The DC Optimal Power Flow

Transcription

1 1 / 20 The DC Optimal Power Flow Quantitative Energy Economics Anthony Papavasiliou

2 The DC Optimal Power Flow 2 / 20 1 The OPF Using PTDFs 2 The OPF Using Reactance

3 3 / 20 Transmission Constraints Lines can carry a limited amount of power Thermal limits Stability limits Voltage drop limits Kirchhoff voltage and current laws Non-linear mapping: power injection in buses power flow in lines We will linearize these Optimal power flow problem (OPF): Maximize welfare (minimize cost) subject to Kirchhoff laws + transmission limits

4 Network Representation 4 / 20 Transmission system is represented as a directed graph N: set of nodes K : set of lines (denoted by k = (m, n)) G n : set of generators located in node n, G = n N G n L n : set of loads located in node n, L = n N L n

5 5 / 20 Two Equivalent Models Decisions: p g : amount of power produced by generator g d l : amount of power consumed by load l Two equivalent models, depending on system state and input data: Model 1 Model 2 System state: nodal injections Input data: power transfer distribution factors (depend on physical characteristics of lines) System state: nodal phase angles Input data: reactance (depend on physical characteristics of lines)

6 Table of Contents 6 / 20 1 The OPF Using PTDFs 2 The OPF Using Reactance

7 Model 1: Power Transfer Distribution Factors 7 / 20

8 Net Injection Hub node: reference node that "absorbs" all injections Injection r n : amount of power shipped from node n to the hub r n = p g g G n l L n d l Not amount of power flowing over line connecting n and hub Conservation of energy: r n = 0 n N 8 / 20

9 9 / 20 Power Flows Power transfer distribution factor (PTDF) (F kn ): amount of power flowing on line k as a result of shipping 1 MW from n to hub F hub,n = 0 PTDF: input data, depend on physical characteristics of lines PTDF depend on choice of hub Flow f k is f k = F kn r n n N Flow can be positive or negative (interpretation?) T k : limit on power that each line can carry T k f k T k

10 10 / 20 Example All lines have identical electrical characteristics 1 F 1 2,1 =?, F 1 2,2 =? 2 Express shipment of 30 MW from 1 to 2 as transaction through hub 3 Compute flow f 1 2 from steps 1, 2 4 Note: r 1 and f 1 hub are different

11 The OPF Using PTDFs 11 / 20 max l L (λ + k ) : f k T k (λ k ) : f k T k dl pg 0 MB l (x)dx g G (ψ k ) : f k F kn r n = 0 n N (ρ n ) : r n p g + d l = 0 g G n l L n (φ) : r n = 0 n N p g, d l 0 0 MC g (x)dx

12 12 / 20 Optimal Solution Denote P g, D l as maximum production/consumption of generators/loads (imposed through domain of objective function) There exists a threshold ρ n for all n such that: If 0 < p g < P g, then ρ n = MC g (p g ). If 0 < d l < D l, then ρ n = MB l (d l ). If p g = P g, then ρ n MC g (P g ). If d l = D l, then ρ n MB l (D l ). If p g = 0, then ρ n MC g (0). If d l = 0, then ρ n MB l (0).

13 13 / 20 Proof: KKT conditions 0 p g MC g (p g ) ρ n(g) 0 0 d l MB l (d l ) + ρ n(l) 0 n(g): node where generator g is located n(l): node where load l is located

14 Sensitivity Helpful in understanding transmission pricing φ: marginal change in welfare from marginal increase in production/marginal decrease in consumption λ + k and λ k : marginal impact of increasing line capacity ρ n : marginal impact of marginal increase of consumption/decrease of generation in node n (what if demand is inelastic?) What sign do we expect for these dual variables? 14 / 20

15 15 / 20 Components of ρ n Useful identity for computing prices: ρ n = φ + k K F kn λ k k K F kn λ + k Proof: KKT conditions

16 Example 16 / 20 Case 1 D 2 = 50 MW, T 1 2 unlimited ρ 1 = ρ 2 = 20 $/MWh Case 2 D 2 = 50 MW, T 1 2 = 50 MW ρ 1 = 20 $/MWh, 20 $/MWh ρ 2 40 $/MWh Case 3 D 2 = 60 MW, T 1 2 = 50 MW ρ 1 = 20 $/MWh, ρ 2 = 40 $/MWh Can you explain multiplier values?

17 Table of Contents 17 / 20 1 The OPF Using PTDFs 2 The OPF Using Reactance

18 Model 2: Reactance 18 / 20

19 Power Flows 19 / 20 Reactance: input data, depends on physical characteristics of lines Independ on choice of hub Flow f k is f (m,n) = B (m,n) (θ m θ n ) Translation of θ results in identical flows, fix θ hub = 0 Conservation of energy: p g + g G n f k = + d l. l L n k=(,n) k=(,n) Input data is independent of network topology: transmission line investment, transmission line outages

20 The OPF Using Reactance 20 / 20 dl pg max MB l (x)dx MC g (x)dx l L 0 g G 0 (ρ n ) : p g f k + d l + f k = 0 g G n l L k=(,n) (γ k ) : f k B k (θ m θ n ) = 0, k = (m, n) (λ + k ) : f k T k (λ k ) : f k T k p g, d l 0 k=(n, )