Multiarea Economic Dispatch Using Evolutionary Algorithms

Department of Electrical Engineering, UIT, Himachal Pradesh University, Shimla, H.P, India Department of Electrical Engineering, National Institute of Technology, Hamirpur, H.P, India School of Electronics, Indian Institute of Information Technology, Una, H.P, India Department of Electrical & Electronics Engineering, Amity University Rajasthan, Jaipur, India Department of Electrical & Electronics Engineering, JK Lakshmipat University, Jaipur, Rajasthan, India


Introduction
e application of economic dispatch (ED) in the operation of the modern power system has a great significance. System demand is economically allocated between different multiarea generators by considering all constraints [1]. ED for multiple areas has paid limited attention. Large and small utilities have many constraints to transmit power through tie lines. All utilities and power pools have different generation characteristics and load patterns in modern power sectors, including spinning reserves. erefore, the main objective of ED is to minimize the fuel cost of all the generators and satisfy all the constraints such as power balance, losses, and generation limits. In the deregulated environment, the generator with the lowest cost should operate with its maximum capacity and transmit more power to those areas consisting of more expensive generating units. MAED is a evolutionary programming (EP) [4], artificial neural network (ANN) [5], ant colony optimization (ACO) [6], particle swarm optimization (PSO) [7], artificial immune system (AIS) [8], differential evolution (DE) [9], bacterial foraging algorithm (BFA) [10], and biogeography-based optimization (BBO) [11], have been successfully applied to have complex ED problem without any limitation in size and condition of cost curves. Shoults et al. [12] have made ED by considering import and export constraints for the singlearea and three-area problems. Yalcinoz and Short [13] have considered transmission constraints for two areas' power systems and applied the ANN approach for the ED problem. Seiffert [14] has used linear programming methodologies. e author calculated the incremental cost for each area, and according to total cost power, cost and tie line values were adjusted. e problem with this method was not feasible on large interconnected power systems. Chen and Chen [15] have used the direct space method to solve MAED problems. e author built an MAGS algorithm to establish a relation between dependability and system security.
e power system of Taiwan has been selected for this work by the author. Manoharan et al. [16] have applied the evaluation algorithm (EA) and Karush-Kuhn-Tucker (KKT) conditions based on optimal confirmation to the MAED problem. KKT-trained variables have been applied to the results obtained by EAs to check optimality. e obtained results using the KKT criterion were compared with linear programming (LP) and dynamic programming (DP) results. e authors concluded that this technique provides better CPU time and standard deviation. Sharma et al. [17] presented differential evolution with a time-varying mutation technique to solve MAED by considering tie line capacity constraints. Venkatakrishnan et al. [18] applied the GWO method to solve the ED problems by considering thermal valves. Evolutionary-based optimization methods are becoming more popular for ED problems due to their advantages, such as the absence of convexity assumptions, better search capability, and simplicity. Many such methods reported in the literature are neural network (NN), tabu search (TS), simulating annealing (SA) [2], particle swarm optimization [7], genetic algorithm (GA) [3], harmony search (HS) [16], ant colony optimization (ACO) [16], and differential evolution (DE) [19]. Some researchers have proposed comprehensive reviews of metaheuristic optimization methods to solve ED problems. ese reviews suggested that PSO and DE techniques are more popular for solving ED problems due to their simplicity, fast convergence rate, greater flexibility to search optimum global points, and easy implementation.
However, all evolutionary techniques required a suitable balance between global search (GS) and local search (LS). Few researchers have focused on convergence time, optimal parameters tuning, premature convergence, and so on. Researchers have attempted to handle these issues using various strategies, such as modified evolutionary techniques [20] and hybridization of algorithms [21,22]. Jain and Pandit [23] have implemented the PSO technique to solve the MAED problem. e authors have modified the PSO technique for the general search to avoid premature convergence. However, AI techniques are becoming popular for nonconvex, multimodal, discontinuous optimization problems for which traditional methods cannot provide a solution. Manoharan et al. [16] have applied EP with the LMO approach to solving the ED problem. e authors reported that the EP-LMO approach has better accuracy and convergence rate than EP. is study focuses on applying DPSO and GWO with optimal mutation to provide an accurate and feasible solution for the ED problems. e main drawback of classical approaches is knocking at local optima and may not offer the best solution. Second, all classical methods are based on the assumption that their objective function to be handled is continuous and differentiable, whereas the practical power system is more complex. Contemporary intelligent techniques have the advantage of being versatile in handling qualitative constraints. Still, their main drawback is that the computational time increases exponentially as the size of the problem increases, and time to convergence is uncertain (convergences are guaranteed). Metaheuristics approaches have been applied to overcome these shortcomings.
Apart from aforementioned papers, there are also some recent studies on multiarea economic dispatch problems, which are hybridization of differential evolution (DE) with immunized ant colony optimization (ACO) [24], electro search optimization approach (ESOA) [25,26], uncertainty of MAED problem using Monte Carlo simulation [27], water wave optimization (WWO) [28], krill herd algorithm (KHA) [29], dynamic dispatch in wind-based power system using chaotic grasshopper optimization algorithm (CGOA) [30], hybridization of chaotic particle swarm optimization (CPSO) and genetic algorithm (GA) is (HCPSOGA) [31], squirrel search optimization (SSO) [32], complete review of metaheuristics on MAED problem [33], salp swarm algorithm (SSA) on stochastic nature of wind [34], improved grasshopper optimization algorithm (GOA) [35], Coulomb's and Franklin's law-based optimizer [36], fast convergence evolutionary programming (FCEP) [37], artificial bee colony (ABC) [38], nature-inspired optimization (NIO) [39], and hybridization of shuffled frog leaping algorithm with PSO considering emissions on MAED problems [40]. e study also compares the solution obtained using the PSO and GWO method to find the appropriate application and accuracy of these techniques. According to the authors' knowledge, proposed DPSO with all variants of PSO and GWO techniques have not been applied yet by considering valve point loading on these test systems.

