ECON 4925 Autumn 2007 Electricity Economics Lecture 11

1. ECON 4925 Autumn 2007Electricity EconomicsLecture 11 Lecturer: Finn R. Førsund Uncertainty

2. Fundamental stochastic elements • Inflows are stochastic • Plus minus 25 TWh within a 90% confidence interval for Norway • Demand is stochastic • Demand for space heating is depending on outdoor temperature • Availability of generating capacity and transmission capacity stochastic • Accidents occur with positive probability (but low values) Uncertainty

3. The general problem formulation with stochastic variables • Stochastic elements: Uncertainty

4. A tree diagram • Impossible to solve from period 1 Inflows Time periods 1 2 3 4 Uncertainty

5. Two periods with stochastic inflow only • Inflow known in period 1 Period 1 Period 2 p1 p2 min max ABC w2min w2max Uncertainty

6. The planning problem Uncertainty

7. Rewriting inequality constraints • Assumption: no spill in period 1 • Follows from the general assumption of no satiation of demand • Implication for a binding reservoir constraint • Implication for the production in period 2 • With no spill e2H is a unique function of e1H Uncertainty

8. The planning problem, cont. • Inserting for the inflow in period 2 • The problem is a function of the production in period 1 only Uncertainty

9. The optimality conditions • The interior solution • The corner solutions Uncertainty

10. Interpreting the optimality conditions • Jensen’s inequality • Increased uncertainty • Mean-preserving spread (Rothchild and Stiglitz, 1970) increases when uncertainty increases Uncertainty

11. Calculating the expected price in period 2 • Discretising the probability distribution • Introducing the frequency for the inflow in interval i Uncertainty

12. The expected water value table • The expected price in period 2, i.e., the water value, is a function of the amount of water transferred to period 2 from period 1 • For each value of R1 we get a value for the expected price in period 2 • The range of R1 Uncertainty

13. Illustration of decision in period 1Use of water value table Expected price p2 as function of water transferred from period 1 p1 Expected price p2 as function of water transferred from period 1 D Expected price p2 as function of water transferred from period 1 N W Energy BW M BD A CD BN CN CW Uncertainty

14. Realised price in period 2 Period 2 p2 Min inflow Energy Max inflow From period 1 Uncertainty

15. A general case for period t Period t-1 Period t pt pt A'A B Mt|t-1 MtC Uncertainty

16. Hydro and thermal with uncertain inflow • Two periods only, period 1 known, period 2 uncertain inflow • The social planning problem Uncertainty

17. Reformulating the planning problem • Eliminating total consumption, using the water accumulating equation for period 2 Uncertainty

18. First-order conditions • The role of thermal in period 2 • For a given R1 we get a relationship between e2Th and w2 via the expected price. Changing R1 shifts this relationship • The water value table contingent upon both the realised R1 and the adjustment rule for thermal: E{p2|R1} Uncertainty

19. Hydro and thermal in period 1 Period 1 p1 p1 c' x1 R1 A' A B M C Thermal Hydro Uncertainty

20. Hydro and thermal in period 2 Period 2 p2 p1=E1{p2} p2 c' R1+w2 A'' A' A C' C Thermal Hydro Uncertainty