for Wind Forecast Correction Valérie 1 Pierre Ailliot 2 Anne Cuzol 1 1 Université de Bretagne Sud 2 Université de Brest MAS - 2008/28/08
Outline 1 2 Linear Model : an adaptive bias correction Non Linear Model : bias and location correction 3 4
Outline 1 2 Linear Model : an adaptive bias correction Non Linear Model : bias and location correction 3 4
Accuracy of wind forecast Wind Energy Management Security and Rescue Forecast errors : intensity and position Satellite Forecast (same date) 2008/5/24 5h55 60 58 56 54 52 50 48 46 44 42 40 16 14 12 10 8 6 4 60 58 56 54 52 50 48 46 44 42 40 16 14 12 10 8 6 4
Forecast and observations 2008/05/25-00 :00 50 2008/05/25-12 :00 50 49 49 48 48 47 47 46 46 45 353 354 355 356 357 358 359 360 45 353 354 355 356 357 358 359 360
Weather forecast correction (state of art) Local Numerical Weather models Purely stochastic Y t = f(y t 1,, Y t k ; θ) + σɛ t Combined models Regression [Lange et al. (2006), von Bremen et al. (2007)] or Y obs t Y obs t = f(y for t = f(y for t, Z for t ; θ) + σɛ t, Y obs t 1,..., Y obs t k ; θ) + σɛ t Data Assimilation [Dee and da Silva (1998), Galanis et al. (2002)] based on state space models
Outline Linear Model Non Linear Model 1 2 Linear Model : an adaptive bias correction Non Linear Model : bias and location correction 3 4
First step : single location Linear Model Non Linear Model Position error phase error 20 15 10 5 observation forecast 0 02/03 02/04 02/05 42 observed : -, forecast : -+ 10 8 6 4 2 52 51 50 49 48 47 46 45 44 43
A first model Linear Model Non Linear Model Linear Gaussian spate-space model db t = α(b t µ)dt + σdw t Y true t Y obs t Error (hidden) = Y for t + B t "True" Y t (hidden) = Y true t + σ obs ɛ t Observed Y t W t : standard brownian motion, ɛ t Gaussian white noise Y t : observed at time t 1,, t K Inference Kalman filter : weighted mean between the predicted state and the observation
Need another model Linear Model Non Linear Model Location or phase error 2008/5/24 5h55 60 58 56 54 52 50 48 46 44 42 40 16 14 12 10 8 6 4 60 58 56 54 52 50 48 46 44 42 40 16 14 12 10 8 6 4
Introduction of a phase correction Linear Model Non Linear Model Model d t = α ( t µ )dt + σ dv t db t = α B (B t µ B )dt + σ B dw t Y true t Y obs t = Y for t+ t + B t Phase error (hidden) Intensity error (hidden) "True" Y t (hidden) = Y true t + σ W ɛ(t) Observed Y t V t, W t : standard brownian motions, ɛ t Gaussian white noise Y t : observed at time t 1,, t K Inference Parameter estimation : EM algorithm (need smoothing) or maximum likelihood by a gradient algorithm [Robbins and Monroe, 1951] θ k = θ k 1 + γ k θ L T (θ) Monte Carlo approximation of L T (θ) and θ L T (θ) based on particular filtering [Coquelin et al., 2007] Breaks in the data due to the assimilations of the observations in the numerical weather model
Outline 1 2 Linear Model : an adaptive bias correction Non Linear Model : bias and location correction 3 4
Wind forecast correction Example of correction for Brest 20 15 10 Obs For Nonlinear Phase Linear Reg Lin 5 0 02/03 02/04 02/05
Comparison Root Mean Square Errors EQM 9 8 7 6 5 4 Linear Nonlinear Phase Numerical Stochastic Reg Lin 3 2 1 0 0 0.5 1 1.5 2 time (day)
Outline 1 2 Linear Model : an adaptive bias correction Non Linear Model : bias and location correction 3 4
Correction of weather forecast by bias and phase correction with dynamic Phase correction does not improve the bias correction rmse Mean error Local phenomena Perspectives : Spatial model Local weather