Optimal trading strategies for a wind-storage power …etd.dtu.dk/thesis/249639/ep09_54_net.pdf · Optimal trading strategies for a wind-storage power system under market conditions

Optimal trading strategies for awind-storage power system under

market conditions

Philip Delff Andersen

Kongens Lyngby 2009

Technical University of Denmark

Informatics and Mathematical Modelling

Building 321, DK-2800 Kongens Lyngby, Denmark

Phone +45 45253351, Fax +45 45882673

[email protected]

www.imm.dtu.dk

IMM-M.Sc: ISSN 1601-233x

ISBN 978-87-643-0547-0

Summary

In this thesis, a model of a system consisting of electric power production onwind turbines combined with a storage device is developed. By use of MonteCarlo simulation, the operation of the system is optimised with respect to twodifferent objective functions. One strategy is to maximise the expected revenuefor the whole delivery period, the other is to minimise the expected regulationcosts. Moreover, two different markets are considered, with different horizonsand duration of the delivery periods.

A passive operation strategy for the electrical energy storage is defined, andhence the delivery to the power net becomes a function of only the productionand the issued contract at the market. Since the production is assumed to beuncontrollable, only the contract is left to optimise.

Three different models of the electrical storage devices are being used. Hence,effects of all limited capacity, charging and discharging efficiencies, and limita-tions on charging and discharging speeds can be observed.

For running the Monte Carlo Simulation, a non-linear estimate of the distri-bution of the future production for each relevant horizon by use of adaptivequantile regression with point forecasts as explanatory variable. From this, theinterdependence of future production at different horizons are estimated. Fromthese two estimates, the scenarios are simulated, and based on these the optimi-sation problems are solved. The simulations are run throughout all in all morethan one year of data.

The results of the optimisation strategies are not as good as expected, assumed

ii

because of too poor estimates of the distributions of the production. However,by use of simulations, a potential gain of the method can be estimated. Thisgain is expected to be realistic if a good model for prediction of the distributionsis found. The results are very depending on optimisation strategy and storagemodel, and the obtained revenues are between -2% and 19% compared to whenusing the point predictions as contracts and the respective storage devices.

Finally, different approaches to improve the method are discussed.

Keywords: Wind power, electrical energy storage, Monte Carlo simulation,adaptive quantile regression.

Resume

I denne rapport udvikles en model af et system bestaende af elproduktion pavindmøller kombineret med et lagringsmedium. Gennem Monte Carlo simu-lering optimeres driften heraf ud fra to forskellige objektivfunktioner. Delsmaksimeres det forventede udbytte af en leveringsperiode pa to forskellige el-markeder, dels minimeres de forventede reguleringsomkostninger. Derudover be-tragtes to forskellige markeder med forskellige horisonter og forskellige varighederaf leveringsperioderne.

En passiv strategi til drift af lagringsmediet defineres, og leveringen til elnettetbliver saledes til en funktion kun af produktion og af den indgaede kontraktpa elbørsen. Idet produktionen antages at være upavirkelig, er kun kontraktentilbage som variabel at optimere.

Tre forskellige modeller af lagringsmedier anvendes. Saledes betragtes virkningerneaf at begrænse kapacitet, virkningsgrader for op- og afladning samt begræn-sninger for op- og afladningshastigheder.

For gennemførsel af Monte Carlo simulering, foretages et ulineært estimat affordelingen af den fremtidige produktion for hver relevant horisont ved hjælpaf adaptiv fraktilregression med punktforudsigelser som forklarende variabel.Herudfra estimeres den interne afhængighed af fremtidig produktion ved forskel-lige horisonter. Med disse to estimater simuleres scenarier af produktionen pabaggrund af hvilke optimeringsproblemet løses. Simuleringerne gennemføres pasamlet set mere end et ars data.

Resultaterne af optimeringsstrategierne er ikke sa gode som forventet, formentligpga. for darlige estimater af fordelingerne af produktionen. Ved hjælp af simu-

iv

leringer kan dog alligevel anslas et potentielt udbytte ved anvendelse af metoden,safremt en bedre model til forudsigelse af fordelingerne findes. Resultaterne ermeget afhængige af optimeringsstrategi og lagringsmodel, og de opnaede udbyt-ter er mellem -2% og 19% sammenlignet med, nar punktforudsigelserne anvendessom kontrakter kombineret med de respektive lagringsmedier.

Til sidst diskuteres forskellige muligheder for at forbedre metoden.

Nøgleord: Vindkraft, lagring af el, Monte Carlo simulering, adaptiv fraktilre-gression.

Preface

This master’s thesis was written at Department of Informatics and Mathemat-ical Modelling at Technical University of Denmark, under supervision of PierrePinson and Henrik Madsen. It was written from September 2008 to March 2009and accounts for 30 ECTS.

The thesis deals with modelling of electricity generated from wind power com-bined with electrical energy storage and the benefits of the latter when tradingon open electricity markets.

The thesis is written in British English.

Kongens Lyngby, March 2009

Philip Delff Andersen

vi

Acknowledgements

I would like to thank my supervisors, Pierre Pinson and Henrik Madsen formaking this work possible. And especially, my thanks go to Pierre for all hishelp and advice.

Thanks to Enfor for letting me use the Adaptive Quantile Regression program,and not less to Christian Viller Hansen for support on the use of it.

My dear friends, Peder and Jon, thank you for spending your time reading andcommenting on my work. Peder, I’m sure our collaboration on the Bacher/DelffR Plotting Tool will lead to even more easy and nice plotting facilities in thefuture. What would this thesis have looked like without it?

In general, I have been overwhelmed by the enormous support and flexibilitythat all of my friends and my family have shown me. I wish that I will have mychance to pay some of it back.

I send my love and special thoughts to the only three persons who have not beenflexible with me. Dear Ingrid who saw your first light only three days before mydeadline. You are wonderful, I can’t wait to hold you again and to get to knowyou. And my brother Kristian and Anna: It’s fantastic, the world is crazy!

viii

Contents

Summary i

Resume iii

Preface v

Acknowledgements vii

1 Introduction 11.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Limitations of the scope . . . . . . . . . . . . . . . . . . . . . . . 31.4 Outline and reader’s guide . . . . . . . . . . . . . . . . . . . . . . 41.5 Abbreviations and notation . . . . . . . . . . . . . . . . . . . . . 6

2 A wind-storage power system under market conditions 92.1 The physical system . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Wind power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 The Nordic electricity market . . . . . . . . . . . . . . . . . . . . 132.4 Formulation of a naive regulation prices predictor . . . . . . . . . 182.5 Electrical energy storages . . . . . . . . . . . . . . . . . . . . . . 182.6 Searching for an optimal contract . . . . . . . . . . . . . . . . . . 272.7 Simulation plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Theory 333.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Quantile regression . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Expectation and variance of random variables . . . . . . . . . . . 373.4 Multivariate normally distributed variables . . . . . . . . . . . . 37

x CONTENTS

3.5 Decision theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . 38

4 Data presentation and preparation 414.1 The considered area . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Production measurements . . . . . . . . . . . . . . . . . . . . . . 424.3 Point forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Probabilistic forecasts . . . . . . . . . . . . . . . . . . . . . . . . 474.5 Market prices and regulation costs . . . . . . . . . . . . . . . . . 484.6 Time handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.7 Partitioning the data . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Scenario generation 555.1 From quantile estimates to full distribution functions . . . . . . . 575.2 Evaluation of probabilistic forecasting . . . . . . . . . . . . . . . 595.3 Using the Adaptive Quantile Regression algorithm . . . . . . . . 625.4 Comparing performance of different models . . . . . . . . . . . . 635.5 Adaptive estimation of the autocorrelation . . . . . . . . . . . . . 69

6 Example on simulations and optimisation 716.1 The example period . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3 Bussiness as usual simulations . . . . . . . . . . . . . . . . . . . . 806.4 Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7 Results 91

8 Discussion 938.1 Simulations without optimisation . . . . . . . . . . . . . . . . . . 938.2 Simulations with optimisation . . . . . . . . . . . . . . . . . . . . 958.3 Discussion of the strategies . . . . . . . . . . . . . . . . . . . . . 988.4 Discussion of the passive operation strategy . . . . . . . . . . . . 1008.5 General aspects about storage . . . . . . . . . . . . . . . . . . . . 100

9 Conclusions 1039.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Bibliography 106

List of figures 110

List of tables 114

A Derivation of Taylor expansions 117

xi

B Results of adaptive quantile regression analysis 119

xii Contents

Chapter 1

Introduction

In the recent years, increasing global focus has been on renewable energy. Envi-ronmental problems caused by exploitation of fossil fuels have gone from beinga political issue mainly argued with moral and aesthetics by a minority in alimited part of the world to be globally discussed as a matter of irreversibleenvironmental and natural disasters. Increasing energy prices and unstable re-lations to suppliers of oil and natural gas have unified political orientations inthe search for alternatives to the use of fossil fuels.

A part from nuclear energy (fission and fusion), these are all more or less de-pendent on climate variables. The production by use of these technologies istherefore dependent on variables of stochastic behaviour. The integration of thisproduction in the existing power net poses the problem of matching a demandside of on short term very little flexibility and price sensitivity with a supplyside of uncontrollable production, and highly fluctuating marginal productioncosts.

Through the nineties, the European electricity markets were deregulated, andopen electricity markets were formed. For securing consistency of supply, thesemarkets - referred to as spot markets - are supplemented by regulation marketswhich balance out demand and supply on short horizons. In Denmark, the costsfor the regulation is held by the producers that are responsible for the imbalancethrough deviations between contract on the spot market and the production.