Problem Formulation
e objective function is to minimize the total fuel cost of generation between all interconnected areas by considering all the constraints. Valve point loading (VPL) directly affects the objective function and produces distortion in heat rate characteristics in ED problems. e introduction of VPL results in the objective function nonconvex, discontinuous, and result in multiple minima of the cost function. erefore, in the present objective part, the VPL is modeled as a sinusoidal function [38] in the input-output cost function to rectify the effect of VPL, and it is given as where P Gij is the power generation of ith to jth units; a ij , b ij , and c ij are the fuel cost coefficients; and e ij and f ij are the fuel cost coefficients of the ith to jth units of the VPL model. Tie line power flow between areas plays a significant role in deciding the operating cost in multiarea power systems. Taking into consideration the cost of transmission through each tie line, the objective function of MAED is given in the following equation.
e objective function of MAED is stated as follows: Constraints: Area power balance: the maximum power generation through all available generators is equal to the demand P Di . In ED, the area power balance constraints, each area power should meet with generation.
Generator constraints: in generating limit constraint, the output of each unit should satisfy the upper and lower limits of generations.
Tie line constraints: the flows through tie line should also be in the maximum and minimum limit range. ese limits of power flow are important and are stated as follows:

Solution Technique
In the modern era, the computer-aided application in power systems has been increased. e application of evolutionary soft computing techniques in power systems has become more popular for solving optimization problems. e popularity of these techniques in complex power systems is increased due to their ease of implementation and reliable operation. Moreover, the modern interconnected power system has mixed types, that is, highly nonlinear cost characteristics functions. Solving these mixed-type functions by classical methods like Newton Raphson and lambda-iteration methods is difficult and inaccurate. erefore, the application of metaheuristic methodologies to solve these types of problems is very popular. Among all evolutionary techniques, PSO [41][42][43][44] is more popular than other methods in the literature. e PSO can handle large dimension, nonconvex, nonlinear, and multiconstraint problems efficiently due to their random search technique. Despite all these advantages of PSO, with an increase in the size and complexity of the problem, PSO is sometimes stuck in local optimum solutions. PSO approach has three tunable parameters, that is, w, C 1 , and C 2 . Here, C 1 and C 2 are mainly random numbers. So, these parameters sometimes faced problems in handling the composite functions and struck at local optima. erefore, a strong variant of PSO is proposed to tackle this problem. DPSO tackled all these problems because all these parameters are tuned dynamically and depend on system parameters like maximum iteration and classical PSO parameters such as velocity, position, gbest, and pbest. e PSO method starts by selecting a population of auxiliary solutions and searching for optima via the aid of modernizing solutions. e particle's velocity has a significant impact on particular social, cognitive, and initial components. e rule for updating particle velocity demands a proper balance between the social and cognitive properties of the swarm required. Initial domination of cognitive part over social part is must to secure by exploration of search space. However, subordination of social part over cognitive is needed to propel all solutions towards global optima to enhance local exploitation. erefore, an explored control equation is propounded for regulating particle velocity dynamically by taking constriction variables e 1 and e 2 . Likewise, the cognitive and social parts are updated by considering RMS experience and preceding experience, respectively [45].
During the application of PSO, the position and velocity of each particle are updated as follows: where ∆t is the time step of 1 second. e inertia weight is given as 3.1. Proposed DPSO. In this study, the proposed DPSO, W, is modified by exponentially decaying function ɳ to avoid premature convergence.
W � e (− η ln k w ) , where k w � (w min /w max ) and η � itr/itr max , and the factor K w be chosen in respect of inertia weight's bounds maximum and minimum limit. In this paper, the value of kw is the ratio of maximum and minimum bound of the inertia weight [45].

