A SDE model with derivative tracking for wind power

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A SDE model with derivative tracking for windpower forecast error: model building, inference and

application

R. Tempone

Alexander von Humboldt Chair of Mathematics for Uncertainty QuantificationRWTH Aachen University, Germany

King Abdullah University of Science and Technology (KAUST), Saudi Arabia.

March 22, 2021

Raúl Tempone SDE model for the wind power error forecast March 22, 2021 1 / 63

https://orcid.org/0000-0001-5114-853X

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Joint work

Joint workA Derivative Tracking Model for Wind Power Forecast Error(https://arxiv.org/abs/2006.15907)

Renzo Caballero , Computer, Electrical and MathematicalSciences and Engineering Division (CEMSE), King AbdullahUniversity of Science and Technology (KAUST), Thuwal23955-6900, Saudi ArabiaAhmed Kebaier , Université Sorbonne Paris Nord, LAGA,CNRS, UMR 7539, F-93430, Villetaneuse, FranceMarco Scavino , Instituto de Estadística, Universidad de laRepública, Montevideo, UruguayRaúl Tempone , Alexander von Humboldt Professor inMathematics for Uncertainty Quantification, RWTH AachenUniversity, 52702 Aachen, Germany and King Abdullah Universityof Science and Technology (KAUST), Saudi Arabia.


https://arxiv.org/abs/2006.15907

https://orcid.org/0000-0003-3220-0923

https://orcid.org/0000-0003-3468-8811

https://orcid.org/0000-0001-5114-853X

https://orcid.org/0000-0003-1967-4446

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1 Introduction and motivation2 Wind power production and forecast providers data in Uruguay3 Model Building: Phenomenological Model

Normalized wind power production model4 State independent diffusion term: Lamperti transform5 Likelihood functions of the forecast error data and optimization al-

gorithmLikelihood in the V�spaceApproximate likelihood in the V�spaceApproximate likelihood in the Z�spaceAlgorithm for the approximate maximum likelihood estimationsInitial guess for the parameters (✓0,↵)

6 Application: Uruguay wind and forecast dataset7 Summary and conclusions8 Main references


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Introduction and motivation

Wind and solar energy are expanding renewable generation capacity,experiencing record growth in the last years.

Installed Wind and Solar Power over the years

2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

Year

0

100

200

300

400

500

600

700

GW

Wind

Solar

Figure 1: Worldwide installed wind and solar power 2010–2019 IRENA, 2020.We recall the importance of accurate forecasts to use green energiesoptimally.


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Reliable wind power generation forecasting is crucial for the follow-ing applications (see, for example, Giebel et al., 2011, Chang, 2014,Zhou et al., 2013):

Allocation of energy reserves such as water levels in dams or oil,and gas reserves.Operation scheduling of controllable power plants.Optimization of the price of electricity for different parties such aselectric utilities, Transmission system operator (TSOs), Electricityservice providers (ESPs), Independent power producers (IPPs),and energy traders.Maintenance planning such as that of power plants componentsand transmission lines.


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In recent years, Uruguay has triggered a remarkable change in itsenergy matrix. In (IRENA, 2019, p.23), Uruguay was among thosecountries showcasing innovation, like Denmark, Ireland, Germany,Portugal, Greece and Spain, with proven feasibility of managingannual variable renewable energy (VRE) higher than 25% inpower systems.According to (REN21, 2019, pp.118–119), in 2018,Uruguay achieved 36% of its electricity production from variablewind energy and solar PV, raising the share of generation fromwind energy more than five-fold in just four years, from 6.2% in2014 to 36% in 2018.Including hydropower, Uruguay now produces more than 97% ofits electricity from renewable energy sources.


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Introduction and motivationAt present, Uruguay is fostering even higher levels of windpenetration by boosting regional power trading with Argentina andBrazil. In this rapidly evolving scenario, it is essential to analyzenational data on wind power production with wind powershort-term forecasting to orientate and assess the strategies anddecisions of wind energy actors and businesses.