2 Introduction

This is mainly a problem when the prediction of the production is subjected touncertainty.

Hence, it is of interest to smooth out deviations between contracts and pro-duction. Introducing electrical energy storage (EES) provides this possibility.However, how to operate the system is not evident. The output from a com-bined production and storage power system is a function of both the productionand the charging/discharging, and the optimisation of the operation is thereforea multivariate optimisation problem. Especially, the spot markets will usuallyhave gate closures hours before the corresponding delivery period, and the de-livery periods are of several hours.

As the prediction errors of power production depend on prediction of meteoro-logical variables the inertia of the latter propagates to the prediction errors.

Even though the components are few, and the constraints simple, such a systemis very complex to analyse analytically. A method for deciding on an operationstrategy on beforehand is the use of Monte Carlo simulation.

1.1 Previous work

Optimisation of trading of electricity generated on wind turbines on the Nordicpower market is dealt with in [15] and [22]. Models for describing the inter-dependence structure in wind power are thoroughly described in [20], and [24]provide a lined out method for simulation of such scenarios including an exampleand advice on how to test that necessary assumptions are fulfilled.

[13] describes integration of electrical storage devices in power nets. It describesmodelling of the devices and guidance on design of such models to meet physicalaspects.

In [5], the operation of a combined wind-storage power plant is optimizedthrough a dynamic optimization formulation. A “state of the art” wind powerprediction model and a diurnal persistence price forecasting model are used togenerate scenarios. Probabilistic forecasts are not used, and interdependence ofprediction errors is not modelled.

In [2] wind power is combined with a mini hydro power plant. The prices forelectricity are modelled as a simple periodic staircase function correspondingto Portuguese wind power renumeration. The system is optimised with respectto the power production and the operation of the electrical energy storage. As

1.2 Problem statement 3

power curve only one 48 hour sequence is used, and this is repeated and scaledwith respect to some “typical” average periodicity to model a whole year. Thisseems to be inappropriate since high/low production is always assumed at thesame time of the day, just like the prices.

1.2 Problem statement

The aim is to do a first step in optimisation of trading on an electricity mar-ket when operating a combined wind-storage power system in Denmark. Theoptimisation should be done through Monte Carlo simulation based on reliableforecasts of the production.

• How can reliable scenarios of electricity production on wind turbines begenerated?

• How can bidding strategies on the electricity market be optimised whenusing an electrical energy storage to smooth out deviations between tradedenergy and production?

• What is the gain from using the model in the considered market models?

• What further steps could be taken to improve the operation strategy?

Different devices must be used to exemplify influences of the parameters max-imum storage level, charging and discharging efficiencies, and limitations ofcharging and discharging speeds.

1.3 Limitations of the scope

In the modelling, different limitations will be used. They will be listed here.

• The production cannot be curtailed. Only the sold and the stored energycan be varied. This means that the produced electrical power must eitherbe dispatched to the power net or stored.

• Investment and maintenance costs regarding the electrical storage deviceare not considered. These could be used as costs when calculating rev-enues, but electrical energy storage is very expensive, and it is not in thescope of this work to evaluate its cost-effectiveness.

4 Introduction

• No reactive power is taken into consideration. The power flow in the powernet consists of both active and reactive power. Only active power will bemodelled here.

• The producer is assumed to be “price taker”. This means that no mat-ter his production or expected production, the prices are assumed to beindependent hereof and given by the Elspot price. The same goes for theregulation prices.

• Only electrical energy storages with constant efficiencies, constant limi-tations on charging/discharging speeds, and constant capacities will beused.

• Regulation prices will be forecasted with a very simple and poorly per-forming prediction model.

• There will only be traded on one market at a time. The only prices thatwill be used for modelling markets to trade on, are Elspot prices.

1.4 Outline and reader’s guide

Here, the purpose of each of the following chapters will briefly be listed.

Chapter 2 is a description of the considered physical system and how it ismodelled. First, the full system is described briefly from above. Then eachcomponent is described independently, and finally it is explained how itwill be tried to optimise the operation of the system, and the simulationplan is outlined. This chapter is important for understanding the modelthat is being used in the rest of the thesis.

Chapter 3 is a presentation of theory used in the mathematical modelling.The reader with a statistical background can skip this chapter. Whenneeded, references will be given to relevant parts of this chapter.

Chapter 4 is a presentation of the data that has been used in the model build-ing and evaluation. It is not necessary to read to understand the mod-elling, but it gives an insight in the behaviour of wind power and themarket prices.

Chapter 5 Describes the method for generating production scenarios. Thisincludes a decision on a model to use for prediction of a probability dis-tribution of electrical energy generation on wind turbines.

1.4 Outline and reader’s guide 5

Chapter 6 The chapter includes a thorough case study where the full mod-elling, decision, and realisation process is described. The chapter gives aninsight in the optimisation procedure that forms the pieces that the fullsimulation experiments are made of.

Chapter 7 briefly lists the results of the optimisations that have been carriedout.

Chapter 8 discusses the results and possibilities of improving the results throughfurther studies.

Chapter 9 concludes on the results and points out paths for further studies ofthe field. It contains some points about the big picture that the studiesshould be seen in.

The thesis contains a lot of notation, and the reader may need to return to thelist of abbreviations and symbols that will be given in Section 1.5. Illustrationsand plots are extensively used throughout the thesis, so the reader may also findthe lists of figures and tables following the conclusions and bibliography usefulas tools for navigation.

Where nothing else is mentioned, the time stamps given in the thesis are inCentral European (Summer) Time.

6 Introduction

1.5 Abbreviations and notation

In general, x is an estimate of x, xt+k|t is a prediction of xt+k based on infor-mation available at time t. x∗ is an optimised value of x.

Abbreviationscdf Cummulated distribution function.CET Central European Time, UTC+1 hour.CEST Central European Summer Time, UTC+2 hours.UTC is the Coordinated Universal Time.EES Electrical energy storage.MAE Mean absolute error.NMAE Normalised mean absolute error.NRMSE Normalised root mean squred error.pdf Probability density function.p.u. production units.RMSE Root mean squared error.r.p. Regulation prices.

Variable namesCd

t ∈ R+0 The regulation cost faced for deviation from contract in the

period [t − 1, t].C+

t ∈ R+0 The down-regulation cost faced for deviation from contract in

the period [t − 1, t].C−

t ∈ R+0 The down-regulation cost faced for deviation from contract in

the period [t − 1, t].Ec

t ∈ R+0 Energy contracted on beforehand to be delivered in the period

[t − 1, t].Ed

t ∈ R The deviation between contracted and delivered energy in theperiod [t − 1, t]. Defined in Equation (2.22).

Eot ∈ R

+0 The energy delivered to the power net in the period [t − 1, t].

Est ∈ R

+0 The energy delivered to the power net from the storage device

the period [t − 1, t].Es,max

ees The maximum possible amount of energy a given storage de-vice can deliver within one sampling period.

Es,minees The minimum possible amount of energy a given storage device

can deliver within one sampling period.Ew

t ∈ [0, 1] The electrical energy generated by wind turbines in the systemin the period [t− 1, t]. Normalised by installed capacity in theinterval. The unit is p.u.

E∼t E∼

t = Ewt − Ec

t

k ∈ R+ Prediction horizonLt The storage level at time t.

1.5 Abbreviations and notation 7

Lmaxt Shorthand notation for Lmax

t|t−1. It is the maximum possiblevalue the storage level can take at time t given the knownstorage level at time t − 1

Lmint Shorthand notation for Lmin

t|t−1. It is the minimum possiblevalue the storage level can take at time t given the knownstorage level at time t − 1

t Time. The sample period is everywhere one hour.T Sometimes used for denoting a sequence of time intervalsRt The total revenue obtained in the interval [t − 1, t].Rc

t The revenue obtained on the spot market for the contract ofenergy to be delivered in the interval [t − 1, t].

πct The price for the contracted energy to be delivered at time t

[e/MWh].π−

t Up regulation price at time t. The price that must be paidfor the contracted energy that is not delivered in the period[t − 1, t] [e/MWh].

π+t Down regulation price t. The value of the energy delivered in

the period [t − 1, t] in surplus to the contract issued.πTSO

t The price paid by the TSO to a producer for balancing thesystem in the period [t − 1, t].

Parametersηchess The effeciency of transforming from to electrical power to

power stored in the EES.ηdchess The effeciency of transforming from power stored in the EES

to electrical power.Lmin

ess

Lminess

L0 The starting storage level when doing simulations.λ The forgetting factor when estimating the autocorrelation by

exponential smoothing.(∇L)max

ees The maximal possible charge within one sampling period for agiven EES

(∇L)minees The minimum possible charge within one sampling period for

a given EES. This is negative, i.e. it can alternatively beinterpreted as minus the maximum possible discharge.

ρ The thickness parameter for the exponential extrapolation ofquantile estimates.

ρair Density of air.

MappingsF Typical notation for a cdf

E (X) The expected value of the random variable X

8 Introduction

EF (X) The expected value of the random variable X under the as-sumption that X has the cdf F

V (X) The variance of the random variable X

C (X, Y ) The covariance of the random variables, X and Y .IRMSE The improvement measure for prediction models based on

RMSE.Γ The Risk of an estimator.Λ The loss function in decision making.∇ The difference operator. ∇Xt = Xt − Xt−1

Φ : R →]0, 1[ The cdf of the standard normal distribution.⌊·⌋ : R → Z The floor operator. ⌊x⌋ is the largest integer smaller than or

equal to x.

