Outlines

Forecasting intraday individual load curveusing functional partial linear model

Mohamed Chaouch

Centre for Mathematics of Human BehaviourDepartment of Mathematics and Statistics

University of Reading, UKemail: m.chaouch@reading.ac.uk

Mohamed Chaouch (University of Reading, UK ) 1 / 36

Outlines

Outlines

1 Introduction and motivations

2 The proposed model: Functional Partial Linear Model

3 Application to individual load curve

Mohamed Chaouch (University of Reading, UK ) 2 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Outlines

1 Introduction and motivations

2 The proposed model: Functional Partial Linear Model

3 Application to individual load curve

Mohamed Chaouch (University of Reading, UK ) 3 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Mohamed Chaouch (University of Reading, UK ) 4 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Forecasting National level load curve

Why forecasting is important for National-level ?

Short-term Forecasting of National-level load curve is very useful to:

improve the global balance between demand/production

Solutions proposed for that task

Several methods have been proposed:

(S)ARIMA, Exponential Smoothing, Non/Semi-parametricRegression, . . .

Neural Network, Random Forest, . . .

Mohamed Chaouch (University of Reading, UK ) 5 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

What about small aggregation ?

Small aggregation: substation, a node in the network, a specific groupin the customers’ portfolio (households with EH, specific professionalactivity, . . . ),

Individual: Residential Household, Small & Medium Enterprises,administration offices, public lighting, . . .

This problem is less studied in literature, especially the intraday one,

The only data we have are 6 months index electricity consumption,

Why do we need that ? (from operational point of view)

Is it possible to solve this problem ? (from Statistical point of view)

Mohamed Chaouch (University of Reading, UK ) 6 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Who needs individual forecast ?

Trading departments

. The Residential Energy market is more and more open to competition

. Energy company should understand the energy consumption behaviorof his clients in order to offer new competitive tariffs

. Help the customer to be an actor in reducing the peak demand inwinter

The Network Energy Manager (NEM)

. New Energy management constraints: Smart Grid

Mohamed Chaouch (University of Reading, UK ) 7 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Mohamed Chaouch (University of Reading, UK ) 8 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Consequences of PV production on the grid

Mohamed Chaouch (University of Reading, UK ) 10 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Mohamed Chaouch (University of Reading, UK ) 11 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

LV network without PV production

The electric charge is decreasing

The electricity cables have decreasing cross sections

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

LV network with PV production

The electric charge becomes multi-directional dependingon consumption and production

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

LV network with PV production

But when PV production becomes important

The LV network cannot resist to that situation

Mohamed Chaouch (University of Reading, UK ) 14 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Solution 1

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Strengthen the LV network

This solution is possible for several hours in the year

If more, we need investments (budget constraints !!)

Mohamed Chaouch (University of Reading, UK ) 16 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Solution 2

Mohamed Chaouch (University of Reading, UK ) 17 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

LV network with PV production

Smart Grid: local balance in the LV network

Energy storage (batteries, EV, hot water storage tank)

Demand shift (utilities monitoring, tariff prompting)

Mohamed Chaouch (University of Reading, UK ) 18 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Problem: Not easy to storage energy and we cannot dothat for long time

Solution:

1 Forecasting demand for each end point in the LV network

2 Forecasting individual PV production

The NEM plays the role of coordinator in order to keepthe local balance in the LV network

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curveGeneral context

Mohamed Chaouch (University of Reading, UK ) 20 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Outlines

1 Introduction and motivations

2 The proposed model: Functional Partial Linear Model

3 Application to individual load curve

Mohamed Chaouch (University of Reading, UK ) 21 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

The prediction problem

. Suppose one observes a square integrable continuous-time stochasticprocess (X = X (t), t ∈ R) over the interval [0,T ],T > 0;

. We want to predict X all over the segment [T ,T + δ], δ > 0

Mohamed Chaouch (University of Reading, UK ) 22 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

The prediction problem

. Divide the interval into subintervals [(l − 1)δ, lδ], l = 1, . . . , n with(δ = T/n);

. Consider a functional-valued discrete time stochastic processZi (t) = X (t + (i − 1)δ), i ∈ N ∀t ∈ [0, δ)

Mohamed Chaouch (University of Reading, UK ) 23 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Functional Partial Linear Model

Functional Partial Linear Model

Let {(Zi ; (Xi1, . . . ,Xip;Zi−1))}i=1,...,n be observations obtained formhistorical data. For all t ∈ [1, 24],

Zi (t) =

p∑j=1

Xijβj(t)︸ ︷︷ ︸linear

+m(Zi−1(t))︸ ︷︷ ︸non-linear

+ εi (t)︸︷︷︸centred

, ∀i = 1, . . . , n

= Xτi β + m(Zi−1(t)) + εi (t), ∀i = 1, . . . , n

where β = (β1(t), . . . , βp(t))τ and Xi = (Xi1, . . . ,Xip)τ

Mohamed Chaouch (University of Reading, UK ) 24 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Estimation: Backfitting approach

Let (X ,T ) ∈ Rp ×F , where F is a functional space and Z ∈ F .

