An Examination of Economic Dispatch Using …brisjast.com/wp-content/uploads/2015/10/MRR-Sept.-13-2015.pdfAn Examination of Economic Dispatch Using Particle Swarm Optimization Mohd

MAGNT Research Report (ISSN. 1444-8939) Vol.3 (8). PP. 193-209

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An Examination of Economic Dispatch Using Particle Swarm Optimization

Mohd Ruddin Ab Ghani1, Saif Tahseen Hussein2, M.T. Mohamad3, Z. Jano4

1, 2, 3 Faculty of Electrical Engineering, Universiti Teknikal Malaysia Melaka, Melaka, Malaysia

2 Ministry of Electricity, Baghdad, Iraq 4Centre for Languages and Human Development, Universiti Teknikal Malaysia Melaka

Abstract: The restructuring of the electrical power industry has given rise to a high degree of vibrancy and competitive market, which changed many features of the power industry. Energy resources become scarce, the cost of power generation increases, environment concerns are raised, and an ever rising demand for electrical energy characterizes this now-altered scenario. Optimal economic dispatch is called for amid this transformation. Practical economic dispatch (ED) problems are known to have an objective function of the nonlinear, non-convex type, and possess constraints of intense equality and inequality. Because of local optimum solution convergence, conventional optimization methods are unable to find a solution to such problems. The last decade saw meta-heuristic optimization techniques, particularly particle swarm optimization (PSO), gaining currency and remarkable recognition as the solution algorithm for such ED problems. This paper summarized the application of PSO in ED problems, known as one of the most complex optimization problems. Keywords: Economic dispatch, Formulation of the problem, Particle swarm optimization (PSO).

1. Introduction In the field of operation and planning of

a power system, economic dispatch problem has evolved as an important task. ED is mainly intended to schedule the outputs of the committed generating units, meeting the requirements of load demand at the least cost while satisfying all operational constraints of the unit and system. Improving unit outputs’ schedules can result in considerable saved costs. ED problem was originally formulated as economic cost dispatch (ECD). Later, when the Clean Air Act was amended in the 1990s, formulating problems as combined emission economic dispatch (CEED) and emission controlled economic dispatch (ECED) resulted from the identification of emission dispatch (EMD); however, optimizing these two contradictory objectives individually will not serve the purpose. The available literature has reported that a number of conventional methods are utilized to solve such problems. These include the following: Bundle method

[1], nonlinear programming [2, 3], mixed integer linear programming [4–7], dynamic programming [8], quadratic programming [9], Lagrange relaxation method [10–13], network flow method [14], and direct search method [15]. ED problem, in practice, is a nonlinear, non-convex type that has multiple local optimal points because of the inclusion of valve point loading effect and multiple fuel options with varied equality and inequality constraints. Traditional methods have failed to surface corresponding solutions because such problems are sensitive to initial estimates and converge into local optimal solution and complex computations. Modern heuristic optimization techniques have been developed which provide a better solution. These techniques, which are based on concepts related to operational research and artificial intelligence: evolutionary programming [16–19], genetic algorithm [20–24], simulated annealing [25–27], ant colony optimization [28, 29], Tabu search [30, 31], neural network


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[32–36] and particle swarm optimization [37–70]. While each method has its intrinsic advantages and disadvantages, PSO is the best suitable solution algorithm. This paper serves as a response note on prior works of applying population-based PSO algorithm as a solution to the different ED problems.

2. Formulation of the problem

Judiciously formulating practical ED problem is a requirement of economic dispatch. Talaq et al. [71] and Lamont and Obessis [72] put forth the suggestion on the different types of ED problem formulation. It could be laid out as single objective and multi-objective problems that are, by nature, nonlinear and non-convex. There are several ways to formulate single objective problems; ECD without valve point loading effect [21, 38,39,41,43,50], ECD with valve point loading effect (ECD-VPL) [34,37,40,52,55,58], ECD with valve point loading effect and multiple fuel option (ECD-VPL-MF) [42,44,52,56], EMD and ECED [59,71,73,74,75]. Multi-objective formulation, on the other hand, can be combined emission economic dispatch CEED [47, 48, 57, 59, 60, 63, 73], multi-area emission economic dispatch MAEED [45, 46], power generation under different utilities [76], and maximization of generated power and irrigation [34].

