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CHAPTER-1

INTRODUCTION


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CHAPTER-1

INTRODUCTION

1.1GENERALThe main objective of electric power utilities is to provide high quality reliable

supply to the consumers at the lowest possible cost while operating to meet the limits

and constraint imposed on the generating units. This formulates the well-known

Economic Load Dispatch (ELD) problem for finding the optimal combination of the

output power of all online generating units that minimizes the total fuel cost, while

satisfying all constraints.

The economic load dispatch (ELD) is an important function in modern power

system like unit commitment, Load Forecasting, Security Analysis, Scheduling of fuel

purchase etc. A bibliographical survey on ELD methods reveals that various

numerical optimization techniques have been employed to approach the ELD

problem.

The Optimal Power Flow (OPF) is an important criterion in todays power

system operation and control due to scarcity of energy resources, increasing power

generation cost and ever growing demand for electric energy. As the size of the power

system increases, load may be varying. The generators should share the total demand

plus losses among themselves. The sharing should be based on the fuel cost of the

total generation with respect to some security constraints. Generally, most of the

approaches apply sensitivity analysis and gradient-based optimization algorithms by

linearizing the objective function and system constraints around an operating point.

Unfortunately, the problems of OPF are highly nonlinear and a multi model

optimization problems, i.e. there exist more than one local optimum. Therefore,conventional optimization methods that make use of derivatives and gradients are, in

general, not able to locate or identify the global optimum.

Heuristic algorithms such as genetic algorithms (GA) and evolutionary

programming have been recently proposed for solving the OPF problem. The results

reported were promising and encouraging for further research in this direction.

Unfortunately, recent research has identified some deficiencies in GA performance.

This degradation in efficiency is apparent in applications with highly epistatic

objective function, i.e. where the parameters being optimized are highly correlated. In


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addition, the premature convergence of GA degrades its performance and reduces its

search capability.

Recently, a new evolutionary computation technique, called Particle Swarm

Optimization (PSO), has been proposed and introduced. This technique combines

social psychology principles in socio-cognition human agents and evolutionary

computations. PSO has been motivated by the behavior of organisms such as fish

schooling and bird flocking. Generally, PSO is characterized as simple in concept,

easy to implement, and computationally efficient. Unlike the other heuristic

techniques, PSO has a flexible and well-balanced mechanism to enhance and adapt to

the global and local exploration abilities.

ELD is solved traditionally using mathematical programming based on

optimization techniques such as lambda iteration, gradient method and so on.

Economic load dispatch with piecewise linear cost functions is a highly heuristic,

approximate and extremely fast form of economic dispatch. Complex constrained

ELD is addressed by intelligent methods. Among these methods, some of them are

genetic algorithm (GA) and, evolutionary programming (EP), dynamic programming

(DP), tabu search, hybrid EP, neural network (NN), adaptive Hopfield neural network

(AHNN), particle swarm optimization (PSO) etc. For calculation simplicity, existing

methods use second order fuel cost functions which involve approximation and

constraints are handled separately, although sometimes valve-point effects are

considered.

Lambda iteration, gradient method can solve simple ELD calculations and

they are not sufficient for real applications in deregulated market. However, they are

fast. There are several Intelligent methods among them genetic algorithm applied to

solve the real time problem of solving the economic load dispatch problem, whereas

some of the works are done by Evolutionary algorithm. Few other methods like tabu

search are applied to solve the problem. Artificial neural network are also used to

solve the optimization problem. However many people applied the swarm behavior to

the problem of optimum dispatch as well as unit commitment problem are general

purpose. However, they have randomness. For a practical problem, like ELD, the

intelligent methods should be modified accordingly so that they are suitable to solve

economic dispatch with more accurate multiple fuel cost functions and constraints,

and they can reduce randomness.


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Intelligent methods are iterative techniques that can search not only local

optimal solutions but also a global optimal solution depending on problem domain

and execution time limit. They are general-purpose searching techniques based on

principles inspired from the genetic and evolution mechanisms observed in natural

systems and populations of living beings. These methods have the advantage of

searching the solution space more thoroughly. The main difficulty is their sensitivity

to the choice of parameters. Among intelligent methods, PSO is simple and

promising. It requires less computation time and memory. It has also standard values

for its parameters. In this, the Particle Swarm Optimization (PSO) is proposed as a

methodology for economic load dispatch.

1.2OPTIMAL POWER FLOW - Literature surveyThe main purpose of an OPF is to determine the optimal operating state of a

power system and the corresponding settings of control variables for economic

operation, while at the same time satisfying various equality and inequality

constraints. The power flow equations are the equality constraints and the inequality

constraints are the limits on control variables and the operating limits of power system

dependent variables. A widely considered objective amongst a number of different

operational objectives that an OPF problem may be formulated is the fuel cost

minimization. Researchers proposed different mathematical formulations of the OPF

problem that can be classified into linear, non-linear or mixed integer linear problem

[1-3].

In its most general formulation, the optimal power flow problem is a nonlinear,

non-convex, large scale, static optimization problem [4, 5]. Many mathematical

programming techniques [6-16] such as linear programming, nonlinear programming,

quadratic programming, Newton method and interior point methods have been

applied to solve the OPF problem successfully.

The interior point methods also have major drawbacks such as improper step

size selection may cause the sub-linear problem to have a solution that is infeasible in

the original nonlinear domain [15]. In addition, a bad initial, termination, and

optimality criterion unable interior point methods to solve nonlinear and quadratic

objective functions [16]. However, these classical optimization methods are limited in

handling algebraic functions.


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In recent years, many heuristic algorithms, such as genetic algorithms [17-20]

and evolutionary programming [21-26], simulated annealing [27], particle swarm

optimization [28-35], chaos optimization algorithm [36, 37], tabu search [38, 40] have

been proposed for solving the OPF problem, without any restrictions on the shape of

the cost curves. A genetic algorithm approach is applied for ac-dc optimal power flow

problem in [17]. In [18], improved genetic algorithm for optimal power flow solutions

under both normal and contingent operation states is proposed. An initialization

procedure in solving optimal power flow by genetic algorithm is proposed in [19]. In

[20], the simple GA with an added set of advanced and problem specific genetic

operators in order to increase its convergence speed and improve the quality of

solutions is applied to OPF. Multi-objective optimal power flow is solved using

strength pareto evolutionary algorithm in [22]. Improved evolutionary programming

is applied for OPF in [23].

Improved evolutionary programming with various crossover techniques is used

in [24] to solve OPF problem. Meanwhile, an improved evolutionary programming

[25] was successfully used to solve combinatorial optimization problems. In [26], a

multi-objective hybrid evolutionary strategy is presented for the solution of the

comprehensive model of OPF.

Optimal power flow subject to security constraints id solved in [29] with a

particle swarm optimizer. In [30], improved particle swarm optimization algorithm for

OPF problems is proposed. In [31], modified particle swarm optimization algorithm is

presented in which particles not only studies from itself and the best one but also from

other individuals. An efficient mixed-integer particle swarm optimization with

mutation scheme for solving the constrained optimal power flow with a mixture of

continuous and discrete control variables and discontinuous fuel cost functions is

presented in [32]. In [33], an improved particle swarm optimization algorithm for

OPF problems, which incorporates non-stationary multistage assignment penalty

function is proposed. Optimal power flow constrained by transient stability is solved

with improved particle swarm optimization in [34]. Authors in [35] used particle

swarm optimization to solve OPF with generator capability curve constraint.

