Optimization and Economics of Power Systems

EPE-821
1
Dr. Muhammad Naeem
Optimization and Economics of Integrated
Power Systems

2
Course Outline
Review of Math
Basics of Optimization
Power systems basics
Review of Matlab
Examples of Optimization in Power systems
Graphical Optimization
Optimization Types Constraint and Unconstraint
Optimization Problem Types Linear NonLinear etc
Linear Optimization and Power Systems Applications
Non Linear Optimization and Power Systems Applications
Integer and Mixed integer programming and Power Systems Applications
Complexity Analysis
Quiz next week

3
MATLAB
 MATLAB is an interactive environment
 Commands are interpreted one line at a time
 Commands may be scripted to create your own functions or
procedures
 Variables are created when they are used
 Variables are typed, but variable names may be reused for different
types
 Basic data structure is the matrix
 Matrix dimensions are set dynamically
 Operations on matrices are applied to all elements of a matrix at
once
 Removes the need for looping over elements one by one!
 Makes for fast & efficient programmes

4
MATLAB
Command
Window
Workspa
ce
Command
History

5
MATLAB
 Variables
 Names
 Can be any string of upper and lower case
letters along with numbers and underscores but
it must begin with a letter
 Reserved names are IF, WHILE, ELSE, END,
SUM, etc.
 Names are case sensitive
 Value
 This is the data the is associated to the
variable; the data is accessed by using the
name.
 Variables have the type of the last thing assigned
to them
 Re-assignment is done silently – there are no
warnings if you overwrite a variable with
something of a different type.
 To assign a value to a variable use the
equal symbol ‘=‘
 >> A = 32
 To find out the value of a variable simply
type the name in

7
MATLAB
 A MATLAB matrix is a rectangular array of numbers
 Scalars and vectors are regarded as special cases of matrices
 MATLAB allows you to work with a whole array at a time
You can also use built in functions to create a matrix
 >> A = zeros(2, 4)
 creates a matrix called A with 2 rows and 4 columns
containing the value 0
 >> A = zeros(5) or >> A = zeros(5, 5)
 creates a matrix called A with 5 rows and 5 columns
You can also use:
 >> ones(rows, columns)
 >> rand(rows, columns)
Note: MATLAB always refers to the first value as the
number of Rows then the second as the number of
Columns

8
MATLAB
 The colon : is actually an operator, that generates a row vector
 This row vector may be treated as a set of indices when accessing a
elements of a matrix
 The more general form is
 [start:stepsize:end]
 >> [11:2:21]
 11 13 15 17 19 21
 >>
 Stepsize does not have to be integer (or positive)
 >> [22:-2.07:11]
 22.00 19.93 17.86 15.79 13.72 11.65

9
MATLAB
Mathematical Operators:
 Add: +
 Subtract: -
 Divide: ./
 Multiply: .*
 Power: .^ (e.g. .^2 means squared)
You can use round brackets to specify the order in
which operations will be performed
Note that preceding the symbol / or * or ^ by a ‘.’
means that the operator is applied between pairs of
corresponding elements of vectors of matrices

10
MATLAB
 Combining this with methods from Accessing Matrix Elements
gives way to more useful operations
>> results = zeros(3, 5)
>> results(:, 1:4) = rand(3, 4)
>> results(:, 5) = results(:, 1) + results(:, 2) + results(:, 3) + results(:, 4)
or
>> results(:, 5) = results(:, 1) .* results(:, 2) .* results(:, 3) .* results(:, 4)

11
MATLAB
Logical Operators:
 Greater Than: >
 Less Than: <
 Greater Than or Equal To: >=
 Less Than or Equal To: <=
 Is Equal: ==
 Not Equal To: ~=
For example, you can find data that is above a certain limit:
>> r = results(:,1)
>> ind = r > 0.2
Boolean Operators:
AND: &
OR: |
NOT: ~

12
MATLAB
 There are a number of special functions that provide useful constants
 pi = 3.14159265….
 i or j = square root of -1
 Inf = infinity
 NaN = not a number
 Passing a vector to a function like sum, mean, std will calculate the
property within the vector
 >> sum([1,2,3,4,5])
 = 15
 >> mean([1,2,3,4,5])
 = 3
 >> max([1,2,3,4,5])
 = 5


13
MATLAB
 The plot function can be used in different ways:
 >> plot(data)
 >> plot(x, y)
 >> plot(data, ‘r.-’)
 In the last example the line style is defined
 Colour: r, b, g, c, k, y etc.
 Point style: . + * x o > etc.
 Line style: - -- : .-
A basic plot
 >> x = [0:0.1:2*pi]
 >> y = sin(x)
 >> plot(x, y, ‘r.-’)
0 1 2 3 4 5 6 7
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1