2.1. Objective function There is one main goal of any ED problem: to bring down the operational costs of the system while fulfilling the load demand within the limit of constraints. Listed below are the different types of objective function formulation.

2.1.1. Single objective problem formulation

Single objective ED problem can have these formulations; as a fuel cost function (A)–(C) or with emission of greenhouse gases (D) as an objective function. Mathematically,

the fuel cost and the emission of greenhouse gases can be expressed as a quadratic polynomial of generated power. For more practical results, fuel cost function is altered by including the valve point loading effect (B) and multiple fuel options (C) (See Fig. 1).

2.1.1.1 Simplified economic cost function

Simplified economic dispatch problem can be expressed as an objective function of quadratic fuel cost. This is described in Eq. (1) [11–13];

where FT: total generating cost; : cost function of ith generating unit; , , : cost coefficients of generator i; : power of generator i; and n: number of generator.

2.1.1.2. Economic cost function with valve

point loading effect

Generating units that come with multiple valves in steam turbines are available. Opening and closing these valves assist in maintaining the active power balance, but these actions also give added ripples in the cost function (See Fig. 1); resulting in the objective function becomes highly nonlinear.

Fig.1. Incremental fuel cost curve for a five-

valve steam turbine unit.

The quadratic cost function (See Fig. 2) increased by adding the sinusoidal functions.


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Fig.2. Piecewise quadratic and incremental fuel cost function.

where and are the coefficients of generator i considering valve point loading effect.

2.1.1.3. Economic cost function with

multiple fuels A thermal generating unit can use

different types of fuels. The fuel cost objective function, therefore, can be expressed as piecewise quadratic function that reflects the impact of fuel type changes.

the nth power level (Fig. 2). 2.1.1.4 Emission function

Since global warming causes the emission of greenhouse gases in the environment, it is an issue of concern for the power industry. As previously discussed, efforts, such as amending the Clean Air Act and developing policies that are environmentally-friendly (the Carbon Credit System), have developed the power sector’s interest towards reducing emissions of NOx, Sox, and CO2 gases. In EMD problem, the emission of greenhouse gases is represented by different mathematical formulations. It can be expressed in quadratic

form [60, 71, 74], adding quadratic polynomial with exponential terms [59, 63, 73], or adding linear equation with exponential terms [77] of generated power

where , , , , , , are the emission function coefficients.

2.1.2. Multi-objective problem formulation

With environmental awareness and deregulation, it is necessary to restructure the operation policies in the power sector. This includes changes in the emission aspects and individual profits as well. Towards this end, the formulation of the ED problem can be stated as a multi-objective problem with two or more competing objectives. Different multi-objective ED problems reported in literature have combined economic emission dispatch (CEED) [47,48,59,60,63,70], maximized generation and irrigation [34], or minimized generation cost under different management structures [76], minimized pool purchase cost and emission [66], minimized cost and all pollutant gases [57], minimized fuel cost-emission and real power loss [48] and minimized multi-area environmental economic dispatch (MAEED) [45,46]. In finding a solution to these conflicting objectives, weighted sum [76] or price penalty factor approach [47] can be used to convert into a single objective function.

2.2. Equality and inequality constraints

(1) System power balance equation: This

equation stands for an equality constraint that


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should be satisfied for a hydrothermal system

where , are the power output from ith thermal unit and jth hydro plant. n,m are the number of thermal and hydro generating units in the system being studied; is transmission loss; and is load demand.

(2) Water balance equation:

(3) Power generation limits:

, , , are the minimum

and maximum limits of generation applied for thermal and hydro plants.

(4) Reservoir storage limits:

, , , represents the

following values respectively: minimum value of reservoir storage, maximum value of reservoir storage, reservoir storage at the starting of time horizon, reservoir storage at end of time horizon.

(5) Hydro water discharge limits:

, stand for the minimum and maximum discharge limit of hydro plants.

Fig.3. Cost functions with prohibited

operating zone.

(6) Generator ramp rate limits: An impractical assumption that is used in conventional economic dispatch problem holds that power output adjustments are instantaneous. In practice, however, the operation of all online units is restricted by the ramp rate limit. Thus, the generation may change (i.e. decreasing or increasing) within the corresponding range in order for the units to be constrained because of the ramp rate limits, as shown below [11–13].

After considering ramp rate limits, modified generation limits are now given as:

where , are the generator’s upward and downward ramp rate limits.

