``Renewable Energy Modelling and...

“Renewable Energy Modelling and Forecasting”

Pierre Pinson

Technical University of Denmark.

DTU Electrical Engineering - Centre for Electric Power and Energymail: [email protected] - webpage: www.pierrepinson.com

23 April 2018

31761 - Renewables in Electricity Markets 1

[email protected]

www.pierrepinson.com

Learning objectives

Through this lecture and additional study material, it is aimed for the students to be ableto:

1 Describe generalities of techniques that can be used to generate renewable energyforecasts

2 Build and estimate fairly simple models to be used as a basis for forecasting

3 Have enough material and motivation to improve your forecast competition entriesfor Assignment 3!


The MIT Technology Review

The MIT Technology Review:

founded at MIT in 1899daily review/analysis of technological innovation worldwideimpact: 580.000 members and 2.400.000 website visitors per month!

The 10 breakthrough technologies2014:

genome editingmicroscale 3D printingneuromorphic chipsbrain mappingetc.

renewable energy analytics (!), andmore particularly forecasting...

[See link:

MIT Technology Review - Smart Wind and Solar Power]


http://www.technologyreview.com/featuredstory/526541/smart-wind-and-solar-power/

Outline

1 Test case and general considerations

2 The benchmark forecast approaches

persistenceclimatologyetc.

3 Going further in a regression-based framework

regression in a few slides (or, how to do it without a Ph.D. in Statistics)regression for forecastingan example: combination of persistence and climatology

4 Digging in the data

measurementsweather forecastsextra features?


1 Test case and general considerations


Basis for the lecture(s)

Wind Energy

Wave Energy (same ideas can be used)

... Also for Solar Energy, the same concepts can be applied!


Test case: the Klim wind farm

The wind farm:

full name: Klim Fjordholmeonshore/offshore: onshoreyear of commissioning: 1996

nominal capacity (Pn): 21 MWnumber of turbines in farm: 35average annual electricity generation: 49 GWh

data available: 1999-2003 (for some researchers)temporal resolution: 5 mins, and hourly averagesweather forecasts: wind speed and direction,temperature

A link to the online description:Vattenfall’s Klim wind farm

The wind farm is being recommissioned these days:NordJyske online article

Remember that we normalize power generation - in practice, yt ∈ [0, 1], ∀t31761 - Renewables in Electricity Markets 7

http://powerplants.vattenfall.com/node/247

http://nordjyske.dk/nyheder/siemens-skal-levere-gigantmoeller/6e13bf3b-f296-4080-81af-bb3f4cd5fd8e/43/1670

General considerations

Forecasting is about the future! Lead times within 0-48 hours, in line withmarket-based operations

When being at time t and aiming to generate a forecast for time t + k, onlyknowledge available at time t can be used...

observations up to time t: power generation, meteorological measurements, etc.

weather forecasts for the period of interest

Since forecasts will always have a part of error, just accept, and try to minimize it


2 The benchmark forecast approaches


No need to make it difficult...

What is the easiest way to predict wind power generation?

Data-free approaches:

making random guesses (it could actuallywork...)

making educated guesses (works fine in certainplaces and seasons, e.g., summer in Crete,all-year-round in Egypt)

Data-based approaches:

persistence

climatology

simple statistical models, etc.








persistence

climatology



The random guess approach

At time t, we make a random guess for all lead times t + k, k = 1, . . . , 48

This translates to

yt+k|t = uk , ∀k,

where uk ∼ U [0, 1]

Right:Example of randomguess forecast forKlim, issued on 28April 2002, 00:00UTC

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lead time [k]

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er [p

.u.]

1 6 12 18 24 30 36 42 48

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0.8

1.0

●

forecastsobservations

Let us apply that forecast strategy for a whole sample year (2002), and analyse itsperformance...


Evaluation of the random guess approach

The quality of the forecasts is summarized in terms of bias, MAE and RMSE

lead time [k]

erro

r [%

of P

n]

1 6 12 18 24 30 36 42 48

−40

−20

020

4060

biasMAERMSE

How does that look like?


The persistence approach

At time t, the persistence forecast (“what you see is what you get”) for all leadtimes t + k, k = 1, . . . , 48 is based on the idea that your best guess is your latestpiece of information...

This translates to

yt+k|t = yt , ∀k,

where yt is the latestmeasurement available

Right:Example of apersistence forecast forKlim, issued on 28April 2002, 00:00UTC

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lead time [k]

pow

er [p

.u.]

