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Economic Dispatch- A Comprehensive Survey
Dr.S.Prabakaran Assistant Professor-EEE
SCSVMV University, Kanchipuram, Tamilnadu, India.
Abstract – This paper reviews the existing developments in Economic Dispatch(ED) problems considering prohibited operating zones, ramp rate limits, capacity limits and power balance constraints. Two very important and related areas of economic dispatch are identified and papers published in the area of economic dispatch are classified in to (i) Economic Dispatch and (ii) Security Constrained Economic Dispatch problems. The author covers various soft computing techniques to address the issues in the area of economic dispatch problems.
Keywords: Economic dispatch, Literature review, Security constrained economic dispatch, Evolutionary programming, Particle swarm optimization, Prohibited operating zones, Ramp rate limits.
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I. Introduction
Economic dispatch (ED) is one of the most significant problems to be solved in the operation of a power system.
Improvement in scheduling the unit outputs can lead to considerable cost savings. The key objective of the economic dispatch problem (EDP) of electric power generation is to schedule the committed generating unit outputs so as to meet the required load demand at least operating cost while satisfying all unit and system equality and inequality constraints [1,2]. This makes the EDP a large-scale highly non-linear constrained optimization problem. Economic dispatch has become a chief function in operation and control of modern power systems. The ED problem can be assured as determining the least cost power generation schedule from a set of online generating units to satisfy the load demand at a given period of time [1].
Though the core objective of the problem is to minimize the operating cost satisfying the load demand, several types of physical and operational constraints make ED highly nonlinear constrained optimization problem, in particular for larger systems [2]. However, precise and intelligent scheduling of the units not only can decrease the operating cost significantly but also can promise higher reliability with improved security and minimum environmental crash [3]. In traditional ED approaches the input–output characteristics of a generator is approximately shown by utilizing a single quadratic function. In exercise, operating conditions of many generating units require
the cost function to be modelled as a piecewise quadratic function[4]. However, higher-order nonlinearities and discontinuities are observed in real input–output characteristics, owing to valve-point loading in fossil fuel burning plants [5]. Besides, due to physical operation limitations units can have prohibited operating regions and generators that operate in these zones may experience
amplification of vibrations in their shaft bearing, which should be prevented in practical applications [6]. Also due to the change restriction in the unit generation output, the units in the actual operation can have ramp rate limits [7].So, ramp rate limits, prohibited operating zones (POZs) and valve loading effects should be considered to solve a realistic ED problem, which makes the finding of the optimum solution extremely hard. Several deterministic optimization techniques were proposed to solve the ED problem The ED problem is a non-convex and nonlinear optimization problem. Due to complex and nonlinear characteristics, it is hard to solve the problem using classical optimization methods. Most of classical optimization techniques such as lambda iteration method, gradient method, Newton‟s method, linear programming, Interior point method and dynamic programming have been used to solve the basic economic dispatch problem [1]. These mathematical methods require incremental or marginal fuel cost curves which should be monotonically increasing to find global optimal solution. In reality, however, the input- output characteristics of generating units are non-convex due to valve-point loadings and multi-fuel effects, etc. Also there are various practical limitations in operation and control such as ramp rate limits and prohibited operating zones, etc. Therefore, the practical ED problem is represented as a non-convex optimization problem with equality and inequality constraints, which cannot be solved by the traditional mathematical methods. Dynamic programming (DP) method [2] can solve such types of problems, but it suffers from so-called the curse of dimensionality. Over the past few decades, as an alternative to the conventional mathematical approaches, many salient methods have been developed for ED problem such as genetic algorithm (GA) , improved tabu search (TS) , simulated annealing (SA), neural network (NN) ,
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evolutionary programming (EP) [34], biogeography-based optimization (BBO) [10],particle swarm optimization (PSO) [13]-[16], differential evolution (DE) [17], and gravitational search algorithm (GSA) [18]. Recently, Kennedy and Eberhart suggested a particle swarm optimization (PSO) based on the analogy of swarm of bird and school of fish. In PSO, each individual makes its decision based on its own experience together with other individual‟s experiences [11].
