myr power system optimization

8/13/2019 MYr Power System Optimization

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Chapter 1

The realm and Concept of Power

System Optimization


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Contents

Introduction

Types of optimization problems

Types of optimization techniques

Nonlinear Programming

Classification of NLP

Unconstrained optimization Techniques

Constrained optimization techniques Modern optimization techniques


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What is Optimization?

Optimizationis the mathematical discipline which is

concerned with finding the maxima and minima of

functions, possibly subject to constraints.

Optimize means make as perfect, effective orfunctional as possible

Used to determine best solution without actually

testing all possible cases


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Need for optimization in power system

Power system operation is required to be

Secure

Economical

Reliable

All operations have to operate at optimum point

Power flow analysis

Economic Dispatch

Reactive power

Load shedding

Configuration of electrical distribution networks etc


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Statement of optimization problem

General Statement of a Mathematical

Programming Problem

Find x which minimize: f(x)Subject to: gi(x) < 0 for i = 1, 2, ..., h

li(x) = 0 for i = h+1, ..., m

f(x), gi(x) and li(x) are twice continuouslydifferentiable real valued functions.

gi(x) is known as inequality constraint

li(x) is known as equality constrain


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X can be a vector with several

variables

Minimization of f(x) is same as

maximization off(x)


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Some terminologies

Design vector

Design constraints

Constraint surface

Objective function

Mathematical programming


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Design vector

Two types of variables

exist

-Pre-assigned variables

-Variables whose value isknown before hand

-Design vector

-Vector of decision

variables-Should be calculated using

techniques

Design space- space of

the design vector

2

1

121

x

xx

x2xx82.9)x(f

x2

x1

Design vectorDesign space


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Design constraints

Restrictions on variables Behavioral constraints

Power cannot be

negative

Geometric constraintsConstraints due to

geometry

Find x which minimize: f(x)

Subject to:

gi(x) < 0 for i = 1, 2, ..., h

li(x) = 0 for i = h+1, ..., m

Where:

gi(x) is inequality constrain

li(x) is equality constraint


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Constraint surface

Set of values which

satisfy a single

constraint

Plot of gi(x)=0 Four possible points

Free and acceptable

Free and unacceptable

Bound acceptable

Bound unacceptable


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Example

Draw constraint surface for problem of minimization

0)250(8

)xx)(10x5.8(

xx

2500

0500xx

2500

8.0x2.0

14x2

x2xx82.9)x(f

2

2

2

2

1

52

21

21

2

1

121

Subject to

2 4 6 8 10 12 140.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x2


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Objective function

The function which gives

the relation between theobjective we want to

achieve and the variables

involved

Single or multiple

Exampleeconomic

dispatch problem

Minimize operating cost

Variablespower outputof each generator

Constraint- system load

demands, generating

capacity of generators


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Objective function

Example: A power

company operates two

thermal power plants A

and B using threedifferent grades of coal

C1, C2 and C3. The

minimum power to be

generated at plants Aand B is 30MWh and

80MWh respectively.

Amount of coal required

Write the objective functionto min cost


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Objective function

surface

Locus of all points

satisfying f(x)=C for someconstant C


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2 4 6 8 10 12 140.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x2

Red lines are

objective function

surfaces for C=50 and

C=30


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Classification of optimization problems

Based on

existence of

constraints

Constrained optimization

Formulation

Min F(X)

subject to

Gj(X)0

Unconstrained optimization

Formulation

Min F(X)


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Classification cont

Based on

nature of

design

variables

Static

Design variables are simple variables

Dynamic

Design variables are function of other

variables


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Classification cont

Based on

expression of

objective

function or

constrains

Geometric programming objective

function and/or constraint are expressed

as power terms

Quadratic programming

Special case of NLP

Objective function is quadratic form


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Classical Optimization techniques

Used for continuous and differentiable functions

Make use of differential calculus

disadvantages

Practical problems have non differentiable objective

functions


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Single variable optimization

Local minima

If for small

positive and negative h

Local maxima

If for small

positive and negative h

Local

minima

Local

maxima


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Single variable cont

Theorem 1

Theorem 2


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Example

5x40x45x12)x(f 345

-1 -0.5 0 0.5 1 1.5 2 2.5 3-100

-50

0

50

100

150

200

250

300

350

400

Soln.

