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Unformatted text preview: RealTime Electricity Markets  2 1.0 Social surplus formulation In the formulation of the previous notes, we minimized cost. In doing so, we are maximizing social surplus under the condition that demand is price insensitive. We can see this by recall the problem being solved by the benevolent dictator is to maximize the sum of consumer utility and suppler profit, i.e., { } ) ( ) ( max P C pP P U P − + (1) or, since U(P)=v(P)+mpP, we see the problem is { } ) ( ) ( max P C pP pP m P v P − + − + (2) Eliminating the pP terms &amp; removing the constant m : { } ) ( ) ( max P C P v P − (3) Under the condition that demand is priceinsensitive, then the demand function v'(P) is an impulse, indicating that the consumer will pay anything to obtain the amount P. This also means that the value to them of obtaining more than P is the same as the value of obtaining P. Therefore, v(P) is constant. This implies that the above maximization problem is just { } ) ( max P C P − ( 4 a ) 1 Or { } ) ( min P C P ( 4 b ) which is what we solved in the last set of notes. Actual markets today have provision for loadserving entities (LSEs) to bid into the market, just as resources can make offers. Resource offers and demand bids are illustrated in Figs. 1a and 1b [1]. Resource offers may be "block" (as shown) or "slope." These offers and bids correspond to the what we have called C'(P) and v'(P) , respectively. Fig. 1a: Resource offer 2 Fig. 1b: Demand bid So we want to account for the possibility that the consumer will desire to adjust their demand as a function of the price. In this case, we must include the consumers' utilities in the objective function, leading us back to the objective function used in the formulation for our present problem, which is: { } ) ( ) ( max P C P v P − ( 5 a ) In the case where we have multiple consumers and suppliers, then (5a) becomes { } { } ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − ∑ ∑ ∈ ∈ GenBuses k k k LoadBuses k k k P P C P v ) ( ) ( max (5b) 3 2.0 LPOPF with Consumer Utility We saw in previous notes that the efficient market occurs when we maximize social surplus, which we expressed as v(P)C(P) , where • v(P) quantifies, in dollars, the satisfaction associated with the amount of energy consumed, P , and is the utility function for energy; • C(P) is the cost of producing the energy P . We assume the network has n buses and m branches, and that there can be both generation and demand at each bus. To simplify, we use a piecewise linear approximation of the cost and utility curves with only 1 "piece" per curve. Thus, each generation unit and each consumer is represented in the objective function by a constant times the MW output for that unit or the MW consumption by that consumer. So here is the formal statement of our problem: 4 { } { } ∑ ∑ ∈ ∈ − + buses load k dk dk buses generator k gk gk P s P s _ _ min (6) 1 Subject to: θ ' B P = ( 7 ) 2 θ × × = ) ( A D P B ( 8 ) 3 max , max...
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This note was uploaded on 11/02/2011 for the course ECON 301 taught by Professor Gandhi during the Spring '01 term at Andhra University.
 Spring '01
 Gandhi

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