CHAPTAR – 1

Introduction

In a practical power system, the power plants are not located at the same distance from the centre of loads and their fuel costs are different. Also under normal operating conditions, the generating capacity is more than the total load demand and losses. Thus there are many options for scheduling the generation. With large interconnection of electrical networks, the energy crisis in the world and continuous rise in prices, it is very essential to reduce the running charges of electrical energy .i.e. reduce the fuel consumption for meeting a particular demand. In an interconnected power system, the objective is to find the real and reactive power scheduling of each power plant in such a way as to minimize the operating cost. This means that the generators real and reactive powers are allowed to vary within certain limits so as to meet a particular load demand with minimum fuel cost. This is called the optimal power flow (OPF) problem. The OPF is used to optimize the power flow solution of large scale power system. This is done by minimizing selected objective functions. While maintaining an acceptable system performance in terms of generator capability limits and output of the compensating devices. The objective functions, also known as cost functions may present economic costs, system security or other objectives efficient reactive power planning enhances operation as well as system security.

ELD or economic load dispatch is a crucial aspect in any practical power network. Economic load dispatch is the technique whereby the

active power outputs are allocated to generator units in the most cost-effective way in compliance with all constraints of the network. The traditional methods for solving ELD include Lambda-Iterative Technique, Newton-Raphson Method, Gradient method, etc. All these traditional algorithms need the incremental fuel cost curves of the generators to be increasing monotonically or piece-wise linear. But in practice the input-output characteristics of a generator are highly non-linear leading to a challenging non-convex optimisation problem. Methods like artificial intelligence, DP (dynamic programming), GA (genetic algorithms), and PSO (particle swarm optimisation) solve non-convex optimisation problems in an efficient manner and obtain a fast and near global and optimum solution. In this project ELD problem has been solved using Lambda-Iterative technique, GA (Genetic Algorithms) and PSO (Particle Swarm Optimisation) and the results have been compared. All the analyses have been made in MATLAB environment.

CHAPTER – 2

Aim of the project

2.1 Overview

By economic dispatch means, to find the generation of the different units in a plant so that the total fuel cost is minimum and at same time the total demand and losses at any instant must be met by the total generation. The classical optimizations of continuous functions have been considered. Various factors like optimal dispatch, total cost, incremental cost of delivered power, total system losses, loss coefficients and absolute value of the real power mismatch are evaluated for a simple system by hand calculation. The MATLAB programs were developed to solve Economic Load Dispatch Problem of an n-unit Plant through lambda iterative method and Particle Swarm Optimization method.

2.2 Aim

In this project our aim was to find optimal solution to the Economic dispatch including losses and generating limits .There are several methods to solve Economic load dispatch problem. Hence we considered one of the conventional methods i.e. lambda iterative method and one of the Artificial Intelligence methods i.e. Particle Swarm Optimization. Lambda iterative method was done by considering a specific lambda value and co-ordination equations were derived. From this equation we got a solution in which inequality constraints imposed on generation of each plant and equality

condition were satisfied. Particle Swarm Optimization is also used to solve the same problem. In this method various steps involved are Initialization, Evaluation etc. Through all the above process the optimal solution was derived. The results of both the Lambda iterative method and Particle Swarm Optimization method were compared and the best method was identified as Particle Swarm Optimization method

2.3 PROJECT MOTIVATION

Revenue loss is a major issue for any country. Conversion of this loss into utilisation would prove to be a huge benefit to the country. Our society wants secure supply of electricity at minimum cost with the least level of pollution in the environment. Through the years many researchers have put forth their ideas to minimise fuel cost and to reduce pollution. Out of the total generation of electricity in India, thermal power plants contribute around 80-85% of the power generation. In view of this fact, the economic load dispatch problem draws much attention. Substantial reduction in fuel cost could be obtained by the application of modern heuristic optimisation techniques for scheduling of the committed generator units.