Find f(x) and then equate it with zero. The

extreme points are x=0,1 and 2

X=0 is inflection point, x=2 is local minima

and x=1 is local maxima


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Multivariable optimization

Without constraint and With constraint

Has similar condition with single variable case

Theorem 3

Theorem 4


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Example: find extreme points of

Find extreme points

Check the Hessian

matrix by determining

second derivatives and

determinants

Function of two variable

Soln.

Evaluate first partial

derivatives and


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Multivariable optimization with equality

constraints

Problem formulation

Find which minimizes F(x) subject to the constraint

gi(x)=0 for i=1,2,3, m where mn

Solution can be obtained using

Direct substitution

Constrained variation and Lagrangian method


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Direct substitution

Converts constrained

optimization to

unconstrained

Used for simplerproblems

Technique

Express the m constraint

variables in terms of the

remaining n-m variables Substitute into the objective

function


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Direct substitution cont

Examplefind the values of x1,x2 and x3 which

maximize

subject to the equality constrain

Soln. Re-write the constraint equation to eliminate any

one of the variables

then

23212 xx1x 232131321 xx1xx8)x,x,x(f


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constrained variation method

Finds a closed form expression for the first order

differential of f at all points where the constraints are

satisfied

Example: minimize

Subject to


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Constrained variation

At a minimum

If we take small

variations dx1and dx2

After Taylor series expansion

Rewriting the equation

Substituting

Necessary condition


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Constrained variation

For general case

Under the assumption that


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Example minimize the following function subject to given constraint

Minimize

Subject to

Soln. The partial

differentials are Using the necessary condition


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Method of Lagrangian multipliers

Problem formulation

s.t.

Procedure:

A function L can be formed as

Necessary conditions for extreme are given by


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Lagrange Multiplier method cont

For a general case L


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Lagrangian method

Sufficient condition is

Has to be positive definite matrix


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Example: find maximum of f given by

Subject to

Soln. The Lagrangian is Giving


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Formulation of multivariable

optimization

When the constraints are inequality constraints, i.e.

It can be transformed to equality by adding slack

variable

Lagrangian method can be used


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Kunh- Tucker conditions

The necessary condition for the above problem is

When there are both equality and inequality constraints, the KTcondition is given us


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Kuhn-Tucker conditions cont

Example: For the following problem, write the KT

conditions

Subject to


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Linear programming

History

George B. Dantzing

1947, simplex method

Kuhn and Tuckerduality

theory

Charles and Cooper -

industrial application

Problem statement

Subject to the constraint


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Properties of LP

The objective function is

minimization type

All constraints are linear

equality type

All decision variables are

nonnegative

Transformations If problem is maximization

usef

If there is negative variable,write it as difference of two

If constraint is inequality, addslack or surplus variables


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Simplex algorithm Objective of simplex algorithm is

to find vector X 0 which

minimizes f(X)

and satisfies equality constraints

of the form

Algorithm

1. Convert the system of

equation to canonical form

2.Identify the basic solution

3. Test optimality and stop ifoptimum

4. If not, select next pivotal

point and re-write

equation5. Go to step 2


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Simplex algorithm


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Example: Maximize

Solution:

Step 1.convert to canonical form

Use tabular method to proceed on the

algorithm

Subject to


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Basic variable means those variables having coefficient 1 in one of the equation and zero in

the rest of the equations


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Various types of solutions

Unbounded solution

If all the coefficients of the entering variable are

negative

Infinite solution

If the coefficient of the objective function is zero at an

optimal solution


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Modifications to simplex method

Two phase methodWhen an initial feasible

solution is not readilyavailable

Phase I is for rearrangingthe equations

Phase II is solving

Revised simplex method

Solve the dual of the basic

solution


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Using MATLAB to solve LP

Example:

Subject to

Soln.

1. Form the matrices containing coefficients of the objective function, constraints

equations and constants separately

2. Use the built in function linprog() [x, fmin]=linprog(f,A,b,[],[],lb);

f=[5 2];

A=[3 4;1 -1;-1 -4;-3 -1];

b=[24;3;-4;-3];

lb=zeros(2,1);

myr power system optimization

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