A hybrid algorithm using the chaos optimization and the linear interior point

algorithm is developed in [36] for optimal power flow. In [37], a hybrid algorithm of

chaos optimization and slp for optimal power flow problems with multimodel


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characteristic is considered. The results reported by the above methods were

promising and encouraging for further research in this direction.

Moreover, to enhance the search efficiency, many hybrid algorithms have been

introduced for solving the power system optimization problems. For instance, a

hybrid tabu search and simulated annealing [39] was applied to solve the OPF with

flexible alternating current transmission systems (FACTS) device problem; a hybrid

evolutionary programming and tabu search or improved tabu search [40] was used to

solve the economic dispatch problem with non-smooth cost functions. An ordinal

optimization theory based algorithm to solve OPF problem is proposed in [41].

Optimal power flow for a system of micro grids with controllable loads is solved with

particle swarm optimization in [42].

In recent past, power full evolutionary algorithm such as differential evolution

techniques are employed for power system optimization problems. Storn and Price

[43] developed the DE, and it is a numerical optimization approach that is easy to

implement, significantly faster and robust. DE can be used to minimize nonlinear and

non-differentiable continuous space functions with real valued parameters.

The most important characteristics of differential evolution is that it uses the

differences of randomly sampled pairs of object vectors to guide the mutation

operation instead of using the probability distribution function as other evolutionary

algorithms. Differential evolution combines simple arithmetic operators with the

classical operators of crossover mutation and selection to evolve from a randomly

generated starting population to a final solution. The fittest of an offspring competes

on-to-one with that of corresponding parent, which is different from other

evolutionary algorithms. This on-to-one competition gives rise to faster convergence

rate.

The differential evolution has been successfully applied to various power

system optimization problems such as generation expansion planning [44],

hydrothermal scheduling [45]. Figueroa and Cederio [46] applied DE for power

system state estimation. Coelho and Mariani [47] used this algorithm for economic

dispatch with valve-point effect. M.Basu [48] applied DE for OPF incorporating

FACTS devices.

In this project report, evolutionary programming and particle swarm

optimization algorithms are developed to effectively solve the optimal power flow


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problem incorporating a set of constraints. Simulations for evolutionary OPF are

carried out on various IEEE test systems with different objective functions.

1.3 SCOPE OF THE THESISThis thesis deals with the development of algorithms for power system

generation cost minimization and real power loss reduction in day-to-day optimal

operation of regulated power systems using EP and PSO techniques.

1.4 ORGANIZATION OF THE THESISThis thesis is organized into five chapters. Chapter 1 presents the literature

survey and overview of optimal power flow by various methods like IPM, EP, PSO in

regulated power system environment. This section gives a brief account of the

information presented in each chapter.

The chapter 2 presents information on economic operation of the power system,

optimal load dispatch, system constraints, load flow studies and their need to power

system operation and different methods of load flow studies are discussed.

In chapter 3, an evolutionary programming and particle swarm optimization

algorithms for solving the optimal power flow problem are discussed.

In chapter 4, the results of IEEE 14-bus and IEEE 30-bus systems have been

presented to show the effectiveness of the OPF algorithms. Comparisons were made

between the approaches in terms of the solution quality and convergence

characteristics. The chapter 5 presents the major contributions of the thesis and

suggestions for further work. The appendix contains the data of the test systems used

for studies.


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CHAPTER 2

LOAD FLOW STUDIES


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CHAPTER 2

LOAD FLOW STUDIES

In power engineering, the power flow study (also known as load-flow study) isan important tool involving numerical analysis applied to a power system. Unlike

traditional circuit analysis, a power flow study usually uses simplified notation such

as a one-line diagram and per-unit system, and focuses on various forms ofAC power

(i.e. reactive, real, and apparent) rather than voltage and current. It analyzes the power

systems in normal steady-state operation. There exist a number of software

implementations of power flow studies. This chapter presents information on load

flow studies and their need to power system operation and different methods of load

flow studies.

2.1 INTRODUCTION

The great importance of power flow or load-flow studies is in the planning the

future expansion of power systems as well as in determining the best operation of

existing systems. The principal information obtained from the power flow study is the

magnitude and phase angle of the voltage at each bus and the real and reactive power

flowing in each line.

Electrical transmission systems operate in their steady state mode under normal

conditions. Three major problems encountered in steady state mode of operations are

listed below in their hierarchical order of importance:

1. Load flow problem2. Optimal load dispatch problem3. Systems control problemThe computational procedure required to determine the steady-state operating

characteristics of a power system network is termed load flow or power flow. The aim

of the power flow calculations is to determine the steady-state operating

characteristics of a power generation/transmission system for a given set of bus bar

loads. Active power generations are specified according to economic dispatching. The

magnitude of generation voltage is maintained at the specified level by an automatic

voltage regulator acting on the machine excitation. Loads are specified by their

constant active and reactive power requirements. The loads are assumed to be
http://en.wikipedia.org/wiki/Power_engineeringhttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/One-line_diagramhttp://en.wikipedia.org/wiki/Per-unit_systemhttp://en.wikipedia.org/wiki/AC_powerhttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/AC_powerhttp://en.wikipedia.org/wiki/Per-unit_systemhttp://en.wikipedia.org/wiki/One-line_diagramhttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Power_engineering


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unaffected by the small variations of voltage and frequency expected during normal

steady-state operation.

The direct analysis of the network is not possible, as the loads are given in terms

of complex powers rather than impedances. The generators behave more like power

sources than voltage sources. The main information obtained from the load flow study

consists of

1. Magnitudes and phase angles of load bus voltages.2. Reactive powers and voltage phase angles at generator buses.3. Real and reactive power flow on transmission lines.4. Power at the reference bus.This information is essential for the continuous monitoring of the current state of

the system. The information is also important for analyzing the effectiveness of the

alternative plans for the future, such as adding new generator sites, meeting increased

load demand and locating new transmission sites.

The single line diagram of a power system having four buses is shown in the

figure. In the power system, the variables defined on each bus are

1. Complex powers, and 2. Complex powers drawn by loads,

,

,

3. Complex voltages, , , and .

Fig 2.1. Single line diagram of a four-bus system


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There results a net injection of power into the transmission system. The

transmission system may be a primary transmission system or sub transmission

system. The primary transmission system transmits bulk power from the generators to

the bulk power stations. The sub-transmission system transmits power from the

substations or some old generators to the distribution systems. The transmission

system has to be designed in such a manner that the power system operation is

reliable and economic and no difficulties are encountered in its operation. The

difficulties are involved, however are:

1. One or more transmission lines becoming over loaded2. Generators becoming over loaded3. The stability margins for a transmission link being too small

There may be emergencies such as

1. The loss of one or more transmission links2. Shut down of some generators which give rise to overloading of other

generators and transmission lines.