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0 6 12 18 24 30 36 42 48

0.0

0.2

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0.8

1.0

●


Obs. at time t

Let us similarly apply that strategy for a whole sample year (2002), and analyse itsperformance...


Evaluation of the persistence approach

Similar scores: bias, MAE and RMSE

lead time [k]

erro

r [%

of P

n]

1 6 12 18 24 30 36 42 48

−10

010

2030

40

biasMAERMSE

Such score values can be explained by the “inertia” in wind power dynamics


A generalization: m-point averaging approach

There might be a gain in considering more than the last observation only...

At time t, the m-point averaging forecast, for all lead times t + k, k = 1, . . . , 48, isbased on an average of recent information

This translates to

yt+k|t =∑m

i=1 yt−i , ∀k,

where yt−i is the i th

latest measurementavailable

Right:Example of a m-pointaveraging (withm = 3) forecast forKlim, issued on 28April 2002, 00:00UTC

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lead time [k]

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er [p

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1.0

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Obs. average at time t



Evaluation of the m-point averaging approach

Focus on RMSE only

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lead time [k]

erro

r [%

of P

n]

1 6 12 18 24 30 36 42 48

010

2030

40● RMSE − m=1 (persistence)

RMSE − m=3RMSE − m=20

There is a compromise to be made between short-term and longer-term forecastquality


The limiting case: Climatology

Climatology is for the case where m→∞At time t, the climatology forecast, for all lead times t + k, k = 1, . . . , 48, is basedon an average of all information ever available (= wind farm capacity factor)

This translates to

yt+k|t =∑∞

i=1 yt−i , ∀k,

where yt−i is the i th

latest measurementavailable

Right:Example of aclimatology forecastfor Klim, issued on 28April 2002, 00:00UTC

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lead time [k]

pow

er [p

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0 6 12 18 24 30 36 42 48

0.0

0.2

0.4

0.6

0.8

1.0

●


Average of all past obs.



Evaluation of the climatology forecast approach

Similar scores: bias, MAE and RMSE

lead time [k]

erro

r [%

of P

n]

1 6 12 18 24 30 36 42 48

−10

010

2030

40

biasMAERMSE

So, it is like random guessing, but somewhat better!


A few conclusions at this stage

Even though these forecasting strategies do not look very smart...

They are difficult to beat!

Especially:

Persistence is difficult to outperform for lead times between 0 and 6 hours ahead

Climatology is difficult to outperform for the furthest lead times (say, after 24 hoursahead)

Still, we may be able to do something better

based on more dynamic approaches

extracting more information within available data


A few conclusions at this stage

Even though these forecasting strategies do not look very smart...

They are difficult to beat!

Especially:

Persistence is difficult to outperform for lead times between 0 and 6 hours ahead

Climatology is difficult to outperform for the furthest lead times (say, after 24 hoursahead)

Still, we may be able to do something better

based on more dynamic approaches



3 Going further in a regression-based framework


What is (linear) regression?

In the simplest case, data is available for:

yi (i = t − n, . . . , t), the response variable, i.e., the variable we will want to predict,eventually

xi (i = t − n, . . . , t), an explanatory variable, i.e., a variable that can help us predict y

At this stage, imagine that xi and yi are your most recent wind speed andcorresponding power observations up to current time t

Example set with thelast n = 120observations of

wind speed xi , and

corresponding powergeneration yi

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time t−i

win

d sp

eed

[m/s

]

t−120 t−100 t−80 t−60 t−40 t−20 t

02

46

810

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time t−i

pow

er [M

W]

t−120 t−100 t−80 t−60 t−40 t−20 t

1030

5070

90 time t


What is (linear) regression? (continued)

The aim is to uncovering some relationship between these explanatory andresponse variables

We first do that visually...

Same example, with the lastn = 120 observations of

wind speed xi , and


In this scatterplot, thereseems to be a (linear)relationship between windspeed and power

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0 2 4 6 8

2040

6080

wind speed [m/s]

pow

er [M

W]

0 2 4 6 8

1030

5070

90


What is (linear) regression? (continued)

Such a linear relationship between x and y can be written as

yi = β0 + β1xi + εi , i = t − n, . . . , t

where

β0 and β1 are the model parameters (called intercept and slope)

εi is a noise term, which you may see as our forecast error we want to minimize

The linear regression model can be reformulated in a more compact form as

yi = β>xi + εi , i = t − n, . . . , t

with

β =

[β0

β1

], x =

[1xi

]

It is often easier to deal with such compact formulations...