14
MATLAB
>> x = [0:0.1:2*pi];
>> y = sin(x);
>> plot(x, y, 'b*-')
>> hold on
>> plot(x, y*2, ‘r.-')
>> title('Sin Plots');
>> legend('sin(x)', '2*sin(x)');
>> axis([0 6.2 -2 2])
>> xlabel(‘x’);
>> ylabel(‘y’);
>> hold off
0 1 2 3 4 5 6
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Sin Plots
x
y
sin(x)
2*sin(x)

15
MATLAB
 For command
 Use a for loop to repeat one or more statements
 The end keyword tells Matlab where the loop finishes
 You control the number of times a loop is repeated by defining the
values taken by the index variable
 This uses the colon operator again, so index values do not need
to be integer
 For example
 >> for i = 1:4
 a(i) = i * 2
 end

16
MATLAB
 The counter can be used to index different rows or columns
 E.g.
 >> results = rand(10,3)
 >> for i = 1:3
 m(i) = mean(results(:, i))
 end
 ..although you could do this in one step
 m = mean(results);

17
MATLAB
 The ‘if’ command is used with logical operators
 Again, the end command is used to tell Matlab where the statement
ends.
 For example, the following code loops through a matrix performing
calculations on each column
 >> for i = 1:size(results, 2)
 m = results(:, i)
 if m > 1
 do something
 else
 do something different
 end
 end

Calculus for Continuous Optimization
Calculus and Analytic Geometry By Thomas and Finney 9th
Edition
Introduction of Optimum Design By Jasbir S. Arora
Some figure from Web
http://www.ece.mcmaster.ca/~xwu/part4.pdf
18

Calculus and Analytic Geometry By Thomas and Finney 9th Edition
Shifting of Graph
19

Shifting of Graph
20

Shifting of Graph
21

Shifting of Graph
22

Slope
23

Slope
24

Tangent and Normal
25
We make use of the fact that if two lines with
gradients m1 and m2 respectively are
perpendicular, then m1 m2 = −1.
The equation of the tangent line to y = f(x) at the point (x1,y1):
( 'm' is the gradient at (x1,y1) )
The equation of the normal line to y = f(x) at (x1,y1) is:
The derivative of y = f(x) at (x1,y1)
gives us the gradient 'm'.

Max/Min
26

Rolle’s Theorem
27

28

Increasing/Decreasing Functions
29

First Derivative
30

First Derivative test
31

Second Derivative Test
32

33
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
-20
-15
-10
-5
0
5
10
15
20
x4
- 4*x3
+ 10
Orig
1st Derv.
2nd Derv.

34

What Derivative Tells us About any Graph
35
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
-20
-15
-10
-5
0
5
10
15
20
x4
- 4*x3
+ 10
Orig
1st Derv.
2nd Derv.
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
-20
-15
-10
-5
0
5
10
15
20
-x4
+ 4*x3
- 10
Orig
1st Derv.
2nd Derv.

36
0 2 4 6 8 10 12 14 16 18 20
-1
0
1
2
3
4
5
exp(x/15)+sin(x)
Orig
1st Derv.
2nd Derv.
0 1 2 3 4 5 6 7
-1000
-500
0
500
1000
1500
x4
+500 sin(x)
Orig
1st Derv.
2nd Derv.
-4 -3 -2 -1 0 1 2 3 4 5
-20
-10
0
10
20
30
40
50
60
70
80
exp(3*x/2)+2x2
Orig
1st Derv.
2nd Derv.

Taylor Series
37

38

39

40

41
Taylor’s expansion for f(x) about the point x* is
where R is the remainder term that is smaller in magnitude than the
previous terms if x is sufficiently close to x*.
Summation notation
For Two Variable
Matrix notation

42
Often a change in the function is desired when x* moves to a
neighboring point x. Defining the change as Δf=f(x)-f(x*)
A first-order change in f(x) at x* (denoted as δf ) is obtained by retaining
only the first term
where δx is a small change in x* (δx=x-x*). Note that the first-order
change in the function given in above equation is simply a dot product
of the vectors rf and δx. A first-order change is an acceptable
approximation for change in the original function when x is near x*.

Gradient
43
The gradient of a line is a specific term for the steepness of a straight line.
gradient =
vertical change
horizontal change
Gradient AB = vertical
horizontal
=
6
4
= 1.5
4
6
A
B
Note: A negative gradient means that the
line is travelling downhill or a decline.