(7) Prohibited operating zone: There are

instances when generators have a certain range where operation becomes restricted because of a number of factors: the machine component’s physical limitation, the steam valve, and vibration in shaft bearing, among others. When prohibited operating zone are considered, discontinuities are created in the cost curve, converting the constraint, as shown below.


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where ; are the lower and upper boundaries of kth prohibited operating zone of unit i, k is the index of prohibited operating zone, and is the number of prohibited operating zone (Fig. 3).

(8) Spinning reserve constraints:

is the system spinning reserve requirement in MW.

(9) Line flow constraints:

where is the real power of line K and L is the number of transmission lines.

(10) Emission constraints:

where , , are the , , and gases emission because of fuel combustion in thermal plants and , , and are the different gases’ maximum limits for emission. 3. Particle swarm optimization (PSO)

The most recent development in the category combinatorial meta-heuristic optimization is the Particle swarm optimization, a population-based stochastic search algorithm [78, 79]. In 1995, Kennedy and Eberhart first introduced PSO as a new heuristic method [64]. Initially, their research aimed to graphically model the social behavior of bird flocks and fish schools. However, only the nonlinear continuous

optimization problems can be handled by this original version. Further leaps in this PSO algorithm have enabled the exploration of global optimal solutions involving complex problems of engineering and sciences. Among its various versions, the most familiar is the one proposed by Shi and Eberhart [75] which involves only two models — Eqs. (19) and (20) — PSO is simplicity itself, which is its key attractive feature. In this algorithm, a possible solution referred to as particles associated with Position and Velocity is represented by the co-ordinates of each particle. The particle moves towards an optimum solution at each iteration and, via its current velocity, it arrives at a personal best solution by itself so far, which the global best solution is reached by all particles. In a physical d dimensional search space, the particle’s position and velocity i are expressed as the vectors of and

in the PSO algorithm. Let be the best position of particle i and

as its neighbors’ best position so far, respectively. Each particle’s modified velocity and position can then be calculated utilizing the current velocity and the distance from and which is shown as follows:

where : velocity of particle i at iteration k; : inertia weight factor; , : acceleration

coefficients; , : uniformly distributed random number between 0 and 1; position of

particle i at k iteration; : best position of particle i until iteration k;


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: best position of the group until iteration k; and K: constriction factor. In these velocity updating processes, the value of the parameters, e.g. , , , and K should be determined ahead. As iteration proceeds, the inertia weight decreases linearly. It is obtained as:

where : final inertia weight; : initial inertia weight; : current iteration number; and : maximum iteration number. 4. PSO reviews in Economic Dispatch

Problem Because of its simplicity in solving the

practical ED problem, population-based PSO has been recognized as the fast growing solution algorithm. PSO shows that the particles’ movement is controlled by its previous own velocity, as well as two acceleration components (i.e. cognitive component and social component). If the cognitive component has a high value compared to the social component, it will result in excessive wandering of particle through search space. Conversely, particles may rush prematurely towards a local solution when the social component has a relatively high value. Cognitive and social components rely on PSO variants , , , and . In a similar vein, each dimension’s particles velocity is clamped to a maximum velocity . Its high value leads to global exploration; low value, on the other hand, encourages local exploitation.

Shi and Eberhart [75] thus, introduce a concept of inertia weight towards a better control between exploitation and exploration. The PSO performance can be improved by the proper control of these variants ( , , ,

, ). Various changes in original PSO have been suggested by several researchers in order to keep a balance between local exploitation

and global exploration, thus improving the quality of the solution with less problem of computational time. A series of yearly reviews (2003–2008) described below have contributed to the field, specifically on PSO application in different kinds of ED problems.

Using PSO, the multi-objective CEED problem has been solved by Selvakumar et al. [47]. They combine two contradictory objectives (emission and economic cost) with the use of penalty factor towards forming a single objective problem. The emission cost and the normal fuel costs are blended with the introduction of the price penalty factor.

In order to find a solution to the bid-based dynamic ED in the context of a competitive electricity market, Zhao et al. [48] utilize PSO with constriction factor and inertia weight (PSO-CF-IW). The maximization of social profit, which is the difference of a customer’s benefit function to a generator’s cost function, is the objective function of their study. Among the topics discussed in their paper are generation bid quantities; power balance; ramp rate limits; customer bid quantities; line limit, and emission as equality and inequality constraints in the optimization process.