Chapter 2

A wind-storage power system

under market conditions

This chapter is a description of the physical system that is modelled. First, theenergy balance in the system is stated. Then each component in the system isdescribed individually. Finally, the revenue of operating the combined system isanalyzed and methods for optimizing the operation are presented, and the planfor investigating the system is outlined.

2.1 The physical system

The energy flows in the considered system are sketched in Figure 2.1. Thephysical components are the wind power production area, the electrical energystorage (EES), and the power net. The electricity production is of a stochasticand non-controllable nature. The storage device is an instrument to control thedelivery to the power net. Finally, the power is traded through an electricitymarket on which the price settlements can also, from a price-taking producer’spoint of view, be modelled as stochastic. As both traded and delivered energycan be controlled under some limitations in the structure of the power net andmarket, the optimisation of the revenue of running the system turns out to bea multidimensional stochastic optimisation problem.

10 A wind-storage power system under market conditions

Wind park Power net

EES

Ew Eo

Es

Figure 2.1 Illustration of the energy balance in the considered system. The possibleenergy flow directions are indicated by arrows.

Before starting the analysis, some notation must be introduced. The availablemeasurements are average power produced in consecutive time intervals. Thesampling period is constant and normalized to 1. Let Pw

t denote the electricalpower generated at time t. The produced energy throughout the time ]t − 1, t]will then be denoted

Ewt :=

∫ t

t−1

Pwτ dτ (2.1)

Notation: When energy quantities are introduced with a time subscript, it willbe implied that they are defined by an integration over the corresponding powerin preceding sample period as in Equation (2.1).

Returning to Figure 2.1, using an EES, the output to the power net consists oftwo terms, the produced energy and the contribution from the EES denoted byEs

t . The energy delivered to the power net within the period ]t − 1, t] is thendenoted by

Eot := Ew

t + Est (2.2)

Notice that Es is positive when it contributes positively to the energy deliveredto the power net. As the storage device can be both charged and discharged,Es can be both positive (EES is being discharged) and negative (EES is beingcharged). The production, Ew on the wind turbines cannot be negative, neithercan the output, Eo, to the power net.

2.2 Wind power 11

2.2 Wind power

The electrical power transformed from the kinetic power in an airflow is de-scribed by the third relation to the wind speed [8]:

P =1

2Cpρairu

3A (2.3)

where P is the electric power, Cp is the power coefficient describing the con-version efficiency of the system, which is dependent on wind speed, u. A is thearea of the disc formed by the rotating blades.

A “power curve” shows the electric power versus wind power. A typical theoret-ical power curve for a wind turbine is sketched with a black curve in Figure 2.2.In summary, the power curve has the following characteristics:

• Up to a certain wind speed, the wind turbine is idle.

• Hereafter the production is increasing non-linearly and then stagnating.

• Until it at a cut-off speed drops to zero.

Wind speed

Ele

ctri

calpow

er

Figure 2.2 Sketch of a power curve for a wind turbine. The curve is highly nonlinear.At a certain wind speed, the turbine is shut down.

The cutting of is necessary to protect the machinery in strong wind. All in all,the power curve is highly non-linear.

The introduction of climate-dependent electricity production technologies alsointroduces stochastic electricity generation. As the infeed in the production isnon-controllable and highly variable, the need for good short-term prediction


tools has emerged in order to implement the stochastic generation in the powernet. [4] provides a good overview of the history of this field.

The non-linearity in the power curve imposes a challenge when modelling. Windis a variable that is unpredictable to some extend. Then it can be interpreted asa random variable. The theoretical power curve is then a function of a randomvariable. In Appendix A, Taylor expansions are made on mappings of randomvariables, to show how the behaviour of the deterministic function affects theexpectation and variance of its response. Here only the results will be given.Let X be an unbiased estimate of X . f is a mapping from and into the realnumbers and is continous and two times differentiable in X . Then a secondorder Taylor expansion gives for the expectation:

E

(

f (X)∣

∣

∣X)

≈ f(

X)

+1

2· f ′′

(

X)

· V

(

X |X)

(2.4)

For the variance, a first order Taylor expansion on the variance gives that

V (f(X)) ≈ f ′(

X)2

· V (X) (2.5)

Ie, the variance of a first order polynomial estimate of the mapping g(X) isproportional to the square of the first derivatives in X. Returning to the powercurve in Figure 2.2, imagine wind predictions with similar error distributions.The variance of the production would be very small at small wind predictions,then increasing, before it would again decrease before the cut-off speed. At somepoint it will be large again, because of the probability that the wind will be toostrong.

An approach that have been widely used in Denmark is the Wind Power Predic-tion Tool (WPPT) described in [18]. It provides a configurable method basedon auto regressive models with exogenous input processes (abbreviated ARXmodels) with meterological forecasts (or nonlinear functions of these) as inputvariables. It also includes a diurnal term, and input variables are square-roottransformed for improving the similarity between the distribution of the errorsand the normal distribution. In the given reference, an upscaling method is alsodescribed for going from prediction of production on seperate wind farms toprediction of the total production of a region. As for other good point predic-tion models, the uncertainty of the predictions from the WPPT model is highlydepending on the forecast horizon.

According to [4], models based on neural networks and fuzzy logic outperformWPPT. Such kind of models are hard to compare once and for all. As they aredesigned by use of different data, they may also perform differently in differentenvironments, maybe in different parts of the world with very different climate.The WPPT is a “state of the art” model.

2.3 The Nordic electricity market 13

In Figure 2.2, along with the power curve, three sketches are shown of differ-ent predictions of distributions of the wind speed. This is only done for anillustrative purpose. If interpreted literally, axis labels lose their interpretation.Think of the colored curves only as illustrations of shapes. Then the curvesintuitively show not only the behaviour of the variance of the power productionat different wind speeds, it also gives an idea about the very different shapes ofthe distributions of the prediction errors of the production can have at differentpredicted wind speeds1. Although it is not the only reason, this small exampleshows the need for a flexible model of the production if it is not to be used forpoint predictions. In the following a technique to get an estimate of the fulldistribution of the production will be used.

2.3 The Nordic electricity market

The deregulation of the Nordic electricity market, which was carried out inthe nineties, was based on the introduction of open electricity markets. Forplanning of a balanced power system, contracts must be issued in advance.As an increasing share of production of stochastic nature is introduced, thisforms a need for other markets in order to insure the system balance on shorterhorizons. Here, a brief presentation of the market mechanisms related to theconsidered production system, will be given. For more information, see [15],www.nordpool.com, and www.energinet.dk.

Nord Pool Spot A/S provides two electricity markets, Elspot and Elbas. Around70% of the consumed electricity in the Nordic countries is traded on these mar-kets. Both markets are based on auction trading, which is that trading takesplace when the supply and demand curves meet.

At the main market, Elspot, the gate closure is at noon for trading electricityto be delivered the next day, i.e. the gate closure is 12 to 36 hours before thedelivery of the concluded contracts. Elspot is by far the largest of the twomarkets. Elbas is a market with a shorter horizon dealing with the periods thathave already been traded on Elspot. The horizon is from one hour ahead. Theprice per unit of energy contracted on beforehand to be delivered in the period[t − 1, t] will be denoted πc

t .

A third market is the regulation market. This is provided by the transmis-

1A wind prediction model with gaussian errors may be very inappropriate because it as-sumes the possibility of negative wind speeds. And if the prediction has normally distributederrors, the distributions of the errors of the power production model would have infinitely highpeaks on both zero and one.


sion system operator (TSO). For western Denmark this is Energinet. If a pro-ducer changes production plan, the TSO balances it by buying at the lowestbid available on the regulation market. This can either be for another producerto increase or decrease production. The price settlements should both reflectproduction expenses and discourage producers to cause imbalances. However,the cost of a deviance from contracts on Elspot or Elbas strongly depends onthe sign of the deviance. Usually, only the producers that add to the systemimbalance will be charged for the imbalance.

In order not to make it profitable to cause imbalances, regulation bids mustfulfil the following two rules.

• An up-regulation bid must at least equal the Elspot price.

• A down-regulation must not exceed the Elspot price.

The up-regulation price per unit energy in the interval [t− 1, 1] will be denotedπ−

t . The down-regulation price at per unit energy in the interval [t − 1, 1] willbe denoted π+

t . The minus and plus sign refer to surplus or lack of energy inthe system if no regulation is done. When the system is up regulated, the down-regulation price equals the Elspot price, π+ = πc. When the system is downregulated, the up-regulation price equals or is very close to the Elspot price,π− = πc.

An example of up-regulation is seen in Figure 2.3. Totally, 400 MWh are orderedby the consumption side. The example involves three producers, called A, B,and C. Within a given hour, Producer A has accepted a contract of 200 MWh,while Producers B and C have agreed to deliver 100 MWh each. It now turnsout that Producer A completely fails to deliver, while Producer C delivers 100MWh more than contracted. The total imbalance in the system is a supply lackof 100 MWh. The TSO balances this out by accepting the cheapest availableoffer on the regulation market, which has been given by Producer B. ProducerA must pay the up-regulation price to Producer B for this. Producer C will getthe Elspot price for the energy delivered in addition to the contract, since thisdeviation helps balancing out the total system.

A summary of the revenue of the producers is seen in Table 2.1. It should benoticed that even though Producer B and C deliver the same amount of energy,they do not obtain the same revenue.

