1. Modelling inflows for SDDP Dr. Geoffrey Pritchard University of Auckland / EPOC

2. Inflows – where it all starts In hydro-dominated power systems, all modelling and evaluation depends ultimately on stochastic models of natural inflow. CATCHMENTS hydro generation thermal generation transmissionconsumption reservoirs

3. Why models? Raw historical inflow sequences get us only so far. - they can’t deal with situations that have never happened before. Autumn 2014 : - Mar ~ 1620 MW - Apr ~ 2280 MW - May ~ 4010 MW Past years (if any) with this exact sequence are not a reliable forecast for June 2014.

4. What does a model need? 1. Seasonal dependence. - Everything depends what time of year it is. Waitaki catchment (above Benmore dam) 1948-2010

5. 2. Serial dependence. - Weather patterns persist, increasing probability of shortage/spill. - Typical correlation length ~ several weeks (but varying seasonally). What does a model need?

6. Iterated function systems (numerical values are only to illustrate the form of the model). Make this a Markov process by applying randomly-chosen linear transformations, as in: Let

17. IFS inflow models Differences from IFS applications in computer graphics: Seasonal dependence - the “image” varies periodically, a repeating loop. Serial dependence - the order in which points are generated matters.

18. Single-catchment version Model for inflow X t in week t : - where (R t, S t ) is chosen at random from a small collection of (seasonally-varying) scenarios. The possible (R t, S t ) pairs can be devised by quantile regression: - each scenario corresponds to a different inflow quantile.

19. Scenario functions for the Waitaki High-flow scenarios differ in intercept (current rainfall). Low-flow scenarios differ mainly in slope. Extreme scenarios have their own dependence structure.

20. Exact mean model inflows We can specify the exact mean of the IFS inflow model. Inflow X t in week t : Take averages to obtain mean inflow m t in week t : where (r t, s t ) are the averages of (R t, S t ) across scenarios. Usually we know what we want m t (and m t-1 ) to be; the resulting constraint on (r t, s t ) can be incorporated into the model fitting process, guaranteeing an unbiased model. Similarly variances. Control variates in simulation.

21. (Model simulated for 100 x 62 years, dependent weekly inflows, general linear form.) Inflow distribution over 4-month timescale.

22. Hydro-thermal scheduling by SDDP The problem: Operate a combination of hydro and thermal power stations - meeting demand, etc. - at least cost (fuel, shortage). Assume a mechanism (wholesale market, or single system operator) capable of solving this problem. What does the answer look like?

23. Week 6 Week 7Week 8 Structure of SDDP

24. Week 6 Week 7Week 8 min (present cost) + E[ future cost ] s.t. (satisfy demand, etc.) Structure of SDDP

25. - Stage subproblem is (essentially) a linear program with discrete scenarios. Week 6 Week 7Week 8 min (present cost) + E[ future cost ] s.t. (satisfy demand, etc.)  ps ps s Structure of SDDP

26. Why IFS for SDDP inflows? The SDDP stage subproblem is (essentially) a linear program with discrete scenarios. Most stochastic inflow models must be modified/approximated to make them fit this form, but... … the IFS inflow model already has the final form required to be usable in SDDP.