Economic Dispatch. Quantitative Energy Economics. Anthony Papavasiliou 1 / 21

Transcription

1 1 / 21 Economic Dispatch Quantitative Energy Economics Anthony Papavasiliou

2 Economic Dispatch 2 / 21 1 Optimization Model of Economic Dispatch 2 Equilibrium Model of Economic Dispatch

3 Outline 3 / 21 1 Optimization Model of Economic Dispatch 2 Equilibrium Model of Economic Dispatch

4 Welfare Maximizing Economic Dispatch Problem 4 / 21 dl pg max MB l (d l ) l L 0 g G (λ) : d l p g 0 g G l L (ν l ) : d l D l (µ g ) : p g P g p g, d l 0 0 MC g (p g ) Set of loads L, set of generators G Increasing marginal cost MC g ( ) Decreasing marginal benefit MB l ( )

5 KKT Conditions 5 / 21 0 p g λ + MC g (p g ) + µ g 0 0 d l MB l (d l ) + λ + ν l 0 0 µ g P g p g 0 0 ν l D l d l 0 0 λ p g d l 0 g G l L

6 6 / 21 System Lambda There exists a threshold λ such that: 1 If 0 < p g < P g, then MC g (p g ) = λ. If 0 < d l < D l, then MB l (d l ) = λ. 2 If p g = 0, then MC g (0) λ. If d l = 0, then MB l (0) λ. 3 If p g = P g, then MC g (P g ) λ. If d l = D l, then MB l (D l ) λ. Proof: KKT conditions System lambda: marginal cost of the marginal generating unit (i.e. the generating unit which will supply the next unit of power at lowest cost)

7 7 / 21 Interpretation of KKT Conditions Optimal solution is matching cheapest generators with consumers who have greatest valuation (can you see why from the KKT conditions?)

8 Graphical Illustration of KKT Conditions 8 / 21 Cost [$/h] Cost [$/h] p C = P C Power [MW] p D = 0 Power [MW] Cost [$/h] Cost [$/h] p A Power [MW] p BPower [MW]

9 Outline 9 / 21 1 Optimization Model of Economic Dispatch 2 Equilibrium Model of Economic Dispatch

10 10 / 21 Path to Deregulation 1 Late 1970s: power systems are operated as vertically integrated regulated monopolies 2 Before 1980s: Premature markets (e.g. Norway) : Chile introduces a sport market : British government privatizes public power sector in England and Wales : Nordic market expands to include Sweden, Finland and Denmark 6 New Zealand and Australia introduced spot markets 7 The United States follow with California (CAISO), Pennsylvania-New Jersey-Maryland (PJM), Texas (ERCOT), New York (NYISO) and the Midwest (MISO)

11 11 / 21 Auctions in Real-Time Markets Real-time markets cannot rely on bilateral negotiations (only takes a few minutes of imbalance for a blackout)... but they can rely on auctions. But do auctions result in the right allocation of power?

12 12 / 21 Definition of Competitive Market A market is competitive if: Agents are price-taking Variable cost is convex and benefit is concave (can you see why this helps?) Agents have access to public information (prices)

13 Aggregate Cost 13 / 21 Aggregate cost is the cheapest way to produce Q MW of power among a collection of producers TC G (Q) = min g G s.t. p g = Q g G pg 0 MC g (x)dx Constraints imposed through domain of objective function What do we know about MC in competitive markets? What is the unit of measurement of TC? MC?

14 Merit Order Curve 14 / 21 Merit order curve = (increasing) system marginal cost curve How do we get aggregate cost from the merit order curve?

15 15 / 21 Aggregate Benefit Aggregate benefit is most beneficial way to consume Q MW of power among a collection of consumers TB L (Q) = max l L s.t. d l = Q l L dl 0 MB l (x)dx What is the graphical analogy of the previous slide?

16 Graphical Solution of Economic Dispatch Problem 16 / 21 [$/MWh] λ Consumer Surplus Producer Surplus MC G (Q) MB L (Q) q G (λ ) = q L (λ ) [MW]

17 Producer and Consumer Surplus, Welfare, Efficiency Suppose price is λ: Producer surplus/profit: profit of producers who are willing to sell λq G (λ) qg (λ) 0 MC G (x)dx, where q G (λ) is quantity sold at price λ Consumer surplus/profit: profit of consumers who are willing to buy ql (λ) 0 MB L (x)dx λq L (λ), where q L (λ) is quantity bought at price λ Welfare: sum of producer and consumer surplus Efficiency: when total surplus is maximized 17 / 21

18 Quantity Adjustment 18 / 21 Price-taking supplier will increase quantity produced if marginal cost price, decrease output otherwise: max λp g (µ g ) : p g P g p g 0 pg 0 MC g (x)dx Price-taking consumer will decrease quantity consumed if marginal benefit price, increase consumption otherwise: dl max MB l (x)dx λd l 0 (µ l ) : d l D l d l 0

19 19 / 21 Price Adjustment When demand exceeds supply, upward pressure on prices When supply exceeds demand, downward pressure on prices Market clearing condition 0 p g d l λ 0 g G l L

20 20 / 21 Equilibrium, Market Clearing Price, Competitive Equilibrium, Competitive Price A market is in equilibrium when no profitable opportunities for trade exist The market clearing price is the price of a market in equilibrium An equilibrium in a competitive market is called a competitive equilibrium The price of a competitive market is the competitive price

21 21 / 21 Competitive Markets Are Efficient Collect KKT conditions of quantity adjustment and market clearing condition of price adjustment: Suppliers: 0 p g λ + µ g + MC g (p g ) 0 0 µ g P g p g 0 Consumers: 0 d l λ + ν l MB l (d l ) 0 0 ν l D l d l 0 Market Clearing: 0 λ p g d l 0 g G l L Identical to KKT conditions of economic dispatch