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Unformatted text preview: EEL 6266 Power System Operation and Control Chapter 6 Generation with Limited Energy Supply © 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 2 Generation with Limited Energy Supply c Hard limits and slack variables r take account of the limits on the takeorpay generating unit r this may be added to the Lagrangian by using two new functions and two new variables, called slack variables c where S 1 j and S 2 j are the slack variables c they may take on any real value including zero r the new Lagrangian max min T T T P P P ≤ ≤ ( 29 ( 29 ( 29 ∑ ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = = + + + +  +  + = max max max max max 1 2 2 min 2 1 2 1 max 1 1 1 1 , 1 1 j j j j T T j j j j T j T j total j j j T j j j N i j T j i j load j N i j i j j j S P P S P P q q n P P P F n L α α γ λ 2 2 min 2 2 1 max 1 and j j T T j j T j T j S P P S P P + = + = ψ ψ © 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 3 Generation with Limited Energy Supply c Hard limits and slack variables r α 1 j and α 2 j are Lagrange multipliers r the first partial derivatives for the k th interval are c when the constrained variable P Tk is within bounds, α 1 j = α 2 j = 0 and S 1 j and S 2 j are nonzero; when P Tk is limited, either S 1 j or S 2 j is zero and the associated Lagrange multiplier is nonzero k k k k k k k k k k T k T k k T k k i k i k k i S L S L P q n P L P F n P L 2 2 2 1 1 1 2 1 2 2 d d d d α α α α α α λ γ λ = = ∂ ∂ = = ∂ ∂ + = = ∂ ∂ = = ∂ ∂ © 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 4 Generation with Limited Energy Supply c Hard limits and slack variables r consider some interval k where P Tk = P max c S 1 k = 0 and α 1 k ≠ 0, then c if the value of α 1 k takes on the value just sufficient to make the equality true d d 1 = + + k T k T k k k P q n γ α λ k T k T k k P q n d d γ λ © 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 5 Generation with Limited Energy Supply c Example r reconsider the 6interval 24hour fuel scheduling c maximum generation on P T is reduced to 300 MW s in the original optimal schedule, P T = 353.3 in the third time period s when the limit is reduced to 300 MW, the gasfired unit burns more fuel in other time periods to meet the 40 M ft 3 gas consumption constraint © 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 6 Generation with Limited Energy Supply c Example r resulting optimal schedule with Ptmax = 300 MW c shadow price, γ : 0.8603 c total cost: $ 122,985 1 183.4 216.6 5.54 5.54 0....
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This note was uploaded on 07/15/2011 for the course EEL 6266 taught by Professor Thomasbaldwin during the Fall '04 term at FSU.
 Fall '04
 THOMASBALDWIN

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