CHAPTER– 3

Literature Review

3.1 Introduction

This chapter discuss about literature review that been collected for this project. It is described generally on power flow analysis problems and the solutions, graphical user interface in MATLAB and power system tool box in market.

3.2 Literature Review

Jerom k Delson and S.M Shahidepour[1] describes the power system economics, planning and operation with the help of linear programming applications. In this paper they shows the power system engineering application of linear programming and to indicate the potentials for its future use They use linear programming applications in three areas: Generation scheduling, Loss minimization through allocation of reactive power supply , and planning of capital investment in generation equipment.

Robert Bacon [2] discuss about the power system restructuring in case of a small system. In his paper he discuss about costs and benefits of privatization, vertical separation, horizontal separation in generation.J.M. Vignolo and R,Zeballons[3] determines the equations for short time optimal point of operation of a power system from

two different perspectives. One is optimizing the system from a global point of

view, other is the individual agents’ behaviours which buys and sells electricity at each of the power system bus bars. By comparing those equations, getting from two different perspectives, they calculate the active and reactive energy prices to optimize the system from two different perspectives. Then they determine the marginal price and the nodal factor from the equations analyses it. Finally they study a Uruguayan system.

M.OloomiBuygi, H.m.Shanechi,G.balzer, M shahidehpour[4] classify the transmission expansion planning in three view point:- Power system uncertainty, Power system horizon and power system structure. They also categorize and summarize the non-deterministic transmission planning approaches. Then they discuss the non- deterministic transmission planning issue in deregulated power system. Finally the abstract of the most interesting publication on the subject of transmission planning under deregulation are presented.

Osamah .Alsayegh[5] gives a brief idea about Kuwait’s electric power system, which vertically integrated and organization owned and operated by government. He suggests that power system should be restructured with the help of private sector due to fast growing demand. He recommends that the government should consider the private sector as partner in operating the power system. But there are some challenges against this type of restructuring and some recommendation are proposed for resolving these challenges, in this paper. The generic algorithm of 14-bus system is discussed in reference [9]. In [10] the emission constraints, unit commitment of a thermal power plant and the nature of electricity markets are

highlighted. M.Zarei, A. Roozegar, R. Kazemzadeh, and J.M. Kauffmann describes an efficient and practical method for economic dispatch problem in one and two area electric power systems with

considering the constraint of the tie transmission line capacity constraint, in their work [10]. They are basically use the Direct Search Method for economical dispatch. A Dynamic program is developed in [12] for solving economic load dispatch problem and iit is compared with an improved Hopfield NN approach (IHN), a fuzzy logic controlled generic algorithm (FLCGA), an advanced engineered conditioning genetic approach (AECGA) and an advance Hopfield NN approach (AHNN). In [13], they calculate the B-coefficients of a system and proposed a method to avoid many limitations associated with traditional loss coefficient calculation. The modelling of the power and voltage control loop of a thermal plant of 180 MW is described in [14].J.H.Park, I.K.Eong, Y.S. Kin, and K.Y.Lee [ 15] proposed Hopfield (neural network) method. Hopfield method solved the ELD problem with the cost function represented as a piecewise quadratic function instead of convex function. It is suitable for large number of generators. The advantage of real time response favours application of hardware.

Po-Hung and Hong-Chan Chang [ 16] applied genetic algorithms to solve the economic load dispatch problem. In case of dispatch on a large scale GA solution time increases with the increase in generator units. This algorithm can be used worldwide. Owing to its flexibility it can deal with ramp-rate limits, restricted zones of operation and losses in the network.

Zee-Lee Gaing [ 16] used PSO to solve ELD. It considers the non-linear characteristics of the generators. The feasibility of PSO was checked and it was found to be superior to Genetic

Algorithms. PSO gives high quality solutions, computational efficiency and better characteristics of convergence.