The example shows a situation of up-regulation. down-regulation is done sim-ilarly. When the system is in positive imbalance (too much energy delivered),


Producer AContract:200 MWh

Producer BContract:100 MWh

Producer CContract:100 MWh

TSO

ConsumersContract:400 MWh

0

100200100

100

400

Figure 2.3 Illustration of up-regulation by the TSO. The flows between the actors areenergy in MWh.

Table 2.1 Summary of the revenues for the producers in the up-regulation exampleillustrated in Figure 2.3.

Producer Revenue

Contracted Total

A 200πct 200πc

t − 200π−t

B 100πct 100πc

t + 100πTSOt

C 100πct 100πc

t + 100π+t = 200πc

t

the TSO accepts the cheapest bid on the regulation market of decreasing pro-duction. The producers that are responsible for the power surplus then only getthe down-regulation prices for the power that they deliver additionally to theircontracts. This price may even be negative meaning that the producers mustpay for over-producing. The regulation prices are known at some point afterthe relevant delivery hour.

Two different market models will be used in this work. One market with deliveryperiods of 6 hours and gate closure just before the delivery period, and onemarket with delivery periods of 24 hours and gate closures 12 hours before thedelivery periods. The idea of using these two market models is to consider theinfluence of the horizon on the system. Therefore, the same prices are usedwhen modelling the two markets. The regulation prices are also the same. In


the model, there will be used data of hourly sampling. That means that ideallyspeaking, new information is available at the moment of the gate closure. It isnot realistic that this data will be available and can be processed to update themodel instantaneously. Therefore it is decided only to use data from one hourbefore gate closures.

time

00:00 06:00 12:00 18:00 00:00

gate closure 1

deliveryperiod 1

gate closure 2

deliveryperiod 2

gate closure 3

deliveryperiod 3

gate closure 4

deliveryperiod 4

hours since last information

0 2 7 2 7 2 7 2 7

(a) A market with six hours delivery periods.

time

12:00

gate closure

00:00 00:00

hours since last information

0 14 37delivery period

(b) A market with 24 hour delivery period like the Elspot market.

Figure 2.4 Illustration of trading and delivery periods at two different markets.

Figure 2.4 illustrates the two spot markets. In Figure 2.4(a) the 6 hour marketis sketched. Each day consists of four delivery periods, one from midnight to 6AM, one from 6 AM to noon, one from noon to 6 PM and finally one from 6PM to midnight. The gate closures are at the beginning of the delivery periods.In the figure there is also another time line indicating how old the data are onwhich the decisions taken on the market are based. At, for instance, 6 AM,“7 hours” is indicated. The gate closure for trading energy delivered in theinterval 5 to 6 AM, is the preceding midnight. For trading at midnight, a modelbased on data from 11 PM is used. At 6 AM, 7 hours have passed since thisinformation was issued. The horizon on the market is therefore 2 to 7 hours.

Figure 2.4(b) sketches the modelled market with 24 hours delivery periods. Thegate closure is at noon, where a model based on data from up to 11 AM isavailable. The delivery period is the whole following day making it a marketwith the horizon 14 to 37 hours.


Whereas the market with delivery periods of 24 hours is thought to model theElspot market, the model with 6 hours delivery periods share properties withboth the Elbas and the regulation market, and is, as already stated, mostly usedto see the difference of modelling markets with different delivery periods andhorizons.

2.3.1 A producer’s revenue

In this work, the trading is limited to take place on either the market of sixhours delievery periods or the market of 24 hours. The regulation prices fromthe regulation market are then used to evaluate the prices of the energy inimbalance. This means that the only relevant prices for calculating the revenueat a time t are the 6 or 24 hour market price depending on which of the two isconsidered (both will be called πc

t ) and the up and down-regulation prices (π−t

and π+t ). Since no trading takes place on beforehand on the regulation market,

πTSOt is irrelevant. While πc

t is known from when the contract is issued, at latestat the gate closure, π+

t , π−t are fixed within the delivery periods.

Since no production costs are taken into consideration, the revenue, RT , through-out the period T consists of the revenue, Rc, from the contracted energy deliveryplus the revenue, Rd, for the deviation between the contracted energy and theactual delivery. This writes

RT = RcT + Rd

T (2.6)

These terms are indeed sums over the considered periods:

RT =∑

t∈T

Rt =∑

t∈T

(Rct + Rd

t ) (2.7)

The income from the contract of energy to be delivered in the period [t− 1, t] is

Rct = πc

t · Ect (2.8)

The regulation price (per amount of energy) at time t is denoted by πdt and

given as

πdt :=

π−t , Ed

t ≤ 0

π+t , Ed

t > 0(2.9)

The regulation revenue at time, t, is now given as

Rdt = Ed

t · πdt =

Edt · π−

t , Edt ≤ 0

Edt · π+

t , Edt > 0

(2.10)


The rules for regulation bids mentioned in Section 2.3 can be formulated as:

Edt < 0 : πd

t = π−t , 0 ≤ πc

t ≤ πdt (2.11a)

Edt ≥ 0 : πd

t = π+t , πd

t ≤ πct (2.11b)

The up-regulation price will usually be greater than or equal to the spot price,and if smaller, very close to the spot price. The down regulation price mustsimilarly be smaller than the spot price.

2.4 Formulation of a naive regulation prices pre-

dictor

Work is being done by Tryggvi Jonsson at IMM, DTU on prediction of regulationprices. This has not been done here. In stead, a naive prediction of regulationprices and costs will be used. The regulation costs denote the absolute differencebetween the spot price and the regulation price and are denoted by a C. Assuperscripts, + and − are used to distinguish between the cost of being inpositive and negative balance with the system (as for the regulation prices). Theregulation costs will simply be predicted as the average up and down-regulationcosts from the decision period. This writes:

C+t+k|t =

1

NTdec

∑

τ∈Tdec

(

π+τ − πc

τ

)

(2.12a)

C−t+k|t =

1

NTdec

∑

τ∈Tdec

(

πcτ − π−

τ

)

(2.12b)

where NTdecdenotes the number of observations in the decission period. The

predictions of regulation prices now follow:

π+t+k|t = πc

t+k − C+t+k|t (2.13a)

π−t+k|t = πc

t+k + C−t+k|t (2.13b)

2.5 Electrical energy storages

Electrical energy storages (EES) can be based on many different technologies.The most common technologies are pumped hydro storage, different kinds of

2.5 Electrical energy storages 19

batteries, compressed air electricity storage and super-conducting magnetic en-ergy storage. These technologies will not be described here. Since the aim ismathematical modelling, some general notes on characteristics relevant in thisprocess will be given. For more on formation on this, see [13]. For a model ofan EES, one needs to consider at least the capacity, the charge and dischargespeeds, efficiencies, and time delay.

Depending on the way, the model is formulated, the capacity can be constant.But, if for instance using a pumped hydro storage, the input supplied from theuser is not the only input. At rainy times of year, the dam may be full evenwithout the EES being charged by the user.

Charge and discharge speeds are usually upper limited. Depending on the phys-ical system, not only the EES it self can account for this limitation. If the EESis located at distance from the production area, the transmission lines may limitthe charging and discharging as well. Both because of wear of the EES and thepossible transmission issue, the speed limits may be functions of time. Whetheror not these rates are linked to the capacity depends on the technology used.Moreover, there can also be a lower limit for charge and/or discharge rates. Takefor instance again a pumped hydro system. To make the turbines perform, acertain flow is needed.

The conversion efficiency is important. Depending on the technology, it can bea non-linear function of both conversion rate, storage level, and time.

An EES may not be possible to control instantaneously. If not, it affects theoptimisation problem a lot, since the charge/discharging plan will have to bebased on predictions.

From an economical point of view, other characteristics are also central. First,one could define costs per each of the mentioned measures, i.e. cost per capacity,cost per maximum charge/discharge and so forth. But also lifetime considera-tions are central to such an assessment.

2.5.1 Example storage devices

Three different storage models will be used in the analysis. Their propertiesare listed in Table 2.2. L0 is the starting energy level that will be used in thesimulations.

• “IEES” stands for “Ideal EES”. It is a storage device without losses and


with unlimited capacity. It differs from summing deviations betweentraded energy and production by having a finite minimum storage level.It is used as a reference mainly to see the effect of the losses in the otherdevices.

• “EES1” is a storage device with limited capacity and losses on both charg-ing and discharging. The capacity is half a production unit (p.u.), and theconversion efficiencies are constant and both equal to 0.85. The conversionrates are unlimited.

• “EES2” is similar to EES1 but with limited conversion rates. The conver-sion is limited to Es ∈ [−0.1, 0.1] p.u.

Table 2.2 The storage devices, IEES, EES1, and EES2, to be used in the analysis. ∗

The maximum charge and discharge rates are limited by Lminees and L

maxees .

Property IEES EES1 EES2

ηchees [-] 1 0.85 0.85

ηdchees [-] 1 0.85 0.85

Lminees [p.u.] 0 0 0

Lmaxees [p.u ] ∞ 0.5 0.5

Es,minees [p.u.]∗ -∞ -∞ -0.05

Es,maxees [p.u.]∗ ∞ ∞ 0.05

L0 [p.u.] 0.25 0.25 0.25

IEES is not thought to be a model of any real storage device. It is a referenceto see the influence on the limitations of the other devices.

EES1 is assumed to model a pumped hydro power system. The charge/dischargeis assumed to be scalable, in the sense that the limit of the flow to the dam isdetermined by the capacity of pumps and pipes, and the discharge of pipes andturbines. For pumped hydro power system, the assumption of no time delay onthe control is considered reasonable.

EES2 can also be thought of as a model of a pumped hydro power system. Butin this model, the charging/discharging rates are limited. This could both bea result of the EES it self or transmission between the production are and theEES.


