optimal placement of capacitor using fuzzy reasoning and genetic algorithm in distribution system

8/14/2019 optimal placement of capacitor using fuzzy reasoning and genetic algorithm in distribution system

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OPTIMAL CAPACITOR ALLOCATION USING FUZZY REASONING AND GENETIC ALGORITHMS FORDISTRIBUTION SYSTEM

UNDER THE ESTEEMEDGUIDANCE OFMs. K.NEELIMA, Assoc. Prof.(EEE

Dept)



Capacitors have been very commonly employedto provide reactive power compensation todistribution systems.

Capacitors are used to minimize the power andenergy losses and to enhance the voltage profile.

We represent an optimization method which usesfuzzy reasoning and genetic algorithm forcapacitor placement.



Fuzzy reasoning finds the sensitive nodes.

Three membership functions are defined for realpower loss, reactive power loss and voltage deviation,respectively.

A high power loss section of the feeder is given alow membership value, while a low power loss sectionof the feeder is given a high membership value.

The voltage deviation can be similarly defined, a buswith high voltage deviation is given a low membershipvalues.



The alpha-cut operation of a fizzy set is used to obtain the

candidate location for capacitor installation.

The genetic algorithm determines the capacitor size andtype for installation.

In genetic algorithm application, the fitness function foreach string of the population is defined as the objectivefunction of the system model, which is composed of the

peak power losses and cost of capacitors added.



The solution procedures start off with performing a load flowstudy to calculate the bus voltages and line losses.

Determination of candidate locations for capacitors sitting isperformed by way of a membership function approach.

If a node whose voltage is not kept within its limits, the solution isdiscarded; otherwise the solution is accepted.

Subsequently the peak power loss and overall annual savings arecomputed and compared with that of the previous selection.

The procedures are repeated until an optimal capacitor

allocation scheme is achieved.



POWER CAPACITOR IN

SERVICE



The power capacitor can be considered to be a

VAR-GEN (reactive power Source), since it actuallysupplies needed-magnetizing current requirementsfor inductive loads.

The fundamental function of power capacitor is toprovide needed reactive power compensation.



Power factor correction

Feeder-Loss Reduction

Release of System capacity

Voltage- Stabilization/RegulationEfficient Power Utilization

Power Quality Enhancement

Power Harmonic Filtering



The Capacitor Location or Placement for low voltagesystems determines capacitor type, size, location andcontrol schemes.

Optimal capacitor placement is generally a hardcombinatorial .optimization problem that can beformulated as a nonlinear/Search Minimization problem.



Almost all the methods to solve capacitor placement problems arebased on the historical data of the load models and associated costof the energy and the cost of capacitor banks.

Cost $/Kvar for Power savings and Losses (Power losses/Energylosses



Historical Data and Load models are uncertain and may

change in reality.

To account for such load model and load pattern/cyclesuncertainties Soft-Computing AI Based algorithms using fuzzysets/Neural networks/Genetic Algorithm can be utilized.



In general, capacitor placement problems canbe solved in two steps:

1.Use of load flow model and find the V,P,Q atall the buses and also the feeder losses

2.Minimize the cost function-Jo-min - subject toconstraints, like practical limits of voltage andcapacitor size!



Line lossevaluation



The power loss of the line section connectingbuses i andi+1 may be computed as

The total power loss of the feeder, PT,Loss is givenby



Overview

Fuzzy setsFuzzy logic and rules An example of fuzzy rules

Uncertainty revisited



Crisp Sets

• A set with a characteristic function iscalled

crisp

• Crisp sets are used to formallycharacterize a

concept, e.g., even numbers

• Crisp sets have clear cut boundaries,hence do

not reflect uncertainty about membership



• Zadeh (1965) introduced “Fuzzy Sets” where he

replaced the characteristic function with membership

• χ S: U → {0,1} is replaced by mS: U → [0,1]

• Membership is a generalization of characteristic

function and gives a “degree of membership”

• Successful applications in control theoretic settings

(appliances, gearbox)

Fuzzy Sets



Example: Let S be the set of people of normal

Height

• Normality is not a crisp concept



• Support in U SupportU(S) = {x ∈ U | mS(x) >

0}

• Containment A ⊆ B if and only if mA(x) ≤

mB(x) for all x ∈ U



• Union mA∪B(x) = max(mA(x), mB(x))

• Intersection mA∩B(x) = min(mA(x), mB(x))

• Complementation mU-A(x) = 1 -mA(x)

• Note that other definitions exist too



• The fuzzy relation R between Sets X and

Y isa fuzzy set in the Cartesian product X×Y

• mR: X × Y → [0,1] gives the degree towhich x

and y are related to each other in R.



