Diffusion processes for operational management of a wind farm Introduction Diffusion processes for operational management of a wind farm Alexandre Brouste Laboratoire Manceau de Mathématiques Institut du Risque et de l’Assurance du Mans Séminaire du Centre Cournot, 23-02-2017
Diffusion processes for operational management of a wind farm Introduction Uncertainties: ◮ wind speed, ◮ vertical extrapolation, ◮ horizontal extrapolation, ◮ power function, ◮ long-term extrapolation, ◮ measures, ◮ cut-out, unavailability. Management: ◮ safe regulation (s, m), ◮ storage, trading (h), ◮ maintenance (d).
Diffusion processes for operational management of a wind farm Introduction Power of the wind stream P w ( v ) = 1 2 ρ Sv 3 . Betz’ limit of wind turbine power P max ( v ) = 16 27 P w ( v ) . This power is transformed and we consider the power (transfer) function P ( v ) (cannot reach in practice 70% of the Betz’ limit). 2000 2000 1500 1500 Power (kW) 1000 1000 500 500 0 0 0 0 5 5 10 10 15 15 20 20 25 25 30 30 Wind speed (m/s)
Diffusion processes for operational management of a wind farm Introduction 0 5 10 15 20 0 20 40 60 80 100 Simulated wind speed dataset of the NREL.
Diffusion processes for operational management of a wind farm Introduction 0 5 10 15 20 0 20 40 60 80 100 Real wind speed dataset in Veulettes-sur-Mer, France.
Diffusion processes for operational management of a wind farm Introduction 1.0 0.8 0.6 NREL Veulettes 0.4 0.2 0.0 0 1 2 3 4 time (days)
Diffusion processes for operational management of a wind farm Introduction Lots of studies of dynamical models for modeling and forecasting: – statistical models (time series, Markov models, neural networks, . . . ) for seconds, minutes, hours; – meteorological models for days and weeks. We are studying two particular diffusion process models and consider their evaluation for modeling and short-term forecasting. They are entries for stochastic control problems for storage and trading.
Diffusion processes for operational management of a wind farm Models We consider homogeneous diffusion processes that are Markov processes for which the transition probability density functions p ( t , y ; x , ϑ ) satisfy the Fokker-Planck-Kolmogorov equation ∂ 2 ∂ t p = − ∂ ∂ ∂ y ( v 0 ( y , ϑ ) p ) + 1 ∂ y 2 ( v 1 ( y , ϑ ) 2 p ) , y ∈ ❘ , t > 0 , 2 with initial condition p ( 0 , y ; x , ϑ ) = δ ( y − x ) . We have evaluated the Cox-Ingersoll-Ross (CIR) model (B. and Ben- soussan 2016) and the 3-parameter marginal Weibull model (B. and Bensoussan, preprint). The transition probability density functions are in closed form for the CIR model (Feller, 1951) but not for the marginal Weibull diffusion model.
Diffusion processes for operational management of a wind farm Models
Diffusion processes for operational management of a wind farm Models
Diffusion processes for operational management of a wind farm Models Construction of the 3-parameter marginal Weibull diffusion model is inspired from (Bibby, Skovgaard et Sorensen, 2003). We compute v 1 such that the stationary distribution is Weibull. Using the F-P-K equation, the stationary distribution f satisfies ∂ 2 − ∂ ∂ y ( v 0 ( y , ϑ ) f ( y )) + 1 ∂ y 2 ( v 1 ( y , ϑ ) 2 f ( y )) = 0 , y ∈ ❘ . 2 and by integration − v 0 ( y , ϑ ) f ( y ) + 1 ∂ ∂ y ( v 1 ( y , ϑ ) 2 f ( y )) = 0 , y ∈ ❘ . 2 Given the stationary distribution f and v 0 , we can find v 2 1 .
Diffusion processes for operational management of a wind farm Models Fixing the linear drift v 0 ( y , ϑ ) = ϑ 1 ( ϑ 2 − y ) and integrating the previous equation leads to � � v 0 ( y , ϑ ) f ( y ) dy = 0 = ⇒ ϑ 2 = y f ( y ) dy . ❘ ❘ It can be shown generally that for the mean-reverting diffusion pro- � � Y y 0 t , t ≥ 0 cess �� � � �� Y y 0 − E Y y 0 Y y 0 s + t − E Y y 0 E s s s + t lim = exp ( − ϑ 1 t ) . var Y y 0 s →∞ s
Diffusion processes for operational management of a wind farm Models 1.0 0.8 0.6 NREL Veulettes Weibull diffusion 0.4 0.2 0.0 0 1 2 3 4 time (days)
Diffusion processes for operational management of a wind farm Models
Diffusion processes for operational management of a wind farm One-step ahead forecasting � p ( 0 ) ( t , · ) y 0 Y ( 0 ) � t Y t p ( t , · )
Diffusion processes for operational management of a wind farm One-step ahead forecasting � � Y y 0 Homogeneous diffusion processes t , t ≥ 0 admit a natural point forecast � � � Y ( 0 ) � Y y 0 = E ϑ = y p ( t , y ; y 0 , ϑ ) dy t t ❘ and a probabilistic forecast with their transition densities p ( 0 ) ( t , · ) = p ( t , · ; y 0 , ϑ ) . � Cox-Ingersoll-Ross diffusion and marginal Weibull diffusion processes with linear drift term v 0 ( y , ϑ ) = ϑ 1 ( ϑ 2 − y ) have an explicit point forecast � � Y y 0 = ϑ 2 + ( y 0 − ϑ 2 ) exp ( − ϑ 1 t ) . E ϑ t
Diffusion processes for operational management of a wind farm One-step ahead forecasting = Y y 0 For a well-specified model setting ( Y obs t ), we have t � t ) 2 � � � 2 = u ( t , y 0 ) − � � 2 ( Y y 0 E Y y 0 E Y y 0 MSE ( t ) = E − t t with u ( t , x ) solving the Feynman-Kac pde, i.e. ∂ x + v 2 ∂ 2 u ∂ u ∂ t = v 0 ( x , ϑ ) ∂ u 1 ( x , ϑ ) 2 ∂ x 2 with u ( 0 , x ) = x 2 .
Diffusion processes for operational management of a wind farm One-step ahead forecasting
Diffusion processes for operational management of a wind farm Estimation in homogeneous diffusion process It is possible to show van Trees inequality in several statistical exper- iments, namely, for any sequence of estimators ( T n , n ≥ 1 ) , ϕ n ( ϑ 0 ) − 1 ( T n − ϑ ) � � �� I ( ϑ 0 ) − 1 2 ξ � � �� lim inf C →∞ lim inf sup E ϑ ℓ ≥ E ℓ n →∞ | ϑ − ϑ 0 | < C ϕ n ( ϑ 0 ) where ξ is a standard Gaussian random variable. We are looking for an asymptotically efficient sequence of estimators that reaches the lower bound.
Diffusion processes for operational management of a wind farm Estimation in homogeneous diffusion process Let Θ ⊂ R d . A family of measures { P n θ , θ ∈ Θ } is called locally asymptotically normal (LAN) at θ 0 ∈ Θ if there exist nondegenerate d × d matrices ϕ n ( θ 0 ) and I ( θ 0 ) such that for any u ∈ R d , the likelihood ratio d P n θ 0 + ϕ n ( θ 0 ) u Z n ( u ) = d P n θ 0 admits the representation � � � u , ζ n ( θ 0 ) � − 1 Z n ( u ) = exp 2 � I ( θ 0 ) u , u � + r n ( θ 0 , u ) , (1) where ζ n ( θ 0 ) → N ( 0 , I ( θ 0 )) , r n ( θ 0 , u ) → 0 (2) in law under P n θ 0 .
Diffusion processes for operational management of a wind farm Estimation in homogeneous diffusion process LAN property of the likelihoods has been established for several sta- tistical experiments: 1. sample of i.i.d. r.v. (second Le Cam’s Lemma); 2. sample of independent but inhomogeneous r.v. with the Lindeberg condition; 3. sequence of an ergodic Markov chain; 4. strictly elliptic and ergodic diffusions; 5. diffusions with observational noise; 6. Levy processes; 7. fractional Gaussian noise; 8. and others . . .
Diffusion processes for operational management of a wind farm Estimation in homogeneous diffusion process Let ( Y t , t ≥ 0 ) be the solution of a (fractional) SDE whose law depends on the unknown parameter ϑ . y 0 0 = t n t n t n ∆ n 0 1 n Our aim is to give asymptotical properties of estimators of ϑ given the observation of the path on a discrete grid 0 < t n 1 < . . . < t n n , as n → ∞ . Asymptotic properties depend on the convergence scheme.
Diffusion processes for operational management of a wind farm Estimation in homogeneous diffusion process Large sample . Here ∆ n = ∆ > 0 is fixed and, under proper assump- tions (smoothness, ergodicity, uniform ellipticity), the LAN property 1 of the likelihoods is satisfied (Roussas 72) with rate ϕ ( n ) = √ n and the Fisher information matrix is equal to � � ∂ log p ∂ I (∆ , ϑ ) i , j = log p · p dy µ ϑ ( dx ) (3) ∂ϑ i ∂ϑ j ❘ ❘ where µ ϑ is the invariant measure of the diffusion process. Conse- quently, the lower bound for the variance of the estimators can be derived precisely for any sequence of estimators ( ϑ n , n ≥ 1 ) , � √ n � �� � � � I ( ϑ 0 ) − 1 ξ C →∞ lim inf lim sup E ϑ ℓ ϑ n − ϑ ≥ E ϑ 0 ℓ n →∞ | ϑ − ϑ 0 | < C √ n with ξ ∼ N ( 0 , I ) , and ℓ is a polynomial cost function.
Diffusion processes for operational management of a wind farm Estimation in homogeneous diffusion process Mixed scheme : Here n ∆ n → ∞ , ∆ n → 0 and the LAN property of the likelihoods have been established (Gobet, 2002) under proper conditions (smoothness, ergodicity, uniform ellipticity) with different rates for ϑ 1 (drift parameter) and ϑ 2 (diffusion coefficient parame- 1 1 ter). Namely ϕ ( n ) 1 , 1 = √ n ∆ n and ϕ ( n ) 2 , 2 = √ n , respectively, and the Fisher information matrix is given by � ∂ ∂ v 0 ( y , ϑ 1 ) · v 1 ( y , ϑ 2 ) − 2 µ ( dy ) I ( ϑ ) i , j = v 0 ( y , ϑ 1 ) ∂ϑ 1 , i ∂ϑ 1 , j ❘ and � ∂ ∂ v 1 ( y , ϑ 2 ) · v 1 ( y , ϑ 2 ) − 2 µ ( dy ) . I ( ϑ ) q + i , q + j = 2 v 1 ( y , ϑ 2 ) ∂ϑ 2 , i ∂ϑ 2 , j ❘
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