1. Constraints1 ECON 4925 Autumn 2007 Electricity Economics Lecture 4 Lecturer: Finn R. Førsund

2. Constraints 2 Modelling a production constraint

3. Constraints 3 The Lagrangian

4. Constraints 4 The Kuhn – Tucker conditions

5. Constraints 5 Impact of a production constraint A binding production constraint implies that the water value will in general be less than the social price  The condition holds with equality since production is positive A binding constraint means less use of water compared with no constraint and therefore the price will be higher

6. Constraints 6 A bathtub illustration: constraint in period 2: peak demand p1p1 p 1 =λ 1 D C A Period 2Period 1 ρ2ρ2 p2p2 λ2λ2 p 2 =λ 2 +ρ 2 B B’B’ B ’’

7. Constraints 7 A bathtub illustration: constraint in period 1:threat of overflow Spill p1p1 p 1 =ρ 1 D C A Period 2Period 1 ρ1ρ1 p2p2 p 2 =λ 2 =γ 1 BB' γ1γ1 λ 1 =0

8. Constraints 8 Must take: Run-of-the-river and wind power Total controllable inflow p1p1 p2p2 BD River flow λ2λ2 e1Re1R p 2 (x 2 ) e2He2H p 1 (x 1 ) C Period 2 Period 1 λ1λ1 e1He1H River flow e2Re2R A M

9. Constraints 9 Degree of filling

10. Constraints 10 Solving for quantities, model with reservoir constraint only Terminal period, two periods The next period (first) going backwards  No threat of overflow

11. Constraints 11 Solving for quantities, cont. If we can determine, then we can solve for production quantities Using the water accumulation equation for period 1 One equation in one unknown, must know inflows and demand functions for both periods

12. Constraints 12 Multiple plants N plants with reservoirs, a single aggregate consumer, x t = total consumption The energy balance The energy balance has to hold as an equality Reservoir constraints only

13. Constraints 13 The social planning problem

14. Constraints 14 The Lagrangian function Eliminating consumption by inserting the energy balance

15. Constraints 15 The Kuhn – Tucker conditions

16. Constraints 16 Backwards induction The terminal period: No satiation of demand → all plants use up all water in the terminal period If all reservoirs are in between full and empty in period T -1: the plants will face the same price equal to the water value in the terminal period also in period T -1

17. Constraints 17 Indeterminacy of plant quantities o pt(xt)pt(xt) xtxt ptpt ptpt Total consumption A

18. Constraints 18 Equality of water values For a water value to change either the reservoir constraint must be binding or the reservoir is emptied Assume that the social price for periods t and t +1 are equal. Can it then be optimal for a plant to have a full reservoir in period t?  Yes it may be optimal, but the shadow price on the reservoir constraint will be zero  The reservoir must be full when the social price increases

19. Constraints 19 Equality of water values, cont. Assume that the social price is the same in period t and t +1. Can it then be optimal for a plant to empty the reservoir in period t ?  Yes, but the water value in period t and t +1 must be the same  The reservoir must be full when the social price increases  The reservoir must be emptied when the social price decreases

20. Constraints 20 The case of pure accumulation If the water value is greater than the social price, then production is zero and all inflows are accumulated A plant may accumulate water and produce zero for many periods The water value will be equal for all zero production periods and equal to the social price in the first period with positive production

21. Constraints 21 Hveding’s conjecture Assume independent hydropower plants with one limited reservoir each, and perfect manoeuvrability of reservoirs, but plant- specific inflows The plants can be regarded as a single aggregate plant and the reservoirs can be regarded as a single aggregate reservoir when finding the social optimal solution for operating the hydropower system.