44
Gradient from co-ordinates
This can be achieved in two ways:
1. Draw the co-ordinates on a grid and use the
previous method.
Gradient = Change in y
Change in x
Vertical
horizontal
=
Pairs (1,2) and (5,18)
Y2 - Y1
X2 - X1
=
m 18 - 2
5 - 1
= =
16
4
= 4
(X2,Y2)
Y2 - Y1
(X1,Y1) X2 - X1

45
 Notations and definitions:
 Let x = (x1,…,xn) be the vector of design variables.
 The gradient vector, f, and Hessian, 2f, of f(x) are the
column vector and nn symmetric matrix defined by
.
,
2
2
2
2
1
2
2
2
2
2
2
1
2
2
1
2
2
1
2
2
1
2
2
1










































































n
n
n
n
n
n
x
f
x
x
f
x
x
f
x
x
f
x
f
x
x
f
x
x
f
x
x
f
x
f
f
x
f
x
f
f









46
Geometrically, the gradient vector is normal to the tangent plane at the
point x*

47

48

49

50
-8 -6 -4 -2 0 2 4 6 8
-20
-15
-10
-5
0
5
X: 0.01681
Y: 0.9999
x
cos(x)
Taylor Approx.

51

52
0
0.5
1
1.5
2
0
0.5
1
1.5
2
-10
0
10
20
30
40
X: 1
Y: 1
Z: 3
9X2
+9XY-18X-6Y+9
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
10
20
30
40
50
X: 1
Y: 1
Z: 3
3 X.3
Y

53

Assignment 1 Problem 1
54
Plot the functions of above examples
and their Taylor approximations in Matlab
For example 4.7 x* = 2
For example 4.8 x* = (1,2)
Due in two weeks

Some Objective Functions
55
 Monotonic and unimodal functions
 Monotonic:
 Unimodal:
ƒ(x) is unimodal on the interval if and only if it is
monotonic on either side of the single optimal point x* in the
interval.
A monotonic increasing function Unimodal function

56
0 0.5 1 1.5 2 2.5 3
0
1
2
3
4
5
6
(i) a discrete function (ii) a discontinuous function
Y
x
(iii) a continuous function
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-40
-20
0
20
40
60
80
100

57
Functions can be categorized as
 Separable or nonseparable – for example, (x+y) vs. xy
 Regular or irregular – for example, x vs. abs(x)
 Unimodal or multimodal – for example, x2 vs. cos x

58
nonseparable, regular, unimodal
Step function:
separable, irregular, unimodal
2
1
]
)
5
.
0
(
[floor
)
( 



n
i
i
x
x
f
Rastrigin:
nonseparable, regular, multimodal





n
i
i
i x
x
n
x
f
1
2
)]
2
cos(
10
[
10
)
( 

59
Rosenbrock:
nonseparable, regular, unimodal



 



1
1
2
2
2
1 ]
)
1
(
)
(
100
[
)
(
n
i
i
i
i x
x
x
x
f
Schwefel 2.22:
nonseparable, irregular, unimodal

 



n
i
i
n
i
i x
x
x
f
1
1
|
|
|
|
)
(
Schwefel 2.26:
separable, irregular, multimodal




n
i
i
i x
x
x
f
1
|
|
sin
)
(

60
Plot the functions in matlab
Step function
Rastrigin
Rosenbrock
Schwefel 2.22
Schwefel 2.26
Due in two weeks

Contour Plot
61
A level curve: Each horizontal plane z=k intersects the surface in a curve.
The projection of this curve on the xy-plane is called a level curve

Contour Plot
62
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
z = x2
+y2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2
-1
0
1
2
-2
-1
0
1
2
0
2
4
6
8
z = x2
+y2
-2
-1
0
1
2
-2
-1
0
1
2
-4
-2
0
2
4
z = -x2
+y2
-
3
-
3
-
3
-
3
-
2
-
2
-
2
-
2
-
1
-
1
-1
-
1
-
1
-1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
z = -x2
+y2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2