Proceeding Experience. Update RMS experience and acceleration coefficients and parameters of constriction factor approach where
in which k is the proposed social and cognitive coefficients. For the identical value of these factors, ɳ � ɳ t . All other factors valued are stated in Table 1. e flowchart for the proposed DPSO and constraint-handling management for MAED are depicted in Figures 1 and 2, respectively (Algorithm 1).

Grey Wolf Optimizer. Grey wolf optimization (GWO)
algorithm is a metaheuristic optimization recently developed by Mirjalili et al. in 2014. e algorithm is inspired by the hierarchy behaviors of grey wolves and imitates the hunting phenomena of grey wolves. Despite the various advantages of metaheuristic algorithms like those applied on nonconvex functions as system complexity increases, the GWO algorithm is free from input parameters initialization. GWO approach has two tunable parameters, a and c. So, exploration and exploitation in search space become faster. In the present algorithm, the first fittest solution is alpha (α), beta (β) is second, delta (δ) is third, and other are followers, that is, omega (ѡ). Wolves follow the behavior of encircling prey, pursuing, hunting, tracking, approaching, and so on [46].

Mathematical Modelling of GWO.
In the present section, the detailed mathematical modelling of the algorithm using the social hierarchy model of wolves and group hunting of prey is presented.

Social Hierarchy Model.
In the social hierarchy model, the fittest solution is assumed as alpha (α) wolf or the leader (first) wolf; the next solution is the beta (β) wolf or secondbest solution, the delta (δ) wolf is the third-best solution among all, and the remaining solutions are omega (ѡ) wolves.

Encircling the Prey.
During hunting, the grey wolves encircle the prey, and the following equations are used to model the process [46].
where t is the present iteration of the objective problem, represents the available position vector of the prey, represents grey wolf's position vector and the coefficient vectors A → , and it is calculated using the following equations.
where r → 1 and r where itr � present iteration and max itr � maximum number of iteration.
Here, a is decreasing linearly from 2 to 0 during each iteration. Generally, the grey wolf updates their position randomly in solution space around the prey using equations (8) and (9).
is concept can be implemented for n dimensions search space.
As In the GWO technique, positions of search agents are updated to correspond to alpha, beta, and delta. ey deviate from searching for prey and assemble to assail prey. For | A → | > 1, the search agents diverge from an optimal local solution to find an optimal global solution. is highlights exploration and permits the GWO algorithm to troll it globally. As the value of a → decrease linearly from [2−0], so the A → mainly emphasizes exploration during initial iterations. But value of C → varies in [0-2] randomly, during initial as well as final iterations. So C → emphasizes exploration in last iterations also for C → > 1.

Hunting.
All grey wolves can capture the site and location of prey during hunting, and the positions of the wolf are updated around the prey using the following equations: where X → α (t), X → β (t), and X → δ (t) are the position of first-, second-, and third-best fitness value. D → a , D → β , and D → δ are determined as above equations.