In system operation and planning, the voltages and powers are kept within certain

limits. The power system networks of today are highly complicated consisting of

hundreds of buses and transmission links. Thus, the load flow study involves

extensive calculations. With the advent of fast digital computers with huge memory,

all kinds of power system studies including the load flow study can now be used

which can be:

1. Accurate or approximate2. Unadjusted or adjusted3. Offline or online4. Single case or multiple cases.

2.2LOAD FLOW PROBLEM

The complex power injected by the source into the bus of a power system is (2.1)

Substituting the value of in the above power equation. ( ) (2.2)

Where

Is the voltage of the bus Is the element of the admittance bus


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Equating the real and imaginary parts

( ) (2.3)

( ) (2.4)Where Is the real power

Is the reactive powerLet ||, =||, ||

Where

|| Is the magnitude of voltage

Is the angle of the voltage Is the load angleSubstituting for , and

|| || || (2.5)Or

|| || || (2.6)Or

= || || || (2.7)Or || ||

||

) (2.8)Separating the real and imaginary parts of the above equation to get real and reactive

powers,

|| || || (2.9) || || || (2.10)Where

(2.11) (2.12)

|| || (2.13)

=|| )|| (i=1,2,3,.........NB) (2.14)Separating real and reactive parts of the above equation,


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|| (i=1,2,3,.........NB) (2.15) || (i=1,2,3,.........NB) (2.16)

Equations 2.15 and 2.16 are called power flow equations. These are NB real and NBreactive power flow equations giving a total of 2 NB power flow equations. At each

bus these are four variables, namely| |, , and , giving a total of 4 NBvariables(for NB buses). If at every bus two variables are specified (thus specifying a

total of 2 NB variables), the remaining two variables at every bus (a total of 2 NB

remaining variables) can be found by solving 2 NB power flow equations. In a

physical system, the variables are specified depending upon what kind of devices are

connected to that bus. In general fur types of buses are defined.

2.2.1 Classification of Buses

Depending upon which quantities have been specified at each bus, buses are

classified into four categories which are given below

1. Slack bus/Swing bus/Reference bus2. PQ bus/Load Bus3. PV Bus /Generator Bus4. Voltage-controlled buses

Slack bus/Swing bus/Reference bus

In a load flow study, real and reactive powers cannot be fixed a priori at all

the buses as the net complex power flow into the network is not known in advance.

So, the system power losses are unknown till the load flow study is complete. It is

therefore necessary to have one bus at which complex power is unspecified so that it

supplies the difference in the total system load plus losses and the sum of the complex

powers specified at the remaining buses. Such a bus is known as slack bus and must

be a generator bus. If slack bus is not specified, the bus connected to the largest

generating station is normally selected as the slack bus.

Usually this bus is numbered 1 for the load flow studies. This bus sets the

angular reference for all the other buses. Since it is the angle difference between two

voltage sources that dictates the real and reactive power flow between them, the

particular angle of the slack bus is not important. However it sets the reference against

which angles of all the other bus voltages are measured. For this reason the angle of


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this bus is usually chosen as 0. Furthermore it is assumed that the magnitude of the

voltage of this bus is known.

Voltage magnitude and voltage phase angle are specified. Normally,

voltage magnitude is set to 1pu and voltage angle is set to zero. The real and reactive

powers are not specified. The known parameters are voltage magnitude || andvoltage angle. The unknown parameters are real power and reactive power.PQ bus/Load Bus

In these buses no generators are connected and hence the generated real power

and reactive power are taken as zero. The load drawn by these buses aredefined by real power and reactive power in which the negative signaccommodates for the power flowing out of the bus. This is why these buses are

sometimes referred to as P-Q bus. The objective of the load flow is to find the bus

voltage magnitude |Vi| and its angle i. In a power system 80% of the buses are P-Q

buses.

A pure load bus is a PQ bus. A load bus has no generating facility (i.e

. At this type of bus, the net real power and the reactive power are known as

=

and

=

(2.17)

Where, are the real and reactive power generations at the bus respectively. , are the real and reactive power demands at the respectively. and areknown from the load forecasting and and are specified The known variableson bus are real power and the reactive power. The unknowns are voltagemagnitude and voltage angle. The PQ buses are the most common comprising almost

85% of all the buses in a given power system.

PV Bus /Generator Bus

These are the buses where generators are connected. Therefore the power

generation in such buses is controlled through a prime mover while the terminal

voltage is controlled through the generator excitation. Keeping the input power

constant through turbine-governor control and keeping the bus voltage constant using

automatic voltage regulator, we can specify constant PGI and | Vi | for these buses.

This is why such buses are also referred to as P-V buses. It is to be noted that the

reactive power supplied by the generator QGI depends on the system configuration and


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cannot be specified in advance. Furthermore we have to find the unknown angle i of

the bus voltage. In a power system 10% of the buses are P-V buses.

A generator is always connected to a PV bus. Hence the net power , knownas

is also known from load forecasting. The knowns are real power

and voltage

magnitude||. The unknowns are reactive power and voltage angle PV busescomprise about 15% of all the buses in o power system.

Voltage-controlled buses

Generally the PV buses and the voltage-controlled buses are grouped

together but these buses have physical difference. The voltage-controlled bus has also

voltage control capabilities, and uses a tap-adjustable transformer and/or a static VAR

compensator a instead of a generator.

Hence, =0 at these buses. Thus =- and = at these buses. Theknowns are real power , reactive power and voltage magnitude. The voltagemagnitude is an parameter.2.2.2 Limits of Power System Hardware and Operating Constraints

For static load flow equations (SLFE) solution to have practical

significance, all the state and control variables must be within the specified practical

limits. These limits are represented by specifications of power system hardware and

operating constraints, and are describes as follows:

Voltage magnitude || must satisfy the inequality|| || || (2.18)

The limit arises due to the fact that the power system equipment is designed to

operate at fixed voltages with in allowable variations of % of ratedvalues.

Certain of the voltage angles (state variables) must satisfy

| | | | (2.19)This constraint limits the maximum permissible power angle of transmission

line connecting buses i and k and is imposed by considerations of stability.

Owing to physical limitations of P and/or Q, generation sources areconstrained.

(i=1,2,.NB) (2.20)

(i=1,2,.NB) (2.21)


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Also the equality constraints are

Where and are system real and reactive power losses respectively.The load flow equations are non-linear algebraic equations and have to be solved

through iterative numerical techniques, etc. At the cost of solution accuracy, it is

possible to linearize load flow equations by making suitable assumptions and

approximations so that fast and explicit solutions become possible.

2.3 METHODS FOR LOAD FLOW STUDIES

1. Gauss-seidel method2. Newton-Raphson method3. Decoupled method4. Fast decoupled method

2.3.1 Gauss Seidel method

This method is a modification to Gauss-iteration method. This modification

reduces the number of iterations. In this method the values of unknowns immediately

replace the previous values in the next step while in case of Gauss method the

calculated values replace the earlier values only at the end of the iteration. Because of

it Gauss-Seidel method converges much faster than the Gauss method, i.e., number of

iterations required to obtain solution is much less in the Gauss-Seidel method

compared to the Gauss method.

Gauss-Seidel method is of the simplest iterative methods and has been in use since

early days of digital computer methods of analysis.

The advantages and disadvantages are given below.Advantages

1. Simplicity of technique2. Small computer memory requirement.3. Less computational time per iteration.

Disadvantages

1. Slow rate of convergence resulting in large number of iterations.2.

Increase in number of iterations directly with the increase in the number ofbuses.


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3. Effect on convergence due to choice of slack bus.Because of the above drawbacks, use of Gauss-Seidel method is limited only to

systems with smaller number of buses.