The individual particles are drawn stochastically towards the position of present velocity of each individual, their own previous best performance, and the best previous performance of their neighbours. It was developed through simulation of a simplified social system, and has been found to berobust in solving continuous non-linear optimization problems [12]. The main advantages of the PSO algorithm are summarized as: simple concept, easy implementation, and computational efficiency when compared with mathematical algorithm and other heuristic optimization techniques. Most of classical optimization techniques such as lambda iteration method, gradient method, Newton‟s method, linear programming, Interior point method and dynamic programming have been used to solve the basic economic dispatch problem [1]. These mathematical methods require incremental or marginal fuel cost curves which should be monotonically increasing to find global optimal solution. In reality, however, the input-output characteristics of generating units are non-convex due to valve-point loadings and multi-fuel effects, etc. Also there are various practical limitations in operation and control such as ramp rate limits and prohibited operating zones, etc.
Therefore, the practical ED problem is represented as a non- convex optimization problem with equality and inequality constraints, which cannot be solved by the traditional mathematical methods. Dynamic programming (DP) method [2] can solve such types of problems, but it suffers from so- called the curse of dimensionality. Over the past few decades, as an alternative to the conventional mathematical approaches, many salient methods have been developed for ED problem such as genetic algorithm (GA) [3], improved tabu search (TS) [4], simulatedannealing (SA) [5], neural network (NN) [6], evolutionary programming (EP) [15],[10],particle swarm optimization (PSO) [35], differential evolution (DE) [17], and gravitational search algorithm (GSA) [18]. Recently, Kennedy and Eberhart suggested a particle swarm optimization (PSO) based on the analogy of swarm of bird and school of fish. In PSO, each individual makes its decision based on its own experience together with other individual‟s experiences [11].
The individual particles are drawn stochastically towards the position of present velocity of each individual, their own previous best performance, and the best previous performance of their neighbours. It was developed through simulation of a simplified social system, and has been found
to be robust in solving continuous non-linear optimization problems [12]. The main advantages of the PSO algorithm are summarized as: simple concept, easy implementation, and computational efficiency when compared with mathematical algorithm and other heuristic optimization techniques. In this paper, a novel approach is proposed to solve the non-convex ED problems such as power loss, ramp rate limits, prohibited operating zones and valve point effect using an efficient PSO (EPSO) technique. The application of Gaussian and Cauchy probability distributions into the PSO is a useful strategy to ensure convergence of the particle swarm algorithm.
II. PROBLEM FORMULATIONS FOR ED PROBLEM
The objective of ED problem is to minimize the total generation cost of thermal generating units, while satisfying various system constraints, including power balance equation, generator power limits, prohibited operating zones and ramp rate limit constraints (19).
The problem of ED is multimodal, non- differentiable and highly nonlinear. Mathematically, the problem can be stated as in (1)
N
T i Gi
i 1
Min F F (P )
(1)
Where
i 1, 2, 3,..., N
Where FT is the total fuel cost, N is the number of generating units in the system. Fi (PGi) is the fuel cost function of unit i and PGi is the output power of unit i.
Generally, the fuel cost of generating unit can be expressed as
2
i Gi i Gi i Gi i
F (P ) a P b P c ($ / hr)
(2)Where ai, bi and ci are the cost coefficients of the unit i subjected to the following constraints.
Real power balance equation: The total generation must be equal to total demand plus transmission losses
N
Gi D L
i 1
P P P
(3)Where PD is real power demand and PL is the transmission loss.
The transmission loss (PL) can be expressed in a quadratic function of generation (Using B-loss coefficient matrix).
N N N
L Gi ij Gj 0i Gi 00
i 1 j 1 i 1
P P B P B P B
(4)Where PGi and PGj are the power generation of ith and jth units and Bij, Boi , B00 are the B – loss coefficients.
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Generator operating limits: The power output of each unit i is restricted by its maximum and minimum limits of real power generation and is given by
Gi min Gi Gi max
P P P
(5)Where PGi min and PGi max are the minimum and maximum generation limits on ith unit respectively.
