This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Power Systems I Economic Dispatch l In practice and in power flow analysis, there are many choices for setting the operating points of generators u in the power flow analysis, generator buses are specified by P and V u generation capacity is more than load demand  generators can produce more than the customers can consume n there are many solution combinations for scheduling generation u in practice, power plants are not located at the same distance from the load centers u power plants use different types of fuel, which vary in cost from time to time l For interconnected systems, the objective is to find the real and reactive power scheduling so as to minimize some operating cost or cost function Power Systems I l General cost function: l Unconstrained parameter optimization, from calculus: u the first derivative of f vanishes at a local extrema u for f to be a local minimum, the second derivative must be positive at the point of the local extrema u for a set of parameters, the gradient of f vanishes at a local extrema and to be a local minimum, the Hessian must be a positive definite matrix (i.e. positive eigenvalues) ( ) C x x x f n = , , , 2 1 L Optimization () = x f dx d () 2 2 > x f dx d , , , or , , 1 2 1 = = = = n i x f x f x f f n i x f L L Power Systems I l The Hessian matrix u a symmetrical matrix u contains the second derivatives of the function f u for f to be a minimum, the Hessian matrix must be positive definite u this condition also requires that all the eigenvalues of the Hessian matrix evaluated at the extrema to be positive ( ) [ ] n i x x x n j i , , 1 eigen 1 L L L = < H ( ) j i n i ij x x x x x f H = 1 2 L L Optimization x Hx x > T Power Systems I Example l Find the minimum of u evaluating the first derivatives to zero results in ( ) 110 32 16 8 3 2 , , 3 2 1 3 2 2 1 2 3 2 2 2 1 3 2 1 + + + + + = x x x x x x x x x x x x x f = = = + = = + + =...
View
Full
Document
This note was uploaded on 10/13/2011 for the course EEL 4213 taught by Professor Thomasbaldwin during the Spring '11 term at FSU.
 Spring '11
 THOMASBALDWIN

Click to edit the document details