Modelling inflows for water valuation

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Modelling inflows for water valuation. Dr. Geoffrey Pritchard University of Auckland / EPOC. Inflows where it all starts. CATCHMENTS. thermal generation. reservoirs. hydro generation. transmission. consumption. We want: inflow scenarios for use with generation/power system models - PowerPoint PPT Presentation

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  • Modelling inflows for water valuationDr. Geoffrey PritchardUniversity of Auckland / EPOC

  • Inflows where it all startsWe want: inflow scenarios for use with generation/power system models - in a form useful for optimization.Historical inflow sequences work for back-testing of a known strategy - but not for optimization (will be clairvoyant).CATCHMENTShydro generationthermal generationtransmissionconsumptionreservoirs

  • Hydro-thermal scheduling 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.

  • SDDP for hydro-thermal schedulingWeek 6Week 7Week 8

  • SDDP for hydro-thermal schedulingWeek 6Week 7Week 8min (present cost) + E[ future cost ]s.t. (satisfy demand, etc.)

  • SDDP for hydro-thermal schedulingThe critical step requires estimating expected cost (the expected future cost for earlier stages); souncertainty (from inflows) must be modelled with discrete scenarios. - lognormal, Pearson III, or other continuous distributions wont do.Week 6Week 7Week 8min (present cost) + E[ future cost ]s.t. (satisfy demand, etc.)S pss

  • Inflow scenarios for a single weekThe historical record gives one scenario per historical year - may be too many scenarios, or too few - historical extreme events can recur, but only in the identical week of the year(1/7/2014 7/7/2014, say)

  • Scenarios by quantile regression Each scenario has its own curve. Any number of scenarios, possibly with unequal probabilities. - computationally intensive models Smooth seasonal variation. - model interpretation

  • Serial dependenceInflow scenarios for successive weeks should not just be sampled independently.

  • (Model simulated for 100 x 62 years, independent weekly inflows.)Serial dependence affects the distribution of total inflow over periods longer than 1 week.

  • Variance inflation Keep the assumed independence of weekly inflows, but modify their distribution to increase its variance. Wrong in two ways, but hopefully the errors cancel. Used e.g. in SPECTRA.

  • Variance inflation Keep the assumed independence of weekly inflows, but modify their distribution to increase its variance. Wrong in two ways, but hopefully the errors cancel. Used e.g. in SPECTRA.

  • (Model simulated for 100 x 62 years, independent weekly inflows with vif=2.2.)With variance inflation, inflow distribution is wrong over 1 week but not bad over 4 months.

  • Explicit serial dependence Inflow is a random linear (or concave) function of inflow (or a state variable) from previous stage(s): Xt = Ft(Xt-1) (Ft random, i.e. scenario-dependent)

    Commonest type is log-autoregressive: log Xt = b log Xt-1 + a + et (et random)

    General linear form (ideal for SDDP):Xt = At + BtXt-1 (At, Bt random)

  • Serially dependent modelsModels fitted to all data, shown for week beginning 2013-08-3162 scenarios derived from regression residuals16 scenarios fitted by quantile regression

  • (Model simulated for 100 x 62 years, dependent weekly inflows, general linear form.)Serially dependent model can match inflow distribution over 1-week and 4-month timescales.

  • A test problemChallenging 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 (EPOCs SDDP code).

  • Results optimal strategy(Quantities are expected averages over full time horizon; probabilities are for any shortage/spill within time horizon.)

  • alpha, beta can be seasonally varying, allowing different dependence structure in different stages. Similarly A_t, B_t.Both models curves vary seasonally. GL model: wetter scenarios differ in intercept (runoff dominated by current precipitation). Drier ones differ in slope (runoff from precipitation in earlier periods).LA model: scenarios differ only in multiplicative constant.At very low (or very high) storage levels, characteristic time scale becomes shorter -> VIF model overstates risk.