1. Trade and transmission1 ECON 4925 Autumn 2007 Electricity Economics Lecture 7 Lecturer: Finn R. Førsund

2. Trade and transmission 2 Trade between Hydro and Thermal The cooperative social planning problem

3. Trade and transmission 3 The Lagrangian function Inserting the energy balances  Export for one country is import for the other

4. Trade and transmission 4 The Kuhn – Tucker conditions

5. Trade and transmission 5 Combining the bathtub diagram and the thermal diagram for two periods θ2θ2 Period 1 Period 2 c'c' c'c' Import Export p 1 Th =p 1 H = p 2 Th =p 2 H = ImportExpor t Hydro Thermal A'A'A M'M' M B'B'B

6. Trade and transmission 6 p 1 Th =p 1 H = 1 Trade Hydro –Thermal with reservoir constraint Period 1 Period 2 c'c' c'c' Import θ2θ2 p 2 Th =p 2 H = 2 A B C D Import Export Hydro Thermal Export γ1γ1 A'A' D'D'

7. Trade and transmission 7 Transmission The model of Lord Kelvin from 1881 (Smith, 1961)  A single production node connected with a single consumption node Assumptions  Voltage at consumption node given  No binding capacity limit on the line Generating node Consumption node Electricity flow

8. Trade and transmission 8 The physical laws of transmission Ohm’s law Symbols P L = loss in kW I = current in amps R = resistance on the line in ohms L = length of line A = area of cross section ρ = specific resistance of the metal

9. Trade and transmission 9 The physical laws of transmission, cont. Constancy of energy Symbols  P i = power produced (kW)  P L = loss on the line (kW)  P o = power received (kW) Kirchhoff’s laws  Current flow into a node must be equal to current flow out (energy cannot be lost)  Voltage drops around any loop sum to zero (relevant for loop flow networks) Ohm’s and Kirchhoff’s laws  Flows distribute within loops proportional to impedance on lines

10. Trade and transmission 10 The connection between voltage and current Definition for AC Symbols Po = power at consumption node in kW Vo = voltage at consumption node I = current in amps cosφ = power factor of the consumer’s load φ = lag between voltage and current variation in an alternating-current circuit

11. Trade and transmission 11 The transmission production function Inserting in the power balance Introducing the weight of the cable K = 2dLA, d= specific weight Renaming P o and P i, x and e, multiplying each term above with K

12. Trade and transmission 12 Substitution between capital and power input Ex ante MRS (marginal rate of substitution) The explicit ex ante production function  Scale properties ex ante and ex post Ex ante: constant returns to scale Ex post (fixed capital): decreasing returns to scale