On the other hand, Victorie and Jeyakumar [69] analyze the implementation of a hybridized PSO with sequential quadratic programming (SQP) designed to solve ECD-VPL problem. Their study considers several factors, namely ramp rate limit; real power balance; voltage at load bus; generation limit; transmission line constraints; and spinning reserve as constraints to the problem. In order to fine-tune the solution region, this method integrates PSO algorithm as the main optimizer with SQP as local optimizer. To find a solution to the problem, PSO is initialized. Then, as the better global best is found, the region undergoes fine-tuning with the use of SQP, accepting this as the technique’s initial searching point. This will


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assist in exploring the global optimal point at the PSO run’s earlier iteration.

Park et al. [65] use a modified PSO (MPSO) that incorporates dynamic search space reduction strategy in order to accelerate the optimization of ECD-VPL and ECD-VPL-MF problems. Their study explores a proper heuristic treatment mechanism in handling the equality constraints (power balance) and a position adjustment strategy in handling the ED problem’s inequality constraints (generation limit). This was done while each particle modifies its PSO algorithm search point. The researchers’ proposed method shows the space reduction strategy as activated when the PSO performance does not improve during the specified iteration number. In such a case, the search space is reduced dynamically as per the distance between the

and the generator’s minimum and maximum output.

A description of PSO as a solution to the multi-objective combined emission economic dispatch problem is presented by Umayal and Kamaraj [57]. In formulating the multi-objective problem, their study considers various competitive objectives fuel cost, NOx, SOx, CO2, and variation of generation mismatch. All these objectives are weighted according to importance, then they are added together in order to form the final objective function. Introducing the inaccuracies and uncertainties in the hydrothermal schedule is a unique feature of this study.

The proposal of Bo et al. [74] was a multi-objective particle swarm optimization (MOPSO) towards solving the ED problem. Three objectives (i.e. emission, fuel cost, and real power loss) need to simultaneously minimize with power balance, generation capacity, and transmission line limit constraints. The proposed method utilizes two repositories (external memories) maintained along with the search population. So far, this is one of the global best individuals and one that contains a single local best for each

swarm member. In order to obtain the different Pareto optimal solutions, a geographically-based approach is utilized. Lastly, a fuzzy-based mechanism is used with linear membership function so as to select a best compromise solution from the trade-off curve.

Park et al. [55] use chaotic particle swarm optimization for the ECD-VPL problem under both active power balance and power generation limit constraints. This method is made more feasible by the valve point loading effect consideration in ED problems, at the expense of an increase in nonlinearity and the number of local optimal points in the solution space. Chaotic logistic map-based inertia weight variation in the velocity update equation prevents the occurrence of premature convergence of PSO and aids in the escape from the local optimal point.

Jeyakumar et al. [42] relates an instance when linearly decreasing inertia weight PSO (PSO-LVIW) has been successfully employed to solve multi-fuel economic dispatch (PQCF), multi-area economic dispatch (MAED), combined emission economic dispatch (CEED), and economic dispatch with prohibited operating zone (ECD POZ). CEED is a problem having multiple objectives, among which is the decrease in fuel expense and NOx emission. The weight sum approach is used to weigh both objectives based on their importance, with the values added to arrive at a final objective function.

Chakraborti et al. [54] adopt PSO with linearly varying inertia weight with constriction factor (PSO-LVIW) to address the ECD-VPL problem. Their work took into account a number of constraints, namely spinning reserve, ramp rate limit, ramp rate limits, and network loss in addition to load balance, operating limits, and network losses.

Yu et al. [38] put forward the global vision of particle swarm optimization with inertia weight (GWPSO), global vision PSO with constriction factor (GCPSO), local vision of


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PSO with inertia weight (LWPSO), and local vision of PSO with constriction factor (LCPSO) to minimize fuel expense under conditions of hydro and thermal power generation limits, power balance, reservoir storage, water balance equation, discharge rate limit, as well as begin and end storage limits. In GWPSO and GCPSO, particles hold input of the entire group, such that the velocity update equation accounts for the global best value in social behavior. Meanwhile, the LWPSO and LCPSO only have input on themselves and their neighbors, hence, the local best value is employed as the social term in this PSO version. The value of neighborhood size is an even number that varies. The initial results of simulation reveal all versions to be faster, with GCPSO, GWPSO decelerating in the latter stages, and LCPSO, LWPSO generating a viable result owing to the significant capability to maintain population diversity.