2.5.2 Electrical energy storages with constant efficiencies

The methodology that is developed in the following describes an EES withconstant efficiencies with respect to both time, amount of stored energy, andabsolute value of transmitted power. It is thus applicable for all of the threeexample devices defined in Section 2.5.1.

Notation: Constant parameters of electrical energy storages are denoted withthe subscript, “ees”. States are denoted with a time index as subscribt.

2.5.2.1 Energy balance

The energy balance for an EES is sketched in Figure 2.5. The circuit betweenthe storage “tank” and the energy Es that goes to the power net illustratesthat the conversion efficiency can, and will in general, depend on the directionof the energy flow. ηch

ees and ηdchees denote the constant charging and discharging

efficiencies.

The energy level in the EES at time t will be denoted by Lt. Notice that thisnotation differs from when referring to energies. Lt refers to the energy levelexactly at time t. Let, in accordance with [17], ∇ be the difference operatorsuch that

∇Lt := Lt − Lt−1 (2.14)

When charging, Es is negative (see Figure 2.1) while ∇L is obviously positive.From Figure 2.5 it is seen that the charging energy balance is then given by

∇L = −ηchees · E

s, Es ≤ 0 (2.15a)

When discharging (∇L negative), the loss is on the energy dispatched from EES.The energy balance then becomes

Es = −ηdchees · ∇L, Es > 0 (2.15b)

Isolating the energy delivered to the power net gives the following relation:

Est =

−ηdchees · ∇Ls

t, ∇Lst ≤ 0

− 1ηchees

∇Lst, ∇Ls

t > 0(2.16)


Storage

Lminees

L

Lmaxees

ηdcheesηch

ees

EES

∇L

Es

Figure 2.5 Illustration of the energy conversion between the storage device and thepower net and the limitations hereof.

2.5.2.2 Flow limitations

An EES will in general have a minimum and a maximum level, Lminees and Lmax

ees

respectively. The charging and discharging speeds can be limited as well. Theseare denoted by Es,min

ees and Es,maxees . These limitations can have different physical

causes. The conversions between electrical power and stored energy will havefinite maximum conversion rates. But for instance a large water dam thatservices a large production area may also be limited by the transmission capacityof the power net. For convenience, another notation for the conversion limitationis introduced. This is the limit in the change in storage level. The maximalpossible increase in storage level is denoted (∇L)max

ees , and the largest dischargeis denoted (∇L)min

ees . Their relations to the corresponding limits on Es,minees and

Es,maxees are:

(∇L)minees := −

1

ηdchees

· Es,maxees (2.17a)

(∇L)maxees := −ηch

ees · Es,minees (2.17b)


Depending on the storage level, the capacity and conversion/transmission speedlimits define the operational limitations. In Figure 2.6 this limitation is illus-trated with a storage device with certain maximal absorption and dispatchingspeeds, (∇L)min

ees and (∇L)maxees . The minimum and maximum obtainable storage

level at time t are denoted Lmint and Lmax

t respectively:

Lmint ≤ Lt ≤ Lmax

t (2.18)

Notice that these limits are shorthand notation, since they are actually condi-tioned on Lt−1.

Three different examples are shown in Figure 2.6. One where the conversionlimitations define both charging and discharging limits, one where the minimumstorage level defines the maximum discharge, and finally one where the storageis almost full, and the maximum storage level limits the maximum possiblecharge.

(a)

L

Lminees

Lta−1

Lmaxees

(∇L)maxees

(∇L)minees

Lmaxta

Lminta

(b)

L

Lmintb

= Lminees

Ltb−1

Lmaxees

(∇L)maxees

(∇L)minees

Lmaxtb

(c)

L

Lminees

Ltc−1

Lmaxtc = Lmax

ees (∇L)maxees

(∇L)minees

Lmintc

Figure 2.6 Illustration of three examples of the limitations of the charging and dis-charging of a storage device. In (a) both charging and discharging is lim-ited by charging speed limits. In (b) only charging is limited by maximumcharging speed while discharging is limited by the storage level. In (c) thestorage is almost full, and charging is limited by the storage level, whereasdischarging is limited by discharge speed limit.

Hence, as seen in the illustrations, the limits for the storage level at time t, using


the information available at time t − 1 are given by

Lmint := Lmin

t|t−1 = max

Lminees , Lt−1 + (∇L)min

ees

(2.19a)

Lmaxt := Lmax

t|t−1 = min Lmaxees , Lt−1 + (∇L)max

ees (2.19b)

Similarly, extremes for the possible amount of energy that can be delivered tothe power net in the period [t − 1, t] can be defined:

Es,mint ≤ Es

t ≤ Es,maxt (2.20)

And the limits are given by:

Es,mint := E

s,mint|t−1 = max

1

ηchees

· (Lt−1 − Lmaxees ) , Es,min

ees

(2.21a)

Es,maxt := E

s,maxt|t−1 = min

ηdchees ·

(

Lt−1 − Lminees

)

, Es,maxees

(2.21b)

Equation (2.21a) expresses that the EES cannot be loaded to more than full,and it cannot be loaded faster than a certain limit. This is relevant for negativevalues of Es

t . Equation (2.21b) expresses the same boundaries for the conversionfrom energy stored in the EES to electrical power. This is relevant for positivevalues of Es

t . The delivered energy can neither exceed what is available in thestorage from time t − 1 passed through the lossy conversion, nor the maximumspeed by which the specific EES can deliver electric power.

2.5.3 Passive operation strategy

An EES should obviously be used to improve the revenue of the production.As deviations in the energy delivery from the contracted amount of energyare generally penalized, a simple way to do this is to, if possible, charge anddischarge as soon as there is a production surplus or lack compared to theconcluded contracts. This deviance from contracts at time t is denoted Ed

t :

Edt := Eo

t − Ect (2.22)

and can be interpreted as the production surplus with respect to the concludedcontract. Without any storing facilities this would simply be the differencebetween production and contracted delivery. Using an EES it writes

Edt = Ew

t + Est − Ec

t (2.23)


The passive operation strategy is to minimize the numeric value of Edt at any

time, t, without paying attention to the future nor to the prices at time t. Theoperation at time t is then

Es*t = arg min

Est

∣

∣Edt

∣

∣

(2.24)

= arg minEs

t

|Ewt + Es

t − Ect | (2.25)

Subjected to the boundaries in Equation (2.20).

The solution consists of five different cases and is given in Equation (2.27).For shorter notation, the production surplus compared to the traded amount ofenergy is denoted

E∼t := Ew

t − Ect (2.26)

Es*t =

Es,maxees , Es,max

ees ≤ E∼t ≤ ηdch

ees ·(

Lt−1 − Lminees

)

ηdchees ·

(

Lt−1 − Lminees

)

, ηdchees ·

(

Lminees − Lt−1

)

≤ E∼t ≤ Es,max

ees

−E∼t , E

s,mint ≤ E∼

t ≤ Es,maxt

1ηchees

· (Lt−1 − Lmaxees ) , 1

ηchees

· (Lmaxees − Lt−1) ≤ E∼

t ≤ −Es,minees

Es,minees , −Es,min

ees ≤ E∼t ≤ 1

ηchees

· (Lmaxees − Lt−1)

(2.27)

Since the solution, (2.27), of the optimization problem (2.25) is a central toolin the modelling, each case will be briefly exlained and illustrated. In the illus-trations, an overview of the conversions are given. Notice that the L and Es

axis are anti-parallel. −E∼ is shown on the Es axis since this is the amount ofenergy from the storage that would balance out the contract and the delivery.The resulting Ed is the length of the red areas on the Es axis. In the thirdcase, there is no difference between the contracted and the delivered energy. Instead, a green area shows the span of the Es that the storage can provide inthis example.

1. In the first case, more energy is contracted to be delivered than is pro-duced. The level of stored energy in the EES is sufficient, but the EEScannot discharge fast enough. The resulting energy delivered is Es,max

ees .Figure 2.7(a).


2. In the second case again, more energy is contracted to be delivered thanis produced. The EES can discharge fast enough, but it is emptied. Theresulting energy delivered is Eo

t = ηdchees · (Lt−1 − Lmin

ees ). Figure 2.7(b).

−E∼t

L

Lminees

Lt−1

Lmaxees

Es

− 1ηchees


0

ηdchees · (Lt−1 − Lmin

ees )

Es,minees

Es,maxees

(a) Illustration of the first line of the solu-tion in Equation (2.27). The storagedevice cannot discharge fast enoughand only E

s,maxees is delivered from it.

−E∼t

L

Lminees

Lt−1

Lmaxees

Es

− 1ηchees


0


ees )

Es,minees

Es,maxees

(b) Illustration of the second line of the solutionin Equation (2.27). The storage is emptied.

Figure 2.7 Illustration of the first two lines of the solution in Equation (2.27). Theseare the cases that result in negative E

dt.

3. In the third case, neither capacity or flow limitations are met, and therewill be no difference between the contracted and the delivered amount ofenergy. Ew

t − Ect can be both positive and negative. Figure 2.8.

−E∼t

L

Lminees

Lt−1

Lmaxees

Es

− 1ηchees


0


ees )

Es,minees

Es,maxees

Figure 2.8 Illustration of the third line of the solution in Equation (2.27). The storagedevice cannot discharge fast enough and only E

s,maxees is delivered from it.

The resulting Ed is the length of the red area on the E

s axis.

2.6 Searching for an optimal contract 27

4. In the fourth case, there is a production surplus compared to what is con-tracted. The EES is fast enough charging, but it is filled up. Figure 2.9(a).