T. A. Albert Victoire, A. E. Jeyakumar [ 17] combined PSO (particle swarm optimisation) and SQP (sequential quadratic programming) to solve the economic load dispatch (ELD) problem. PSO acts as the main optimiser and SQP adjusts the refinement in every solution of the PSO. SQP is a non-linear programming technique used to solve constrained optimisation problem. It showed high efficiency and accuracy. The property of convergence is not strained; it depends on incremental-cost-function. The combination PSO-SQP offers fast convergence characteristics and high quality solutions. This method is more practical as it can be employed in prohibited zone and with the consideration of network losses and valve-point effects.

Authors Yi Da, and G. Xiurun [ 18] proposed SA (simulated annealing) to improve PSO. They introduced the idea of SAPSO-based-ANN. Three-layer feed-forward neural network was employed. It consists of one hidden, one input and one output layer. SAPSO-based ANN proved to be superior to PSO-based ANN. Owing to its flexibility it was employed to solve many other problems.

So in recent time the restructuring and rescheduling of power system is necessary, due fast growing demand. Also the cost of generation is another important factor. So here we discuss about the economic operation of the power system and develop some Mat Lab program to solve economic dispatch problem. Here I have use two different methods (classical and Newton-Raphson) for solving those problem. Then the loss coefficients are also calculated by Mat Lab program. By developing program, I schedule the generators economically of IEEE-14, IEEE-30 and IEEE-57 bus systems, for different loads. Finally try to study the stability of a small thermal power plant.

CHAPTER - 4

Economic L oad Dispatch

4.1 Introduction

The sizes of Electric power system are increasing rapidly to meet the energy requirement. A number of power plants are connected in parallel to the supply system. With the development of integrated power system it becomes necessary to operate the plant units most economically. The economic scheduling of generators aims to guaranty at all times the optimum conditions of generator connected to the load demand. The economic despatch problem involves two separate steps namely the unit commitment and the online economic dispatch.

The Economic Dispatch can be defined as the process of allocating generation levels to the generating units, so that the system load is supplied entirely and most economically. For an interconnected system, it is necessary to minimize the expenses. The economic load dispatch is used to define the production level of each plant, so that the total cost of generation and transmission is minimum for a prescribed schedule of load. The objective of economic load dispatch is to minimise the overall cost of generation.

4.2 OPERATING COST OF GENERATOR

Generators can be categorised as: nuclear, hydro and fossil (coal, oil or gas). Nuclear power plants tend to operate at output levels which are constant. For hydro-stations, the storage of energy is free apparently and so the operating cost does not infer any meaning. So only the cost of fuel burnt in fossil plants contributes to the dispatching procedure. The cost of operation of generator includes fuel, labour and cost of maintenance. Labour cost, costs of supplies and maintenance are not charged since these are a fixed percentage of incoming fuel cost. So only the fuel cost needs to be considered. Figure shows a simple model of thermal power plant.

FIGURE 1: SCHEMATIC DIAGRAM OF A THERMAL PLANT.

The input in a thermal power plant is expressed in Btu/h. Active power output is expressed in MW. In fossil plants the power output is increased sequentially by opening the valves at the inlet of steam-turbine. When a valve is just open, the throttling losses are large and when it is fully opened, throttling losses are small. The fuel cost curve is modelled as a quadratic function of real power or active power. The fuel-cost curve as a function of active power takes the form:

Where are cost coefficient for unit, is the total

cost of generation, is the generation of plant.

Output power(MW)

Figure 2: Ideal fuel cost characteristic

The ideal fuel cost curve is a monotonically increasing convex function. But in practice fuel cost curve has many discontinuities owing to extra boilers, steam condensers or other

equipments. and are the maximum and minimum limits on real power generation of the committed units. The incremental fuel cost is expressed by piece-wise linearization in the range of continuity.

4.2.1 Active Power balanced Equation

For power balance, an equality constraint should be satisfied. The total generated power should be the same as total load demand plus the total transmission line loss.

Where is the total load demand and is the total line loss.

Where is the B coefficient constraints.