Abido et al. [73] propose the use of the novel multi-objective PSO to address the CEED problem under generation limits and power balance constraints. This approach puts forward novel definitions of the local and global best individuals in multi-objective optimization problems to extend the single-objective PSO. An average linkage-based hierarchical clustering technique is likewise employed to address the Pareto optimal set size, whereas a fuzzy-based mechanism is employed to determine the best compromise solution. According to the results, this proposed method could generate multiple Pareto optimal solutions in a single simulation run.

Lee et al. [61] employ iteration PSO to minimize fuel and outage costs under conditions of power generation limit, active power balance, up and down time of generation unit, spinning reserve, and ramping speed. This method integrates an additional new index called iteration best (best value attained by any particle in current iteration) to

control particle movement through the velocity updating equation.

Titus and Jeyakumar [53] propose an enhanced PSO to decrease fuel, start-up and start down costs under the constraints of the prohibited operating zone, hydro and thermal generation limit, and water flow equation. An unusual concept is added to the original PSO to gain the characteristics of diversity, periodicity, and algorithmic stochastic behavior to enhance solution quality while reducing time and preventing premature convergence.

Cohelo and Mariani [37] propose the use of a chaotic PSO hybridized with implicit filtering technique to address the ECD-VPL problem under the constraints of power balance and generation limit. This work integrates a hanon chaotic mapping into PSO to enhance the global convergence and implicit filtering technique in order to make the chaotic PSO more efficient by running in sequential. This method has the capability of escaping from the local minima point.

Jiejin et al. [41] put forward two versions of chaotic PSO (CPSO), namely CPSO1 and CPSO2, to address the ECD problem under the constraints of generation limits, power balance, prohibited operating zone, ramp rate limits, and line flow limits. The first method is the two-phase iterative method where the PSO with adaptive inertia weight factor (AIWF) is used for global exploration. The second method is the further chaotic map local search, which is used for local exploitation. The CPSO1 and CPSO2 methods are somewhat alike, their difference being in the mapping of decision variables into chaotic variables. In CPSO1 and CPSO2, the logistic and text mapping approaches are respectively used for local exploitation.

Selvakumar and Thanushkodi [44] employ a new PSO (NPSO) to address the three ED problem types, which include the ED with prohibited operating zone (ED-POZ), ED with valve point loading (ECD-VPL), and ED with


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valve point and multiple fuel option problems. In the proposed method, the cognitive behavior is divided into the good and bad experience components, such that the particle is able to recall its best and worst positions. NPSO can identify the promising area but is unable to exploit the promising area to obtain a good quality solution for complex multi-minima functions. Thus, the local random search is integrated into the NPSO for the exploration of the promising region.

Lee [49] put forward multi-pass iteration PSO (MIPSO) to minimize fuel and start-up costs under the constraints of generation limit, power balance, water storage limit, water discharge limit, spinning reserve, ramp rate limit, wind turbine generation percentage limit, minimum up and down time, and generation limit of wind system. The conventional PSO is degraded by low computational efficiency with the increase in the time horizon of scheduling. Thus, solving such problems with the use of the proposed method could enhance computational efficiency. MIPSO is initiated with the random time stage and searching space and then further refines the time interval between two time stages and the search spacing pass by pass (iteration). A novel iteration best index (best amongst all particles in previous iteration) is likewise employed to control particle movement.

Jayabarthi et al. [80] analyze a hybrid differential evolution (HDE) and particle swarm optimization method to address the ECD problem. The objective function is designed such that fuel cost is minimized under the constraints of water discharge limit, active power balance, thermal and hydro generation limit, reservoir storage limit, and water balance equation. Both methods are tested through application to the test system, and the convergence rate was found to be much smoother and faster in the case of PSO despite the similarity in the optimal global solution in the case of HDE and PSO.

