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Stochastic processes for hydrological optimization problems

Explore the application of Markov processes in hydro scheduling to optimize water releases over time for maximal value. Develop stochastic models of inflows to improve forecasting and decision-making for hydrological systems.

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Stochastic processes for hydrological optimization problems

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  1. Stochastic processes for hydrological optimization problems Geoffrey Pritchard University of Auckland

  2. Prologue: Iterated Function Systems Consider the following Markov process in the plane: (Each step is randomly chosen from a finite list of affine transformations, independently of previous steps.)

  3. (Is this useful for anything, besides (maybe) computer graphics?)

  4. Hydro scheduling – an optimal control problem random inflows state variables: stored energy control variables: outflows Control water releases over time to maximize value. • As a workably competitive market would do.

  5. Hydro scheduling – an optimal control problem statistics OR How not to do it: 1. Develop a stochastic model of inflows. 2. Optimize releases versus the given inflow-generating process.

  6. Why develop a model of inflows? Why not just use the historical data non-parametrically? • Small dataset. e.g. 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. • A model allows events to be more extreme than anything in the data The worst event ever observed is not the worst possible

  7. Time/information structure Week t Week t+1 Week t-1 min (present cost) + E[ future cost ] s.t. satisfy demand, etc. with stored energy + random inflow Xt E • Each stage subproblem is a random optimization problem. • Stage t subproblem is solved with knowledge of Xt, but not of the future. • Weekly stages might be good, continuous time worse.

  8. Optimization Week t Week t+1 Week t-1 minu (present cost)(u) + gt(u) s.t. satisfy demand, etc. with (stored energy)(y) + random inflow Xt E gt-1(y) = • Let gt(u) = expected cost of consequences after week t of doing u in week t. • Essential observation is that is convexity-preserving. • so all optimizations can be of convex functions, i.e. tractable. • Computationally, convex subproblems -> linear programs.

  9. Model me this... (the upper Waitaki catchment) • ~ 20% of NZ electricity derives from precipitation in this region • Rainfall + summer snowmelt from Southern Alps. Tekapo A Tekapo B Ohau A Ohau B Ohau C Benmore Aviemore Waitaki

  10. Inflow data Waitaki catchment above Benmore dam, weekly, 1948-2010 Strong seasonal dependence • 3:1 ratio between midsummer high/midwinter low.

  11. Serial dependence • Weather patterns persist • increases probability of shortage/spill. • Typical correlation length ~ several weeks (but varying seasonally). • convenient for optimization (cf. e.g. Brazil).

  12. Extreme values Hydro-scheduling is sensitive to extremes of inflow (in both tails). • Low inflow -> reservoirs run dry (the most momentous thing that can happen) • High inflow -> economic loss (spill); removes risk of shortage. • Beware discrete approximations to the distribution!

  13. De-seasonalization inflow via regression: A convenient normalization, but does not make (Qt) stationary!

  14. Suggestion: an autoregressive model The AR(1) model ACF of (Qt) that is, seems reasonable. (Life should be so simple.)

  15. Stagewise independence Week t Week t+1 Week t-1 minu (present cost)(u) + gt(u) s.t. satisfy demand, etc. with (stored energy)(y) + random inflow Xt E gt-1(y) = • Stage t subproblem is solved with knowledge of Xt, but not of the future. • stagewise independence, i.e. (Xt) an independent sequence. • Inflows are not stagewise independent. • Suggested model is Markov.

  16. From independent to Markov inflows Week t Week t+1 Week t-1 minu (present cost)(u) + gt(u) s.t. satisfy demand, etc. with (stored energy)(y) + (inflow)(y, Wt) E gt-1(y) = • Make inflow a function of • what happened last week (y) • a random innovation Wt – with (Wt ) independent That’ll work – if we can express it as a linear program.

  17. LP-compatible autoregressive processes • We’re allowed a process with with an independent sequence, and a linear function. • But what we had in mind was which is nonlinear. • It’s concave, though, so admits a piecewise linear approximation.

  18. LP-compatible autoregressive processes • We’re allowed a process with with an independent sequence, and a linear function. • But what we had in mind was which is nonlinear. • It’s concave, though, so admits a piecewise linear approximation.

  19. LP-compatible autoregressive processes • We’re allowed a process with with an independent sequence, and a linear function. • But what we had in mind was which is nonlinear. • It’s concave, though, so admits a piecewise linear approximation.

