the proposed hybrid algorithm based FAPSOTVAC. 0 100 200 300 400 500 600 700 800 900 1000 Generation 104 105 106 107 108 109 1010 Best Cost ($/h) BBO-PSOTVAC BBO FA FA-PSOTVAC 100101102103 3.2 3.25 3.3 3.35 3.4 3.45 3.5 104 Fig. 4: Comparison of convergence characteristics of the proposed four algorithms for test system 1. 0 20 40 60 80 100 120 140 160 180 200 Generation 105 106 107 108 109 1010 Best Cost ($/h) Fig. 5: Convergence characteristic of BBO-PSOTVAC for test system 1. 20 40 60 80 100 120 140 160 180 200 Generation 105 106 107 108 109 Best Cost ($/h) BBO-PSOTVAC FA-PSOTVAC Fig. 6: Convergence characteristics of FA-PSOTVAC and BBO-PSOTVAC for test system 1. 012345678910 Unit Nember -10 0 10 20 30 40 50 60 70 80 Ramp Up Rate Distribution Fig. 7: Distribution of Ramp Up violation for 50 trials for test system 1. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 464 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Tab. 1: Best solution of FA for test system 1. H Pg1 Pg2 Pg3 Pg4 Pg5 Pg6 Pg7 Pg8 Pg9 Pg10 Cost 1 150.0000 309.5755 73.0000 60.0000 122.8639 122.4782 76.0961 47.0000 20.0000 55 28745 2 150.0000 309.5507 73.0999 60.0000 122.8885 149.3127 93.1691 47.0000 49.9658 55 30391 3 226.5853 309.6643 153.0568 60.0000 123.0489 140.4734 123.1521 47.0000 20.0000 55 33563 4 226.5508 309.5670 185.0731 100.5493 172.7406 159.9138 129.5695 47.0000 20.0159 55 36629 5 226.6134 309.5774 238.8934 120.4456 172.8045 159.9993 129.6077 47.0000 20.0537 55 38338 6 303.2963 309.6235 297.3994 120.4101 222.6227 122.9657 129.6770 47.0000 20.0000 55 41054 7 379.7484 309.5613 296.1865 120.2595 222.2314 122.4294 129.5911 47.0000 20.0000 55 42688 8 379.9163 309.5878 331.9346 120.3543 222.5364 160.0000 129.5783 47.0463 20.0414 55 44682 9 456.5803 309.5402 340.0000 170.3125 235.8224 160.0000 129.6444 47.0412 20.0000 55 48408 10 457.0786 389.4996 339.9167 220.2939 223.6654 159.9001 129.7003 47.0000 49.9592 55 52014 11 456.4682 460.0000 323.9527 241.3500 222.6343 160.0000 129.6314 76.9459 20.0000 55 53694 12 456.5319 460.0000 339.6460 291.1819 222.5952 160.0000 129.6202 85.3100 20.0792 55 55437 13 456.6231 459.9778 297.3991 241.2082 172.7557 154.0301 129.6521 85.3238 20.0016 55 51712 14 456.4936 396.7729 294.7512 191.2087 172.4486 122.5401 129.6818 85.1015 20.0000 55 47899 15 379.8663 396.7837 233.4647 180.8780 172.5976 122.5602 129.5828 85.2840 20.0000 55 44832 16 302.8660 316.7899 185.2285 130.9164 172.7174 155.4687 129.5770 85.4114 20.0461 55 40103 17 226.7340 309.5333 200.4338 120.5870 172.8071 160.0000 129.6007 85.2751 20.0000 55 38265 18 303.2308 309.5587 229.6569 120.5972 222.5535 122.5015 129.5881 115.2475 20.0000 55 41774 19 379.9181 309.4884 297.1465 119.7770 222.6016 122.4900 129.5685 120.0000 20.0000 55 44552 20 456.3717 389.4182 319.4344 169.7228 222.5600 159.8888 129.5937 120.0000 49.9539 55 51945 21 456.7414 309.6307 306.0718 181.1537 222.7390 122.9915 129.6789 120.0000 20.0000 55 48011 22 379.8686 229.6340 267.7906 131.2140 172.7559 122.2274 129.6219 119.8901 20.0000 55 41927 23 302.1687 222.1867 187.9891 81.3181 122.8433 120.9311 99.6632 119.8957 20.0000 55 35496 24 226.0974 222.2598 178.4508 60.0000 122.5708 80.0712 129.5932 89.9053 20.0000 55 32081 Total Cost ($) 1024240 5. Simulation Results In this study a comparative analysis is elaborated to validate the robustness of the proposed hybrid algorithm in solving the DED considering several practical constraints. Four algorithms are investigated, FA, BBO, PSOTVAC, FA-PSOTVAC, and BBO-PSOTVAC. Two test power systems are investigated to validate the efficacy of the proposed algorithms and in particular the hybrid method named FA-PSOTVAC. 