4.2.2 SYSTEM CONSTRAINTS

Two types of constraints exist:

1. Equality Constraints2. Inequality Constraints

4.2.2.1 EQUALITY CONSTRAINTS

Equality constraints are the basic power balance equations in compliance with the fact that total generation equals total demand minus transmission losses.

4.2.2.2 INEQUALITY CONSTRAINTS

GENERATOR CONSTRAINTS

The kVA loading of the generator must not exceed a preset thermal limit. The maximum real power generation is restricted by the thermal constraint so that rise in temperature remains within limits.

The maximum value of reactive power generation is restricted by overheating of the rotor and the minimum limit is due to the machine’s stability limit. So the generator reactive power should be

within the range as stated by the inequality constraint:

CONSTRAINTS ON VOLTAGE

The values of voltage magnitude and phase angle at different nodes must be within a specific range. The power angle of transmission must lie between 30 degrees and 45 degrees to comply with transient stability. Higher the power angle, lower is the stability in case of faults. The lower limit of operating angle assures optimum use of transmission capability that is available.

4.3 Economic Dispatch Problem Formulation Without Regarding Loss

Consider a lossless system consisting of NG thermal generating units connected to a single bus bar serving electrical load with the aim to minimize total power generating cost.

Subject to the simplified power balance constraint,

Or

Where is load demand and NG is the number of generating units. Thus Lagrange function can be given as

The optimality condition is

The necessary condition for the existence of a minimum cost operating point of a thermal power system is that the incremental cost of all units are equal to some undetermined value, lambda(λ), known as lagrangian multiplier.

In addition, the power output of each unit must not exceed its operating limit. That is :

Whereby and are generator minimum and maximum

operating limit. With respect to the generators operating limits,

the necessary condition is slightly expended to:

4.4 Economic Dispatch Considering Transmission losses

Considering that all thermal power generating systems are

connected to an equivalent load bus through a transmission

network, the economic dispatching problem becomes slightly

more complicated than discuss in the calculation neglecting the

losses. With respect to the network losses, the objective

function is the same with the following load balance constraint

equation.

Where is the total real power loss.

The Lagrange function can be set up

The Optimality condition is,

Where is the penalty factor.

CHAPTER – 5

Particle Swarm Optimization

5.1 Introduction

Electrical power systems are designed and operated to meet the continuous variation of power demand. In power system minimizing, the operation cost is very important. Economic Load Dispatch (ELD) is a method to schedule the power generator outputs with respect to the load demands, and to operate the power system most economically, or in other words, we can say that main objective of economic load dispatch is to allocate the optimal power generation from different units at the lowest cost possible while meeting all system constraints Over the years, many efforts have been made to solve the ELD problem, incorporating different kinds of constraints or multiple objectives through various mathematical programming and optimization techniques. The conventional methods include Newton-Raphson method, Lambda Iteration method, Base Point and Participation Factor method, Gradient method, etc . However, these classical dispatch algorithms require the incremental cost curves to be monotonically increasing or piece-wise linear . The input/output characteristics of modern units are inherently highly nonlinear (with valve-point effect, rate limits etc) and having multiple local minimum points in the

cost function. Their characteristics are approximated to meet the requirements of classical dispatch algorithms leading to suboptimal solutions and therefore, resulting in huge revenue loss over the time. Consideration of highly nonlinear

characteristics of the units requires highly robust algorithms to avoid getting stuck at local optima . The classical calculus based techniques fail in solving these types of problems. In this respect, stochastic search algorithms like

genetic algorithm (GA) , evolutionary strategy (ES) , evolutionary programming (EP) , particle swarm optimization (PSO) and simulated annealing (SA) may prove to be very efficient in solving highly nonlinear ELD problem without any restrictions on the shape of the cost curves. Although these heuristic methods do not always guarantee the global optimal solution, they generally provide a fast and reasonable solution (sub optimal or near global optimal).

