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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
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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