Excerpt

## TABLE OF CONTENTS

**Chapter One**

1.1 Introduction

1.2 Literature survey

1.3 Methodology in brief

1.4 Organization of the Report

**Chapter Two**

2.1 Introduction to Economic Dispatch

2.1.1 Generator operating cost:

2.2 Mathematical Analysis

2.2.1 Analytical method

2.2.2 Gradient method

2.3 Valve Point Loading

2.4 Problem Formulation

**Chapter Three**

3.1 Evolutionary Algorithm

3.2. Ant Colony Optimization

3.3 Particle Swarm Optimization

3.4 Over view of Particle Swarm Optimization

3.5 Implementation of PSO method in ED

3.5.1 Advantages of PSO

**Chapter Four**

4.1 Introduction to various PSO techniques

4.2 Adaptive Particle Swarm Optimization

4.2.1 The procedure of Adaptive PSO

4.3 Chaotic Particle Swarm Optimization

4.3.1 CPSO methods for EP

4.4 New Particle Swarm Optimization

**Chapter Five**

5.1 Introduction

5.1.1 Organization of the result

5.2 The Test Bus System in Detail

5.3 Results obtained by using the PSO

5.3.1 Parameters

5.3.2 Overall Report

5.4 Results obtained by using the APSO

5.4.1 Parameters

5.4.2 Overall Report

5.5 Results obtained by using the CPSO

5.5.1 Parameters

5.5.2 Overall Report

5.6 Results obtained by using the NPSO

5.6.1 Parameters

5.6.2 Overall Report

5.7 Analysis of four PSO techniques

5.8 Comparison of graphs

**Chapter Six**

6.1 Analysis of different pso techniques

6.2 Conclusion

List of Tables

5.2 Input data for IEEE 13 generator system

5.3 Result obtained by using PSO algorithm

5.4 Results obtained by using the APSO

5.5 Results obtained by using the CPSO

5.6 Results obtained by using the NPSO

5.7 Analysis of four PSO techniques

List of Figures

5.3. A Graph of G_{best}versus iterations for best solutions of PSO

5.4. B Graph of G_{best}versus iterations for best solutions of APSO

5.5. C Graph of G_{best}versus iterations for best solutions of CPSO

5.6. D Graph of G_{best}versus iterations for best solutions of NPSO

5.8 Comparison of four optimization techniques graphs

## Chapter one

### 1.1 Introduction

The main purpose of Economic Load Dispatch is to minimize the total generation cost of the plant by considering the generator limits. In power generation fuel cost plays major role. Factors which influence power generation at minimum cost are operating efficiencies of generator, fuel cost and transmission losses.

Efficient generator in the system does not generate minimum cost as it may be located in an area where fuel cost is high. If the plant is located far from the load centre, transmission losses may be higher and the plant may be uneconomical.

The main aim is to identify the generation of different plants, such that the total operating cost is minimum. The major component of generator operating cost is the fuel input/hour and the maintenance cost contributes very less.

Total operating cost includes the fuel cost, cost of labour, maintenance. These costs are assumed to be a fixed percentage of the fuel cost. After neglecting the transmission losses in economic load dispatch we are considering only the generator units but not as the system.

We are neglecting the transmission line losses, line impedance etc., for analysis system is having only one bus with all generations and load are connected. As there are no transmission losses, the total load demand (Pd) is the sum of all generations.

### 1.3 Methodology in Brief

The coding of the algorithms was done on MATLAB 6.5, and the test system is the IEEE 13 generator system. Each algorithm was run for specified number of iterations and the best value obtained was recorded, along with the graph for the average and minimum value against the number of iterations.

The time of execution for all four algorithms were measured and recorded. Each algorithm was executed ten times and the best and the worst value were found, the graph for these executions were plotted.

The optimization techniques used are PSO, CPSO, NPSO and APSO, a random population is initialized and the fitness value of each is calculated. This population is sent through a selection process where the probability of the member of the population being selected into the matting population is directly proportional to the previously measured fitness.

Velocity limits of the generators are initialized and it has been carried out by initializing the generating velocities, besides those iterations are started and gbest values are found out by continuously updating the population and also finding the fitness of the present population.

### 1.4 Organization of the Report

Chapter Two contains the basic theory on Economic Load Dispatch along with some elementary mathematical background. The various available algorithms are discussed in Chapter Three, Chapter Four explains in detail the algorithm used, Chapter Five and Six are result and conclusion.

