1. Polyhedral Risk Measures Vadym Omelchenko, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic.

2. The presentation’s structure 1.Definition of polyhedral risk measures (Two-stage) 2.Definition of polyhedral risk measures (Multi-stage) 3.Applications in the energy sector (CHP)

3. Definition of Polyhedral Risk Measures (Two-Stage)

4. Polyhedral Risk Measures

5. Polyhedral Probability Functionals

6. Linear Reformulation

7. Example

8. Example

9. Theorem Rockafellar and Wets

10. Popular examples CV@R is a polyhedral risk measure. Every linear combination of CV@Rs are polyhedral risk measures V@R is not polyhedral.

11. Properties of Polyhedral Functionals

13. Definition of Polyhedral Risk Measures (Multi-Stage)

14. Polyhedral Multi-Period Acceptability Functionals

15. Conditions for Supremal Values

16. Note on Multi-Stage The dual solutions that correspond to the constraint is the slope of the R. This problem is solved by means of cost-to-go functions and bellman’s equation.

17. Note on Multi-Stage The dual solutions that correspond to the constraint is the slope of the R. This problem is solved by means of cost-to-go functions and bellman’s equation.

18. Note on V@R If we use V@R, many problems will cease to be linear and convex. However, replacing V@R with CV@R enables us to preserve the convexity of the underlying problem because this measure is polyhedral.

19. Applications in the Energy Sector (CHP)

20. Liberalization/Deregulation of the Energy Markets The deregulation of energy markets has lead to an increased awareness of the need for profit maximization with simultaneous consideration of financial risk, adapted to individual risk aversion policies of market participants. More requirements on Risk management.

21. Liberalization/Deregulation of the Energy Markets Mathematical modeling of such optimization problems with uncertain input data results in mixed- integer large-scale stochastic programming models with a risk measure in the objective. Often Multi-Stage problems are solved in the framework of either dynamic or stochastic programming. Simultaneous optimization of profits and risks.

22. Applications of polyhedral Risk Measures

23. Specification of the Problem The multi-stage stochastic optimization models are tailored to the requirements of a typical German municipal power utility, which has to serve an electricity demand and a heat demand of customers in a city and its vicinity. The power utility owns a combined heat and power (CHP) facility that can serve the heat demand completely and the electricity demand partly.

24. Stochasticity of the Model Sources: 1.Electricity spot prices 2.Electricity forward prices 3.Electricity demand (load) 4.Heat demand.

25. Stochasticity of the Model Multiple layers of seasonality 1.Electricity spot prices (daily, weekly, monthly) 2.Electricity demand (daily, weekly, monthly) 3.Heat demand (daily, weekly, monthly) The seasonality is captured by the deterministic part.

26. Interdependency between the Data (prices&demands) Prices depend on demands and vice versa Tri-variate ARMA models (demand for heat&electricity and spot prices). Spot prices AR-GARCH. The futures prices are calculated aposteriori from the spot prices in the scenario tree. (month average)

27. Parameters

28. Decision Variables

29. Objective

30. Objective – Cash Values Cash values are what we earn from producing heat and electricity. We of course take into account technical constraints.

31. Objective

32. Simulation Results The best strategy is to not use any contracts. Minimizing without a risk measure causes high spread for the distribution of the overall revenue. The incorporation of the (one-period) CV@R applied to z(T) reduces this spread considerably for the price of high spread and very low values for z(t) at time t<T.

33. Simulation Results

39. Conclusion Polyhedral risk measures enable us to incorporate more realistic features of the problem and to preserve its convexity and linearity. Hence, they enable the tractability of many problems. V@R is a less sophisticated risk measure, but many problems cannot be solved by using V@R unlike CV@R.

40. Bibliography A. Philpott, A. Dallagi, E. Gallet. On Cutting Plane Algorithms and Dynamic Programming for Hydroelectricity Generation. Handbook of Risk Management in Energy Production and Trading International Series in Operations Research & Management Science, Volume 199, 2013, pp 105-127. A. Shapiro, W. Tekaya, J.P. da Costa, and M.P. Soares. Risk neutral and risk averse Stochastic Dual Dynamic Programming method. 2013. G. C Pflug, W. Roemisch. Modeling, Measuring and Managing Risk. 2010. A. Eichhorn, W. Römisch, Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures. 2005