Prohibited operating zones constraint: The generators may have the certain range where operation is restricted due to the physical limitation of steam valve, component, and vibration in shaft bearing etc,. The consideration of Prohibited Operating Zones (POZ) creates a discontinuity in fuel cost curve and converts the constraint as below
i
l
Gi min Gi Gi,1
u l
Gi Gi,k 1 Gi Gi,k
u
Gi,z Gi Gi max
P P P
P P P P
P P P
(6) k = 2,3,….zi and i=1,2,…...N
Where, PlGi,k and PuGi,k are the lower and upper boundary of kth prohibited operating zone of unit i, k is the index of the prohibited operating zone, and zi is the number of prohibited operating zones is shown in Figure 1.
Figure 1 Cost function with prohibited operating zones
Ramp rate limits constraint: The generator constraints due to ramp rate limits of generating units are given as
(i) As generation increases
P
Gi(t) P
Gi(t 1) UR
i (7)(ii) As generation decreases
Gi(t 1) Gi(t) i
P
P DR
(8)Therefore the generator power limit constraints can be modified as
Gi min Gi(t 1) i Gi(t) Gi max Gi(t 1) i
Max(P , P
DR ) P Min(P , P
UR )
(9)
From equation (9), the limits of minimum and maximum output powers of generating units are modified as
Gi min Gi min Gi(t 1) i
P Max (P , P
DR )
(10)Gi max Gi max Gi(t 1) i
P Min (P , P
UR )
(11)Where PGi(t) is the output power of generating unit i in the time interval (t), PGi(t-1), is the output power of generating unit i in the previous time interval (t-1), URi is the up ramp limit of generating unit i and DRi is the down ramp limit of generating unit i. The ramp rate limits of the generating units with all possible cases are shown in Figure 2.
Figure 2 Ramp-rate limits of generating units
III. Recent Research Works: A Brief Review Classical Optimization Techniques: Diverse classical optimization techniques were used to solve the ED problems. In the conventional ED problem, the cost function for each generator was represented by a single quadratic function and was solved using a lot of mathematical programming based optimization techniques.
Lambda Iteration and Lagrange Relaxation Methods: A Lambda iteration method propagated by Wood &
Wollenberg (2) was found useful in solving the ED problem.
Since the lambda iteration method requires a consistent problem formulation, it cannot be openly applied to ED problems with discontinuous prohibited operating zones. In the units with prohibited operating zones, the zones divided the operating region between the minimum and maximum limits into sub-regions to form non-convex sub-regions and the associated ED problem were thus non-convex optimization problems. Thus, it was found that the conventional Lagrange Relaxation approach cannot be directly constructive.
Dynamic Programming and Quadratic Programming Methods: Ringlee et al (53) projected a DP method to resolve the non-convex ED problem, but had disadvantage, namely, that the computational necessities of the DP based method depended on the size of the distinctive capacity step (10 MW, 20 MW). With a capacity step of 1 MW, which is the common accuracy required in the ED schedule, a number of states at each stage
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Balamurugan & Subramanian (26) suggested an Improved Dynamic Programming (IDP) approach to explain the economic power dispatch problem considering the ramp-rate limits, prohibited operating zones and generating capacity limit constraints. In this approach, the transmission losses are augmented with the objective function using price factor and the method is validated on 6, 15 and 40 unit sample systems. Then the proposed method results get compare with the results of the GA and PSO methods reported in the literature concerned. Test results show that the IDP approach can acquire better quality solutions with enhanced performance.
Reid & Hasdorff (55) proposed the Quadratic Programming (QP) method for solving the economic dispatch problem considering both equality and inequality constraints. In this method, the QP method does not demand the use of penalty factors or the determination of gradient step size which can cause convergence difficulties. The effectiveness of the algorithm was validated on 5, 14, 30, 57 and 118 bus test systems.