## Chapter two

### 2.1 Introduction to Economic Load Dispatch

Scarcity of energy resources, increasing power generation cost and ever-growing demand for electric energy necessitates optimal economic dispatch in today’s power systems. Optimal system operation involves consideration of economy of operation, system security, emission at fossil fuel plants, and optimal release of water at hydro generation.

Economic dispatch problem is to minimize the total cost of generating real power (production cost) at various stations while satisfying the loads and losses in the transmission lines. In load flow problems, two variables are specified at each bus and solutions is obtained for the variables.

In a practical power system, power plants are not loaded at the same distance from the center of loads and their fuel cost is different. The generation capacity is more than the demand and losses. So there is a need to schedule the generation. In an interconnected power system, the objective is to find the real and reactive power scheduling of each power plant in such a way to minimize the operation cost. The generators real and reactive powers are allowed to vary within certain limits to meet a particular load demand with minimum fuel cost.

Electrical energy can not be stored, but is generated from natural sources and delivered as demand arises. A transmission system is used for the delivery of bulk power over considerable distance and a distribution system is used for local deliveries. An interconnected power system consists of mainly three parts :

1. The generator, which produce electrical energy

2. The transmission line which transmit it to far away places

3. The load which use it

Such a configuration applies to all inter connected networks, where the number elements may vary. The transmission networks are interconnected through ties so that utilities can exchange power, share reserves and render assistance to one another in times of need. Since the sources of energy are so diverse , the choice of one or the other is made on economic, technical or

geographic basic. As there are few facilities to store electric energy, the net production of utility must clearly track its total load for an inter connected system, the fundamental problem is one of minimizing the source expenses. The economic dispatch problem is to define the production level of each plant so that the total cost of generation and transmission for a prescribed shecdule of loads.

-Forecasting includes determining the peak rate of supply i.e, energy demand for both long-term investment decisions and short-term operating decisions.

-Operating applications include allocation of out put, unit start-up selection, hydro thermal co ordinations and maintenance scheduling.

-The investment planning applications cover the generation and transmission system.

#### 2.1.1 Generator operating cost:

Factors which influence power generation at minimum cost are operating efficiencies of generator, fuel cost and transmission losses.

Efficient generator in the system does not generate minimum cost as it may be located in an area where fuel cost is high. If the plant is located far from the load centre, transmission losses may be higher and the plant may be uneconomical.

The main aim is to identify the generation of different plants, such that the total operating cost is minimum. The major component of generator operating cost is the fuel input/hour and the maintenance cost contributes very less.

### 2.2 Mathematical Analysis

He we are considering only the generating units, but not as the system. We are neglecting the transmission line losses, line impedance etc. for analysis, the system is having only one bus with all generations and load are connected.

As there is no transmission losses, the total demand P_{D}is the sum of all generation. For each plant assume the cost function FC_{i}

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Subject to the constraint

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The power output of any generator should not exceed the its rating nor should it be below that necessary for stable turbine operation thus, the generations are restricted to lie within given minimum and maximum limits. The problem is to find the real power generation for each plant such that the objective function as defined by (2) is minimum, subject to the constrain given by (3) and the inequality constraints given by

where *i* = 1,2,3,N_{G}

- minimum and maximum generating limits

FCtotal – total production cost

FCi – production cost of ith plant

Pi – power generation of ith plant

PD – total load demand

NG – total number of generating units

Using Lagrange Multipliers

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The minimum value will be obtained at the poing where the partials of the function to its variables are zero

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There fore optimal dispatch condition is ,

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When losses are neglected, for most economic operation all plants must operate at equal incremental production cost.

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This is the coordination equation which is a function of

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The value of has to be substituted in

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to obtain the optimal scheduling of generation

To get the economical values of Pi, it has to undergo iterative process. Using gradient method, we get the solutions quickly

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Expanding the left hand side in Taylors Series above an operating point

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Economic load dispatch problems can be solved theoretically by the following two methods they are as follows:

1. Analytical method

2. Gradient method

#### 2.2.1 Analytical method:

In this method the λ is determined by solving the given parameters

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α , β , γ – Cost coefficients

i – Index of the generator

P_{D}– total load demand

N_{G}– total number of generating units

#### 2.2.2 Gradient method:

In this method λ value is assumed (λ= 0 to 1)

**[...]**

- Quote paper
- Vikramarajan Jambulingam (Author), 2008, Economic dispatch with valve point effect using various PSO techniques, Munich, GRIN Verlag, https://www.grin.com/document/270063

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