Implementation of GWO for MAED Problem.
e implementation of the GWO algorithm to solve the ELD complex problem with VPL is described in Figure 3 as follows (see Algorithm 2).  Step 1: enter the system data and initialize all control parameters and new particles randomly.
Step 2: check the feasibility of the current particle; if it is not feasible, then run the constraint management algorithm.
Step 3: make an increment in population count by 1. Now check the population, if it is less than its maximum value, go back to Step 1.
Step 6: repeat Steps 1 and 2. Update preceding for the current particle. en repeat Step 4.
Step 8: make an increment in iteration count by 1. If iteration did not reach its maximum value, repeat Step 8.
Step 9: print final results.  Step 1: assign grey wolf population X k (k � 1, 2, . . ., N); N � no of generators Step 2: initialize a, A, and C Step 3: compute the objective function values for each search agent Step 4: find all solutions and initializ the best solution with them Step 5: X α � select first leader (archive) X β � select second leader (archive) X δ � select third leader (archive) z � 1; Exclude alpha from the archive temporarily to avoid selecting the same leader X β � select leader (archive) Exclude beta from the archive temporarily to avoid selecting the same leader X δ � select leader (archive) Add back alpha and beta to the archive t � t + 1 end while Step 12: return archive ALGORITHM 2: Algorithm for handling MAED problem by grey wolf optimizer.

Results and Discussion
e constraints considered in this study made MAED problem much more complex and difficult to solve than the classical ED problem. DPSO and GWO techniques are used and tested for the MAED problem on three systems having different sizes and complexities. e performance of both DPSO and GWO variants is compared. e data for the test system is taken from [1]. In Table 2, results are taken by varying load demand, and results are compared from the classical method, that is, lambda-iteration method.

Description of the Test
It is revealed from the results that all constraints are satisfied within their limits. In the above case study, power violation � zero means all constraints are satisfied and no power loss occurs. From Figure 5, it is seen that the GWO approach converges faster compared to the PSO approach.

Test System 2: Two-Area Problem with 1 Tie Line.
e two areas with four generator units are tied through a single tie line shown in Figure 6. Generator cost coefficients adopted from literature [17,45] are as follows: fixed cost for 4 generators (a) is 561, 310, 78, and 250; running cost (b) is 7.92, 7.85, 7.97, and 7.5; and maintenance cost (c) is 0.001562, 0.00194, 0.00482, and 0.00181 respectively. is case study's upper and lower generator limit is [600, 400, 200, and 340] and [150, 100, 50, and 70]. In this case study, the load is varied, and the corresponding tie line limit is also varied. e initial value for C 1 and C 2 is 1.8 and 0.2, respectively. e final value for C 1 and C 2 is 0.2 and 1.9, respectively. e results are concluded after 500 iterations for both methods.
From Table 3, it is seen that power mismatch is zero, and the system satisfies all the constraints within the prespecified limits. (Table 4).. e results show the effectiveness of the GWO technique over DPSO and ABCO techniques. e generation cost and execution time are less in the case of GWO as compared to DPSO and ABCO. e DPSO saves Rs. 12.4 over ABCO approach and in the case of GWO Rs. 12.7 over ABCO.
From Figure 7, it is seen that the GWO approach converges faster compared to the PSO approach.

Test System 3: Four-Area
System. In this system, four areas with ten generator units in each area are considered for generation, including all constraints. All the generating units included valve point loading coefficients. e areas are fully interconnected, that is, the power can flow between any two areas. Hence, the system has four areas, each consisting of 10 generators and connected with three tie lines, as shown in Figure 8. e total demand for this case is 10,500 MW. In this case study, Area 1 shares 15% load demand, Area 2 shares 40% load demand, Area 3 shares 30% load demand, and Area 4 shares 15% load demand. e tie line limit from Area 1 to Area 2, from Area 1 to Area 3, and from Area 2 to Area 3 or vice versa is taken as 200 MW and that for the remaining, each tie line is taken as 100 MW. Along with this, other data related to this case study is mentioned in Table 1. e cost coefficient data for 40 generators are adapted from [47].
e dispatch of the current MAED problem consists of power generation of each generator for every area and the power flowing through the tie line given in the system. e dispatch schedule of the system, for the best run with minimum cost, is presented below in Table 5.
e DPSO and GWO are successfully applied to MAED in MATLAB. e scheduled dispatch of the problem specified earlier is recorded for twenty-five test runs. e total cost and CPU time taken for each dispatch have been presented in Table 6. e analysis of these runs is also done, and the results obtained are compared with other methods reported in Table 7.
e Wilcoxon rank-sum test is performed on cost values of both the approaches that are obtained in 25 runs in MATLAB using the command rank-sum and that command prompt return the p value of a two-sided Wilcoxon rank-sum test equal to 2.4170 × 10 −6 and h value return equal to 1 that indicates a rejection of the null hypothesis at the 5% significance level. e struct format returns zval and rank-sum values as 4.7150 and 881, respectively, for the obtained cost values. Table 8 shows the power flow between each pair of areas. Every entry corresponds to the power flowing in the respective tie line. As can be seen below, the diagonal entries will always be zero because no tie line flows possible within the area. e MATLAB program has been run for various combinations of iteration count and population size. e cost convergence curve is a plot of fuel cost obtained versus iteration count. e curve has been plotted for iteration count and population size being one thousand and one hundred, respectively. is curve has been shown in Figure 9.
It is visible that the convergence curves obtained by solving Test System 3 using DPSO and GWO are shown in Figure 9. It is initially observed that the rate of decrease of the cost value is significant but slows down later, and GWO shows better results after complete iterations. From Table 7, it is concluded that the GWO approach is better in terms of operating cost, execution time, and higher efficiency, keeping in mind all the constraints so that the power mismatch and violation are zero.