Linear Programming and Interior Point Methods:
Somuah & Khunaiz (57) presented the Linear Programming (LP) method to solve the economic dispatch problem taking into consideration of spinning reserve and other security constraints. In this paper, two methods are hand-me-down in the solution of the problem. The first method is a QP technique combined with a LP re-dispatches technique. The second method uses a LP formulation of the dynamic dispatch problem based on the Dantzig-Wolfe was tested on four different test systems and the results are compared with other methods.
Irisarri et al (41) reported of an Interior Point (IP) method to solve the economic dispatch problem considering generator limits, reserve constraints, line flow limits and ramp-rate limit constraints. The proposed method was tested on the IEEE-30 bus network over 168 hours and the convenience of the structured interior point formulation was renowned.
Evolutionary Computing Methods: Evolutionary computing techniques are a recent class of algorithms to solve nonlinear optimization problems. Methods such as the genetic algorithm, particle swarm optimization, evolutionary programming, etc. are used by many researchers to solve ED problems.
Particle Swarm Optimization: Zwe-Lee Gaing (60) suggested a Particle Swarm Optimization (PSO) algorithm for solving the economic dispatch problem considering ramp -rate limits, prohibited operating zones, power balance equation and generator limits constraints. The schedule of the proposed PSO method was tested on three different test
systems and the result obtained was compared with the Genetic Algorithm (GA) in terms of the solution quality and computation efficiency. The experimental result shows that the proposed PSO method was better than GA for solving ED problems.
Mohammadi-Ivatloo et al (50) presented Iteration PSO with Time Varying Acceleration Coefficients (IPSO-TVAC) for solving non-convex economic dispatch problem taking into consideration the valve-point effects and prohibited operating zones. The proposed IPSO-TVAC method was validated on standard three different test systems and the results were compared with other optimization algorithms which are presented in the references.
Park et al (53) reported the concept of PSO to an economic dispatch problem with non smooth cost functions. The recommended method provides high probability solutions for a 3 unit test system and quasi- optimums for a 40 unit test system. The proposed methodology was shown to be superior compared with conventional numerical methods, Artificial Neural Network (ANN) and EP methods
Evolutionary Programming: Hong-Tzer et al (40) recommended the EP technique to solve economic dispatch problems considering non-smooth fuel cost functions. The recommended method was demonstrated on two example power systems and the results were compared with dynamic programming, simulated annealing and GA methods.
Nidul Sinha et al solved the ED problem using various EP techniques considering power balance equation, generator limits and valve point loading effects. The main objective of this paper was to expand and study the various EP techniques to ELD problem (i) Classical EP with Gaussian mutation (CEP) (ii) Cauchy-mutation-based EP called fast EP (FEP) (iii) mean of Gaussian and Cauchy mutations with scaled cost (MFEP) (iv) Gaussian and Cauchy mutations with scaled cost (IFEP). Then the proposed techniques are tested on 3, 13 and 40 unit test systems and the results were compared in terms of solution quality.
Jayabarathi et al (42) accessible the Evolutionary Programming (EP) method to solve ED problem with ramp-rate limits, prohibited operating zones and spinning reserve constraints. The proposed method was implemented for solving the economic dispatch problems.
The result obtained by this approach was compared with those obtained using traditional methods. The revise results showed that the approach developed was feasible and competent.
Firefly and Cuckoo Search Algorithms: Xin-She Yang et al (59) suggested Firefly Algorithm (FA) for solving non-
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convex economic dispatch problems taking into account the power balance equation, ramp rate limits, prohibited operating zones and generation capacity limit constraints.
The proposed algorithm was tested on 3, 13, 15 and 40 generating unit test systems and the results obtained were compared with other methods which are obtainable in the references.
Adriane B.S. Serapiao (20) solved the Economic Load Dispatch (ELD) problem using Cuckoo Search (CS) algorithm considering power balance equation, generator limits and transmission system losses. The effectiveness of the CS algorithm was tested on 3 and 6 unit test systems and the results were compared with other algorithms presented in the reference in terms of solution quality.