• Two fuzzy relations R in X × Y and S in Y

× Z

can be composed into R°S in X × Z as

mR°S(x,z) = maxy∈Y[min[mR(x,y),

mS(y,z)]]



• “Probability of cold weather tomorrow”

• U = {x1, x2, …, xn}, p is a probability density,

A is a fuzzy set (event) in U

P( A) =Σi=1 to n mA ( xi) p( xi)



• Finding a single representative for a fuzzy set

A in U = {xi|i in {1,…n}}• Max: x in U such that mA(x) is maximal

• Center of gravity:



• A is a fuzzy set in U

• Aa = {x | mA(x)≥a } is the a-cut of A in U

• Strong a-cut is Aa = {x | mA (x)> a }

• Alpha cuts are crisp sets



• Different views

– Foundation for reasoning based on uncertain

statements

– Foundation for reasoning based on uncertain

statements where fuzzy set theoretic tools are used

(original Zadeh)

– As a multi valued logic with operations chosen in a

special way that has some fuzzy interpretation



• Generalization of proposition over a set

• Let Xs:U → {0,1} denote the characteristic

function of the set S

• Recall that in “crisp” logic

I(p(x)) = p(x) = XT(p)(x)

where p is a proposition and

T(p) is the corresponding truth set



We extend the proposition

p:U → {0,1}

to be a fuzzy membership

p:U → [0,1]• The fuzzy set associated with

p corresponds to the truth set T(p) and

p(x) is the degree of truth of p for x

• We extend the interpretation of logical formulaeanalogously

to the crisp case



• Basic operations:

–I(p(x)) = p(x)

– I(α ∨ β) = max (I(α),I(β))– I(α ^ β) = min (I(α),I(β))

– I(~ α) = 1 – I(α)



• “If x in A then y in B” is a relation R

betweenA and B

• Two model types

– Implicative: (x in A → y in B) is an upperbound

– Conjunctive: (x in A ^ y in B) is a lowerbound



• Fuzzy sets can be said to model inherent

vagueness

Bob is "tall" -vagueness in the meaning

of "tall", not in Bob's height



Mathematical ProgrammingMathematical Programming

Network AnalysisNetwork Analysis

Branch & BoundBranch & Bound

Genetic AlgorithmGenetic Algorithm

Simulated AnnealingSimulated Annealing

Tabu Search Tabu Search



42

Genetic Algorithms (GA)OVERVIEW

A class of probabilistic optimization algorithms

Inspired by the biological evolution process

Uses concepts of “Natural Selection” and “Genetic Inheritance” (Darwin 1859)

Originally developed by John Holland (1975)

Genetic Algorithms follow the idea of SURVIVAL OF THE FITTEST-Better and better solutions evolve from previous generationsuntil a near optimal solution is obtained.



43

GA overview (cont)Particularly well suited for hard problems where

little is known about the underlying search spaceWidely-used in business, science and engineering

Based on Darwinian’s principle of evolution



A process called natural selection, ‘selects’ individualsbest adapted to the environment.

Those fittest survive longest.

Characteristics, encoded in genes are transmitted tooffspring and tend to propagate into new generations.

In sexual reproduction, the chromosomes of offspringare a mix of their parents.

An offspring’s characteristics are partially inheritedfrom parents and partly the result of new genes createdduring the reproduction process.



Chromosome often encoded as a bit string, represent a

candidate solution in the population.

Genes are either single bits or short blocks of adjacentbits that encode a particular element of the candidatesolution.

Alleles are 0’s or 1’s in a bit string.