Conclusion
In this paper, the DPSO and GWO have been applied successfully to model and solve the multiarea economic dispatch in three different test cases. First, the MAED problem with nonlinear cost function is solved on a single-area test system consisting of 3 thermal generators. Second, DPSO and GWO are applied on two-area test systems with four thermal units with single tie lines. ese are, after that, employed on a multiarea test system consisting of 40 thermal units and six tie lines. Optimum demand sharing of power generating units is evaluated using DPSO and GWO optimization techniques. e simulation results reveal that GWO techniques produce qualitative cost solutions without any constraint violation. A significant improvement in the cost results has been obtained compared to other optimization techniques discussed in the literature. In the future, this work can be extended to work in deregulated, stochastic, and contingent environments. e losses can also be calculated using the B-coefficients of thermal generators in future research work. Ramp rate is also an important constraint that makes power system problems more realistic and can also be considered in future research work.
Nomenclatures a ij , b ij , and c ij : e cost coefficients of the jth generator in area i (Rs/hr), (Rs/hr MW −1 ), and (Rs/hr MW −2 ) C 1 and C 2 : Acceleration coefficients for the best and social experience of PSO C 1b and C 1p : Acceleration coefficients for best and preceding experience e ij and f ij : e valve point effect coefficients of the jth generator in area i (Rs/hr, MW −1 ) gbest t : e best particle during tth iteration grms t : Root mean the square experience of the swarm during tth iteration itr: Current iteration count itr max : Maximum iteration count P Gij : e real power output of the jth generator in area i (MW) P min G ij /P max G ij : Minimum/maximum generation limits of jth generator in area i (MW) P min Tim /P max Tim : Minimum/maximum tie line power limit from area i to area m (MW) preceding n : Preceding position of nth particle achieved based on its just previous experience P Tim : Tie line real power flow from area i to area m (MW) rand 1 () and rand 2 () random numbers in [0, 1] V t n : e velocity of nth particle at tth iterationW: e velocity of nth particle at tth iterationW:Inertia weight PSO: Particle swarm optimization GWO: Grey wolf optimizer RCGA: Real codec genetic algorithm ABCO: Artificial bee colony optimization MAED: Multiarea economic dispatch P Gi : Total real power generation in area i (MW) k: e ratio of dynamic cognitive and social acceleration coefficients: e ratio of dynamic cognitive and social acceleration coefficientsk w : e ratio of maximum and minimum bound of the inertia weight M: Number of areas N Gi : Number of generating units in the system in area i P Di : e total real power demand of area i (MW) Pbest n : e best position of nth particle achieved based on its own experience PD: e total actual power demand of the system (MW) W min /W max : Minimum/maximum value of inertia weight ∆t: Time step (s) ζ 1 and ζ 2 : Exponential constriction functions ɳ: e ratio of the current and maximum iteration count ɳ t : e value of g at which cognitive and social behavior equalizes µ: Constant µ 1   Evolutionary programming DPSO: Dynamically controlled particle swarm optimization ED: Economic dispatch.
Data Availability e data used in this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.