Coelho and Lee [39] put forward a chaotic and Gaussian-based PSO to address the ECD problem with consideration of the prohibited operating zone, ramp rate limits, transmission loss, and line slow constraints. This work employs a Gaussian or chaotic sequence random number generation between 0 and 1 for the cognitive and social aspects of the velocity update equation. The following are some of the available PSO versions: Original PSO (PSO1), Gaussian distribution in cognitive part and uniform distribution in social part (PSO 2), Gaussian distribution in social part and uniform distribution in cognitive part (PSO3), Gaussian distribution in social and cognitive part (PSO 4), uniform distribution in cognitive part and logistic map Chaotic distribution in social part (PSO 5), logistic map chaotic distribution in cognitive part and uniform distribution in social part (PSO 6), Gaussian distribution in cognitive and Logistic chaotic distribution for Social part (PSO 7) and logistic chaotic distribution for cognitive part and Gaussian distribution for social part (PSO 8). All PSO versions are tested on both a 15 thermal unit system and 20 thermal unit system. PSO 4 is found to have a good result for the 15-unit system, whereas PSO 3 has an optimal result for the 20-unit system.

Panigrahi et al. [56] analyze an adaptive PSO (APSO) method to reduce the smooth (ECD) and non-smooth cost function (ECD-MF) accounting for transmission loss, prohibited operating zone, ramp rate limits, power balance, and power generation limit. This new algorithm is revised for the control of particle movement in a swarm and population re-initialization after a specific iteration. A highly fitted particle will move slowly when compared with the low fitted particle by simply defining the inertia weight between to base on the particle rank in the swarm.

Chandrasekar et al. [43] employ PSO with inertia weight and constriction factor (PSO-


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CF-IW) to reduce fuel cost taking into account the transmission loss, power balance, discharge limit, hydro generation limit, thermal and hydrogenation limit, and water balance equation. This work separately accounts for the hydro generation, thermal generation, discharge limits and reservoir volume as possible particles in the PSO algorithm. Results reveal that cost is best minimized by selecting reservoir volume as the particle.

Selvakumar et al. [52] utilize anti-predatory PSO to address the ECD-VPL and ECD-VPL-MF problems. In this method, the PSO relies on both foraging and anti-predatory activity. This work puts forward new PSO variants for anti-predatory activity to aid the swarm in evading predators. This method divides the cognitive and social behavior of the particles into the best and worst experiences. By including the worst experiences, the swarm is endowed with enhanced exploration capacity.

Yuan et al. [50] propose an enhanced PSO (EPSO) to reduce fuel cost under the constraints of hydro and thermal generation limit, power balance, reservoir storage volume, hydro discharge limit, initial and terminal storage limit, and water balance equation. This method incorporates three additional properties to the original PSO. These properties include the concept of repellor opposite to attractor (means particle learns from its worst position), chaotic logistic map random number generation, and feasibility-based rule to address both the equality and inequality constraints in the scheduling problem.

Wang and Chanan [45] employ the linearly varying inertia weight PSO (PSO-LVIW) to address the MAEED problem. The objective function is designed to distribute power to different areas by reducing operational costs (fuel cost and transmission cost) while minimizing pollutant emission. The multi-objective PSO is employed to

address this problem, and a Pareto optimal solution is derived.

Wang and Chanan [46] employ PSO with local search to reduce operational costs (fuel cost and transmission cost) of a multi-area system while minimizing emissions. This MAEED problem is designed to have generation limits, area power balance, and area spinning reserve, and tie line constraints as both equality and inequality constraints.

After population initialization, the proposed method can calculate the fitness values. The concept of Pareto-dominance is applied to the fitness value, whereas the non-dominated components are archived. The further fuzzy global best scheme is employed to determine the global best, whereas synchronous particle local search (SPLS) is adopted to have the personal best values. Parallel niching and fitness process are also implemented to improve solution diversity. Particle velocity and position are updated using Eqs. (19) and (20), respectively. The turbulence factor (mutation operator), which use a random value between -1 and 1, is integrated into the updated positions to prevent a condition of being stuck in the local optimal solution.

Dobakhshari and Soroudi [58] propose a hybrid gradient search PSO (HGPSO) method to address the ECD-VPL problem under the constraints of power balance, transmission loss, and power generation limits. When this method is employed, the parallel execution of GSM along with PSO enables the use of the best results derived from the gradient search as for PSO.

Chuanwen and Etorre [51] propose a self-adaptive chaotic PSO for solving the ED problem under a deregulated environment. In this problem, profit generation is set as an objective function under the constraints of the hydro generation limit, water balance equation, discharge rate, and reservoir volume. Random numbers , and the self-adaptive inertia weight scale in the


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original PSO are generated to enhance the performance by introducing the logistic map chaotic sequence.

