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lecture7 - Power Systems I Economic Dispatch l In practice...

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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) ( 29 C x x x f n = , , , 2 1 L Optimization (29 = x f dx d (29 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 ( 29 [ ] n i x x x n j i , , 1 eigen 1 L L L = < H ( 29 j i n i ij x x x x x f H = 1 2 L L Optimization x Hx x 2200 T Power Systems I Example l Find the minimum of u evaluating the first derivatives to zero results in ( 29 110 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 = = =- + = =- + + =...
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This note was uploaded on 11/27/2011 for the course EEL 4213 taught by Professor Thomasbaldwin during the Spring '11 term at FSU.

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lecture7 - Power Systems I Economic Dispatch l In practice...

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