1. Wind power scenario generation Geoffrey Pritchard University of Auckland by regression clustering

2. Scenarios for stochastic optimization Uncertain problem data represented by a probability distribution. For computational tractability, need a finite discrete distribution, i.e. a collection of scenarios. Make decision here ?

3. Power system applications Wind power generation, 2 hours from now. Inflow to hydroelectric reservoir, over the next week. Typical problems solved repeatedly: –Need a procedure to generate scenarios for many problem instances, not just one.

4. Situation-dependent uncertainty Scenarios represent the conditional distribution of the variable(s) of interest, given some known information x. Different problem instances have different x.

5. Change in wind power over next 2hr Tararua/Te Apiti 28/5/2004-31/3/2010

6. Change in wind power over next 2hr Tararua/Te Apiti 28/5/2004-31/3/2010 Change in wind power: 7 discrete scenarios Each scenario is a function of the present wind power x.

7. Change in wind power over next 2hr Tararua/Te Apiti 28/5/2004-31/3/2010 Change in wind power: 7 discrete scenarios Each scenario is a function of the present wind power x.

8. Have data x i and y i for i=1,…n x y Scenarios by quantile regression

9. Have data x i and y i for i=1,…n Want scenarios for y, given x. x y Scenarios by quantile regression

10. Have data x i and y i for i=1,…n Want scenarios for y, given x. Quantile regression: choose scenario s k () to minimize  i  k ( y i – s k (x i ) ) for a suitable loss function  k (). x y

11. Quantile regression fitting For a scenario at quantile ,   is the loss function  

12. Scenarios as functions Choose each scenario to be linear on a feature space: s k (x) =  j  jk b j (x) Typically b j () are basis functions (e.g. cubic splines). The quantile regression problem is then a linear program.

13. Change in wind power over next 2hr Tararua/Te Apiti 28/5/2004-31/3/2010 Change in wind power: 7 discrete scenarios Equally likely scenarios, modelled by quantiles 1/14, 3/14, … 13/14.

14. Quantile regression: pros and cons Each scenario has its own model. Scenario models are fitted separately. Fitting is computationally easy. Scenarios have fixed probabilities. Events with low probability but high importance may be left out.

15. Another way to choose scenarios … choose scenarios to minimize expected distance of a random point to the nearest scenario. (Wasserstein approximation.) Robust to general stochastic optimization problems. Given one probability distribution …

16. Scenarios for conditional distributions Have data x i and y i for i=1,…n Want scenarios for y, given x. x y

17. Scenarios for conditional distributions Have data x i and y i for i=1,…n Want scenarios for y, given x. Wasserstein: minimize  i min k | y i – s k (x i ) | over scenarios s k () chosen from some function space. x y

18. Scenarios as functions Choose each scenario to be linear on a feature space: s k (x) =  j  jk b j (x) Typically b j () are basis functions (e.g. cubic splines). The Wasserstein approximation problem is then a MILP with SOS1 constraints (not that that helps).

19. Algorithm: clustering regression Let each observation (x i,y i ) be assigned to a scenario k(i). Choose alternately the functions s k the assignments k(i) to minimize  i | y i – s k(i) (x i ) |, until convergence (cf. k-means clustering algorithm).

20. Clustering regression Let each observation (x i,y i ) be assigned to a scenario k(i). Choose alternately the functions s k the assignments k(i) to minimize  i | y i – s k(i) (x i ) |, until convergence (cf. k-means clustering algorithm). For univariate y, a median regression problem

21. Example: wind power Example: wind power, next 2 hours

26. Scenario probabilities Each scenario gets a probability: that of the part of the distribution closest to it. Given one probability distribution …

27. Probability p k (x) of scenario k must reflect the local density of observations (x i, y i ) near (x, s k (x)). Multinomial logistic regression: probabilities proportional to exp( j  jk b j (x)) where  jk are to be found. Conditional scenario probabilities

28. Wind: scenarios and probabilities

29. 9% 7% 3% 90% 33% 70% 41% 26% 21% Wind: scenarios and probabilities

30. The End

31. Wind power 2hr from now: lowest scenario, conditional on present power/wind direction

32. Wind power 2hr from now: lowest scenario, conditional on present power/wind direction