2.3.2. Fast-Decoupled load flow methodThe fast decoupled load flow method is a very fast and efficient method of

obtaining power flow problem solution. In this method, both, the speeds as well as the

sparsity are exploited. This is actually an extension of Newton-Raphson method

formulated in polar coordinates with certain approximations which result into a fast

algorithm for power flow solution. This method exploits the property of the power

system where in MW flow-voltage angle and MVAR flow-voltage magnitude are

loosely coupled. In other words a small change in the magnitude of the bus voltage

does not affect the real power flow at the bus and similarly a small change in phase

angle of the bus voltage has hardly any effect on reactive power flow. Because of this

loose physical interaction between MW and MVAR flows in a power system, the

MW- and MVAR-V calculations can be decoupled. This decoupling results in a very

simple, fast and reliable algorithm. The sparsity feature of admittance matrix

minimizes the computer memory requirements and results in faster computations.

2.3.3 Newton-Raphson method

The power flow problem can also be solved by using Newton-Raphson method. In

fact, among the numerous solution methods available for power flow analysis, the

Newton-Raphson method is considered to be the most sophisticated and important.

Many advantages are attributed to Newton-Raphson approach.

Gauss-Seidel is a simple iterative method of solving n number load flow equations

by iterative method. It does not require partial derivatives. Newton- Raphson method

is based on Taylors series and partial derivatives. The N-R method is recent and

needs less number of iterations to reach convergence.

Advantages

1. It needs less number of iterations to reach convergence.2. N-R method is more accurate.3. It is insensitive to the factors like slack bus selection, regulating transformers

etc.


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4. The number of iterations required in this method is almost independent of thesystem size.

5. N-R method is suitable for large power systems.Disadvantages

1. Difficult solution technique.2. More calculations in each iteration resulting in large computer time per

iteration and the large requirement of computer memory but the last drawback

can be overcome through a compact storage scheme.

2.3.4. Algorithm for Newton-Raphson method

1. Read data

NB is the number of buses; NV is the number of PV buses.

and for slack bus, (2,3,NB) for PQ and PV buses.(i= NV+1, NV+2,..NB for PQ buses), (2,3,NV for PV buses) , (i= NV+1, NV+2,NB for PQ buses). , (i=2,3,.NV for PV buses).R (maximum number of iterations) , (tolerance in convergence).

2. Form the 3. Assume initially (i=NV+1, NV+2NB) and (i=2, 3NB)4. Set the iteration count, r=0.

5. Compute , using the equations. (i=2,3,.NB)

(i=2,3,..NB)

Compute (i=NV+1,NV+2, ..NB)

6. If maximum { }then GOTO step15.

7. Compute Jacobian matrix elements using the equations

When i=k


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= , + - , = =

When i

= = = = = =

= = 8. Compute

and

[ ]

= []

9. Modify and , =

= + 10. Set bus count i=2.

11. If PQ bus then check the limits of and set. = if = if

12. If PV bus then compute and check the limits of and set.

=

if

= if If the limits are violated then PV bus is temporarily converted to PQ bus. So, compute

and With updated values of , and . Then calculate the change in voltage i.e

=

And specified voltage magnitude of PV bus is updated as


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= + 13. Increment the bus count, i=i+1

If i NB, then GOTI step11.

14. Advance the count, r=r+1

If r R then GOTO step5 and repeat.

15. Compute active and reactive power on slack bus i.e

= =

16. Calculate lone flows using equations

={ }

={ } Where = 17. Stop.

2.4 CONCLUSION

In this chapter, information on load flow studies, their need to power system

operation and different methods of load flow studies were presented.


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CHAPTER 3

OPF BY EVOLUTIONARYCOMPUTATION TECHNIQUES


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CHAPTER 3

OPF BY EVOLUTIONARY COMPUTATION

TECHNIQUES

Optimal power flow is an optimizing tool for operation and planning of

modern power systems. This OPF problem involves the optimization of various types

of objective functions while satisfying a set of operational and physical constraints

while keeping the power outputs of generators, bus voltages, shunt capacitors/reactors

and transformers tap settings in their limits. This chapter presents information on OPF

methodologies and evolutionary computation techniques.

3.1 INTRODUCTION

In past three decades, various optimization techniques have been proposed to

solve the optimal power flow (OPF) problem. They range from improved

mathematical techniques to more efficient problem formulation. According to difficult

models in use, the OPF methods can be classified as non-compact methods where

network sparsity is retained, or compact ones in which the state variables are

expressed in terms of control variables using various sensitivities. Based on the

applied mathematical optimization, the OPF methods can be categorized as

1. Nonlinear Programming (NLP),

2. Successive Linear Programming (SLP), and

3. Non-conventional techniques.

The gradient methods, using only first order information, were initially used

for the solution of OPF problem. These methods were characterized by slow and

unreliable convergence soon after, the quadratic programming (QP) approaches were

proposed, which use the second-order derivatives to improve the convergence of the

gradient methods. Their distinct feature is that they use the Quasi-Newton process to

iteratively approximate the Hessian matrix and, thus avoid the difficulty in explicitly

calculating the second derivatives of the load flow equations. However, the reduced

Hessian so created is dense, which may make these methods too slow as the number

of control variables becomes very large

As the demand increases, faster and more stable techniques grow more

accurate representation of the second-order information became essential. Lagrangian

techniques with the exact Hessian matrix regained engineers interest. Although these


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methods were proposed as earlier as 1960s, few were either reliable or fast until SUN

introduced a Newton approach combined with Lagrangian techniques and penalty

functions. The major difficulty in this algorithm development was turned out to be the

efficient identification of binding inequality constraints. In some cases the

convergence is affected by the chosen initial point. Sometimes the problem becomes

divergence due to chosen initial point. The initial point should be the feasible point.

To overcome the above difficulties Karmarkar introduced a method called

Interior point Method. He introduced a Fiacco &McCormics logarithmic barrier

method for optimization with inequalities, Lagranges method for optimization with

inequalities and Newton method for solving the nonlinear equations of Karush-Kuhn

Tucker (KKT) optimality conditions. With their nice polynomial complexity plus

computational efficiency, interior point methods have proved much faster than the

traditional methods.

The name interior point comes from LP notation. Namely, IPM move

through the interior of the feasible region towards the optimal solution. In past fifteen

years, researches on Interior Point (IP) methods experience an awesome expansion.

Both Interior Point theory and computational implementation have evolved extremely

fast. Interior point method variants are being extended to solve all kinds of problems

from linear to nonlinear and from convex to non convex (the later with no guarantee

regarding convergence). In the same way they are also being applied to solve all sorts

of practical problems. Optimization of power system operation is one of the areas

where Interior Point (IP) methods are being applied extensively due to size and

special features of these problems.

3.2 EVOLUTIONARY COMPUTATION TECHNIQUE

Evolutionary programming is one of the four major evolutionary algorithm

paradigms. It was first used by Lawrence J. Fogel in the US in 1960 in order to use

simulated evolution as a learning process aiming to generate artificial intelligence.

Fogel used finite state machines as predictors and evolved them. Currently

evolutionary programming is a wide evolutionary computing dialect with no fixed

structure or (representation), in contrast with some of the other dialects. It is

becoming harder to distinguish from evolutionary strategies. Some of its original

variants are quite similar to the later genetic programming, except that the program

structure is fixed and its numerical parameters are allowed to evolve.
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In artificial intelligence, an evolutionary algorithm (EA) is a subset of

evolutionary computation, a generic population-based metaheuristic optimization

algorithm. An EA uses some mechanisms inspired by biological evolution:

reproduction, mutation, recombination, and selection. Candidate solutions to the

optimization problem play the role of individuals in a population, and the fitness

function determines the environment within which the solutions "live". Evolution of

the population then takes place after the repeated application of the above operators.

