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ECON 4925 Autumn 2007 Electricity Economics Lecture 2

ECON 4925 Autumn 2007 Electricity Economics Lecture 2. Lecturer: Finn R. Førsund. Non-linear programming. Problem formulation (Sydsæter et al., 1999) The Lagrangian function Kuhn – Tucker necessary conditions. Applying Kuhn - Tucker. The Lagrangian. Applying Kuhn – Tucker, cont.

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ECON 4925 Autumn 2007 Electricity Economics Lecture 2

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  1. ECON 4925 Autumn 2007Electricity EconomicsLecture 2 Lecturer: Finn R. Førsund Reservoir constraint

  2. Non-linear programming • Problem formulation (Sydsæter et al., 1999) • The Lagrangian function • Kuhn – Tucker necessary conditions Reservoir constraint

  3. Applying Kuhn - Tucker • The Lagrangian Reservoir constraint

  4. Applying Kuhn – Tucker, cont. • The necessary first-order conditions Reservoir constraint

  5. Interpreting the Lagrangian parameters • Shadow prices on the constraints • A shadow price tells us the change in the value of the objective function (value function) when the corresponding constraint is changed marginally • t : shadow price on stored water, water value, the benefit of one unit of water more • t : shadow price on the reservoir constraint, the benefit of one unit increase in capacity Reservoir constraint

  6. Interpreting the Kuhn – Tucker conditions • Inequality in the first condition • Social price lower than the shadow price on stored water • Nothing will be produced • Inequality in the second condition • Water value in the next period lower than the sum of water value in the present period and the shadow price on the reservoir constraint • May have overflow or just reaching reservoir capacity Reservoir constraint

  7. Qualitative interpretations • Reasonable assumption • Positive production in each period → • Social price equal to the water value • Assuming reservoir capacity is not reached→ • Water value in the present period will be equal to the water value in the next period Reservoir constraint

  8. Qualitative conclusion • For consecutive periods with the reservoir constraint not binding the water values are all the same • The social price is the same for all periods with non-binding reservoir constraints and equal to the common water value Reservoir constraint

  9. Hourly price variation four days in 2005 Reservoir constraint

  10. 800 700 600 NOK/MWh 500 400 300 200 100 0 1 1000 2000 3000 4000 5000 6000 7000 8000 8760 Hours Price-duration curve Norway 2005 Reservoir constraint

  11. 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 Reservoir constraint

  12. The bathtub diagram for two periods Period 2 Period 1 p2 p1 p1(e1H) p2(e2H) λ2 λ1 e1H e2H e2H C D A B M e1H Total available water Ro + w1 + w2 Reservoir constraint

  13. Backwards induction, Bellman • Start with the terminal period • Assumption: no satiation of demand • Implication • No threat of overflow in period 1 Reservoir constraint

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