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lecture7

# lecture7 - Economic Dispatch l In practice and in power...

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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

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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 ( ) 0 = x f dx d ( ) 0 2 2 > x f dx d 0 , , , or , , 1 0 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 ˆ ˆ ˆ eigen 0 1 L L L = < H ( ) j i n i ij x x x x x f H = ˆ ˆ ˆ 1 2 L L Optimization 0 x Hx x > 0 T

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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 = = = + = = + + = = + = 5 2 3 ˆ ˆ ˆ 32 16 8 ˆ ˆ ˆ 6 1 0 1 4 1 0 1 2 0 32
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lecture7 - Economic Dispatch l In practice and in power...

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