5.1. Test System 1 The first test system consists of 10 units, system data is takem from [35] and [17]. The optimized active power of thermal units during 24 H is achieved considering valve point effect, prohibited zones and ramp rate limits. For fair comparison between different methods, the population size for all methods is set to 50. Table 1 and Tab. 2 show the details of the optimized active power of 10 thermal units during 24 H. The FA achieves the best solution 1024200 $at 500 iterations, the corresponding execution time is 41.8955 min, the convergence characteristics are shown in Fig. 4, the BBO achieves the best total cost 1044000 $which is higher than FA, also this algorithm requires large number of iterations (1000), at a relatively reduced execution time (7.1236 min) compared to FA. 012345678910 Unit Number 0 10 20 30 40 50 60 70 80 Ramp Down Rate Distribution Fig. 8: Distribution of Ramp Down violation for 50 trials for test system 1. Table 3 depicts details about the performances of several algorithms in solving DED in terms of the best, the mean and the maximum value. Figure 4 shows the convergence behavior for total cost minimization for a period of 24 h for all proposed methods. As well shown in Fig. 5, the hybrid algorithm named BBO-PSOTVAC allows to achieve a total cost of 1055000 $at a competitive time (0.9350 min). On the other side, the proposed hybrid algorithm based on combining the FA and PSOTVAC achieves a remarkable total cost of 1024163 $at a reasonable execution time (8.4934 min), It is also important to confirm that the proposed algorithm is found to be better than other standard and combined algorithms in terms of c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 465 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Tab. 2: Best solution of FA-PSOTVAC algorithm for test system 1. H Pg1 Pg2 Pg3 Pg4 Pg5 Pg6 Pg7 Pg8 Pg9 Pg10 Cost 2 150.0000 222.2977 119.9927 60.0000 172.7551 133.2690 129.6194 47.0684 20.0000 55 30220 3 226.6829 222.2649 199.8623 60.0000 172.7874 124.6362 129.7631 47.0000 20.0000 55 33106 4 303.2261 222.1884 273.8517 60.0000 172.7244 122.4197 129.5903 47.0000 20.0000 55 36316 5 379.9085 222.2685 297.4001 60.0000 122.8779 145.7868 129.7293 47.0000 20.0276 55 37912 6 379.9504 302.2682 300.9859 60.2773 172.7347 160.0000 129.6819 47.0126 20.0874 55 41268 7 379.9384 309.6140 317.8722 110.1765 172.7422 160.0000 129.6391 47.0000 20.0155 55 43089 8 379.7673 309.5442 339.5965 120.4278 222.6159 122.4767 129.6386 47.0000 49.9280 55 44764 9 456.4904 309.5325 297.2723 170.0277 222.2960 154.8207 129.5225 76.9843 52.0546 55 48421 10 468.5611 309.5358 340.0000 220.0197 223.7258 160.0000 129.8466 85.3120 80.0000 55 52581 11 456.5008 389.5146 339.9296 241.2453 236.7129 160.0000 129.7207 85.3177 52.0588 55 53679 12 456.5777 460.0000 340.0000 258.8141 222.6798 159.9631 129.5956 85.3123 52.0564 55 55608 13 456.4978 396.7024 297.2402 284.6130 222.5896 122.3767 129.6034 85.3192 22.0572 55 51452 14 456.5069 316.7464 300.2911 241.3478 222.6032 126.4383 129.7427 55.3223 20.0000 55 48255 15 456.4983 309.2706 251.9156 191.4026 222.5805 122.5846 99.7451 47.0000 20.0000 55 45245 16 379.8921 309.5309 182.2315 172.1561 172.6454 122.4838 93.0579 47.0000 20.0000 55 39961 17 303.2201 309.5034 180.3546 176.7481 172.6892 122.4267 93.0445 47.0000 20.0100 55 38312 18 303.2842 309.9226 260.3178 181.1381 172.7265 125.5862 123.0291 47.0000 49.9962 55 41938 19 379.8756 309.5352 328.4486 181.1280 172.7430 122.6920 129.5852 76.9929 20.0000 55 44915 20 456.5191 389.5333 328.8065 224.6494 222.5961 160.0000 129.5758 85.3189 20.0000 55 51778 21 379.9298 460.0000 317.9828 180.9005 172.7697 122.4799 129.6152 85.3121 20.0090 55 48296 22 303.2117 396.7645 297.3459 130.9016 122.8333 87.0071 129.6583 85.2567 20.0201 55 41694 23 226.6244 316.7646 251.9903 81.1457 73.1616 122.4624 129.5912 55.2591 20.0000 55 35449 24 150.0000 309.5334 185.1918 60.0000 73.0000 154.6855 129.5901 47.0000 20.0000 55 31501 Total Cost ($) 1024163 Tab. 3: Best solution of FA, BBO, PSOTVAC, FA-PSOTVAC, BBO-PSOTVAC for test system 1. Method Pop Size Max Best Worst Mean Value Min-Max