Lecture 8. Lecturer: Finn R. Førsund. Transmission 1

Transcription

1 ECON 4930 Spring 011 Elecriciy Economics ecure 8 ecurer: Finn R. Førsund Transmission 1

2 A sysem ransmission nework Two nodes wih one line he simples case Convering variables o energy unis (kwh) The energy balance x e e, 1,.., T x e consumpionin kwh e producion in kwh lossin kwh Transmission

3 A sysem ransmission nework, con. Expressing loss on a line supplying a single consumer node from a single hydropower generaor node e ( x) e ( x) x x e e ( x ), 0, 0, 1,.., T oss is increasing i in consumpion, and wih a posiive second order derivaive, according o Ohm s law A hermal capaciy on he line: x x Transmission 3

4 The social planning problem for he wo node case for wo ime periods x max p ( z) dz 1 z0 subjec o R R w e R 1 R x e e e e ( x ) x x R, x, e, e 0, 1, R,, given o R x Transmission 4

5 The agrangian funcion x 1 1 z0 p ( z) dz ( R R 1 w e ) 1 1 ( R R) ( x e ( x) e ) 1 ( x x) Transmission 5

6 The Kuhn Tucker condiions e p ( x ) 0( 0for 0) x e R x x 0( 0fore 0) 1 0( 0forR 0) 0( 0 for R R w e ) 1 0( 0for R R) 0( 0 for x x), 1, Transmission 6

7 Inerpreaion of he firs order condiions Assuming posiive producion in boh periods The shadow prices on he energy balance will hen be posiive and equal o he waer values e p( x) 0, 1, xx The difference beween he social price and he waer value e p ( x ) 0, 1, x Transmission 7

8 Inerpreaions, con. Difference in social price beween periods e e 1 ( ) 1 ( 1 ) 1 ( 1 ) x x 1 p x p x Difference even if waer values are equal (reservoir consrain no binding) e e x p x p ( ) ( ) ( ) ( ) 1 p x x1 Transmission 8

9 A bahub illusraion wihou congesion Period 1 Period p 1 λ oss 1 e e 1 ( ) 1( 1) ( ) x x1 p x p x e p1( x1) x 1 1 p λ oss A A' B M C D' D Toal available waer R o + w 1 + w Transmission 9

10 A bahub illusraion wih binding reservoir consrain, bu wihou congesion Period 1 Period p p 1 λ 1 λ 1 A A' B C Toal available waer R o + w 1 + w D' D Transmission 10

11 Bahub wih congesion Period 1 Period p p 1 λ x λ A A' B C D' D Toal available waer Transmission 11

12 Three nodes and wo periods Generaing node 1 Generaing node Elecriciy flow Elecriciy flow Consumpion node Transmission 1

13 The social planning problem x max p ( z) dz 1 z0 subjec o R R w e R j j, 1 j j R j j xj ej ej x x j 1 j e e ( x ) x j j j j x j R, x, x, e, e 0 j j j j w, R, R, x given, R free, j 1,, 1, j jo j j j j Transmission 13

14 The agrangian funcion j 1 1 z0 1 j1 x j p ( z) dz ( R R w e ) 1 j1 1 j 1 j j j, 1 j j ( R R ) j j j ( x e ( x ) e ) 1 j1 j j j j j ( x x ) j j j Transmission 14

15 The Kuhn Tucker condiions x j e j R j e p ( x ) 0( 0for 0) j j j xj j xj 0( 0fore 0) j j j j j, 1 j 0( 0forRj 0) 0( 0 for R R w e ) j j j, 1 j j 0( 0 for R R ) j j j 0 ( 0 for x x ), j 1,, 1, j j j Transmission 15

16 Inerpreaions of he firs order condiions Assumpions: Posiive producion in he firs period a boh plans (boh empy he reservoirs in he second period) No hrea of overflow in he firs period Waer values for a plan he same for boh periods Difference beween consumer price and waer values e j ( ), 1,, 1, p x j j j j x j Transmission 16

17 Inerpreaions, con. Difference beween waer values e j ( ) j j j x j p x e e x1 x e e ( ), 1, x x1 The plan wih highes sum of marginal loss and congesion will have he lowes waer value Transmission 17

18 Inerpreaions, con. Differences beween consumer prices e e p ( x ) p ( x ) ( ), j 1, j j1 1 1 j j j j1 x j x j1 The highes consumer price will be in he period wih he highes value of he sum of marginal lloss and congesion erm Transmission 18

19 Nodal prices Prices and waer values are specific o each node Consumer price greaer han waer values for each ime period Waer values differ due o loss and congesion beween plans for each ime period Marginal loss evaluaed a waer values plus congesion is equal for each ime period Transmission 19

20 Congesion in he wo period case Assumpions: A line is a mos congesed in he high demand period only There is no lock in of waer due o congesion R jo wj1 wj xj, j 1 1, Period 1 is he low demand period and period he high demand period Immediae implicaion: a leas one plan mus produce morein hehigh demandhigh period han he low demand period Transmission 0

21 Producion levels for he wo periods Boh plans will produce more in he high demand period and less in he low demand period due o marginal loss increasing in energy e e (1 ) (1 ), 1, x1 x Disregarding congesion if plan 1 produces more in he high demand h dperiod so mus plan Inroducing congesion does no change his siuaion Transmission 1

22 Implicaion of ransmission for uilisaion of he hydro plans Transmission causes higher price in he highdemand period and leads o a relaively greaer use of waer in period 1 han compared o no ransmission The plan wih relaively less marginal loss will shif hf producion from he low demand period o he high demand period, and opposie for he plan wih relaively greaer marginal loss Transmission

23 Implicaions, con. The plan wih relaively higher marginal loss plus congesion in one period will also have a relaively higher loss plus congesion in he oherperiod (plan waer value consan) The plan wih he lowes waer value is used relaively l more in he low demand period, and he plan wih he highes waer value relaively more in he high demand period Transmission 3

24 oop flows (meshed nework) Generaing node 1 Generaing node Elecriciy flow Elecriciy flow Elecriciy flow Consumpion node Transmission 4