5. In the last case, there is again a production surplus compared to what isthe. contracted production surplus exceeds the maximum charging speedof the EES. The two last cases result in a negative Ed

t .

−E∼t

L

Lminees

Lt−1Lmax

ees

Es

− 1ηchees


0


ees )

Es,minees

Es,maxees

(a) Illustration of the fourth line of the solutionin Equation (2.27).

−E∼t

L

Lminees

Lt−1

Lmaxees

Es

− 1ηchees


0


ees )

Es,minees

Es,maxees

(b) Illustration of the fifth line of the so-lution in Equation (2.27).

Figure 2.9 Illustrations of line 4 and 5 in Equation (2.27). Situations where the EEScannot charge enough to balance out the contract and the delivery

Keeping in mind that Lminees , Lmax

ees , Es,minees , Es,max

ees , ηchees, and ηdch

ees are fixed pa-rameters, Es*

t is a function of only Lt−1, Ect , and Ew

t . This is why the operationstrategy is called “passive”.

2.6 Searching for an optimal contract

The obvious objective is to maximise the revenue of the trading. But as thecontracts are concluded on beforehand, this leads to a problem of stochasticoptimisation. As the Law of Large Numbers says that the average of indepen-dent identically distributed random variables converges in probability to theirexpected value [26], maximisation of the expectation of the revenue is indeedmaximisation of the long term revenue.

By combining Equations (2.7), (2.8), and (2.10) with the energy balances in


Equations (2.22) and (2.23), the total revenue in the period, T , can be written

RT =∑

t∈T

(

Ect · πc

t + (Eot − Ec

t ) · πdt

)

(2.28)

=∑

t∈T

(

Ect · πc

t + (Ewt + Es

t − Ect ) · π

dt

)

(2.29)

=∑

t∈T

(

Ect ·(

πct − πd

t

)

+ Est · π

dt + Ew

t πdt

)

(2.30)

The last expression is ordered so that the last term is independent of the con-tracted energy. With the passive storage operation strategy Ed

t is a function ofEc

t and Ewt , but Ew

t is assumed to be uncontrollable.

Let the time where the contract is issued be denoted tc. Therefore the infor-mation available is assumed to be the realizations known at time tc − 1. Theσ-algebra for the production at time t is denoted Ft and contains the informationgiven by all production measurements up to and including time, t. Optimisa-tion of the expectation of the revenue during the period T given the informationavailable at time tc − 1 can be formulated as

Ec*t t∈T = argmax

Ect ,∀t∈T

E

(

∑

t∈T

(

Ect ·(

πct − πd

t

)

+ Est · π

d + Ewt πd

t

)

|Ftc−1

)

= argmaxEc

t ,∀t∈T

E

(

∑

t∈T

(

Ect ·(

πct − πd

t

)

+ Est · π

d)

|Ftc−1

)

(2.31)

and the solution to this problem is a vector of as many elements as there are inT . The reason that the last term in the sum disappears is that the contractedenergy does not have any influence on it, so the two optimization problems areequivalent. But the stored energy is a function of the contracted energy and thegenerated amount of electrical energy, and it must therefore be included in theobjective function.

Figure 2.10 illustrates the total revenues of different contracts but with theenergy delivered to the power net fixed. A case with up regulation and a casewith down regulation are seen. In both cases the maximum revenue is obtainedby avoiding a deviance. It is noticed that even though in both cases both positiveand negative deviances give a loss in revenue, it is the sign of the deviance thatis of importance. In other words, if the regulation state of the power net isknown on beforehand, trading would be the easy task of not having a devianceof the same sign as the power surplus in the system that the regulation balancesout.

2.6 Searching for an optimal contract 29

0

up regulationdown regulationpred. reg. prices

Edt

Rt

or

Rt|

t c−

1

Figure 2.10 Total revenues for a fixed production versus deviation, Edt, from contract.

The last example shown in Figure 2.10 is a case with regulation prices predic-tions. In this case the predicted revenue is heavily depending on the deviation,no matter the sign.

We will now consider the simpler system where no storage is available. Theenergy delievered to the power net then equals the generated energy, and theonly way the revenue can be optimized is by controlling the contracts. Thedeviation from the contract at time t is then

Edt = Ew

t − Ect (2.32)

and the total revenue from the production at time t:

Rt = Ect · πc

t + Edt · πd

t

= (Ewt − Ed

t ) · πct + Ed

t · πdt

= πct · E

wt + Ed

t · (πdt − πc

t ) (2.33)

Since the power generation in the first term is uncontrollable, the maximisationof the total revenue is then equivalent to the maximisation of Ed

t · (πdt − πc

t ).It is seen from Equations (2.11) that when Ed is negative, πd − πc is non-negative, and when Ed is positive, πd − πc is non-positive. The last term inEquation (2.33) is negative and is interpreted as the cost of regulation. Theregulation cost for the deviance between contracted and delivered energy in the


period ]t − 1, t] is therefore introduced as

Cdt := −Ed

t · (πdt − πc

t ) (2.34)

which is non-negative.

It is proved in [15] that the optimal contract in such a case is given by

FEt|tc−1

−1(

π+t

π+t + π−

t

)

(2.35)

where FEt|tc−1 is the cumulated distribution function of the energy generation at

time t given the the information available at time tc − 1.

When introducing an EES, the optimisation problem does not reduce this easily.That was already seen from Equation (2.31) where two terms are left dependingon the contracts. However, it may be a good idea to focus on the regulationcosts. The idea behind this is that maximising the revenue within a period isperfectly succeeded when the storage device is completely emptied at the endof the period and the energy is sold at highest possible price. But this approachobviously has two problems. First, it will leave the task more dificult for thefollowing period, since the end storage level for each period is expected to tendto zero. Then additionally, this expected tendency to empty the storage mayresult in a large variance of the regulation costs, since some realizations will leadto a power lack and hence the need to pay up-regulation costs. Because of theseconsiderations it has been decided also to investigate the benefits from minizingthe expected regulation costs. This writes

Ec,∗T = arg min

EcT

E(

CdT

∣

∣Ftc−1

)

= arg minEc

T

E(

−Edt · (πd

t − πct )∣

∣Ftc−1

)

= arg minEc

T

E(

|Edt | · |π

dt − πc

t |∣

∣Ftc−1

)

= arg minEc

T

E(

|Eot − Ec

t | · |πdt − πc

t |∣

∣Ftc−1

)

= arg minEc

T

E(

|Ewt + Es

t − Ect | · |π

dt − πc

t |∣

∣Ftc−1

)

(2.36)

Where the conditioning on Ftc−1 denotes that the decision must be made solelybased on information available before time the market gate closure, tc. With thisstrategy the market game of intentionally charging and discharging for savingproduction for times with better prices on the market disappears.

2.7 Simulation plan 31

2.7 Simulation plan

The three example storage devices described in Section 2.5.1 will all be testedwith the two different optimisation strategies described in Section 2.6, exceptfor IEES which will only be tested with the “maximisation of the expectedrevenue” strategy. They will be tested on both 6 and 24 hour market scenariosas described in Section 2.3 with common spot and regulation prices in the twomarket scenarios. This gives 10 optimisation problems.

Since the prediction model for the regulation costs is expected to perform poorly,maximisation of the expected revenue will also be done with perfect knowledgeabout the regulation prices. This is done to have a reference on how much thatcould be gained on improving this part of the model. This is also done forall of the three storage devices for both market setups, resulting in another 6optimisation problem.

Only full days without any missing prediction, prices or measurements will beused in the optimisations. It shall later be seen that this is 384 full days, or 1.5366 hours periods. When one or more days are skipped, everything will be keptconstant until next full day of data, and the next optimisation is carried out.Each optimisation process will be carried out using 500 generated scenarios.

The main reference for the revenues is the value of the generation traded atElspot in the considered period with perfect predictions used as contracts. Thiswill be the unit for all revenues of operation throughout the whole simulationperiod. This is not the same as maximum possible revenue, since it may bebetter to store production for selling at a higher price later. It should thereforenot be interpreted as a theoretical limit.

As other references, the WPPT predictions and the estimate of the expecta-tion calculated from an estimated distribution of the generation will be used ascontracts on the 6 and 24 hours markets. These will be used without storagedevice and with each of the three different EES as well. These are calculatedfor improving the possibility to distinguish between different effects on the totalrevenues.

All algorithms have been implemented in R. The optimisation algorithm thathas been used is nlminb which can perform multivariate constrained optimi-sation. It is based on the PORT library which can perform several differentoptimisation algorithms. It was chosen because it was considerably faster thanthe standard implementation, optim, of among others the L-BFGS-B algorithmfor constrained optimisation. The optimisation algorithms have not been infocus in this work.


Chapter 3

Theory

In this chapter, theory that will be used is described. The reader who is familiarwith the concepts given in the names of the sections, can skip them. Whenthe theory is used, references will be given, and the chapter can be used as areference.

3.1 Random variables

A random variable, X , is a mapping from a sample space, Ω, to a measurablespace. Here, it will always be into the real numbers, R.

X : Ω → R (3.1)

X is said to be measurable on the σ-algebra, F , if for all x ∈ R, ω ∈ Ω :X(ω) ≤ x ∈ F . A σ-algebra is a family of subsets of Ω containing all definableevents (and their complements). The construction of the mapping P : F → [0, 1]is therefore possible and it describes the probabilities of the events in F . With(Ω,F , P) being the probability space, the distribution function, F is the function

F : R → [0, 1], F (x) = P (X(ω) ≤ x) (3.2)

34 Theory

For a shorter notation, ω is often omitted and X(ω) is simply written X . For amore systematic presentation of these concepts, consult [27].