of Balance Time Iteration Solution Solution Demande Violation (min) FA 50 500 1024200 8161000 1254900 0.0213-0.0658 41.8955 BBO 50 1000 1044000 53566000 4909400 0.0198-0.0162 7.1236 PSOTVAC 50 1000 - - - Violation - FA-PSOTVAC 50 200 1024163 13793000 1794400 0.0023-0.0049 8.4934 BBO-PSOTVAC 50 200 1055000 1065600 1060000 0.0222-0.0150 0.9350 speed of convergence, standard deviation of generation cost, and computational time. Figure 6 shows the convergence characteristics of FA-PSOTVAC and BBO-PSOTVAC. Figure 7 and Fig. 8 show that the constraints related to ramp up and ramp down are verified. Figure 9 shows the ditribution of the best cost for 50 trials, this test demonstrates the robustness of the proposed hybrid method named FA-PSOTVAC. 5.2. Test System 2 In order to demonstrate the efficacy and performances of the proposed hybrid methods such as FA-PSOTVAC and BBO-PSOTVAC a large scale test system is considered. This second test system consists of 30 units, system data is takem from [11]. For fair comparison with other methods cited in the literature, only two constraints are considered, the valve point effects and ramp rate limits. The best total cost achieved using the proposed algorithms are compared to various methods cited recently in the literature such as Evolutionary Programming 0 5 10 15 20 25 30 35 40 45 50 Distribution Cost for 50 Trials 0 2 4 6 8 10 12 14 Best Cost ($/h) 106 Fig. 9: Distribution of the best cost for 50 trials for test system 1. (EP) [36], Differential Evolution (DE) [37], Criss Cross Optimization algorithm (CSO) [11], Harmony Search (HS) [38] and a modified hybrid EP-SQP approach (MHEP-SQP) [35], as well depicted in Tab. 4, it is found that by using the proposed hybrid method BBOPSOTVAC the best total cost achieved is 3105700 $. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 466 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Tab. 4: Best solution of FA, BBO, PSOTVAC, BBO-PSOTVAC and FA-BBO for test system 2. Method Pop Size Max Best Worst Min-Max of Balance Time (min) Iteration Solution Solution Demande Violation FA 50 200 3119200 3153700 0.0982-0.0565 11.769 BBO 50 200 3192700 7297400 0.0121-0.0562 3.2391 PSOTVAC 50 1000 - - Violation - BBO-PSOTVAC 50 1000 3105700 3122200 0.0313-0.0360 3.4018 FA-BBO 50 200 3141960 3166200 0.1323-0.0623 20.001 CSO [11] 30 1000 3051260 3054960 - 1.797 EP [37] - - 3164531 - NA NA DE [38] - - 3163000 3173100 NA 0.52 HS [39] - - 3143254 NA NA NA MHEP-SQP [36] - - 3151445 3157738 NA NA It is also important to note that the obtained results were achieved at a competitive time. 6. Conclusion In this study, four algorithms the FA, PSOTVAC, BBO, FA-PSOTVAC, BBO-PSOTVAC have been adapted and applied to solve the DED considering three practical constraints simultaneously such as the valve point effect, prohibited zones and ramp rate limits. The performances of the standard algorithms such as FA and BBO in terms of solution quality and number of generations required have been improved by hybridization. The main idea introduced in this study is to exploit the best properties of FA and PSOTVAC, the BBO and PSOTVAC by creating flexible balance between diversification and intensification during search process. The performances of the hybrid methods were validated on two practical test with 10 units and 30 units to solve the DED considering three practical constraints. The total cost achieved using the hybrid method named FA-PSOTVAC is competitive in terms of solution quality and convergence characteristics. Due to the competitive aspect of the proposed hybrid method, authors will strive to develop an extended hybrid variant to solve DED of modern power system characterized by the large integration of various types of renewable sources and FACTS devices. References [1] WANG, C. and S. M. SHAHIDEHPOUR. 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