Figure 2: Renewables: Top Ten countries according to REN21 in 2018.


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Data descriptionWind power production data in Uruguay between April and December2019, normalized with respect to the maximum installed wind powercapacity (1474 MW). Each day, wind power production recordings areavailable every ten minutes. Data from three different forecast providers,available each day starting at 1 pm.

1PM

3PM

5PM

7PM

9PM

11PM

1AM

3AM

5AM

7AM

9AM

11AM

1PM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

124-Apr-2019 and 25-Apr-2019

Forecast Provider A

Real Production

1PM

3PM

5PM

7PM

9PM

11PM

1AM

3AM

5AM

7AM

9AM

11AM

1PM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

126-Apr-2019 and 27-Apr-2019

Forecast Provider A

Real Production

Figure 3: Two 24-hour segments with the normalized wind power realproduction in Uruguay (blue line) recorded every ten minutes, and the hourlywind power production forecasted by provider A (black line).


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Wind production forecast error histograms


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Curtailment

Figure 4: A real headache: Example of a day with curtailment

In Figure 4, we plot the real and corrected data corresponding to theday 19/01/2019. We observe from this figure that this day shows signsof curtailment at the beginning. The data "Real" represents the raw realproduction and "Real_corrected" is the real production with a tentativecorrection made by the provider to curb the curtailment. The days con-taining curtailment are removed from the dataset as they induce error.


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Forecast error, no curtailment (147 daily segments)


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Forecast error transition histograms

Figure 5: Forecast error transition histograms, applying the first-orderdifference operator to the forecast errors.


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Phenomenological model

Let X = {Xt , t 2 [0,T ]} be a [0, 1]-valued stochastic process that rep-resents the normalized wind power production, defined by the followingItô stochastic differential equation (SDE):

⇢dXt = a (Xt ; pt , pt ,✓) dt + b (Xt ; pt , pt ,✓) dWt , t 2 [0,T ]X0 = x0 2 [0, 1] (1)

where:a (·, pt , pt ,✓) : [0, 1] ! R denotes a drift function,b (·; pt , pt ,✓) : [0, 1] ! R+ is a diffusion function,✓ is a vector of unknown parameters,(pt)t2[0,T ] is the given forecast, taking values in [0,1] and (pt)t2[0,T ]is its time derivative,(Wt)t2[0,T ] is a standard real-valued Wiener process.


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Specification of the drift function

Time-dependent drift function that features the mean-reverting propertyas well as derivative tracking:

a (Xt ; pt , pt ,✓) = pt � ✓t (Xt � pt) (2)

where (✓t)t2[0,T ] is a positive deterministic function, whose range de-pends on ✓, that controls the speed of reversion.Observe: Given E [X0] = p0, apply Itô’s lemma on the forecast error,Vt = Xt � pt , yielding

dVt = dXt � ptdt = �✓t Vtdt + btdWt ,

and taking expectations yields, for t > 0,

dE [Vt ]

dt= �✓t E [Vt ]

implying E [Vt ] = 0 for t > 0. [Centering property]Raúl Tempone SDE model for the wind power error forecast March 22, 2021 14 / 63

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At this stage, the process defined by (1) with drift (2) satisfies the twofollowing properties:

it reverts to its mean pt , with a time-varying parameter ✓t ,it tracks the time derivative pt .

Obs: A mean-reverting model without derivative tracking shows a de-layed path behavior.

Example: Consider the diffusion model (1) with

a(Xt ; pt ,✓) = �✓0(Xt � pt) , ✓0 > 0.

Then, given E [X0] = p0, this diffusion has mean

E [Xt ] = pt � e�✓0tZ t

0pse✓0sds 6= pt . [Not Centered]


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Models with and without derivative tracking

1PM

3PM

5PM

7PM

9PM

11PM

1AM

3AM

5AM

7AM

9AM

11AM

1PM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

124-Apr-2019 and 25-Apr-2019

99% confidence interval



Forecast

Real production

1PM

3PM

5PM

7PM

9PM

11PM

1AM

3AM

5AM

7AM

9AM

11AM

1PM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

124-Apr-2019 and 25-Apr-2019




Forecast

Real production

Figure 6: Pointwise confidence bands fitted, for the same daily segment,through diffusion models without derivative tracking (plot on the left) and withderivative tracking (plot on the right).


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Specification of the diffusion functionLet ✓ = (✓0,↵), and choose a state-dependent diffusion term that avoidsthe process exiting from the range [0, 1] as follows:

b (Xt ;✓) =p

2↵✓0Xt (1 � Xt) (3)

where ✓0 > 0, ↵ > 0 is an unknown parameter that controls the pathvariability.This diffusion term belongs to the Pearson diffusion family,in particular, it defines a Jacobi type diffusion.Recall (Forman and Sørensen, 2008) that a Pearson diffusion is a sta-tionary solution to a stochastic differential equation of the form

dXt = �✓(Xt � µ)dt +q

2✓�aX 2

t + bXt + c�dWt (4)

where ✓ > 0, and a, b, and c are parameters such that the square rootis well defined when Xt is in the state space.These parameters, together with µ, determine the state space of the diffusionas well as the shape of the invariant distribution.


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Normalized wind power production model

Normalized wind power production model

⇢dXt = (pt � ✓t (Xt � pt))dt +

p2↵✓0Xt (1 � Xt)dWt , t 2 [0,T ]

X0 = x0 2 [0, 1](5)

To ensure that Xt is the unique solution to (5) 8t 2 [0,T ] with statespace [0,1] a.s., the mean-reversion time-dependent function ✓tmust satisfy the condition:

✓t � max

✓↵✓0 + pt

1 � pt,↵✓0 � pt

pt

◆. (6)


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Theorem (Existence and Uniqueness)

Assume that

8t 2 [0,T ], 0 pt + ✓tpt ✓t , and supt2[0,T ]

|✓t | < +1. (A)

Then, there is a unique strong solution to (5) s.t. for all t 2 [0,T ],Xt 2 [0, 1] a.s.


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Truncated prediction function

Issue: If we choose the equality in (6), then ✓t becomesunbounded when pt = 0 or pt = 1.Our approach: Introduce a truncation parameter, 0 < ✏ << 1.Consider the following truncated prediction function

p✏t =

8<

:

✏ if pt < ✏pt if ✏ pt < 1 � ✏

1 � ✏ if pt � 1 � ✏

that satisfies p✏t 2 [✏, 1 � ✏] for any 0 < ✏ < 1

2 and t 2 [0,T ],implying that ✓t is bounded for every t 2 [0,T ].

TheoremTake 0 < ✏ < 1/2 and let (6) hold. Once we truncate p into p✏, thesolution X to (5) does not reach the boundary of [0, 1] a.s.


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Forecast error of the normalized wind powerproduction

Model for the forecast error of the normalized wind powerproductionThe model for the forecast error of the normalized wind powerproductionV = {Vt , t 2 [0,T ]}, Vt = Xt � pt , 8t 2 [0,T ] is defined bythe following Itô stochastic differential equation (SDE):⇢

dVt = �✓tVtdt +p

2↵✓0 (Vt + pt) (1 � Vt � pt)dWt , t 2 [0,T ]V0 = v0 2 [�p0, 1 � p0]

(7)


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Lamperti transformJohn Lamperti (Lamperti, 1964) first showed that the use of Itô’s for-mula on a well-chosen transformation of a diffusion process is again adiffusion process solving a SDE with unit, constant diffusion coefficient.(Nonlinear) Lamperti transform with unknown parameters:

Zt = h(Vt , t ;✓) =Z

dv�(v)

��v=Vt

=1p

2↵✓0

Z1p

(v + pt)(1 � v � pt)dv

��v=Vt

= �

s2

↵✓0arcsin(

p1 � Vt � pt)

(8)

By Itô’s lemma, if h(v , t) is C2([�pt , 1 � pt ]) for v and C1([0,T ]) for t ,then:

dZt =

✓@th + @v h (�✓tVt) +

12@2

v h �2◆

dt + @v h � dWt .


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SDE with state independent unit diffusion term

Zt satisfies the SDE with constant, unitary diffusion coefficient,

dZt =

2

664

pt � ✓t

✓1 � pt � sin2

⇣�q

↵✓02 Zt

⌘◆

p2↵✓0 cos

⇣�q

↵✓02 Zt

⌘sin⇣�q

↵✓02 Zt

⌘

�14

p2↵✓0

✓1 � 2 cos2

⇣�q

↵✓02 Zt

⌘◆

cos⇣�q

↵✓02 Zt

⌘sin⇣�q

↵✓02 Zt

⌘

3

775 dt + dWt

=

2pt � ✓t(1 � 2pt) + (↵✓0 � ✓t) cos(�

p2↵✓0Zt)p

2↵✓0 sin (�p

2↵✓0Zt)

�dt + dWt .

(9)


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Z-Forecast error transition histograms after Lamperti T.

Figure 7: Lamperti transformed forecast error transition histograms betweenApril and December 2019 without wind power production curtailment:low-power (upper-left plot), mid-power (upper-right plot), high-power(lower-left plot), and the global range of power (lower-right plot).


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Likelihood in the V�space (1/2)

M non-overlapping paths of the continuous-time Itô process V .Each path is sampled at N + 1 equispaced discrete points with agiven interval length �.We denote this random sample by

V M,N+1 =n

V N+1t1 ,V N+1

t2 , . . . ,V N+1tM

o,

where tj is the start time of the path j andV N+1

tj =n

Vtj+i�, i = 0, . . . ,No

, 8j 2 {1, . . . ,M}.

Let ⇢(v |vj,i�1;✓) be the conditional probability density of Vtj+i� ⌘ Vj,igiven Vj,i�1 = vj,i�1 evaluated at v , where ✓ = (✓0,↵) are the unknownmodel parameters.


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Likelihood in the V�space (2/2)

The Itô process defined by the SDE (7) is Markovian.The likelihood function of the sample V M,N+1 can be written asfollows:

L⇣✓;V M,N+1

⌘=

MY

j=1

( NY

i=1

⇢⇣

Vj,i |Vj,i�1; p[tj,i�1,tj,i ],✓⌘)

where tj,i ⌘ tj + i� for any j = 1, . . . ,M and i = 0, . . . ,N.

Obs: We have used an independence assumption over the index j inthe likelihood above.


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Moment matching technique

Closed-form expression for the transition densities of V ,⇢�Vj,i |Vj,i�1;✓

�are rarely available (Egorov et al., 2003).

Approximate likelihood methods(Särkkä and Solin, 2019, Chapter 9).Moment matching technique:

assume a surrogate transition density for V .match the conditional moments of the surrogate density for V withthe conditional moments of the SDE models (7).

m1(t) ⌘ E [Vt |Vtj,i�1 = vj,i�1] = e�

R ttj,i�1

✓sdsvj,i�1, for any t 2 [tj,i�1, tj,i [,

j = 1, . . . ,M and i = 1, . . . ,N .

For k � 2, let mk (t) ⌘ E⇥V k

t |Vtj,i�1 = vj,i�1⇤

apply Itô’s lemma ong(Vt) = V k

t , yielding


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Moment matching technique

dmk (t)dt

= �k(✓t + (k � 1)↵✓0)mk (t)

+ k(k � 1)↵✓0(1 � 2pt)mk�1(t)+ k(k � 1)↵✓0pt(1 � pt)mk�2(t). (10)

with initial conditions mk (tj,i�1) = vkj,i�1 .

For any t 2 [tj,i�1, tj,i [, the first two moments of V , m1(t) and m2(t),solve the following ODE system

8><

>:

dm1(t)dt = �m1(t)✓t

dm2(t)dt = �2(✓t + ↵✓0)m2(t) + 2↵✓0(1 � 2pt)m1(t)

+2↵✓0pt(1 � pt)

(11)

with initial conditions m1(tj,i�1) = vj,i�1 and m2(tj,i�1) = v2j,i�1 .


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Approximate log-likelihood in the V�space

For any t 2 [tj,i�1, tj,i [, approximate the transition densities of theprocess V using a Beta distribution (the invariant distribution ofthe Jacobi type processes) with parameters ⇠1 and ⇠2.

⇠1(t) = �(µt + 1 � ✏)(µ2

t + �2t � (1 � ✏)2)

2(1 � ✏)�2t

,

⇠2(t) =(µt � 1 + ✏)(µ2

t + �2t � (1 � ✏)2)

2(1 � ✏)�2t

,

(12)

where µt = m1(t) and �2t = m2(t)� m1(t)2 .

The approximate log-likelihood ˜(·; vM,N+1) of the observedsample vM,N+1 :


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Approximate log-likelihood in the V�space

˜�✓; vM,N+1�

=MX

j=1

NX

i=1

log

(1

2(1 � ✏)

1B(⇠1(t�j,i ), ⇠2(t�j,i ))

✓vj,i + 1 � ✏

2(1 � ✏)

◆⇠1(t�j,i )�1

⇥✓

1 � ✏� vj,i

2(1 � ✏)

◆⇠2(t�j,i )�1), (13)

where the shape parameters ⇠1(t�j,i ) and ⇠2(t�j,i ), according to (12),depend on the left limit moments, µ(t�j,i ;✓) and �2(t�j,i ;✓), as t " tj,i .These are computed solving numerically the initial-value problem (11).B(⇠1, ⇠2) denotes the Beta distribution with parameters ⇠1 and ⇠2.


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Approximate likelihood in the Z�space

The transition density of the process Z , which has been defined throughthe Lamperti transformation (8) of V , can be conveniently approximatedby a Gaussian surrogate density.The drift coefficient a(Zt ; pt , pt ,✓) of the process Z that satisfies (9) isnonlinear. After linearizing the drift around the mean of Z , µZ (t) ⌘E [Zt ], we obtain the following system of ODEs to compute, for any t 2[tj,i�1, tj,i [, the approximations of the first two central moments of Z , sayµZ (t) ⇡ E [Zt ] and vZ (t) ⇡ Var [Zt ]:

(d µZ (t)

dt = a�µZ (t); pt , pt ,✓

�

dvZ (t)dt = 2a0�µZ (t); pt , pt ,✓

�vZ (t) + 1

(14)


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Approximate likelihood in the Z�spacewith initial conditions µZ (tj,i�1) = zj,i�1 and vZ (tj,i�1) = 0 , and where

a0 (µZ (t); pt , pt ,✓)

=(↵✓0 � ✓t)� cos(

p2↵✓0Zt)[✓t(1 � 2pt)� 2pt ]

sin2 (p

2↵✓0Zt).

The approximate Lamperti log-likelihood ˜Z�·; zM,N+1� for the observed

sample zM,N+1 is given by

˜Z⇣✓; zM,N+1

⌘

=MX

j=1

NX

i=1

log

8<

:1q

2⇡vZ (t�j,i ;✓)exp

�

(zj,i � µZ (t�j,i ;✓))2

2vZ (t�j,i ;✓)

!9=

; , (15)

where the limits µZ (t�j,i ;✓) and vZ (t�j,i ;✓) are computed solving numeri-cally the initial-value problem (14).


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Initial guess for (✓0,↵)

We use least square minimization and quadratic variation over the datato find an initial guess (✓⇤0,↵

⇤).We consider the observed data vM,N+1 with length between observa-tions �, where i 2 {0, . . . ,N � 1} and j 2 {1, . . . ,M}.

For any t 2 [tj,i , tj,i+1[, the random variable (Vj,i+1|vj,i) has aconditional mean that can be approximated by the solution of thefollowing system:

⇢dE[V ](t) = �✓tE[V ](t)dtE [V ] (tj,i) = vj,i

in the limit t " tj,i+1, i.e., E [V ] (t�j,i+1).

If we assume that ✓t = c 2 R+ for all t 2 [tj,i , tj,i+1[, thenE [V ] (t�j,i+1) = vj,i e�c�.


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Given M ⇥ N transitions, we can write the regression problem forthe conditional mean with L2 loss function as:

c⇤ = argminc�0

2

4MX

j=1

N�1X

i=0

⇣vj,i+1 � E[V ]

⇣t�j,i+1

⌘⌘23

5

= argminc�0

2

4MX

j=1

N�1X

i=0

⇣vj,i+1 � vj,i e�c�

⌘23

5

⇡ argminc�0

2

4MX

j=1

N�1X

i=0

�vj,i+1 � vj,i(1 � c�)

�2

3

5 (16)


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Least square minimizationAs equation (16) is convex in c, then

c⇤ ⇡PM

j=1PN�1

i=0 vj,i�vj,i � vj,i+1

�

�PM

j=1PN�1

i=0�vj,i�2

Set ✓⇤0 = c⇤ .


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Quadratic variationWe approximate

the quadratic variation of the ltô’s process V is[V ]t =

R t0 b(Vs;✓, ps)2ds

where b(Vs;✓, ps) =p

2↵✓0 (Vs + ps) (1 � Vs � ps)withthe discrete process quadratic variation :

P0<tj,it

�Vtj,i+1 � Vtj,i

�2.

Initial guess for the diffusion variability coefficient ✓0↵:

✓⇤0↵⇤ ⇡

PMj=1PN�1

i=0�vj,i+1 � vj,i

�2

2�PM

j=1PN�1

i=0�vj,i+1 + pj,i+1

� �1 � vj,i+1 � pj,i+1

�

where � is the length of the time interval between two consecutivemeasurements.


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Model specification with the additional parameter �

To ensure that E(Xt) = pt at all times, we need E [V0] = 0. For mostdays, the forecast error at time tj,0 = 0 is not zero.

1 Assume that there is a time in the past tj,�� < tj,0, such that theforecast error is zero, Vj,�� = 0.

2 Extrapolate backward linearly the truncated prediction function toget its value at time tj,��, pj,��, and set vtj,�� = 0.Given the parameters (✓0,↵), find � by maximizing the likelihood ofinitial transitions:

argmax�

L�

⇣✓, �; vM,1

⌘= argmax

�

MY

j=1

⇢0�vj,0|vj,��;✓, �

�, (17)

where L� is the approximated ��likelihood.Now assume that the initial transition density has a Betadistribution and apply the moment matching technique.


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Model specification with the additional parameter �

The approximated complete likelihood Lc , which estimates the vector(✓0,↵, �), is given by

Lc

⇣✓, �; vM,N+1

⌘= L

⇣✓; vM,N+1

⌘L�

⇣✓, �; vM,1

⌘, (18)

where L�✓; vM,N+1� is the non-log version of (13). As we can provide

initial guesses for ✓ and �, we have a starting point for the numericaloptimization of the approximated complete likelihood (18).


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Application: Uruguay wind and forecast dataset

Partition the 147 segments of normalized wind power production, each24-hours long. Select 73 non-contiguous segments for the models’ cal-ibration procedure, assigning them to the training set. The other 74non-contiguous segments compose the test set.

Optimal parameters in the V -space: (✓V0 ,↵

V ) = (1.93, 0.050)Optimal parameters in the Z -space: (✓Z

0 ,↵Z ) = (1.87, 0.043)


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Model comparison and assessment of the forecast providers.Model 1: (Elkantassi et al., 2017, p.383): This model does notfeature derivative tracking:⇢

dXt = �✓0(Xt � pt)dt +p

2↵✓0Xt(1 � Xt)dWt , t 2 [0,T ]X0 = x0 2 [0, 1],

(19)

with ✓0 > 0, ↵ > 0.

Model 2: This model features derivative tracking and time-varyingmean-reversion parameter, ✓t = max

⇣✓0,

↵✓0+|pt |min(pt ,1�pt )

⌘,

⇢dXt =

�pt � ✓t(Xt � pt)

�dt +

p2↵✓0Xt(1 � Xt)dWt , t 2 [0,T ]

X0 = x0 2 [0, 1],(20)

with ✓0 > 0, ↵ > 0 and ✓t satisfying condition (6).


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Table 1: Model comparison.

Model ForecastProvider Method Product

✓0↵AIC BIC

Model 1 Provider A Gaussian Proxy 0.105 -58226 -58211Shoji-Ozaki 0.104 -58226 -58211Beta Proxy 0.104 -58286 -58271

Provider B Gaussian Proxy 0.105 -58226 -58211Shoji-Ozaki 0.104 -58226 -58211Beta Proxy 0.104 -58288 -58273

Provider C Gaussian Proxy 0.105 -58226 -58211Shoji-Ozaki 0.104 -58226 -58211Beta Proxy 0.104 -58286 -58271

Model 2 Provider A Beta Proxy 0.097 -73700 -73685Provider B Beta Proxy 0.098 -73502 -73487Provider C Beta Proxy 0.108 -72518 -72503


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The optimal estimates of the parameters of Model 2, for the three fore-cast providers, with Beta surrogates for the transition density:

Table 2: Optimal parameters for the three different forecast providers usingModel 2 with Beta proxies.

Forecast Provider Parameters (✓0,↵) Product ✓0↵

Provider A (1.93, 0.050) 0.097Provider B (1.42, 0.069) 0.098Provider C (1.38, 0.078) 0.108


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Figure 8: Two days with five simulated wind power production paths.

Given optimal estimates of the parameters of the complete likelihood forModel 2, obtain empirical pointwise confidence bands for wind powerproduction (5000 simulations per day).


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Figure 9: Empirical pointwise confidence bands for the wind power productionusing the approximate MLEs for Model 2.


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Summary and conclusions

A methodology is developed to assess the short-term forecast ofthe normalized wind power, which is agnostic of the wind powerforecasting technology.We built a phenomenological stochastic differential equation modelfor the normalized wind power production forecast error, with time-varying mean-reversion parameter and time-derivative tracking ofthe forecast in the linear drift coefficient, and state-dependent andtime non-homogenous diffusion coefficient.The Lamperti transform with unknown parameters provides a ver-sion of the proposed model with a unit diffusion coefficient.We used approximate likelihood-based methods for models’ cali-bration.The incorporation of an early transition with an additional parame-ter accounts for the forecast’s uncertainty at the beginning of eachfuture period.


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Summary and conclusions

We obtained a robust procedure for synthetic data generation that,using the available forecast input, embraces future wind power pro-duction paths through empirical pointwise bands with prescribedconfidence.Application to the wind power production and three forecast providersdataset in Uruguay between April and December 2019.An objective tool is available for forecast assessment and compar-ison through model selection.This work contributes toward the efficient management of renew-able energies.


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Main references

Alfonsi, A. (2015). Affine Diffusions and Related Processes: Simulation,Theory and Applications (Vol. 6). Springer.

Badosa, J., Gobet, E., Grangereau, M., & Kim, D. (2018). Day-aheadprobabilistic forecast of solar irradiance: A stochastic differen-tial equation approach (P. Drobinski, M. Mougeot, D. Picard, R.Plougonven, & P. Tankov, Eds.). In P. Drobinski, M. Mougeot,D. Picard, R. Plougonven, & P. Tankov (Eds.), Renewable en-ergy: Forecasting and risk management, Cham, Springer Inter-national Publishing.

Caballero, R., Kebaier, A., Scavino, M., & Tempone, R. (2020). A deriva-tive tracking model for wind power forecast error. arXiv preprintarXiv:2006.15907.

Carlsson, J., Moon, K.-S., Szepessy, A., Tempone, R., & Zouraris, G.(2010). Stochastic differential equations: Models and numerics.Lecture notes.


