Application of a Novel Modified Hybrid Algorithm for Solving Dynamic Economic Dispatch Problem with Practical Constraints

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.


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.
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 Modi-fied 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.

2.
Nomenclature: The unit i production cost at time t.DED: Dynamic Economic Dispatch.FA: Firfly algorithm.ng: The number of generation units.n i : The number of prohibited operating zones in the ith generating unit.P d (t): Load demand at time t.P min i , P max i : The maximum and the minimum production of unit i. P it : Power output of unit at time t.P loss : Transmission losses.PSOTVAC: Particles Swarm Algorithm with a variable acceleration coefficient.T : The total number of hours in the operation period.
The ramp up and the ramp down rate limit's respectively.

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]: where C it is the cost of ith generating unit at time t, ng is the number of generation units and P it is the power output of it unit at time t.T is 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]: where a i , b i , c i , e i , f i are the cost coefficients of i th power generating units.This objective function should be minimized considering the following equality and inequality constraints [22].

3.3.
Inequality Constraints [23] and [24] P min i , P max i are the minimum and the maximum of unit's production.

2) Prohibited Operation Zone
The Prohibited Operation Zones [26] and [27] are mathematically expressed by the following equation: where: n i is the number of prohibited operating zones in the i th generating unit.k is the index of the prohibited operating zones of the i th generating unit.P L iK , P U iK are the lower and upper bounds of k th prohibited operating zones of unit i.

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:

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).
Ensure: : Initialize population of m fireflies, then return Move firefly i towards j (eq 13) end if end for end for Update Light intensity f (x i ) 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: where r is the distance between any two fireflies, B 0 is the initial attractiveness at r = 0, and γ is an absorption coefficient which controls the decrease of the light intensity.

2) Distance
During the search process, the distance between two fireflies i and j at location x i and x j can be defined by the following expression: c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING where r ij is the distance between two fireflies and d is the dimension of the problem.

3) Movement
The movement of a firefly i which is attracted by a more attractive firefly j is given by the following equation: 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 j are two variables which reflect the light intensity that is associated with a specified fitness function of particles to be evaluated [30].

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: 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: 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 λ and emigration rate µ.A good solution enjoys a higher µ and 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 k th island as [34].

S max
Fig. 2: Species model of an island.
when E = I, the immigration and emigration rates can be related as:

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 FA-PSOTVAC 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.

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.

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.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 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.

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 (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 BBO-PSOTVAC the best total cost achieved is 3105700 $.It is also important to note that the obtained results were achieved at a competitive time.

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 PSOT-VAC, 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.
Belkacem MAHDAD was born in Biskra, Algeria.He received his B.Sc. degree in Electrical Engineering from Biskra University, Algeria in 1990, and the Magister and Ph.D. degrees from Annaba University and Biskra University in 2000 and 2010, respectively.He is an assistant professor at Biskra University.His research interests include power system optimization, FACTS modelling and integration in practical power system, optimization methods, power system stability and system protection coordination.

Fig. 8 :
Fig. 8: Distribution of Ramp Down violation for 50 trials for test system 1.

Fig. 9 :
Fig. 9: Distribution of the best cost for 50 trials for test system 1.
(10)(w max − w min ) * (iter max − iter min ) iter max + w min ,(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.w is the inertia factor presented by 0.4 ≤ w ≤ 0.9.iter is the iteration number.iter max is the maximum iteration number.α 1 , α 2 are respectively the cognitive and the social factors.C 1i , C 2i , C 1f , C 2f 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.
Tab. 1: Best solution of FA for test system 1.
SRAIRI was born in Batna, Algeria, in 1967.He received a B.Sc. degree in Electrical Engineering in 1991 from the University of Batna, Algeria; an M.Sc.degree in Electrical and Computer Engineering from the National Polytechnic Institute of Grenoble, France, in 1992; and a Ph.D. degree also in Electrical and Computer Engineering from the University of Nantes, France, in 1996.After graduation, he joined the University of Biskra, Algeria, in 1998 where he is a Professor in the Electrical Engineering Department.His main research interests power system planning and control, analysis, design, and magnetic modeling.Nabil MANCER was born in Ouargla, Algeria.He received his M.Sc.from Biskra University in 2013 and a Ph.D. degree also in Electrical Engineering from the University of Biskra, Algeria, in 2017.He is an assistant professor at Constantine 1 University.His research interests include power system optimization, FACTS devices and power system protection coordination.