Improved Harmony Search and Water Cycle Algorithms: Leandro dos Santos Coelho & Viviana (47) presented an Improved Harmony Search (HIS) algorithm for solving the ELD problem. The proposed HIS algorithm does not require derivative information, but uses stochastic random search instead of a gradient search. This paper reflected on a new IHS algorithm based on the exponential distribution for solving ED problems is proposed by considering the valve-point loading effects. The proposed IHS algorithm was tested on 13 unit test system and the numerical results show that the proposed method has the fine convergence property.
Mostafa & Hamdi (52) presented a Water Cycle Algorithm (WCA) to solve the economic load dispatch problem considering the valve point effect and ramp rate limit constraints. The algorithm was tested on a system of 3 and 15 units with a load demand of 850 (MW) and 2630 (MW). The results obtained by the proposed method were compared with other methods presented in the references.
Hybrid Evolutionary Techniques: Hybrid techniques integrate two or more optimization techniques in order to unite their strength and conquer one another‟s weakness in solving the optimization problems.
Evolutionary Programming based Levenberg- Marquardt Optimization (EP-LMO): Manoharan &
Kannan (48) projected the Evolutionary Programming based Levenberg-Marquardt Optimization (EP-LMO) technique to solve the economic dispatch problem considering the value- point effects and multiple fuel options. In this proposed method, the EP method was applied as a base level search to find the direction of the optimal global region and LMO method was used as an excellent tuning to determine the optimal solution. This novel EP approach was applied to two benchmark problems and the results were compared with
improved PSO, improved GA, a Modified Hop Field Neural Network and the EP methods.
Hybrid EP and SQP: Pathom et al (54) suggested a hybrid EP and Sequential Quadratic Programming (SQP) method for the solving dynamic economic dispatch with non- smooth fuel cost function. In this proposed approach, the EP was applied as a base level search which could give a good direction to the optional global region and the SQP was used as a fine tuning to determine the optimal solution at the final stage. To reveal the effectiveness of the proposed method, a 10-unit test system was considered and the results obtained were compared with EP and SQP methods.
Hybrid PSO-SQP: Arul doss & Ebenezer Jeyakumar (24) applied the hybrid PSO-SQP method for solving the economic dispatch problem with the value - point effects. In this proposed hybrid method, PSO is the main optimizer and the SQP was used to fine tune every improvement in the solution of the PSO run. The PSO is a derivative free optimization technique which produces results quickly and proves itself fit for solving large-scale complex economic dispatch problems. SQP is a nonlinear programming method which starts from a single searching point and finds a solution using the gradient information. The proposed method was tested on 3, 13, and, 40 unit systems and the results were compared with GA, EP, and other methods.
Hybrid PSO – DSM: Chen & Jan (29) used the hybrid algorithm for the solution of economic dispatch considering valve-point loading effects. The novel hybrid algorithm is a combination of Particle Swarm Optimization (PSO) and Direct Search Method (DSM). In the proposed method, a new inertia weight mechanism was integrated into PSO to advance its search capacity that led to a higher probability of obtaining the global optimal solution. The DSM algorithm was used as a fine tuning to conclude the eventual global optimal solution with a reduced computing time. The proposed hybrid method was validated with standard 3 and 13 unit test systems.
Hybrid GA- PSO - SQP: Alsumait et al (22) suggested a hybrid GA-PS-SQP method to solve power system problems with value-point effects. In this proposed method, GA was hybridized with Pattern Search (PS) and SQP to overcome the drawback of PS and SQP method that required a suitable starting point by the user. In this hybrid method, the GA generates the initial good solution automatically and PS was used with SQP to fine tune the search. For validating the hybrid method, it was applied to 3, 13 and 40 generators and the results were compared to the EP, EP-SQP, PSO, PSO- SQP, PS and DE method
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IV. Conclusion
This paper has tried to review some of the publications in the area of economic dispatch problems.. Non-linear characteristics of the generators such as prohibited operating zones and ramp-rate limits constraints are considered. From this study, it can be concluded that the various approaches has been discussed in the area of Economic Dispatch problems for finding a better solutions for the non linear economic dispatch problems
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