Mandal et al. [62] employ a PSO with inertia weight and constriction factor (PSO-IW-CF) to address the ECD-VPL problem under the constraints of hydro power and thermal generation limits, water balance, discharge limits, power balance, reservoir volume, as well as initial and final reservoir limits.

Roy and Ghoshal [40] adopt a canonical PSO (CPSO), linearly decreasing inertia weight PSO, constriction factor PSO, velocity update relaxation momentum factor induced PSO (VURMFIPSO) and crazy PSO to address the ECD-VPL problem. In a case like bird flocking or fish schooling, a bird or fish sometimes change its movement direction suddenly. The proposed crazy PSO method integrates this natural behavior as the craziness factor and then models it in the PSO algorithm to enable the particle to have the predefined craziness probability required to maintain the particle diversity. The fuel cost is reduced through the valve point loading effect and transmission loss under the constraints of power generation limit, active power balance, and ramp rate limits. This method is validated using three types of objective functions: the first has positive values for , (refers to conventional ECD problem), the second has positive values for , but negative for (refers to auction based load dispatch) and the third has positive values for , but both positive and negative values for (not reported so far).

Giang et al. [67, 68] propose the use of linearly decreasing inertia weight PSO to address the ECD problem under the constraints of generation limit, power balance, prohibited operating zone, ramp rate limits, spinning reserve, and line flow. The generation power output of each unit is directly taken as a gene, a number of which comprises an individual. Each individual

within the population represents a candidate solution for the ECD problem. To determine the personal best ( ) and accelerated iteration procedure convergence, the evaluation value is normalized into a range between 0 and 1. This evaluation function is a reciprocal of the constraints of fuel cost and power balance.

Sriyaanyong [70] integrates the Gaussian mutation into the traditional PSO for solving the ECD-VPL problem. Swarm diversity, along with good quality solutions, is achieved by including the Gaussian mutation. Particle modification is achieved with the use of heuristic rules.

Alrashidi and Hawary [59] employ PSO to address the multi-objective CEED problem. This work proposes new methods by which to address equality constraints. The weight sum approach is used to convert two objectives, namely economic fuel cost and emission, into a single objective. These two competitive objectives are assigned with different weights, and PSO provides one solution in the Pareto optimal set for each assigned weight value.

5. Suggested PSO modification to address

the Economic Dispatch Problem PSO is a population-based meta-heuristic

optimization approach wherein particle movement is controlled by the two stochastic acceleration components (cognitive and social components) and the inertia component. Studies in extant literature use different PSO versions. A number of these versions are the PSO with time varying inertia weight (PSO-TVIW), PSO with constriction factor (PSO-CF), PSO with Gaussian or chaotic based random number generation, introduction of craziness factor to boost the diversity of particle in PSO algorithm, assimilation of PSO with implicit filtering technique or direct search method or mutation operator, and others. Despite having the capacity to generate good quality results at a faster rate, their capability to enhance the optimum solution is


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comparatively weak compared with other evolutionary optimization methods primarily because of the lack of diversity at the end of the search [PSO selforganising-8]. Thus, PSO with time varying acceleration coefficient (PSO-TVAC) [81], centroidal voronoi tessellation-based initialization of swarm [82], active target PSO (APSO) [83], exponential varying inertia weight PSO [84], PSO combined with either Gaussian or sobol mutation [85,86], guaranteed convergence PSO (GCPSO) [87], self-adaptive PSO [87,89], introducing new metropolis coefficients as learning coefficients [90], or PSO with division of work strategy (PSOwDOW) [91] can be employed to enhance the capability to reach a global optimal solution at a later stage.

6. Conclusions

Regulations requiring environment friendliness, intense competition amongst power generating companies, as well as the rapidly emerging differences between supply and demand have required the design of appropriate operation policies for power-generating companies. Such policies can only be realized when the ED problem is mathematically formulated, taking into consideration all the objectives and constraints possible. PSO has been used to provide the solution to such problems because of its ability in avoiding getting stuck in the local optimal solution, avoiding the curse of dimensionality, exhibiting slow convergence, and dependable initial variables. PSO is able to obtain more reduced computational times than the tradition optimization method. In this work, the extant literature on PSO in the field of economic dispatch has been reviewed. However, further improvement is required for PSO algorithms, given that existing versions of PSO exhibit slower convergence at later stages and fail to generate an optimal solution for real-time scheduling problems.

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