1. ECON 4925 Autumn 2007 Electricity Economics Lecture 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
Uncertainty

5. Two periods with stochastic inflow only
Inflow known in period 1
Period 1
Period 2 p1 p2
min
max
A B C
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 1 Use 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 BD A CD BN M BW CN CW
Uncertainty

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

15. A general case for period tpt pt
A' A B Mt|t Mt C
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 1c' x1 R1
A' A B M C
Thermal
Hydro
Uncertainty

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