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ECON 4925 Autumn 2007 Electricity Economics Lecture 7 Lecturer: Finn R. Førsund Trade and transmission 1 Trade between Hydro and Thermal The cooperative social planning problem T max [ xtH t 1 z 0 ptH ( z ) dz xtTh ptTh ( z ) dz c (etTh )] z 0 subject to xtH etH eThXI,t eHXI,t xtTh etTh eThXI,t eHXI,t T H e t W t 1 etTh e Th xtH , etH , eThXI,t , eHXI,t 0 T ,W , e Th given , t 1,.., T Trade and transmission 2 The Lagrangian function Inserting the energy balances Export for one country is import for the other T XI XI etH eTh ,t eH ,t t 1 z 0 L [ XI XI etTh eTh ,t eH ,t ptH ( z ) dz ptTh ( z ) dz c(etTh )] z 0 T t (etTh e Th ) t 1 T ( etH W ) t 1 Trade and transmission 3 The Kuhn – Tucker conditions L H H H p ( x ) 0 ( 0 for e t t t 0) H et L H H Th Th XI p ( x ) p ( x ) 0 ( 0 for e t t t t H ,t 0) XI eH ,t L Th Th Th Th p ( x ) c '( e ) 0 ( 0 for e t t t t t 0) Th et L H H Th Th XI p ( x ) p ( x ) 0 ( 0 for e t t t t Th ,t 0) XI eTh ,t T 0 ( 0 for etH W ) t 1 t 0 ( 0 for etTh e Th ) Trade and transmission 4 Combining the bathtub diagram and the thermal diagram for two periods Period 2 Period 1 θ2 p2Th=p2H= p1Th=p1H= c' c' Export Import A' A Expor t Thermal M' M B' Hydro Trade and transmission B Import Thermal 5 Trade Hydro –Thermal with reservoir constraint Period 2 Period 1 θ2 γ1 p1Th=p1H=1 c' c' Import Export A' A Thermal p2Th=p2H=2 Export B C D' D Hydro Trade and transmission Import Thermal 6 Transmission The model of Lord Kelvin from 1881 (Smith, 1961) A single production node connected with a single consumption node Consumption node Generating node Electricity flow Assumptions Voltage at consumption node given No binding capacity limit on the line Trade and transmission 7 The physical laws of transmission Ohm’s law PL I R 2 2L R A Symbols PL = 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 Trade and transmission 8 The physical laws of transmission, cont. Constancy of energy Pi PL Po Kirchhoff’s laws Symbols Pi = power produced (kW) PL = loss on the line (kW) Po = power received (kW) 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 Trade and transmission 9 The connection between voltage and current Definition for AC Po Vo I cos Po I Vo cos 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 Trade and transmission 10 The transmission production function Inserting in the power balance 2 Po 2 L Po Pi PL Pi I R Pi Vo cos A 2 Introducing the weight of the cable K = 2dLA, d= specific weight Renaming Po and Pi , x and e, multiplying each term above with K 2 4 L d 2 F ( x, e, K ) K (e x) kx 0 , k (Vo cos )2 Trade and transmission 11 Substitution between capital and power input Ex ante MRS (marginal rate of substitution) dK K MRS 0 de e x The explicit ex ante production function K ke 12 x f (e, K ) (1 4 ) 1 2k K Scale properties ex ante and ex post Ex ante: constant returns to scale Ex post (fixed capital): decreasing returns to scale Trade and transmission 12