ECON 4925 Autumn 2007 Electricity Economics Lecture 11 Lecturer: Finn R. Førsund Uncertainty

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

The general problem formulation with stochastic variables Stochastic elements: Uncertainty

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

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

The planning problem Uncertainty

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

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

The optimality conditions The interior solution The corner solutions Uncertainty

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

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

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

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

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

A general case for period t pt pt A' A B Mt|t-1 Mt C Uncertainty

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

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

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

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

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