5.2 Features of Particle Swarm Optimization Method

In 1995, Kennedy and Eberhart developed a PSO based on the analogy of a swarm of birds and a school of fish . The PSO is a mathematical modelling and simulation of the activities of a swarm of birds while searching for food. In the multidimensional space where the optimal solution is sought, each particle in the swarm is moved toward the optimal point by adding a velocity with its position. The velocity of a particle is influenced by 3 components, namely inertial, cognitive, and social components. The inertial component simulates the inertial behaviour of the bird to fly in a previous direction. The cognitive component models the memory of the bird about its

previous best position, and the social component models the memory of the bird about the best position among the particles. The particles move around the multidimensional search space until they find the food (optimal solution).

5.3 Method used to solve particle Swarm optimization

Particle Swarm optimization method has been used many times for optimization tasks including solving economic load dispatch for a

power system. This stochastic and iteration base method, by following a defined algorithm tries to lead particles and finally the swarm to obtain an optimum region and after that to obtain best point, in the search space. A completely randomized particles is selected where their velocity and location in the next iterations will be updated, based on the existing formulas. After a limited number of iterations, the particles will obtain the best solution for the problem and all particles will converge to the best point. This method does not need any information about the derivative of function and has a higher flexibility to solve any constraint optimization problem and thus can be used to optimize a wide range of functions with various constraints. This method is compared considered superior to other classical methods, since it does not need to solve complex mathematical formulas in finding the best solution for the problem. Various modifications have been applied to increase the efficiency and performance of this method in recent years. Based on the above discussion, the mathematical model for PSO is as follows.

In a PSO system particles fly around in a multi dimensional search space. During flight each particle adjust its position according to its own experience and experience of neighbouring particle, making use of best position encounter by itself and its

neighbour. The swarm direction of a particle is guided by the set of neighbouring particles and its historical experience.

In the hybrid PSO – ED algorithm the set of particles is represented as follows:

Where is a matrix representing set of individual search. More specifically, in this case, it is a set of real power

generators. The sub matrix is the set of the current position of particle j representing the real power generation of the

generator connected to bus i( ). Each particle is used to solve quadratic programming optimal power flow(POPF) and

the best previous position of the particle is recorded and represented as:

The index of the best particle among all the particles in the

group is represented by . The velocity rate of particle j is represented as:

The modified velocity and position of each Particle can be calculated using current velocity and the distance from the pbest to gbest as shown in the following formulas.

The velocity update equation is given by:

Where is the velocity of individual i at iteration k, w is the

weight parameter, and are the weight factors, and are random numbers between 0 and 1, the position of

individual i at iteration k, the best position of individual i until iteration k, and is the best position of group i until iteration k.

Each individual moves from the current position to the next one by the modified velocity equation by using the following equation:

Where,

M=number of particle in a group,

NG=number of members in a particle,

K=pointer of iterations (generations),

=velocity of particle j corresponding to at iteration k,

=current position of particle j corresponding to at

iteration k.

The inertia weight factor linearly declines from its maximum to

the minimum value as the no of iterations increases from 0 to

.

The inertia weight factor at iteration k is updated as follow:

Where and are maximum and minimum weight factor,

respectively and is the maximum number of iterations.

PROPOSED METHODOLOGY

Where,

is the maximum generating cost among all individuals in

the initial population, is the minimum generating cost among all individuals in the initial population,

BG is the number of buses with generation.

To limit the evaluation value of each population within a feasible limit ,before simulation ,the generated power output must satisfy all constraints. If one individual satisfies all constraints, it is a visible individual and has a small value. Otherwise the value of individual is penalized and a very large positive constants.

5.4 Comparison of PSO and GA

The algorithm of PSO offers a variety of benefits like easy concept, simple execution, robustness to parameters of control and efficiency in computation.

