1. Reservoir constraint1 ECON 4925 Autumn 2006 Resource Economics Hydro with reservoir constraint Lecturer: Finn R. Førsund

2. Reservoir constraint 2 Single plant with reservoir constraint The dynamics of water accumulation The reservoir constraint The objective function: consumer + producer surplus

3. Reservoir constraint 3 The social planning problem

4. Reservoir constraint 4 The Lagrangian function

5. Reservoir constraint 5 Necessary first-order conditions

6. Reservoir constraint 6 Qualitative characterisations of the optimal solution Assumptions  Unique solution  Positive production of electricity in all periods Implication λ:The shadow price of water, the water value

7. Reservoir constraint 7 Backwards induction Start with terminal period Assumption Implication

8. Reservoir constraint 8 The bathtub diagram for two periods Total available water λ2λ2 M e2He2H B e2He2H p 2 (e 2 H )p 1 (e 1 H ) D C Period 2 Period 1 λ1λ1 A e1He1H p1p1 p2p2 e1He1H

9. Reservoir constraint 9 Threat of overflow p 2 (e 2 H ) e2He2H p 1 (e 1 H ) D C Period 2 Period 1 λ1λ1 λ2λ2 B A e1He1H Total available water γ1γ1 p2p2 p1p1

10. Reservoir constraint 10 Degree of filling

11. Reservoir constraint 11 Hourly price variation four days in 2005

12. Reservoir constraint 12 Price-determining events s+1 1 s tt+1 tT-1T Low price Threat of overflow Scarcity Highest priceHigh price u+1u No scarcity or overflow No scarcity or overflow No scarcity or overflow TimeTime

13. Reservoir constraint 13 In between empty and full M λ u+1 =p T p u p u+1 Period u Period u+1 BC D A Water to period u+2

14. Reservoir constraint 14 Threat of overflow M Total available water B Period s+1 γsγs e H s+1 p s+1 (e H s+1 ) p s (e s H ) DC p s =λ s p t min A esHesH p t = λ s+1 p t max Water saved to period s+2 Period s psps p s+1 M

15. Reservoir constraint 15 p u p u+1 Period u Period u+1 C D A Water to period u+2 B

16. Reservoir constraint 16 Must take: Run-of-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