The distribution function is also called the cumulated distribution function(cdf) distinguishing it from the probability density function (pdf), f , givenby [9]:

f : R → R+0 , f(x) =

dF (x)

dx(3.3)

Equation (3.2) implies that the cdf is an increasing function with boundaryvalues zero for x going to −∞ and one for x going to ∞, respectively. Whendealing with more than one random variables, the names of the variables areoften used as subscripts when referring to their cdf’s or pdf’s, eg. FX .

In relation to this present work, random variables with strictly increasing cdf’sare of special interest. These variables are, no matter their distributions, easilytransformed into random variables with other distributions. The distributionfunction is a transformation into a random variable following the uniform dis-tribution. The proof of this is simple. Let X follow the strictly increasingdistribution, FX , and remember that ∀ω ∈ Ω : FX(X(ω)) ∈ [0, 1]. Then foru ∈ [0, 1]:

P (FX(X) ≤ u) = P(

X ≤ F−1X (u)

)

= FX(F−1X (u)) = u (3.4)

which shows that U := FX(X) follows the uniform distribution. This is funda-mental [20] in simulation of random variables from non-parametric distributions,because it can also be used for transforming uniformly distributed variables intovariables from any other distribution. Formally, let

U ∼ unif(0, 1),

P (X ≤ x) = F (x), ∀x ∈ ΩX ,

F (x1) < F (x2), ∀x1 < x2 ∈ ΩX

Then

X = F−1X (U), (3.5a)

U = FX(X) (3.5b)

3.1.1 Quantiles

A distribution function can also be expressed by quantiles. The α’th quantileof the random variable, X , is: [14]

q(α) := F−1X (α) = infx : FX(x) ≥ α, 0 < α ≤ 1 (3.6)

3.2 Quantile regression 35

This is the smallest value that the random variable, X , is larger than or equal towith probability α. The infimum in (3.6) is sometimes let out, but it ensures theuniqueness of the quantile in cases where the distribution function is not strictlyincreasing. The infimum however precludes the existence of q(0) which would beinfinitely small. This quantile has to be defined alternatively. With assumptionsfor α = 0 and α = 1, quantiles are uniquely defined by Equation (3.6) for randomvariables with strictly increasing distribution functions.

Often, only a limited number of quantiles is considered. If the values of theα’s are equidistant, they can be given names referring to the number of α’s.Common examples are quartiles (4-quantiles) and percentiles (100-quantiles).The 0.25th quantile may thus be referred to as the first quartile or the 25th

percentile. The 0.5th quantile, the second quartile, or the 50th percentile, isspecially called the median.

By definition, the probability that a random variable takes a value betweentwo quantiles equals the difference between the quantile fractions. This is forα1 ≤ α2:

PF

(

X ∈]

q(α1), q(α2)])

= PF

(

X < q(α2))

− PF

(

X < q(α1))

= F(

q(α2))

− F(

q(α2))

= α2 − α1 (3.7)

This means that a random variable takes a value between a set of quantiles ofequidistant fractions with equaling probabilities. This result is closely relatedto the one given in Equations (3.5).

3.2 Quantile regression

Quantile regression is thoroughly described in [14]. The basic idea here is alinear model of the quantile function. Each quantile is modelled independently.Extending the notation from Equation (3.6) to n explanatory variables, this is

q(α)k : R

n → R, q(α)k (xk) = xT

k βk(α) + r(α)k (3.8)

The subscript, k, indicates that each quantile is modelled independently for eachhorizon, k. The estimation is an optimization problem with the piecewice linearcost function depending on α:

ρα(r) =

α · r r ≥ 0

(α − 1) · r r < 0(3.9)

36 Theory

Illustrations of the cost function for different values of α are seen in Figure 3.1.

r

ρα(r)

α = 0.1

α = 0.5

α = 0.7

Figure 3.1 Illustration of the cost function in the quantile regression estimation.

The estimate, for the the horizon, k, using N data points is

βk(α) = argminβk(α)

N∑

i=1

ρα(ri) (3.10)

In [14] this is reformulated for solving using linear programming.

In [19] the problem is extended to one depending on time, making the estimateadaptive. Furthermore, as linearity is a bad assumption for the power curve,splines are suggested as base functions in stead. Using only the point forecastas explanatory variable, the model now changes to

qαk (ǫp, t) = βk,0,t(α) +

K−1∑

j=1

bk,j(pt) · βk,j,t(α) (3.11)

where ǫp denotes the prediction error from the point forecast, b1, . . . , bK−1are the spline base function, and K is the number of knots in the spline. Theadaptivity in the method comes from the substitution of the data points usedin the estimation. For information on this, see [19].

The method leaves several choices to the user.

• The spline base function.

• The number of knots and their placements.

• The number of bins and their placements.

• If the method is adaptive, the number of observations in each bin.

3.3 Expectation and variance of random variables 37

3.3 Expectation and variance of random vari-

ables

When modelling stochastic processes, sums of random variables often appear.The expectation of the sum of two random variables is simply the sums of theirexpectations. Expressed more generally, expectation is a linear operator (onexpectation, see [9], on linearity see [3]):

E (a · X + b · Y ) = a · E (X) + b · E (Y ) (3.12)

The variance is on the contrary a non-linear mapping, and the variance of thelinear combination of the two random variables, X and Y , is given by [9]

V (a · X + b · Y ) = a2 · V (X) + b2 · V (Y ) + 2 · a · b · C (X, Y ) (3.13)

where C (X, Y ) denotes the covariance of the random variables, X and Y .

3.4 Multivariate normally distributed variables

The second order representation of a multivariate consist of the expectation vec-tor, µ, and the covariance matrix, Σ, which is by definition symmetric [7]. Withthese known, one can simulate drawings from the autocorrelated multivariatenormal distribution by use of the Cholesky factorisation of the covariance ma-trix. The Cholesky factorisation leads to the unique lower triangular matrix, C,which satisfies

CCT = Σ (3.14)

Let Z follow the multivariate normal distribution with means zero and theidentity matrix as covariance matrix, i.e. it consists of independently standardnormally distributed variables:

Z ∼ N(0, I)

Then [25]:

µ + CZ ∼ N(µ,Σ)

Hence, simulations can be done of multivariate normally distributed variableswith any valid covariance structure.

38 Theory

3.5 Decision theory

Equations (2.31) and (2.36) were presented as two different ways to optimisethe trading on an electricity market using an EES with the passive operationstrategy are given. Both objective functions are expectations of random vari-ables. As already stated, the Law of Large Numbers justifies this. The problemwith this way of optimising is that the variance is not controlled, meaning thatlarge losses on the short term are not seeked to be avoided. Even though thethe long-term revenue is optimised, it may be beneficial to use a risk aversion(RA) strategy instead. With a loss function, Λ, the risk, Γ, is defined as [26]:

Γ(θ, θ) = Eθ

(

Λ(

θ, θ))

(3.15)

where the subscript, θ reads “under assumption that θ is correct”. An exampleof a common loss function is the mean squared error (mse). Some function ofthe risk is now minimised. A classical choice is to minimize the maximum risk.This is called the “minimax” rule. θ is the minimax rule if and only if it satisfies[26]

supθ

Γ(

θ, θ)

= infθ

supθ

Γ(

θ, θ)

(3.16)

i.e., it gives the infimum of the “maximum risk”. Another method is the min-imising the Bayes risk. This will not be discussed here.

3.6 Monte Carlo Simulation

The analysis of sequences of random variables very easily gets too complex to doanalytically. In stead, the Monte Carlo method can then be used. The systemis simulated a number of times, and parameters or function of such are thenestimated by use of statistical methods.

Take as an example the minimisation of the expectation of a function, f , of arandom variable, X , with a parameter, θ. What is wanted is

θ∗ = argminθ

E (f(X, θ)) (3.17)

= argminθ

∫

Ω

f(X(ω), θ) dP (ω) (3.18)

The problem is that the density function is too complex to calculate, and hencethe integral cannot be evaluated. Based on N simulations the minimum is

3.6 Monte Carlo Simulation 39

simply estimated as

θ∗ = arg minθ

1

N

N∑

i

X(ωi; θ) (3.19)

The calculation of this estimate is a deterministic optimisation problem thatcan be solve with an appropriate optimisation algorithm. For optimisation al-gorithms, see e.g. [12].

40 Theory

Chapter 4

Data presentation and

preparation

In this chapter, data from external sources is described. Data has been collectedfrom different sources and in different formats. Where the merging of data hasposed problems, the workarounds will also be described here.

Data consists of the following parts: power production measurements, pointforecasts calculated with the WPPT model, probabilistic forecasts calculatedwith adapted re-sampling, and market prices from Energinet. Market priceswere downloaded from www.energinet.dk.

4.1 The considered area

The production measurements are of electricity generation from wind power ina subregion of Denmark, namely the North of Jutland. Figure 4.1 is a sketchof the division of Denmark into subareas, and the concerned area is denoted asArea 1. Due to confidentiality, information about the installed capacity has notbeen provided.

http://www.energinet.dk

42 Data presentation and preparation

Figure 4.1 Western Denmark divided into five regions. Data come from productionin region 1.

4.2 Production measurements

The production measurements are hourly and denote the average electric powergenerated during the preceding hour normalized with the available capacity inthat hour. This is the series called Ew

t . As installed capacity is unknown, thevariance of the available capacity was of interest, since large fluctuations in thecapacity would increase the production variance. According to Pierre Pinsonthis effect can be neglected, and the variance of the measurements is assumedto be independent of it.