Evolutionary algorithms often perform well approximating solutions to all

types of problems because they ideally do not make any assumption about the

underlying fitness landscape; this generality is shown by successes in fields as diverse

as engineering, art, biology, economics, marketing, genetics, operations research,

robotics, social sciences, physics, politics and chemistry.

Apart from their use as mathematical optimizers, evolutionary computation

and algorithms have also been used as an experimental framework within which to

validate theories about biological evolution and natural selection, particularly through

work in the field ofartificial life. Techniques from evolutionary algorithms applied to

the modeling of biological evolution are generally limited to explorations of

microevolutionary processes, however some computer simulations, such as Tierra and

Avida, attempt to model macroevolutionary dynamics.

In most of real applications of EAs, computational complexity is a prohibiting

factor. In fact, this computational complexity is due to fitness function evaluation.

Fitness approximation is one of the solutions to overcome this difficulty.

Another possible limitation of many evolutionary algorithms is their lack of a

clear genotype-phenotype distinction. In nature, the fertilized egg cell undergoes a

complex process known as embryogenesis to become a mature phenotype. This

indirect encoding is believed to make the genetic search more robust (i.e. reduce the

probability of fatal mutations), and also may improve the evolvability of the

organism. Such indirect (aka generative or developmental) encodings also enable

evolution to exploit the regularity in the environment. Recent work in the field of

artificial embryogeny, or artificial developmental systems, seeks to address these

concerns.

Its main variation operator is mutation; members of the population are viewed

as part of a specific species rather than members of the same species therefore eachparent generates an offspring, using a ( + )survivor selection.
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Evolutionary Programming has developed into an extraordinary subject to

study in the field of Artificial Intelligence. Evolutionary Programming is the study of

programs that use simulations of biological functions in order to evolve to a specified

environment. Scientists have used Evolutionary Programming techniques in order to

find solutions to very complex problems. These evolving programs find good

solutions to these complex problems, but not perfect solutions. The perfect solutions

would take even the most advanced super computer decades to figure out.

Evolutionary Programming is a method for finding solutions in smaller intervals of

time, because the solutions themselves evolve over time.

An initially random population of individuals (trial solutions) is created.

Mutations are then applied to each individual to create new individuals. Mutations

vary in the severity of their effect on the behavior of the individual. The new

individuals are then compared in a "tournament" to select which should survive to

form the new population. EP is similar to a genetic algorithm, but models only the

behavioral linkage between parents and their offspring, rather than seeking to emulate

specific genetic operators from nature such as the encoding of behavior in a genome

and recombination by genetic crossover. EP is also similar to an evolution strategy

(ES) although the two approaches developed independently. In EP, selection is by

comparison with a randomly chosen set of other individuals whereas ES typically uses

deterministic selection in which the worst individuals are purged from the population.

Evolutionary algorithms, such as evolutionary programming (EP), evolution strategies

(ES), genetic algorithms (GA), and genetic programming (GP), have attracted

considerable interest as optimization heuristics during the past 10-15 years. Because

of the exponential increase in computer power during the last decade, they are able to

deal with real-world problems and new application domains are still arising. Although

evolutionary algorithms are easy to implement, the underlying process is complicated

and stochastic, depending on the fitness function and the free parameters controlling

variation and selection. The analysis of these stochastic processes seems to be much

more difficult than the analysis of randomized algorithms for special purpose.

Evolutionary algorithms (EAs) are stochastic optimization methods that are

based on principles derived from natural evolution. From a more general perspective,

EAs are one instance of bio-inspired search heuristics. Other examples include Ant

Colony Optimization (ACO) and Particle Swarm Optimization (PSO), where thesearch behaviors of ant colonies or insect swarms inspired a randomized search


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technique. Since the underlying ideas of bio-inspired search are easy to grasp and easy

to apply, EAs and different bio-inspired search heuristics are widely used in many

practical disciplines, mainly in computer science and engineering. It is a central goal

of theoretical investigations of search heuristics to assist practitioners with the tasks of

selecting and designing good strategy variants and operators. Due to the rapid pace at

which new strategy variants and operators are being proposed, theoretical foundations

of EAs and other bio-inspired search heuristics still lag behind practice. However, EA

theory has gained much momentum over the last few years and has made numerous

valuable contributions to the field of evolutionary computation. Much of this

momentum is due to the Dagstuhl seminars on ``Theory of Evolutionary Algorithms'',

which have been held biannually since 2000.

The theory of EAs today consists of a wide range of different approaches.

Runtime analysis, schema theory, analyses of the dynamics of EAs, and systematic

empirical analysis consider different aspects of EA behavior. Moreover, they employ

different methods and tools for attaining their goals, such as Markov chains, infinite

population models, or ideas based on statistical mechanics or population dynamics. In

the most recent seminar, more recent types of bio-inspired search heuristics were

discussed. Results regarding the runtime have been generalized from EAs to ACO and

PSO. Although the latter heuristics follow a different design principle than EAs, the

theoretical analyses reveal surprising similarities in terms of the underlying stochastic

process. Theoretical studies of EAs in continuous domain have recently evoked

interest of people working in the field of classical numerical optimization. Although

stochastic and deterministic optimization algorithms address optimization of different

types of problems---mainly convex and smooth for deterministic algorithms and

noisy, multimodal, irregular for stochastic algorithms---the focuses of both fields

became closer and closer: on the one hand many hybridizations of stochastic search

and gradient-based algorithms have been proposed, on the other hand, derivative-free

optimization is now a well established part of the research in the classical

optimization community.

3.3 IMPLEMENTATION OF BIOLOGICAL PROCESSESUsually, an initial population of randomly generated candidate solutions

comprises the first generation. The fitness function is applied to the candidate

solutions and any subsequent offspring. In selection, parents for the next generationare chosen with a bias towards higher fitness. The parents reproduce one or two
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offspring (new candidates) by copying their genes, with two possible changes:

crossover recombines the parental genes and mutation alters the genotype of an

individual in a random way. These new candidates compete with old candidates for

their place in the next generation (survival of the fittest). This process can be repeated

until a candidate with sufficient quality (a solution) is found or a previously defined

computational limit is reached.

3.4 METHODOLOGY OF EVOLUTIONARY PROGRAMMING

Evolutionary Programming (EP) is an optimization technique based on the

natural generation. It involves random number generation at the initialization process.

The generated random numbers represent the parameters responsible for the

optimization of the fitness value. In addition, EP also involves statistics, fitness

calculation, mutation and the new generation will be bred by mode of selection.

Evolutionary programming, termed as EP is a mutation based evolutionary

algorithm. EP belongs to a class of population based search strategies. EP is a

methodology not an algorithm. EP is a stochastic optimization strategy, which places

emphasis on the behavior linkage between parents and their offsprings. EP is a

computational intelligence method in which the optimization algorithm is the main

engine for the process of three steps namely natural selection, mutation and

competition. According to the problem each step can be modified and configured in

order to achieve the optimal result. EP is a global optimization technique that starts

with the population of randomly generated candidate solution and evolves a better

solution over a number of generations or iterations. It is more suitable to effectively

handle non-continuous and non-differentiable function. The main stage of this

technique includes initialization, mutation, competition and selection.

3.4.1. Reproduction

Reproduction (or procreation) is the biological process by which new

"offspring" individual organisms are produced from their "parents". Reproduction is a

fundamental feature of all known life; each individual organism exists as the result of

reproduction. The known methods of reproduction are broadly grouped into two main

types: sexual and asexual.

