economic operations
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EE 220
Economic Operation
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Economic Operation of Power System
One of the objective of power system plannersand operators is to minimize the cost of operatinga power system
A power system is composed of severalcomponents:
Generators Transmission Lines Transformers Capacitors, Inductors Other devices such as breakers, synchronous
condensers etc.
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Economic Operation of Power System
However generator units have the majorcontribution in operating costs since fuel isneeded
Fuel can be oil, coal, uranium, natural gas Fuel prices are volatile and dictated by market
forces Although hydro plants are cheaper but its
availability is inferior than those plants thatutilizes conventional fuel
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Objectives of Economic OperationStudy
Optimize certain controllable power systemvariables to achieve a desired objective
The common objectives are: Minimize generator operating cost Minimize copper loss (I 2R) Optimize transmission/distribution configuration
(advance)
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Controllable Variables
Generator active power output Generator reactive power output
Transmission/distribution configurationthrough breaker configuration Status of power system components: ON/OFF
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Optimization Refers to choosing the best element from some set of
available alternatives [1] In the simplest case, this means solving problems in
which one seeks to minimize or maximize a realfunction by systematically choosing the values of realor integer variables from within an allowed set
Although brute -force method can be employed tofind the optimal solution to a problem, however alarge-scale and realistic system has several or hundredsof variables that must be taken into consideration
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Optimization Techniques Conventional Optimization Methods
Unconstrained Optimization Nonlinear Programming (NLP)
Linear Programming (LP) Quadratic Programming (QP) Generalized Reduced Gradient Method Newton Method Network Flow Programming (NFP) Mixed Integer Programming (MIP) Interior Point Programming
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Optimization Techniques
Intelligent Method Neural Networks (NN) Evolutionary Programming (EP) Particle Swarm Programming (PSO)
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Optimization Techniques
Optimization with Uncertainties Probabilistic Optimization Fuzzy Set Analytic Hierarchal Process
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Conventional Optimization Methods
Unconstrained Optimization Serves as basis for constrained optimization
formulation No constraints i.e. transmission limit, generator
limit Approaches gradient method, line search,
Lagrange multiplier method
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Conventional Optimization Methods
Linear Programming Linearization of nonlinear equations are necessary Has two major components:
(1) Objective (2) Constraints
Has reliable convergence Very easy to formulate once linearization is performed Very fast However due to the linearization properties some nonlinear
properties introduce approximation inaccuracies i.e. line losses However its solution/precision is generally acceptable for most
applications Trivia: The Philippine Wholesale Electricity Spot Market uses LP
solution to optimize schedules and derive process
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Conventional Optimization Methods
Nonlinear Programming Directly handles non-linear equations in the
problem solution However it requires a good approximation of a
starting point to start (aids in finding the globalextreme points)
More accurate than LP since little or no
information is lost This is generally slower than LP More complicated to formulate
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Conventional Optimization Methods
Interior Point Programming Can handle linear and non-linear equations Accuracy is greater than LP Although is harder to formulate
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Intelligent Methods
Non-traditional method of finding optimalsolution
Usually simulates a natural event or phenomenon Example of a natural event: Evolution which was
used as a pattern for Evolutionary algorithm(mutation, reproduction, selection etc.).
Usually used for academic and research sincecommercial systems utilizes conventionalmethods
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Optimization with Uncertainties Several events/parameters are probabilistic/uncertain
in nature For Power System: Real-time demand is uncertain but
can be forecasted This type of optimization considers several parameters
as probabilistic inputs to determine a solution Probabilistic inputs are usually modeled using
Probability Distribution Functions (PDF) i.e. NormalCurve
Usually helpful when analyzing possibilities anduncertainties
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Unconstrained Optimization
Extreme point of a function f (X) defineseither a maximum or a minimum of thefunction f .
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f ( x )
x x
1 x
2 x
3 x
4 x
5 x
6a b
A point X 0 = ( x 1 , , x j , , x n) is a maximum if
00 XhX
f f
for all h = ( h1 , , h j , , hn) such that | h j | is sufficiently small for all j .
A point X 0 = ( x 1 , , x j , , x n) is a minimum if
00 XhX f f
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Necessary and Sufficient Conditionsfor Extrema
Assuming that the first and second partial derivatives of f (X) are continuous at every X,
A necessary condition for X0 to be an extreme point of f (X)is that
A sufficient condition for a stationary point X0 to beextremum is that the Hessian matrix H = 2 f (X) evaluated at X0 is
Positive definite when X0 is a minimum point Negative definite when X0 is a maximum point
0X 0 f
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Example
Consider the function
f ( x 1, x 2, x 3) = x 1 + 2 x 3 + x 2 x 3 x 12
x 22
x 32
The necessary condition
0X 0 f
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Example
Solution to 3 unknowns and 3 equations:
022
02
021
323
232
11
x x x f
x x x f
x x f
34
32
21
0 ,,X
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Lagrange Method
A method that provides a strategy in findingthe maximum/minimum of a function subjectto constraints
Named after Joseph Louis Lagrange, an Italianborn mathematician
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Lagrange Equation
Where is a Lagrange Multiplier
( , ) ( ) ( ) L x f x g x c
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Lagrange Method
f(x): is a function representing controllablevariables i.e. cost-function that depends onnumber of controllable output product
g(x): A function representing a set ofconstraints i.e. maximum limit a controllablevariable can be delivered
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Karush Kuhn Tucker conditionsOptimality Conditions
Are necessary conditions for the solution of anonlinear optimization problem to be optimal
Originally developed by Harold W. Kuhn andAlbert W. Tucker and later developed byWilliam Karush
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Karush Kuhn Tucker conditionsOptimality Conditions
0 0 01. , , 0 1i
Li N
x x
02. 0 1i g i N x
03. 0 1i g h i N x
0 00
4. 01
0
i i g
i
g i N
x
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Karush Kuhn Tucker conditionsOptimality Conditions
Condition 1 partial derivative of Lagrangefunction must equal zero at the optimum.
Conditions 2 and 3 restatement ofconstraint conditions.
Condition 4 complementary slacknesscondition. Slack variables or excess is equal tozero in optimal condition
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Example
Consider two thermal plants feeding a given power demand. Thefuel costs of each is related to the output power as follows:
The objective is to minimize the total cost of operation whilesatisfying the equality constraint
2222
2111
03.02
01.04
P P F
P P F
D P P P 21
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Example
The modified cost is
The necessary conditions for minimization are
2 21 2 1 2 1 2 1 2( , , ) 4 2 0.01 0.03 D L P P P P P P P P P
1 21
1
1 22
2
1 21 2
( , , )4 0.02 0
( , , )2 0.06 0
( , , ) D
L P P P
P
L P P P
P
L P P P P P
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Example
For this simple system we are able to eliminate l and P2 to obtain a singleequation in P1 given by
006.0208.0 1 D P P
P D P1 P2
50 12.5 37.5 4.25100 50 50 5
200 125 75 6.5250 162.5 87.5 7.25
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Reference [1] , Wikipedia. [Online] [Cited: May 2, 2010.]
http://en.wikipedia.org/wiki/Optimization_%28mathematics%29.
[2] Zhu, Jizhong., Optimization of Power SystemOperation. New Jersey : John Wiley & Sons, 2009.
[3] Nerves, Allan C., "EE358: Economic Operation ofPower System (Lecture Notes)." Diliman : EEEI,University of the Philippines-Diliman, 2005. Lecture 1.
[4] , Joseph Louis Lagrange. Wikipedia. [Online] [Cited:May 2, 2010.]http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange.