1. Reservoir and production constraints1 ECON 4925 Autumn 2007 Electricity Economics Lecture 3 Lecturer: Finn R. Førsund

2. Reservoir and production constraints 2 The bathtub diagram Only two periods Available water in the first period Available water in the second period Adding together the two water-storage equations

3. Reservoir and production constraints 3 The bathtub diagram for two periods Total available water R o + w 1 + w 2 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

4. Reservoir and production constraints 4 Explaining price change Binding reservoir constraint in period 1 No overflow due to the non-satiation assumption Implication of binding reservoir constraint

5. Reservoir and production constraints 5 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 γ1γ1 p2p2 p1p1 Total available water R o + w 1 + w 2

6. Reservoir and production constraints 6 Time Price-determining events, general case Price-determining events  Scarcity and threat of overflow s+1 1 s t+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

7. Reservoir and production constraints 7 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

8. Reservoir and production constraints 8 Reservoir full in period s 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

9. Reservoir and production constraints 9 Production constraint A constraint on power (turbines, dimension of pipes)  Implies a constraint on energy production  Constraint independent of period A production constraint may become binding when  Preventing overflow  Trying to satisfy peak demand

10. Reservoir and production constraints 10 The manoeuvrability of the system Perfect manoeuvrability:  All available water can be produced within the period Limited manoeuvrability:  Takes more than a period to empty the reservoir  Key parameter: the minimum number of periods t o it takes to empty a full reservoir  Manoeuvrability depends on the choice of time period here, but is a real time problem

11. Reservoir and production constraints 11 Locking in of water Unavoidable loss of water due to a production constraint  Starting with an empty reservoir the inflows are such that even processing maximal water in each period overflow is in the end unavoidable