In asexual reproduction, an individual can reproduce without involvement

with another individual of that species. The division of a bacterial cell into two

daughter cells is an example of asexual reproduction. Asexual reproduction is not,
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however, limited to single-celled organisms. Most plants have the ability to reproduce

asexually and the ant species Mycocepurus smithii is thought to reproduce entirely by

asexual means. Sexual reproduction typically requires the involvement of two

individuals or gametes, one each from opposite type ofsex.

3.4.2. Initialization

Initial population is one of the deciding factors for reaching the optimum, it should be

carefully generated. The initial population is composed by K parent individuals. The

elements of the parent are the randomly created permutation of the input variables of

the generated units. Each element in a population is uniformly distributed within its

feasible range.

3.4.3. MutationThe most commonly used evolutionary operator is mutation. Mutation is the random

occasional alteration of the information contained in the individual. It is performed on

each element by adding a normally distributed random number with mean few and

standard deviation.

In molecular biology and genetics, mutations are changes in a genomic

sequence: the DNA sequence of a cell's genome or the DNA or RNA sequence of a

virus. They can be defined as sudden and spontaneous changes in the cell. Mutations

are caused by radiation, viruses, transposons and mutagenic chemicals, as well as

errors that occur during meiosis or DNA replication. They can also be induced by the

organism itself, by cellular processes such as hypermutation.

Mutation can result in several different types of change in DNA sequences;

these can either have no effect, alter the product of a gene, or prevent the gene from

functioning properly or completely. Studies in the fly Drosophila melanogaster

suggest that if a mutation changes a protein produced by a gene, this will probably be

harmful, with about 70 percent of these mutations having damaging effects, and the

remainder being either neutral or weakly beneficial. Due to the damaging effects that

mutations can have on genes, organisms have mechanisms such as DNA repair to

remove mutations.

Therefore, the optimal mutation rate for a species is a trade-off between costs

of a high mutation rate, such as deleterious mutations, and the metabolic costs of

maintaining systems to reduce the mutation rate, such as DNA repair enzymes.

Viruses that use RNA as their genetic material have rapid mutation rates, which can
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be an advantage since these viruses will evolve constantly and rapidly, and thus evade

the defensive responses of e.g. the human immune system.

3.4.4Competition and Selection

The offsprings produced from the mutation process were combined with theparents to undergo a selection process in order to identify the candidates to be

transcribed into the next generation. Two selection strategies were tested namely the

priority selection and pair wise comparison. In priority selection strategy, the

populations were sorted in descending order according to their fitness values since the

objective function is to obtain the total loss.

The selection is used to determine the individuals that will be represented in

the next generation. It includes competition in which each individual in the combined

population has to compete with some other individuals to get chance to be transcribed

to the next generation. The 2k individuals compete with each other for selection.

Natural selection is the process by which traits become more or less common

in a population due to consistent effects upon the survival or reproduction of their

bearers. It is a key mechanism ofevolution.

The natural genetic variation within a population of organisms may cause

some individuals to survive and reproduce more successfully than others in their

current environment. For example, the peppered moth exists in both light and dark

colors in the United Kingdom, but during the industrial revolution many of the trees

on which the moths rested became blackened by soot, giving the dark-colored moths

an advantage in hiding from predators. This gave dark-colored moths a better chance

of surviving to produce dark-colored offspring, and in just a few generations the

majority of the moths were dark. Factors which affect reproductive success are also

important, an issue which Charles Darwin developed in his ideas on sexual selection.

Natural selection acts on the phenotype, or the observable characteristics of an

organism, but the genetic (heritable) basis of any phenotype which gives a

reproductive advantage will become more common in a population (see allele

frequency). Over time, this process can result in adaptations that specialize

populations for particular ecological niches and may eventually result in the

emergence of new species. In other words, natural selection is an important process

(though not the only process) by which evolution takes place within a population of

organisms. As opposed to artificial selection, in which humans favor specific traits, in
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natural selection the environment acts as a sieve through which only certain variations

can pass.Natural selection is one of the cornerstones of modern biology. The term was

introduced by Darwin in his influential 1859 bookOn the Origin of Species, in which

natural selection was described as analogous to artificial selection, a process by which

animals and plants with traits considered desirable by human breeders are

systematically favored for reproduction. The concept of natural selection was

originally developed in the absence of a valid theory of heredity; at the time of

Darwin's writing, nothing was known of modern genetics. The union of traditional

Darwinian evolution with subsequent discoveries in classical and molecular genetics

is termed the modern evolutionary synthesis. Natural selection remains the primary

explanation for adaptive evolution.

3.4.5. Genetic recombination

Genetic recombination is a process by which a molecule of nucleic acid

(usually DNA, but can also be RNA) is broken and then joined to a different one.

Recombination can occur between similar molecules of DNA, as in homologous

recombination, or dissimilar molecules, as in non-homologous end joining.

Recombination is a common method ofDNA repair in both bacteria and eukaryotes.

In eukaryotes, recombination also occurs in meiosis, where it facilitates chromosomal

crossover. The crossover process leads to offspring's having different combinations of

genes from those of their parents, and can occasionally produce new chimeric alleles.

In organisms with an adaptive immune system, a type of genetic recombination called

V(D)J recombination helps immune cells rapidly diversify to recognize and adapt to

new pathogens. The shuffling of genes brought about by genetic recombination is

thought to have many advantages, as it is a major engine ofgenetic variation and also

allows sexually reproducing organisms to avoid Muller's ratchet, in which the

genomes of an asexual population accumulate deleterious mutations in an irreversible

manner.

In genetic engineering, recombination can also refer to artificial and deliberate

recombination of disparate pieces of DNA, often from different organisms, creating

what is called recombinant DNA. A prime example of such a use of genetic

recombination is gene targeting, which can be used to add, delete or otherwise change