Z = Xτβ + m(T ) + ε, where E (ε | X,T ) = 0F . (1)

Subtract Xτβ from (1) ⇒

E{Z − Xτβ | T} = m(T ). (2)

Step1: Estimate the parametric coeff. β by LS regression of Z on X,Step2: Plugging Xτ β into (2) yields now a classic nonparametricregression problem of the regression operator m(•),Step3: use m(T ) to update β, E (Z − m(T ) | X) = XτβStep4: update then m(T ), etc.

Mohamed Chaouch (University of Reading, UK ) 25 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Estimation: Backfitting approach

Estimation of β and m(•)

βh =(XτhXh

)−1XτhYh

mh(Zn) =n−1∑m=1

wn,h(Zn,Zm)(Zm+1 − Xτm+1βh

)where

Xi = (Xi1, . . . ,Xip)τ and X = (X1, . . . ,Xn)τ

Y = (Z1(t), . . . ,Zn(t))τ

For any n × p-dimensional matrix A, Ah = (In −Wh)A

Wh = (wn,h(Zi ,Zj))1≤i ,j≤n, wn,h(Z ,Zi ) = K(d(Z ,Zi )/h)∑nm=1 K(d(Z ,Zm)/h)

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Day ahead forecasting

Zn+1(t) = Xτn+1β + mh(Zn(t)), ∀t ∈ [1, 24]

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Estimation procedure: wavelet transform

Each segment Zi (t) is broken up into two terms: a smooth approximationSi (t) (lower freqs) and a set of details Di (t) (higher freqs) using DTW

Zi (t) =2j0−1∑k=0

c(i)j0,kφj0,k(t) +

J∑j=1

2j−1∑k=0

d(i)j ,kψj ,k(t)

where φj ,k and ψj ,k the scaling and wavelet functions associated to anMRA.

. the parameter j0 controls the separation. We set j0 = 0.

Mohamed Chaouch (University of Reading, UK ) 28 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Estimation procedure

Step1: Dissimilarity between segments

Search the past for segments that are similar to the last one. For tweobserved series of length 2J say Zm and Zl we set for each scale j ≥ j0:

distj(Zm,Zl) =

2j−1∑k=0

(d(m)j ,k − d

(m)j ,k )2

1/2

Then, we aggregate over the scales taking into account the number ofcoefficients at each scale

D(Zm,Zl) =J−1∑j=j0

2−j/2distj(Zm,Zl)

Mohamed Chaouch (University of Reading, UK ) 29 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Estimation procedure

Step2: Kernel regression

• The weights wi are now given by

wi =K (D(Zn,Zm)

hn)∑n−1

m=1 K (D(Zn,Zm)hn

)

• K : R→ R is symmetric function, centered at zero, (the so-calledkernel function), such that K (x) ≥ 0, (x)dx = 1 and∫x2K (x)dx <∞ and,

• hn is a tuning parameter (the so-called bandwidth) which controls theeffective number of segments for which wi is positive.

Mohamed Chaouch (University of Reading, UK ) 30 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Outlines

1 Introduction and motivations

2 The proposed model: Functional Partial Linear Model

3 Application to individual load curve

Mohamed Chaouch (University of Reading, UK ) 31 / 36

Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

The data

. Hourly data from 01/10/1996 to 30/06/1999

. We forecast the last year (01/07/1998 to 30/05/1999)

. Validation criteria: for each day i , Relative Mean Absolute Error(RMAE)

RMAEi =

∑24h=1 |Zi (h)− Zi (h)|∑365

i=1

∑24h=1 Zi (h)

, i = 1, . . . , 365

The forecasted mean temperature of the day (of the nearest weatherstation)

The load curve of yesterday

Heating Degree Day (HDDi ):= max(Tref − Tmean,i , 0)

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Perspectives

1 Extend this work to several customers (classification/profiles isrequired)

2 Try other covariates

3 Improve forecasts by classifying days and fit a model for each cluster4 Massive data analysis: clustering and forecasting

issues (smart meters)

. e.g.: to store 10mn electricity consumption of 30 Million of residentialcustomers we need 100 terabit (1 terabit=1 000 gigabits)!!

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Introduction and motivationsThe proposed model: Functional Partial Linear Model

Application to individual load curve

Thank you for your attention

Mohamed Chaouch (University of Reading, UK ) 36 / 36