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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);