  20. One linear piece Approximate the model by linearizing about How to fit this?

  21. Inference 1. Auto-regression on log-inflows: (ignores the linear approximation step) 2. Auto-regression on Qt-1 : (ignores the structure of errors) 3. Or we could do it right: a max-likelihood fit on the actual model:

  22. Stochastic dual dynamic programming (SDDP) Week t Week t+1 Week t-1 minu (present cost)(u) + gt(u) s.t. satisfy demand, etc. with (stored energy)(y) + random inflow Xt E gt-1(y) = • The leading algorithm for problems of this type. • Essential step (backward pass): • evaluate the expectation for given y, using current estimate of gt. • Use dual variables from optimization to form a cut (linear lower bound), which improves estimate of gt-1.

  23. The importance of being discrete Week t Week t+1 Week t-1 minu (present cost)(u) + gt(u) s.t. satisfy demand, etc. with (stored energy)(y) + random inflow Xt Ss ps gt-1(y) = s • If random elements have a discrete joint distribution: • solve the optimization problem for each atom. • Otherwise, need a (Monte Carlo?) discrete approximation • with not too many discrete scenarios, please (computation time is (at least) proportional to number of scenarios)

  24. Sample average approximation (SAA) Objective function is an expectation, over a continuous distribution. Only way to evaluate it is by Monte Carlo sampling. Fix a sample, optimize the resulting approximation.

  25. A catalogue of errors Our efforts to model inflows have incurred • model mis-specification error • inflows might not really be AR-1 • inferential sampling error (finite data) • parameters may be wrong • sample average approximation error • optimization is vs. a discrete approximation of inflow process discrete (data) continuous (AR-1 model) discrete (SAA approx to transition kernel) Can we obtain, in one step, a good representation of the data by a model of the final form required?

  26. The final form required Model for inflow Qt in week t : (et discretely distributed) Or more generally - where (Rt, St) is chosen at random from a small collection of (seasonally-varying) scenarios. A linear iterated function system (IFS) Markov process.

  27. The final form required Model for inflow Qt in week t : (et discretely distributed) Or more generally - where (Rt, St) is chosen at random from a small collection of (seasonally-varying) scenarios. A linear iterated function system (IFS) Markov process.

  28. Linear IFS Markov inflow model

  29. Fitting a model to data: quantile regression • Have data xi and yi for i=1,…n y x

  30. Fitting a model to data: quantile regression • Have data xi and yi for i=1,…n • Want to represent the distribution of y|x by finitely many scenarios. y x

  31. Fitting a model to data: quantile regression • Have data xi and yi for i=1,…n • Want to represent the distribution of y|x by finitely many scenarios. • Quantile regression: choose scenario sk() to minimize Sirk( yi – sk(xi) ) for a suitable loss function rk(). y x

  32. Quantile regression fitting • For a scenario at quantilet (0 < t < 1) , rt is the loss function t-1 t • For each scenario, the quantile regression problem is a linear program.

  33. Fitting a model to data: quantile autoregression Quantiles (t) 0.02 0.1 0.2 0.35 0.5 0.65 0.8 0.9 0.98 Scenario probabilities 0.06 0.07 0.125 0.15 0.15 0.15 0.125 0.07 0.06 For each of a fixed collection of quantiles, fit a scenario (Rt, St) by quantile regression: Scenario probabilities determined from quantiles; can be unequal.

  34. Fitting a model to data: quantile autoregression • Scenarios should not cross. • Dependence (slope St) can vary across the probability distribution. High-flow scenarios differ in intercept (current rainfall). Low-flow scenarios differ mainly in slope. Extreme scenarios have their own dependence structure.

  35. Continuous ranked probability score (CRPS) • A method of judging the merit of a prediction made in the form of a probability distribution. • Given prediction distribution F and actual outcome y, F y

  36. Fitting a model to data: CRPS M-estimation • CRPS can also be used as an estimation method for multi-scenario regression. • Given x, scenarios for y are s1(x) … sm(x) with probabilities p1 ... pm • Choose sk() and pk to y x • This is the most computationally challenging method (global optimization, not LP or least-squares).

  37. Fitting a model to data: CRPS M-estimation • Scenarios may cross. • Scenario probabilities are optimized in model fitting, instead of being arbitrarily chosen • Quite small numbers of scenarios seem possible.

  38. Multivariate inflow models • Need to capture spatial as well as temporal correlations. • Generalize models: • Autoregressive: need discrete approx. to multivariate error. • Quantile regression: no natural generalization of quantile. • CRPS M-estimation: generalization to energy score.

  39. A test problem • Challenging fictional system based on Waitaki catchment inflows. • Storage capacity 1000 GWh (cf. real Waitaki lakes 2800 GWh) • Generation capacity 1749 MW hydro, 900 MW thermal • Demand 1550 MW, constant • Thermal fuel $50 / MWh, VOLL $1000 / MWh • Test problem: a dry winter. • 35 weeks (2 April – 2 December) • Initial storage 336 GWh • Initial inflow 500 MW (~56% of average) • Solved with Doasa 2.0 (EPOC’s SDDP code).

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