Economic Operations

8/10/2019 Economic Operations

1/30

EE 220

Economic Operation


2/30

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.


3/30

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


4/30

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)


5/30

Controllable Variables

Generator active power output Generator reactive power output

Transmission/distribution configurationthrough breaker configuration Status of power system components: ON/OFF


6/30

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


7/30

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


8/30

Optimization Techniques

Intelligent Method Neural Networks (NN) Evolutionary Programming (EP) Particle Swarm Programming (PSO)


9/30

Optimization Techniques

Optimization with Uncertainties Probabilistic Optimization Fuzzy Set Analytic Hierarchal Process


10/30

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


11/30


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


12/30


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


13/30


Interior Point Programming Can handle linear and non-linear equations Accuracy is greater than LP Although is harder to formulate


14/30

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


15/30

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


16/30

Unconstrained Optimization

Extreme point of a function f (X) defineseither a maximum or a minimum of thefunction f .


17/30

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


18/30

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


19/30

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


20/30

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


21/30

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


22/30

Lagrange Equation

Where is a Lagrange Multiplier

( , ) ( ) ( ) L x f x g x c


23/30

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


24/30

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


25/30


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


26/30


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


27/30

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


28/30

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


29/30

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


30/30

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.

Date post:	02-Jun-2018
Category:	Documents
Upload:	uplbseles
View:	224 times
Download:	0 times

Economic Operations

Documents