11-04-2011, 11:52 AM
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ABSTRACT
Economic load dispatch is the process of allocating the required load demand between the available generation in power system in naturally predominant in determining allocation of generation to each generation for various systems load levels.
Economic load dispatch (ELD) is an important function in power system planning and operation.ELD solutions are found by solving the conventional load flow equations while at the same minimizing fuel costs.
This project aims to find ELD for small systems. There have been many algorithms proposed for economic dispatch such as separable convex linear programming, Genetic algorithm and reduced gradient with linear constraints, steepest ascent Descent Gradient and efficient method etc.
The main aim of this project is an application of ELD to power systems using based on genetic algorithm. The developed algorithm is tested with small systems.
The model is developed by using MATLAB 7.0 package and results are shown.
INTRODUCTION
For any practical systems generators are to be operated in order to meet the load, But it is not economical to run all the units available at all the time .To determine the units of a plant that should operate for a particular load is the problem of unit commitment (UC) which is assumed to be known. Else the optimum allocation of generators at each generating station status load levels including the load sharing among committed generators is first problem of power system.
For determining the most efficient, low cost and reliable operation of a power system by dispatching the available electricity generations resources to supply the load on the system, we go for Economic Load Dispatch. The primary objective of Economic dispatch is to minimize the total cost of generation while honoring the operational constraints of the available generation’s resource. The operation of generation facilities to produce energy at the lowest cost to reliably serve consumers, recognizing any operational limits of generation and transmission facilities.
Planning and operation of power systems under existing conditions, its improvement and also future expansion requires the load flow studies, short circuit studies and stability studies. However the load flow studies are very important for planning ,control and operations of existing systems as well as planning, control and operations of existing systems as well as planning its future expansion as the satisfactory operation of the system depends upon knowing the effects of inter connection ,new loads, new generating stations or new transmission lines etc .before they are installed.
Load flow calculations can be made by using admittance matrix or bus impedance matrix. However, it is simpler to use the bus admittance matrix in these studies since it can be generated by inspection .There are many methods used in load flow studies. Load flow studies are the first step in solving Economic dispatch problem and optimum power flow.
Newton Raphson method have been implemented for solving load flow studies for a twenty six bus system and that results are utilized as input for Genetic Algorithm which solves Economic load dispatch problem. It linearizes the non linear objective function. It minimizes the fuel cost .For that many methods have been implemented to solve the nonlinear objective function. In this project, We presented Genetic Algorithm as one of the best method for solving. This method involves binary conversion, Sorting and many more .For that we required to know some terms involved in Genetic Algorithm
Genetic Algorithm has implemented in MATLAB 7.0 package and the results of the twenty six bus system are shown in this project.
Economic Load Dispatch
2.2.1Economic load dispatch:
Scarcity of energy resources, increasing power generation costs and ever growing demand for energy necessitate optimal economic dispatch in modern power systems. The main objective of economic dispatch is to reduce the total power generation cost while satisfying various equality and inequality constraints. Tradionally≤, in economic dispatch problems, the cost function for generating units has been approximated as a quadratic function.
The economic load scheduling of a power systems is perhaps the most exciting branch for the power system engineering .Of course this topics not very important in the beginning when there were small power generations for each locality such as urban power systems. But now, with the continuity of supply to the consumers to the consumers under normal conditions, have forced the power system engineers to develop grid systemize. Interconnections of different generating stations located at different places .For each such systems, the optimum scheduling of the different generating plants in the system has become increasingly important.
A wide variety of optimization techniques have been applied to solving Economic Load Dispatch (ELD) . Some of these techniques are based on classical optimization methods, such as linear programming or quadratic programming to solve ELD problems.
Such classical optimization methods are highly sensitive to starting points and Often converge to local optimum or diverge altogeather.Linear programming methods are fast and reliable but have a disadvantage associated with the piecewise linear cost approximation . Non linear programming methods have known problems of convergence and algorithm complexity.Newton based algorithms have difficulty with handling a large number of inequality constraints .Methods based on artificial intelligence techniques ,such as artificial neural networks ,have also been applied successful. Lately, many heuristic search techniques ,such as particle swam optimization ,have been considered in the context of ELD problems
By economic load scheduling we mean to determine the generations of different plants such that total operating cost is minimum and at the same time the total demand and the losses at any instant is met by the total generation. The operating cost of thermal plants is mainly the cost of fuel. It is given as a function of generation .This cost function is defined as a nonlinear function of plant generation’s .Normally graph is given between the heat value of fuel and power generation and knowing the cost of fuel .We can definitely determine the fuel cost as a function of generations for each thermal plant.
This project aims to find solutions for ELD problems using Genetic Algorithm and is tested on small systems and an attempt is made to find the solution for large practical systems.
2.2 FORMULATION OF ELD
The ELD problem is considered as a general minimization problem with constraints and can be written in the following form :
Minimize f(x) (1)
Subject to: g(x) =0 (2)
h(x) ≤ 0 (3)
f(x) is the objective function, g(x) and h(x) are respectively the set of equality and inequality constraints . x is the vector of control and state variables. The control variables are generator active and reactive power outputs, bus voltages, shunt capacitors/rectors and transformers tap setting .The state variables are CLASSIC ECONOMIC LOAD DISPATCH PROBLEM
The objective of the ELD problem is to minimize the total fuel cost at thermal plants
n
OBJ = ∑ Fi (Pi)
i=1
Subject to the constraint of equality in real power balance
n
∑ Pi – PL – PD = 0
i=1
The inequality constraints of real power limits of the generation outputs are
Pi min < Pi < Pi max
Where
Fi (Pi) is the individual generation production in terms of its real power generation
Pi, Pi the output generation for unit i, n the number of generators in the system
Pd the total current system load demand, and Pl the total system transmission losses.
The thermal plant can be expressed as input-output models (cost function), where the input is the fuel cost and the output the power output of each unit, in practice, the cost function could be represented by a quadratic function.
Fi (Pi) = Ai * Pi2 + Bi * Pi + Ci
The incremental cost curve data are obtained by taking the derivative of the unit input-output equation resulting in the following equation for each generator:
dFi (Pi) / dPi = 2 Ai * Pi + Bi
Transmission losses are a function of the unit generations and are based on the system topology. Solving the ELD equations for a specified system requires an iterative approach since all unit generation allocations are embedded in the equation for each unit. In practice, the loss penalty factors are usually obtained using on line power flow software. This information is updated to ensure accuracy. They can also be calculated directly using the Bmn matrix loss formula.
PL = Pi Bij Pj
Where Bij are coefficients, constants for certain conditions.
voltage and angle of load buses.
OBJECTIVE FUNCTION:
The objective function for the ELD reflects the costs associated with generating power in the system .The quadratic cost is model are used. The objective function for the entire power system can then be written as the sum of the quadratic cost model for each generator:
f(x)=
[$/h]
Where ng is the number of thermal units, Pgi is the active power generation at unit I and ai, bi and ci are the cost coefficients of the ith generator .
EQUALITY CONSTRAINTS:
The equality constraints g(x) of the ELD problem are represented by the power balance constraint ,where the total power generation must cover the total power
demand and the power loss. This implies solving the load flow problem, which has equality constraints on active and reactive power at each bus as follows.
Where i= 1,2,3,……. N and injected active and reactive power at bus I; Pdi,Qdi: active and reactive power demand at the bus I; Vi, : bus voltage magnitude and angle at bus I;Gij,Bij : conductance and suspectance of the (I,j)element in the admittance matrix.