an organism's genes. This technique is important to biomedical researchers as it

allows them to study the effects of specific genes. Techniques based on genetic
http://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/On_the_Origin_of_Specieshttp://en.wikipedia.org/wiki/Artificial_selectionhttp://en.wikipedia.org/wiki/Heredityhttp://en.wikipedia.org/wiki/Darwinismhttp://en.wikipedia.org/wiki/Classical_geneticshttp://en.wikipedia.org/wiki/Molecular_geneticshttp://en.wikipedia.org/wiki/Modern_evolutionary_synthesishttp://en.wikipedia.org/wiki/Adaptive_evolutionhttp://en.wikipedia.org/wiki/Nucleic_acidhttp://en.wikipedia.org/wiki/DNAhttp://en.wikipedia.org/wiki/RNAhttp://en.wikipedia.org/wiki/Homology_%28biology%29http://en.wikipedia.org/wiki/Homologous_recombinationhttp://en.wikipedia.org/wiki/Homologous_recombinationhttp://en.wikipedia.org/wiki/Non-homologous_end_joininghttp://en.wikipedia.org/wiki/DNA_repairhttp://en.wikipedia.org/wiki/Bacteriahttp://en.wikipedia.org/wiki/Eukaryoteshttp://en.wikipedia.org/wiki/Meiosishttp://en.wikipedia.org/wiki/Chromosomal_crossoverhttp://en.wikipedia.org/wiki/Chromosomal_crossoverhttp://en.wikipedia.org/wiki/Chimera_%28genetics%29http://en.wikipedia.org/wiki/Allelehttp://en.wikipedia.org/wiki/Adaptive_immune_systemhttp://en.wikipedia.org/wiki/V%28D%29J_recombinationhttp://en.wikipedia.org/wiki/Pathogenhttp://en.wikipedia.org/wiki/Genetic_variationhttp://en.wikipedia.org/wiki/Muller%27s_ratchethttp://en.wikipedia.org/wiki/Genomehttp://en.wikipedia.org/wiki/Asexual_reproductionhttp://en.wikipedia.org/wiki/Populationhttp://en.wikipedia.org/wiki/Genetic_deletionhttp://en.wikipedia.org/wiki/Genetic_engineeringhttp://en.wikipedia.org/wiki/Recombinant_DNAhttp://en.wikipedia.org/wiki/Gene_targetinghttp://en.wikipedia.org/wiki/Biomedical_researchhttp://en.wikipedia.org/wiki/Biomedical_researchhttp://en.wikipedia.org/wiki/Gene_targetinghttp://en.wikipedia.org/wiki/Recombinant_DNAhttp://en.wikipedia.org/wiki/Genetic_engineeringhttp://en.wikipedia.org/wiki/Genetic_deletionhttp://en.wikipedia.org/wiki/Populationhttp://en.wikipedia.org/wiki/Asexual_reproductionhttp://en.wikipedia.org/wiki/Genomehttp://en.wikipedia.org/wiki/Muller%27s_ratchethttp://en.wikipedia.org/wiki/Genetic_variationhttp://en.wikipedia.org/wiki/Pathogenhttp://en.wikipedia.org/wiki/V%28D%29J_recombinationhttp://en.wikipedia.org/wiki/Adaptive_immune_systemhttp://en.wikipedia.org/wiki/Allelehttp://en.wikipedia.org/wiki/Chimera_%28genetics%29http://en.wikipedia.org/wiki/Chromosomal_crossoverhttp://en.wikipedia.org/wiki/Chromosomal_crossoverhttp://en.wikipedia.org/wiki/Meiosishttp://en.wikipedia.org/wiki/Eukaryoteshttp://en.wikipedia.org/wiki/Bacteriahttp://en.wikipedia.org/wiki/DNA_repairhttp://en.wikipedia.org/wiki/Non-homologous_end_joininghttp://en.wikipedia.org/wiki/Homologous_recombinationhttp://en.wikipedia.org/wiki/Homologous_recombinationhttp://en.wikipedia.org/wiki/Homology_%28biology%29http://en.wikipedia.org/wiki/RNAhttp://en.wikipedia.org/wiki/DNAhttp://en.wikipedia.org/wiki/Nucleic_acidhttp://en.wikipedia.org/wiki/Adaptive_evolutionhttp://en.wikipedia.org/wiki/Modern_evolutionary_synthesishttp://en.wikipedia.org/wiki/Molecular_geneticshttp://en.wikipedia.org/wiki/Classical_geneticshttp://en.wikipedia.org/wiki/Darwinismhttp://en.wikipedia.org/wiki/Heredityhttp://en.wikipedia.org/wiki/Artificial_selectionhttp://en.wikipedia.org/wiki/On_the_Origin_of_Specieshttp://en.wikipedia.org/wiki/Biology


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recombination are also applied in protein engineering to develop new proteins of

biological interest.

3.5 EVOLUTIONARY PROGRAMMING ALGORITHM

i. An Initial population of parent vectors is considered as the trial solutionii. From these parents off springs are created by mutation, hence off springs areobtained

iii. By combining the parents and off springs, 2 solutions are obtainediv. Through competition and selection, first optimal solutions are selectedv. The selected solutions are considered as parents for the next iteration

vi. After the required number of iterations, the best optimal solution is obtained.

3.6 FLOW CHART FOR IMPLEMENTATION OF EP

NO

YES

Fig 3.1. Flow Chart for Implementation of EP

start

Generate random number

Calculate fitness

Selection

Calculate fitness

Mutation

Convergence

test

End
http://en.wikipedia.org/wiki/Protein_engineeringhttp://en.wikipedia.org/wiki/Protein_engineering


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3.7 PARTICLE SWARM OPTIMIZATION

Particle swarm optimization (PSO) is a population based stochastic

optimization technique developed by Dr.Eberhart and Dr. Kennedy in 1995, inspired

by social behavior of bird flocking or fish schooling. PSO shares many similaritieswith evolutionary computation techniques such as Genetic Algorithms (GA). The

system is initialized with a population of random solutions and searches for optima by

updating generations. However, unlike GA, PSO has no evolution operators such as

crossover and mutation. In PSO, the potential solutions, called particles, fly through

the problem space by following the current optimum particles. The detailed

information will be given in following sections. Compared to GA, the advantages of

PSO are that PSO is easy to implement and there are few parameters to adjust. PSO

has been successfully applied in many areas: function optimization, artificial neural

network training, fuzzy system control, and other areas where GA cannot be applied..

3.7.1 Back ground of artificial intelligence

The term "Artificial Intelligence" (AI) is used to describe research into human-

made systems that possess some of the essential properties of life. AI includes two-

folded research topic

a) AI studies how computational techniques can help when studying biologicalphenomena.

b) AI studies how biological techniques can help out with computationalproblems.

The focus of this report is on the second topic. Actually, there are already lots of

computational techniques inspired by biological systems. For example, artificial

neural network is a simplified model of human brain; genetic algorithm is inspired by

the human evolution. Here we discuss another type of biological system - social

system, more specifically, the collective behaviors of simple individuals interacting

with their environment and each other. Someone called it as swarm intelligence. All

of the simulations utilized local processes, such as those modeled by cellular

automata, and might underlie the unpredictable group dynamics of social behavior.

Some popular examples are bees and birds. Both of the simulations were created to

interpret the movement of organisms in a bird flock or fish school. These simulations

are normally used in computer animation or computer aided design. There are two

popular swarm inspired methods in computational intelligence areas: Ant colony


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optimization (ACO) and particle swarm optimization (PSO). ACO was inspired by the

behaviors of ants and has many successful applications in discrete optimization

problems. The particle swarm concept originated as a simulation of simplified social

system. The original intent was to graphically simulate the choreography of bird of a

bird block or fish school. However, it was found that particle swarm model could be

used as an optimizer.

3.8 BASIC PARTICLE SWARM OPTIMIZATION

PSO simulates the behaviors of bird flocking. Suppose the following scenario:

a group of birds are randomly searching food in an area. There is only one piece of

food in the area being searched. All the birds do not know where the food is. But they

know how far the food is in each iteration. So what's the best strategy to find the food

is to follow the bird, which is nearest to the food. PSO learned from the scenario and

used it to solve the optimization problems. In PSO, each single solution is a "bird" in

the search space. We call it "particle". All of particles have fitness values, which are

evaluated by the fitness function to be optimized, and have velocities, which direct the

flying of the particles. The particles fly through the problem space by following the

current optimum particles. PSO is initialized with a group of random particles

(solutions) and then searches for optima by updating generations. In every iteration,

each particle is updated by following two "best" values. The first one is the best

solution (fitness) it has achieved so far. (The fitness value is also stored.)

Swarm behavior can be modeled with a few simple rules. Schools of fishes

and swarms of birds can be modeled with such simple models. Namely, even if the

behavior rules of each individual (agent) are simple, the behavior of the swarm can be

complicated. Reynolds utilized the following three vectors as simple rules in the

researches on boid.