Genetic algorithms have been applied quite efficiently to solve hard problems of optimisation. But research in recent times has revealed a number of drawbacks of GA. The breakdown in efficiency of GA is prominent in applications where objective functions are highly epistatic i.e. the parameters to be optimised are correlated highly. The genetic operators of crossover and mutation are unable to ensure

better fitness or competitiveness of offspring owing to the fact that chromosomes that are present in the population have nearly close structures and the fitness on an average becomes high as the evolution cycle approaches the end. Apart from these, premature convergence in case of GA deteriorates its efficiency and decreases search capability which leads to greater probability of achieving local optimum.

PSO offers high quality solutions, limited time of computation and stable convergence.

5.5 Drawback of PSO

PSO has got the disadvantage of getting caught up in local minimum while it handles heavy constraints in problems of optimisation due to restricted local or global search capabilities.

5.6 Steps of implementation of PSO in ELD

The PSO algorithm is utilised to determine the optimum allocation of power among committed units to minimise the total generation cost. In PSO each particle is a potential solution for the ELD (economic load dispatch) problem. The particles

are generated for each generating unit considering all the constraints.

The sequence of steps applied to solve the ELD problem using PSO is as follows.

1. The fitness function i.e. the reciprocal of the cost of generation is initialised.

2. The parameters of PSO i.e. c1, c2, population size, ,

, error gradient, etc. are initialised.

3. Input data is fed, which includes cost functions, MW limits of generators, B-coefficient matrix and load-demand.

4.At the beginning of execution of the algorithm a large number of active power vectors which satisfy MW limits of generators are allocated at random.

5. The value of fitness function for each vector of active power is evaluated. The values which are obtained in a single iterative step are compared so as to decide pbest. All the fitness function 6. values for the whole population are compared which decides the gbest. These pbest and gbest values are updated at each iterative step.

7.In each iteration the error gradient is checked and gbest is plotted till it comes within the pre-specified range.

8.The gbest value so obtained is the minimum cost. Active power vector determines the optimum ELD (economic load dispatch) solution.

5.7 Flow chart of PSO

CHAPTAR – 6

6.1 Case Study

For the three generator system, the fuel cost coefficients and the operating generator limits are given in following table. The B-coefficients for transmission loss are given in another table. Determine the economic schedule for loads 160 MW and 210 MW.

Table for fuel cost coefficient and operating generator limits

Generator ai(Rs/MW2h) bi(Rs/MWh) ci(Rs/h)

1. 0.006085 10.04025 136.9125 5 150

2. 0.005915 9.760576 59.1550 15 100

3. 0.005250 8.662500 328.1250 50 250

Table for B-coefficient in

0.0001363 0.0000175 0.0001839

0.0000175 0.0001545 0.0002828

0.0001839 0.0002828 0.0016147

6.2 Results

6.2.1 ELD using Lambda Iteration (without loss)

PD(MW)

160210

6.2.2 ELD using PSO (without loss)

PD(MW)

160

210

6.2.3 ELD using Lambda Iteration (with loss)

PD(MW)

160

210

6.2.4 ELD using PSO (with loss)

PD(MW)

160

210

6.2.5 Result Discussion

CHAPTER – 7

Conclusion:-

PSO is a population-based evolutionary technique that has many advantages over other optimization techniques. PSO is a derivative-free algorithm unlike many conventional techniques and is less sensitive to the nature of the objective function. Viz... .convexity or continuity. These swarm intelligence based methods have few parameters to adjust and escape local minima. The proposed method is easy to implement and program with basic mathematical and logical operations. It can also handle objective functions with stochastic nature and does not require a good initial solution to start its iteration process.

Now a day the demand of power all over the world grows very rapidly. So it is important to restructure the power system to fulfil this demand. To make such type of restructured power system, government should have to take the help of private sector. Economic operation of power system is also needed to minimize the cost of generation. Due to the privatization, there is a huge competition in the market, to supply the power in nominal cost. For this reason economic operation is very much effective. Here we solve some economic dispatch problem by IFC method. It is seen that this method takes less number of step to optimize the system. So, this method is faster, as well as efficient than the Classical method.

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