Contour Plot
63
-2
-1
0
1
2
-2
-1
0
1
2
-1
-0.5
0
0.5
1
z = cos(x2
+y2
)
-0.8
-0.8
-0.8
-0
.8
-0.8
-
0
.
8
-0.8
-
0
.
8
-
0
.
8
-0.8
-0.6
-0.6
-0
.6
-
0
.
6
-0.6 -0.6
-
0
.
6
-0.6
-
0
.
6
-0.4
-0.4
-0.4
-0.
4
-0.4
-0.4
-0.4
-0.4
-0
.2
-0.2
-0.2
-0.2
-
0
.2
-0.2
-0.2
-
0
.
2
0
0
0
0
0
0
0 0
0
0
.2
0.2
0.2
0
.
2
0.2
0.2
0
.
2
0.
4
0.4
0.4
0
.
4
0.4
0.4
0
.
4
0.6
0.6
0.6
0
.
6
0.6
0.6
0.8
0.8
0
.
8
0.8
0.8
z = cos(x2
+y2
)
-1.5 -1 -0.5 0 0.5 1
-1.5
-1
-0.5
0
0.5
1
-2
-1
0
1
2
-2
-1
0
1
2
-1
-0.5
0
0.5
1
z = sin(x2
+y2
)
-0.8
-0.6
-0.6
-0.6
-
0
.
4
-0.4
-0.4
-
0
.
2
-
0
.
2
-0.2
-0.2
0
0
0
0
0
.
2
0.2
0.2
0.2
0.2
0
.
2
0
.2
0
.
4
0.4
0
.
4
0.4
0
.
4
0.4
0
.
4
0.4
0
.
4
0.6
0
.
6
0.6
0
.
6
0.6
0
.
6
0
.
6
0
.
6
0.6
0.
6
0.6
0
.
6
0.8
0
.8
0
.
8
0.8 0.8
0
.
8
0.8
0
.
8
0.8
0.8
0.8
0.8
0.8
0
.
8
z = sin(x2
+y2
)
-1.5 -1 -0.5 0 0.5 1
-1.5
-1
-0.5
0
0.5
1

Contour Plot
64
-2
-1
0
1
2
-2
-1
0
1
2
0
2
4
6
8
10
z = exp(x)+exp(y)
1
1
2
2
2
2
3
3
3
3
4
4
4
4
4
5
5
5
6
6
7
8
z = exp(x)+exp(y)
-1.5 -1 -0.5 0 0.5 1
-1.5
-1
-0.5
0
0.5
1
-4
-2
0
2
4
-4
-2
0
2
4
-20
-10
0
10
20
z = xy
-10
-10
-
5
-5
-
5
-5
0
0
0
0
0 0
5
5
5
5
10
10
z = xy
-4 -3 -2 -1 0 1 2 3 4
-4
-3
-2
-1
0
1
2
3
4

Contour Plot
65
-
1
.
5
-1
.5
-
1
.
5
-1
.5
-1
.5
-1
.5
-
1
-1
-
1
-
1
-1
-
1
-
1
-
1
-
0
.
5
-0.5
-0.5
-
0
.
5
-0.5
-0
.5
-0.5
-0.5
0
0
0
0
0
0
0
0
0
0
0
0.5
0.5
0
.
5
0.5
0
.
5
0.5
0.5
1
1
1
1
1
1
.
5
1.
5
1
.
5
1.5
Z = sin(X)+cos(Y)
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
-4
-2
0
2
4
-4
-2
0
2
4
-2
-1
0
1
2
Z = sin(X)+cos(Y)
-4
-2
0
2
4
-4
-2
0
2
4
-2
-1
0
1
2
3
Z = sin(X)+cos(Y)+0.25*rand
-
1
.
5
-1.5
-1.5
-
1
.
5
-1
-1
-1
-
1
-1
-
1
-1
-
1
-
0
.
5
-0.5
-0
.5
-
0
.
5
-0.5
-
0
.
5
-0.5
-0.5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.5
0.5 0.5
0.5
0
.5
0.5
0
.
5
1
1
1
1
1
1
1
.
5
1.5
1
.
5
1
.
5
2
2
2
Z = sin(X)+cos(Y)+0.25*rand
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3

66
Plot the contour of Step function
Rastrigin
Rosenbrock
Schwefel 2.22
Schwefel 2.26
In Matlab
Due in two weeks

Necessary and Sufficient Conditions
67
Example: If a number is divisible by 4(call this H), then it is divisible by 2
(call this C). H implies C but C is not strong enough to imply H for example
6 is divisible by 2 but not by 4
Therefore C is necessary for H but not sufficient for H.
Example: In Euclidean geometry, a triangle has equal sides (call this H) if
and only if the triangle has equal angles (call this C). This means
H if C: H is necessary for C since C implies H
H only if C: C is necessary for H since H implies C
H if and only if C: H and C imply each other and are both necessary and
sufficient conditions for each other.

68
Unconstraint Optimization
One-dimensional unconstrained optimization
– Analytical method
– Newton’s method
– Golden-section search method
Multidimensional unconstrained optimization
– Analytical method
– Gradient method—steepest ascent (descent) method
– Newton’s method

Newton’s Method
70

71
Newton’s Method

Title
72

Title
73

Optimization and Economics of Power Systems

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