Step away from the nearest agent Go toward the destination Go to the center of the swarm

The behavior of each agent inside the swarm can be modeled with simple vectors. The

research results are one of the basic backgrounds of PSO.

Boyd and Richardson examined the decision process of humans and developed

the concept of individual learning and cultural transmission. According to their

examination, people utilize two important kinds of information in decision process.


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The first one is their own experience; that is, they have tried the choices and know

which state has been better so far, and they know how good it was. The second one is

other peoples experiences, i.e., they have knowledge of how the other agents around

them have performed. Namely, they know which choices their neighbors have found

most positive so far and how positive the best pattern of choices was.

Each agent decides its decision using its own experiences and the experiences

of others. The research results are also one of the basic background elements of PSO.

According to the above background of PSO, Kennedy and Eberhart developed PSO

through simulation of bird flocking in a two-dimensional space. The position of each

agent is represented by its x, y axis position and also its velocity is expressed by (the velocity of x axis) and

(the velocity of y axis). Modification of the agent

position is realized by the position and velocity information.

Bird flocking optimizes a certain objective function. Each agent knows its best

value so far (pbest) and its x, y position. This information is an analogy of the

personal experiences of each agent. Moreover, each agent knows the best value so far

in the group (gbest) among pbests. This information is an analogy of the knowledge

of how the other agents around them have performed. Each agent tries to modify its

position using the following information:

The current positions (x, y), The current velocities ( , ) The distance between the current position and pbest The distance between the current position and gbest

This modification can be represented by the concept of velocity (modified value for

the current positions). Velocity of each agent can be modified by the following

equation:

( ) (3.1)where kiv is velocity of agent i at iteration k, w is weighting function, c1

and c2 are weighting factors, rand1 and rand2 are random numbers between 0 and 1,

kis is current position of agent i at iteration k, pbesti is the pbest of agent i, and gbest

is gbest of the group. Namely, velocity of an agent can be changed using three vectors

such like boid. The velocity is usually limited to a certain maximum value. PSO using

eqn. (3.1) is called the Gbest model.


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The following weighting function is usually utilized in eqn. (3.1):

(3.2)Where

maxw is the initial weight,

minw is the final weight, itermax is maximum iteration

number and iter is current iteration number. The meanings of the right-hand side(RHS) of eqn. (3.1) can be explained as follows. The RHS of eqn. (3.1) consists of

three terms (vectors). The first term is the previous velocity of the agent. The second

and third terms are utilized to change the velocity of the agent. Without the second

and third terms, the agent will keep on flying in the same direction until it hits the

boundary. Namely, it tries to explore new areas and, therefore, the first term

corresponds with diversification in the search procedure. On the other hand, without

the first term, the velocity of the flying agent is only determined by using its currentposition and its best positions in history. Namely, the agents will try to converge to

their pbests and/or gbest and, therefore, the terms correspond with intensification in

the search procedure. As shown below, for example,max

w andmin

w are set to 0.9 and

0.4. Therefore, at the beginning of the search procedure, diversification is heavily

weighted, while intensification is heavily weighted at the end of the search procedure

such like simulated annealing (SA). Namely, a certain velocity, which gradually gets

close to pbests and gbest, can be calculated. PSO using eqns.(3.1) & (3.2) is calledinertia weights approach (IWA).

Fig 3.2: concept of modifications of a searching point by PSO

=current searching point= modified searching point= current velocity= modified velocity

= velocity based on pbest

= velocity based on gbest


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The current position (searching point in the solution space) can be modified by the

following equation (3.3):

(3.3)Fig3.2 shows a concept of modification of a searching point by PSO, and it shows asearching concept with agents in a solution space. Each agent changes its current

position using the integration of vectors as shown in Fig3.2.

3.9 PSO ALGORITHM

Step 1:Generation of initial condition of each agent. Initial searching points ( 0is ) and

velocities ( 0iv ) of each agent are usually generated randomly within the

allowable range. The current searching point is set to pbest for each agent. The

best evaluated value of pbest is set to gbest, and the agent number with the

best value is stored.

Step 2:Evaluation of searching point of each agent. The objective function value is

calculated for each agent. If the value is better than the current pbest of the

agent, the pbest value is replaced by the current value. If the best value of

pbest is better than the current gbest, gbest is replaced by the best value and

the agent number with the best value is stored.

Step 3: Modification of each searching point. The current searching point of each

agent is changed using eqns. (3.1), (3.2), and (3.3).

Step 4: Checking the exit condition. The current iteration number reaches the

predetermined maximum iteration number, then exits. Otherwise, the process

proceeds to step 2.


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3.10 FLOW CHART FOR IMPLEMENTATION OF PSO

NO

YES

Fig 3.3 Flow Chart for Implementation of PSO

The features of the searching procedure of PSO can be summarized as follows:

As shown in eqns. (3.1), (3.2), and (3.3), PSO can essentially handlecontinuous optimization problems.

Start

Initialize particles with random position and velocity

vectors

For each particle position (p) evaluate the

fitness

If fitness (p) is better than fitness o (pbest) then P best=p

Set best of pbest as g best

Update particle velocity and position

If gbest is the

optimal

solution

end


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PSO utilizes several searching points, and the searching points gradually getclose to the optimal point using their pbests and the gbest.

The first term of the RHS of eqn. (3.1) corresponds with diversification in thesearch procedure. The second and third terms correspond with intensification

in the search procedure. Namely, the method has a well-balanced mechanism

to utilize diversification and intensification in the search procedure efficiently.

The above concept is explained using only the x, y axis (two-dimensionalspace). However, the method can be easily applied to n-dimensional problems.

Namely, PSO can handle continuous optimization problems with continuous

state variables in an n-dimensional solution space.

Shi and Eberhart tried to examine the parameter selection of the above parameters.

According to their examination, the following parameters are appropriate and the

values do not depend on problems:

The values are also proved to be appropriate for power system problems. The

basic PSO has been applied to a learning problem of neural networks and Schaffer f6,

a famous benchmark function for GA, and the efficiency of the method has been

observed.

3.11 CONCLUSION

In this chapter, the methodology, algorithms and flow charts of EP/PSO

methods for solving optimal power flow problems has been presented.


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CHAPTER 4

MATHEMATICAL

FORMULATION OF OPFPROBLEM

AND

SIMULATION RESULTS


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CHAPTER 4

MATHEMATICAL FORMULATION OF OPF PROBLEM

AND SIMULATION RESULTS

The OPF procedure consists of using mathematical methodology to find the

optimal operation of a power system under feasibility and security constraints. It has

been consider as basic tool for determining secure and economic operating conditions

of power systems. The objective of the work in this chapter is to find out the solution

of nonlinear OPF problem by using PSO algorithm.

4.1 INTRODUCTION

Evolutionary Programming (EP) based approach is proved to be quiteencouraging in solving OPF problem. The EP technique is a stochastic optimization

method in the area of evolutionary computation, which uses the mechanics of

evolution to produce optimal solutions to a given problem. It works by evolving a

population of candidate solutions towards the global minimum through the use of a

mutation operator and selection scheme. The EP technique is particularly well suited

to non-monotonic solution surfaces where many local minima may exist.

Another evolutionary computation technique, called Particle Swarm

Optimization (PSO), has been proposed and introduced. This technique combines

social psychological principles in socio-cognition human agents and evolutionary

computations. PSO has been motivated by the behavior of organisms such as fish

schooling and bird flocking. Generally, PSO is characterized as simple in concept,

easy to impl

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