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lecture10- Generation with Limited Energy Supply

lecture10- Generation with Limited Energy Supply - EEL 6266...

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EEL 6266 Power System Operation and Control Chapter 6 Generation with Limited Energy Supply
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© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 2 Generation with Limited Energy Supply circle6 Hard limits and slack variables rhombus4 take account of the limits on the take-or-pay generating unit rhombus4 this may be added to the Lagrangian by using two new functions and two new variables, called slack variables circle6 where S 1 j and S 2 j are the slack variables circle6 they may take on any real value including zero rhombus4 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
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© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 3 Generation with Limited Energy Supply circle6 Hard limits and slack variables rhombus4 α 1 j and α 2 j are Lagrange multipliers rhombus4 the first partial derivatives for the k th interval are circle6 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 0 2 0 d d 0 d d 0 α α α α α α λ γ λ = = = = - + - = = - = =
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© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 4 Generation with Limited Energy Supply circle6 Hard limits and slack variables rhombus4 consider some interval k where P Tk = P max circle6 S 1 k = 0 and α 1 k 0, then circle6 if the value of α 1 k takes on the value just sufficient to make the equality true 0 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
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© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 5 Generation with Limited Energy Supply circle6 Example rhombus4 reconsider the 6-interval 24-hour fuel scheduling circle6 maximum generation on P T is reduced to 300 MW square4 in the original optimal schedule, P T = 353.3 in the third time- period square4 when the limit is reduced to 300 MW, the gas-fired unit burns more fuel in other time periods to meet the 40 M ft 3 gas consumption constraint
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© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 6 Generation with Limited Energy Supply circle6 Example rhombus4 resulting optimal schedule with Ptmax = 300 MW circle6 shadow price, γ : 0.8603 circle6 total cost: $ 122,985 1 183.4 216.6 5.54 5.54 0. 2 350.0 300.0 5.94 5.86 0.08 3 500.0 300.0 6.30 5.86 0.44 4 245.4 254.6 5.69 5.69 0.
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