Nature ComputerIndividual

Population

Fitness

ChromosomeGene

Crossover and

MutationNatural Selection

Solution to a problem

Set of solutions

Quality of a solution

Encoding for a solutionPart of the encoding of a

solution

Search operators

Reuse of good (sub-)

solutions



Genetic AlgorithmGenetic AlgorithmBased on Darwinian Paradigm

Intrinsically a robust search and optimization mechanism

ReproductionCompetition

SelectionSurvive



A Population of chromosomes.

A Fitness Function.Genetic Operators

- Selection

- Crossover

- Mutation



Alternate solutions are too slow or overlycomplicated

Need an exploratory tool to examine newapproaches

Problem is similar to one that has already beensuccessfully solved by using a GA

Want to hybridize with an existing solution

Benefits of the GA technology meet key problemrequirements



Multiple solutions can be obtained without extra effort.

GAs are implicitly parallel and can be implemented onparallel machines.

GAs are quite successful in locating the regionscontaining optimal solution(s), if not the optimumsolution itself.

GAs can solve problems involving large time domain.



GAs work with a population of candidate solutions andnot a single point.

GAs work with coding of parameters instead of

parameters themselves.GAs do not require any domain knowledge (gradientinformation etc.) and just use the payoff information.

GAs are stochastic methods, i.e., use probabilistic

transition rules and not deterministic ones.Applies to a variety of problems and not works in arestricted domain.



Initialise and evaluate a population

While (termination condition not met) doo Select sub-population based on fitness

o Produce offspring of the population using crossover

o Mutate offspring stochastically

o Select survivors based on fitness

Flow Diagram of theFlow Diagram of the



Flow Diagram of theFlow Diagram of the

Genetic AlgorithmGenetic Algorithm

ProcessProcessDescribe

Problem

GenerateInitialSolutions

Test: is initialsolution goodenough?

Stop

Select parentsto reproduce

Apply crossover processand create a set of offspring

Apply random mutation

Step 1

Step 2

Step 3

Step 4

Step 5

Yes

No



PROBLEM FORMULATION

The total power loss is given by

Considering shunt capacitors, there exists a finitenumber of standard sizes which are integer multiples of the smallest size Qc . Besides, the cost per kvar varies

from one size to another. In general, larger-sizecapacitors have lower unit prices than smaller oneshave. The available capacitor size is usually limited to



where L is an integer. Therefore for each installation

location, we have L capacitor sizes to choose from.

Following the description aforementioned, the total annual costfunction due to capacitor placement and power loss change iswritten as:

The available capacitor size is usuallylimited to



where CL is the equivalent annual cost per unit of and j=1,2,...,J are the indices of buses selected for

compensation. The objective is to minimize the cost functionsubject to

where Vmin and Vmax are the permissibleminimum and maximum bus voltages, respectively, and n is thetotal number of buses. The problem mentioned above is an nonlinear optimization problem. In this case, the objective function and

constraints can be computed for each possibility. Then, the globaloptimum solution is simply the one which satisfies the constraintswith the least cost. However, as the number of possibilitiesincreases the computational requirements become unacceptable.



THE PROPOSED

METHOD• Three membership functions are defined to enablethe proposed method to apply fuzzy technique.

• The membership function Uvi for voltage of bus

i(Vi,pu) is defined as



where Mv is a constant, and αγ is the level for the α-cut operation of the fuzzy set.

The second membership function UPi defined for real power lossof the line section between buses i and i+l (Pi,Loss) is depicted as



where CP is a constant, and is the level for the α-cut operation of thefuzzy set.

The third membership function UQi defined for the reactive

power loss of the line section between buses i and i+1(Q,,Loss) isdepicted



where CQ is a constant, and is the level for the

-cut operation of the fuzzy set.



FIS Editor:Untitled



FIS Editor



Membership funtioneditor



Membership funtion editor



RULE EDITOR FOR FUZZY GUI INMATLAB



Rule viewer



1.Optimize the cost using GA in MATLAB

2.SOLVING LOAD-FLOW USING C Language(or)MATLAB



The repeated simulation results could be used todevelop a Model using any artificial intelligencetechnique which can accurately predict the locationand size of capacitor for any load conditions whichgives a great promise for practical

implementation of the proposed technique.

GENETIC



GENETIC

ALGORITHMS

optimal placement of capacitor using fuzzy reasoning and genetic algorithm in distribution system

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