Transmission Expansion Planning in Deregulated Power ... - tuprints

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Appendix D: An Example on Computing Shadow Price Consider the following optimization problem: Max J = 6 x x s . t. : 3x1 + 4 x2 = 18 Lagrangian is equal to: L 1 = 6 x1 x2 − λ ( 3x1 + 4 x2 −18) 2 λ is called Lagrange multiplier or dual variable. Necessary conditions for extremum are: ⎧ ∂L ⎪ = 0 ⎪ ∂ x1 ⎪ ∂L ⎨ = 0 ⇒ ⎪∂ x2 ⎪∂L ⎪ = 0 ⎩∂ λ ⎧6 x2 − 3λ = 0 ⎪ ⎨6 x1 − 4λ = 0 ⎪ ⎪⎩ 3x1 + 4 x2 = 18 Let to rewrite above constraint as follows: 3x 1 + 4 x2 = b then the Lagrangian is as bellows: L = 6 x1 x2 − λ ( 3x1 + 4 x2 − b) * * * * ⇒ x = 3, x = 2. 25, λ = 4. 5, J = 40. 5 Shadow price of a constraint is equal to the change in objective function per unit change in right hand side of the constraint, assuming all other constraints remain unchanged [65]-[66]. Shadow price of above constraint is equal to: ∂J ∂L = = ∂b at optimal poin. t ∂b at optimal poin. t If b increases by 0.1 then: * λ * ∆ J = ∆b λ = 0. 45 ⇒ J = 40. 95 If the optimization problem is solved directly with b = 18.1, the same result will be archived. 1 2
Symbols k A i appropriateness degree of plan k versus ith risk assessment criterion k A i appropriateness degree of plan k versus ith stakeholders’ desire k ANCC annual congestion cost of the network after adding plan k k ANIC annual investment cost of plan k k ANLP annual load payment after adding plan k k ANOC annual operation cost of the network after adding plan k B linearized Jacobian matrix C D vector of load bids C G vector of generator bids C d i, C g i, j j bid of load i in the jth iteration of Monte Carlo simulation bid of generator i in the jth iteration of Monte Carlo simulation k C i ith risk assessment criterion f plan k k C i criterion for measuring ith stakeholders’ desire cc i congestion cost of line i k cc i, j congestion cost of line i after addition of plan k in the jth iteration of Monte Carlo simulation k DANOC decrease in annual operation cost of the network after adding plan k k F ap fuzzy appropriateness degree (index) of plan k versus combination of all decision criteria in risk assessment k F ap fuzzy appropriateness degree (index) of plan k versus combination of all decision criteria in measuring stakeholders’ desires F fuzzy appropriateness index of plan k scenario l k, l ap
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Transmission Expansion Planning in
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Acknowledgements This work was star
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Contents 1 Introduction 1 1.1 Resea
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Contents III 8.2.1 Fuzzy Regret and
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1 Introduction 2 benefits. Consumer
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2 Transmission Expansion Planning A
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3 Probabilistic locational Marginal
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4 Market Based Criteria The main ob
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4 Market Based Criteria 33 cc i wit
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4 Market Based Criteria 35 In the r
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4 Market Based Criteria 37 Fig. 4.4
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4 Market Based Criteria 39 k 1 k µ
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4 Market Based Criteria 41 with: k
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4 Market Based Criteria 43 k ANOC a
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4 Market Based Criteria 45 28.8 235
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4 Market Based Criteria 47 σ σ σ
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5 Market Based Transmission Expansi
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Page 82 and 83: 5 Market Based Transmission Expansi
Page 89 and 90: 6 Fuzzy Risk Assessment In chapter
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Page 93 and 94: 6 Fuzzy Risk Assessment 85 6.4.1 Fu
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Page 97 and 98: 6 Fuzzy Risk Assessment 89 plan 5-2
Page 99 and 100: 7 Stakeholders’ Desires Restructu
Page 101 and 102: 7 Stakeholders’ Desires 93 Flexib
Page 103 and 104: 7 Stakeholders’ Desires 95 Table
Page 105 and 106: 7 Stakeholders’ Desires 97 7.5 Ca
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Page 110 and 111: 8 Transmission Expansion Planning U
Page 117 and 118: 9 Conclusions The main goal of this
Page 119: 9 Conclusions 111 of future scenari
Page 122 and 123: Appendix A: Examples on Scenario Te
Page 124 and 125: Appendix A: Examples on Scenario Te
Page 126 and 127: Appendix B: An Example on Decision
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Page 130 and 131: Appendix C: Examples for Locational
Page 134 and 135: Symbols 126 F fuzzy appropriateness
Page 136 and 137: Symbols 128 µ mean of load at bus
Page 139 and 140: References [1] G. Latorre, R. D. Cr
Page 141 and 142: References 133 [25] H. Sun, and D.
Page 143 and 144: References 135 [53] H. Rudnick, R.
Page 145 and 146: Zusammenfassung Das Hauptziel diese
Page 147: Zusammenfassung 139 Übereinstimmun
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