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: [email protected]
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
Mohamed Chaouch (University of Reading, UK ) 9 / 36
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
Mohamed Chaouch (University of Reading, UK ) 12 / 36
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
Mohamed Chaouch (University of Reading, UK ) 13 / 36
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
Mohamed Chaouch (University of Reading, UK ) 15 / 36
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
Mohamed Chaouch (University of Reading, UK ) 19 / 36
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)
Mohamed Chaouch (University of Reading, UK ) 26 / 36
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]
Mohamed Chaouch (University of Reading, UK ) 27 / 36
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)
Mohamed Chaouch (University of Reading, UK ) 32 / 36
Introduction and motivationsThe proposed model: Functional Partial Linear Model
Application to individual load curve
Mohamed Chaouch (University of Reading, UK ) 33 / 36
Introduction and motivationsThe proposed model: Functional Partial Linear Model
Application to individual load curve
Mohamed Chaouch (University of Reading, UK ) 34 / 36
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)!!
Mohamed Chaouch (University of Reading, UK ) 35 / 36
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
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