Least Squares (LS) estimation

Now we need to find the best value of β that describes this cloud of point

Under a number of assumptions, which we overlook here, the (best) modelparameters β can be readily obtained with Least-Squares (LS) estimation

The Least-Squares (LS) estimate β of the linear regression model parameters is given by

β = argminβ

∑i εi

2 = argminβ

∑i

(yi − β>xi

)2= (X>X)−1X>y

with

β =

[β0

β1

], X =

1 xt−n

1 xt−n+1

......

1 xt

, y =

yt−n

yt−n+1

...yt

Even better: some functions in R/Matlab can do it for you!


The resulting (linear) regression

For the same example setwith the last n = 120observations of

wind speed xi , and


The LS-estimate of themodel parameters is:

β =

[−3.99.2

],

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0 2 4 6 8

2040

6080

wind speed [m/s]

pow

er [M

W]

0 2 4 6 8

2040

6080

0 2 4 6 8

1030

5070

90

y = − 3.9 + 9.2x

This type of model and estimation can then be incorporated within in a forecastingapproach


Forecasting in a (linear) regression framework

At a given time t, you (as a forecaster) identified a good model:

yi = β0 +∑m

j=1 βjxj,i + εi , i = t − n, . . . , t

(or, equivalently: yi = β>xi + εi )

whereyi is still your response variable (say, wind power generation) observed at time i

xj,i is the observation at time i for the jth explanatory variable (j = 1, . . . ,m)

βj is the model parameter for the jth explanatory variable


Based on the last n observations, you obtain an LS-estimate βt , valid at time t

And you can issue forecast using these estimates βt , for any new values of theexplanatory variables, i.e.

yt+k|t = β>t xt+k

Potential problem here: we do not know future values of the x variable (e.g.,wind speed)!


Example application: combining persistence and climatology

Persistence and climatology were shown to be good benchmarks (difficult tooutperform), though

persistence is good for short lead times

climatology is good for longer lead times

Why no combining them, as function of the lead time k?

Reminder of the qualityof the persistence andclimatology forecasts,

in terms of RMSE

as a function of thelead time k

lead time [k]

erro

r [%

of P

n]

1 6 12 18 24 30 36 42 48

010

2030

40

RMSE − persistenceRMSE − climatology


Proposal of a combination model and estimation

Our proposal model has the following form, for a given lead time k,

yi = βk,persy(p)i|i−k + βk,climy

(c)i|i−k + εi , i = t − n, . . . , t

where

y(p)i|i−k

and y(c)i|i−k

are the persistence and climatology forecasts, issued at time i − k for

time i

βk,pers and βk,clim are intercept and the weights to be given to persistence andclimatology forecasts, respectively


It therefore combines persistence and climatology forecasts

The weight given to each of these forecasts can change with the lead time k


Estimation of the model coefficients

For the example of the 28 April 2002 (as in first slides),

the necessary vectors and matrices are formed, with n = 200 last values

LS estimates βk,pers and βk,clim are computed for every lead time k (k = 1, . . . , 48)

Right:Evolution of theestimated modelparameters βk,pers andβk,clim as a function ofthe lead time k

persistence is given lessweight for further leadtimes

climatology is givenmore weight instead

lead time [k]

coef

ficie

nt v

alue

[p.u

.]

0 6 12 18 24 30 36 42 48

−0.

20.

00.

20.

40.

60.

81.

0

βpers

βclim


The resulting forecast

For the example of the 28 April 2002 (as in first slides),

LS estimates βk,pers and βk,clim are used to combine the available persistence andclimatology forecasts

The combination is different for every lead time k (k = 1, . . . , 48)

Right:Example of acombined forecast forKlim (persistence andclimatology), issued on28 April 2002,00:00UTC

lead time [k]

pow

er [M

W]

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0.0

0.2

0.4

0.6

0.8

1.0

●

persistenceclimatologycombinedobservations

Let us similarly apply that strategy for a whole sample year (2002), and analyse itsperformance


Evaluation of the combined forecasts

RMSE only, for persistence, climatology, and the combined forecasts

lead time [k]

erro

r [%

of P

n]

1 6 12 18 24 30 36 42 48

010

2030

40

RMSE − persistenceRMSE − climatologyRMSE − combined

With this combination strategy, we are getting the best out of the original simplebenchmarks!


A few (more) conclusions at this stage

We have now learned to handle more variables and data

The forecasting approaches do not look impressive still

What could we do?


go further than using simple linear relationships only (well, we will not do it today...)