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Main references

Chang, W.-Y. (2014). A Literature Review of Wind Forecasting Methods.Journal of Power and Energy Engineering, 2(4), 161–168. https://doi.org/10.4236/jpee.2014.24023

Egorov, A. V., Li, H., & Xu, Y. (2003). Maximum likelihood estimation oftime-inhomogeneous diffusions. Journal of Econometrics, 114,107–139.

Elkantassi, S., Kalligiannaki, E., & Tempone, R. (2017). Inference andSensitivity in Stochastic Wind Power Forecast Models. In M. Pa-padrakakis, V. Papadopoulos, & G. Stefanou (Eds.), 2nd EC-COMAS Thematic Conference on Uncertainty Quantification inComputational Sciences and Engineering (pp. 381–393). RhodesIsland, Greece, Eccomas Proceedia UNCECOMP 2017. https://doi.org/10.7712/120217.5377.16899

Forman, J. L., & Sørensen, M. (2008). The Pearson diffusions: A classof statistically tractable diffusion processes. Scandinavian Jour-nal of Statistics, 35(3), 438–465.


https://doi.org/10.4236/jpee.2014.24023

https://doi.org/10.4236/jpee.2014.24023

https://doi.org/10.7712/120217.5377.16899

https://doi.org/10.7712/120217.5377.16899

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Main references

Giebel, G., Brownsword, R., Kariniotakis, G., Denhard, M., & Draxl,C. (2011). The state-of-the-art in short-term prediction of windpower: A literature overview. ANEMOS. plus.

Iacus, S. M. (2008). Simulation and Inference for Stochastic DifferentialEquations: With R Examples. New York, Springer.

IRENA. (2019). Innovation landscape for a renewable-powered future:Solutions to integrate variable renewables. Abu Dhabi.

IRENA. (2020). Renewable Energy Statistics 2020 The InternationalRenewable Energy Agency. Abu Dhabi.

Jang, H. S., Bae, K. Y., Park, H.-S., & Sung, D. K. (2016). Solar powerprediction based on satellite images and support vector ma-chine. IEEE Transactions on Sustainable Energy, 7(3), 1255–1263.

Karatzas, I., & Shreve, S. E. (1998). Brownian motion. In Brownianmotion and stochastic calculus (pp. 47–127). New York, NY,Springer New York. https://doi.org/10.1007/978-1-4612-0949-2_2


https://doi.org/10.1007/978-1-4612-0949-2_2

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Main references

Kuo, H. (2006). Introduction to stochastic integration. universitext, SpringerNew York.

Lamperti, J. (1964). A simple construction of certain diffusion processes.J. Math. Kyoto Univ., 4(1), 161–170. https://doi.org/10.1215/kjm/1250524711

Møller, J. K., Zugno, M., & Madsen, H. (2016). Probabilistic Forecastsof Wind Power Generation by Stochastic Differential EquationModels. Journal of Forecasting, 35(3), 189–205.

Rana, M., Koprinska, I., & Agelidis, V. G. (2016). Univariate and mul-tivariate methods for very short-term solar photovoltaic powerforecasting. Energy Conversion and Management, 121, 380–390.

REN21. (2019). Renewables 2019 Global Status Report. Paris.Särkkä, S., & Solin, A. (2019). Applied Stochastic Differential Equations.

Cambridge University Press.


https://doi.org/10.1215/kjm/1250524711

https://doi.org/10.1215/kjm/1250524711

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Main references

Wu, Y.-K., Chen, C.-R., & Abdul Rahman, H. (2014). A novel hybridmodel for short-term forecasting in pv power generation. Inter-national Journal of Photoenergy, 2014.

Zhou, Z., Botterud, A., Wang, J., Bessa, R., Keko, H., Sumaili, J., &Miranda, V. (2013). Application of probabilistic wind power fore-casting in electricity markets. Wind Energy, 16(3), 321–338. https://doi.org/10.1002/we.1496


https://doi.org/10.1002/we.1496

https://doi.org/10.1002/we.1496

Documents

A SDE model with derivative tracking for wind power