4 Digging in the data


What to look for in the data?

What do we have here...?

measurements, i.e.,

power measurements (yt) - Remember that only past measurements can be used!

weather forecasts, i.e.,

wind speed (ut+k|t)

wind direction forecasts (θt+k|t)

temperature forecasts (Tt+k|t)

different variations of those could be used since the relationship betweenmeteorological variables and power is nonlinear, e.g.,

power of wind speed: u2t+k|t , u

3t+k|t , etc.

harmonics of wind direction: cos

(2πθt+k|t

360

), sin

(2πθt+k|t

360

), etc.

we also know the hour of the day (ht), or the lead time k, which could be useful...(though not used here)

Let us call all these variables xj (j = 1, . . . ,m), and also nickname them “features”


We can still write a linear regression...

Remember that a linear relation between the xj variables and y can be written as

yi = β0 +∑m

j=1 βjxj,i + εi , i = t − n, . . . , t

(or, equivalently: yi = β>xi + εi )

whereyi is still your response variable (say, wind power generation) observed at time i

xj,i is the corresponding value for the jth explanatory variable (j = 1, . . . ,m, examplewind speed forecast used as input)

βj is the model parameter for the jth explanatory variable


This linear regression model can be reformulated in a more compact form as

yi = β>xi + εi , i = t − n, . . . , t

with

β =

β0

β1

· · ·βm

, x =

1x1,i

· · ·xm,i


Estimation and feature selection

We need to find the best value of β that describes this cloud of point, but alsousing a minimum of variables (parsimony principle)

LS-estimation is not very good for that, as the number of variables becomes high...the LASSO version should be used instead

The LASSO estimate β of the linear regression model parameters is given by

β = argminβ

1√nλ

∑i εi

2 +∑

j |βj |

with λ a so-called regularization parameter, and

β =

β0

β1

· · ·βm

, X =

1 x1,t−n . . . xj,t−n

1 x1,t−n+1 . . . xj,t−n+1

...... . . .

...1 x1,t . . . xj,t

, y =

yt−n

yt−n+1

...yt

As before, some functions in R/Matlab can do it for you!


Example based on a set of features

Our proposal model has the following form, for a given lead time k,

yi = βk,0 + βk,1ui|i−k + βk,2u2i|i−k + βk,3u

3i|i−k

+βk,4 cos

(2πθt+k|t

360

)+ βk,5 sin

(2πθt+k|t

360

)+βk,6Ti|i−k + βk,7yi|i−k + εi

where

we have 8 model parameters to estimate, for each lead time k

the weight given to each of these features therefore varies with the lead time k



Estimation of the model coefficients

For the example of the 28 April 2002 (as for the other examples),


LASSO estimates of the estimates βk,j s are computed for every lead time k(k = 1, . . . , 48)

Right:Evolution of theestimated modelparameters as afunction of the leadtime k

Only a few featureshave parameterssignificantly differentfrom 0: ut+k|t , u

2t+k|t

and yt

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● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ●

lead time [k]

coef

ficie

nt v

alue

[p.u

.]

0 6 12 18 24 30 36 42 48

−0.

20.

20.

61.

01.

41.

8

● β0

β1

β2

β3

β4

β5

β6

β7


The resulting forecast

For the example of the 28 April 2002 (as before),


Right:Example of aforecast for Klimwith our moreadvanced model,issued on 28 April2002, 00:00UTC

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lead time [k]

pow

er [p

.u.]

1 6 12 18 24 30 36 42 48

0.0

0.2

0.4

0.6

0.8

1.0

●


Let us similarly apply that strategy for a whole sample year (2002), and analyse itsperformance


Evaluation of more advanced forecasts

Various criteria: bias, MAE, RMSE (and RMSE comparison with our previous bestforecasts)

lead time [k]

erro

r [%

of P

n]

1 6 12 18 24 30 36 42 48

−10

010

2030

40

biasMAERMSERMSE − ref.

It seems we have substantially improved... Could still do better!!


Final remarks

What else may you need to forecast, in an electricity market context?

power generation from other renewables

electric load

market prices

imbalance sign

offers from other (important) players

import/export (for your market zone of interest)

clearing outcomes of neighboring markets, etc.

And, in terms of more advanced forecasting:

make the forecast probabilistic

build more complex models (i.e., not relying on regression only)

dig more into the data...


Thanks for your attention! - Contact: [email protected] - web: pierrepinson.com


``Renewable Energy Modelling and...

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