The measurements are given in UTC time starting at 2006-01-01 at 2:00 AmUTC, ending at 2007-10-26, 7:00 Pm UTC, which is a period of 15930 hours oralmost 23 months. No data points are missing throughout the period, and allmeasurements are within 0 and 0.828 p.u. The time series is shown in Figure 4.2.

A histogram of the production measurements is shown in Figure 4.3. The over-all general tendency is a decaying density with increasing generation. Whencomparing with the theoretical power curve for a wind turbine in Figure 2.2,the peak in density at zero generation is explained by the failure to generateelectrical power at wind powers up to a certain speed. But as was explainedin Section 2.2, a single wind turbine will produce at its maximum power for an

4.2 Production measurements 43

0.0

0.2

0.4

0.6

0.8

Jan 06 Apr 06 Jul 06 Oct 06 Jan 07 Apr 07 Jul 07 Oct 07

Time

Mea

sure

dpow

er,E

w t[p

.u.]

Figure 4.2 The production measurements normalised to the available capacity.

0.0 0.2 0.4 0.6 0.8

050

010

0015

0020

00

Power [p.u.]

Counts

of15930

Figure 4.3 A histogram of the production measurements recorded hourly between 2006-01-01, 2:00 am and 2007-10-26, 7:00 pm, both in UTC.

interval of wind speeds lower than the cut-off speed. A peak at full productionis therefore expected for a single wind turbine. But since this is data from awhole area, the histogram is different for larger generation. All in all, the whole


area has failed to generate more than 0.83 of the available power installationthroughout the considered period.

4.3 Point forecasts

Point forecasts made by the Wind Power Prediction Tool (WPPT) presentedin Section 2.2 are are available for the per hour horizons 1 to 43 hours. Thesesequences are referred to as Ew

t+k|t meaning that it is an estimate of theexpected value of Ew

t+k given the information available at time t.

In accordance with [16], a point forecast and the forecast error are given by

Ewt+k = E

w

t+k|t + ǫwt+k|t (4.1)

The time index in the error term, ǫ, only serves as a name. It does not make senseto condition random variables on anything. The prediction error is thereforedefined as

ǫwt+k|t := Ewt+k − Ew

t+k|t (4.2)

The performance of the point forecasts is highly depending on the horizon.In this framework though, different horizons are of interest in single models.Because of the dependence of the quality on the prediction horizon, one wouldalways prefer as recent predictions as possible. For example, when trading onElspot, 24 different horizons are used, because one would only use the mostrecent sequence of predictions. On the other hand, since many sequences ofpredictions will be issued before the next gate closure, other predictions willnever be used. The question about which predictions to use in the evaluation istherefore not trivial. One could argue to extract only the predictions that will beused and solely base the evaluation on those. But as trading on different marketswith different gate closures are of interest, the set of relevant gate closures varybetween different setups. Moreover, the predictions come from the same modeland are assumed to have properties with little dependency on issue time. It ischosen to use all available predictions for the evaluation assessment.

As explained in Section 2.5.3, the EES will be used to smoothen out predictionerrors. Bias is therefore of special interest, since it will lead to a higher need foreither charging or discharging depending on the sign. Figure 4.4 shows meanbias for the different prediction horizons. The mean bias is positive for allhorizons. Especially for horizons larger than 35 hours, the mean bias is rapidly

4.3 Point forecasts 45

increasing. Before this point, the bias does not exceed 0.45%. The plot alsocontains some horizontal lines indicating mean biases for the sequences that areused in the trading modelling. The horizontal rows of dots indicate means ofalternative sequences of data with the same horizons, i.e. they start at differenttimes of day.

0 10 20 30 40

0.00

30.

004

0.00

50.

006

0.00

7

Single horizonsUsed 6 h sequencesAlternative 6 h sequencesUsed 24 h sequencesAlternative 24 h sequences

Horizon [hours]

Mea

nbia

s

Figure 4.4 Bias versus horizons.

The reason for the 6 and 24 hour sequences generally being more biased thanaverage is can be understood from Figure 4.5. Here, the bias is consideredas function of time. The figure shows cumulated prediction errors for severalhorizons, and for the 6 and 37 hour sequences. As expected, the tendency is thatthe larger horizon, the bigger fluctuations in the cumulated sum of predictionerrors. It is also seen that the prediction errors are largest later in the consideredperiod (i.e. the cumulated sum changes the most). We shall see in Section 4.7that this is where the used sequences are found. That is why the 6 and 24 hourssequences are more biased than average.

As a last consideration of the bias, Figure 4.6 shows the bias versus both mea-sured and predicted generation. As a function of predicted generation, the biasis numerically quite small until around Ew = 0, 75 p.u. Hereafter it drops dra-matically, which is because the predictions span from 0 to 1 p.u. while as seenin Section 4.2, the maximum generation is only at around 0.83 p.u.

In [16] the normalized bias (NBIASm), the normalized mean absolute error(NMAEm), and the normalized root mean squared error (NRMSEm) are pro-posed as a minimal set of evaluation tools for point predictions. The subscript,

46 Data presentation and preparation−

200

2040

6080


2 hours7 hours14 hours37 hours6 hour sequences24 hour sequences

Time

Cum

ula

ted

sum

ofpre

dic

tion

erro

rs.

Figure 4.5 BIAS versus time.

0.0 0.2 0.4 0.6 0.8

−0.

15−

0.10

−0.

050.

000.

050.

10 BIAS(Ew)BIAS(Ew)

Electrical energy generation

Mea

nbia

s

Figure 4.6 BIAS versus power.

m means that the normalization is done by the average measured production, ienot by the installed capacity, which would be 1 p.u. in this case. The following

4.4 Probabilistic forecasts 47

definitions will herefore be used (the subscript is skipped).

NBIAS(k) :=

∑Nt=1 ǫt+k|t∑N

t=1 pt

(4.3)

NMAE(k) :=

∑Nt=1 |ǫt+k|t|∑N

t=1 pt

(4.4)

NRMSE(k) :=

√

N ·∑N

t=1 ǫt+k|t∑N

t=1 pt

(4.5)

Before calculating such evaluation measures, considerations about missing datapoints must be done. In these predictions, 390 sequences of predictions aremissing. What is meant by a full sequence is a sequence of predictions withcommon issue time but with all of the different prediction horizons. A part fromthese full sequences, no predictions are missing. This corresponds to a share ofmissing points of approximately 2.45%, which is considered as a sufficiently lowrate of missing points.

The benefits from using a set of point forecasts are measured by a compari-son to the persistence model, also called the “naive prediction” [16]. It is theassumption of no change in production:

Ew,reft+k|t = Ew

t (4.6)

Based on RMSE, the improvement measure is [16]

Iref,RMSE(k) =RMSEref(k) − RMSE(k)

RMSEref(k)(4.7)

In Table 4.1 some performance measures are summarized for the used pointpredictions. The performance measures are given for some fixed horizons andfor the specific series of forecasts used in the modelling.

The measures are in general larger than but though comparable to what is seenin [16].

4.4 Probabilistic forecasts

Probabilistic forecasts of the production were provided by Pierre Pinson. Theyare based on fuzzy logic and adapted re-sampling, and the method is developed


Table 4.1 Evaluation properties for the point forecasts from WPPT.

Property Fixed horizons Sequences

2 hours 7 hours 14 hours 37 hours 6 hours 24 hours

Missing share [%] 2.455 2.455 2.455 2.455 2.828 2.513BIAS [%] 0.364 0.382 0.264 0.473 0.520 0.450NBIAS [%] 1.678 1.761 1.218 2.177 2.335 2.025MAE [%] 4.062 4.521 5.055 7.325 4.193 5.986NMAE [%] 18.712 20.828 23.290 33.746 18.810 26.941RMSE [%] 5.822 6.510 7.319 10.478 5.932 8.599NRMSE [%] 27.492 30.745 34.569 49.485 27.383 3.414IRMSE [%] 4.147 56.038 64.029 58.785 48.376 63.822

in [21]. They will be evaluated together with probabilistic forecasts that will bebased on adaptive quantile regression in Chapter 5.

4.5 Market prices and regulation costs

Elspot prices and regulation prices are available at www.energinet.dk. Theyare all available throughout the whole considered period with only few miss-ing data points. They are apparently given in official Danish time which is acombination of Central European Time (CET) and Central European SummerTime (CEST). This leads to a classical data problem where a linear time scale isstrongly preferred. The time stamps themselves show this problem, since theyare constantly increasing with one hour from line to line. In CET/CEST, oneday a year (last Sunday in March) only has 23 hours, while the last Sunday inOctober has 25 hours. Therefore there are obviously problems with data aroundshifts between normal time and summer time. At time shifts a varying numberof observations is missing, and since no explanation is given, data around thesetime shifts are considered unreliable. It is therefore decided to neglect the datafrom days where the time is changing. This is an easy and reliable solutionsince no data points are removed in the preceding days. In Figure 4.7 threetime series, Elspot prices (πc

t), the down-regulation prices (π+t ), and the

up-regulation prices (π−t ) are plotted versus time. In the bottom of each plot,

red marks indicate where data has been either missing or removed.

Figures 4.8 and 4.9 show histograms of both the regulation prices and costs forup-regulation and down-regulation, respectively. Since these prices will only beused for simulations, only the data from the period used for simulations is shown.

http://www.energinet.dk

4.5 Market prices and regulation costs 49

050

100

150

200


−10

00

100

200


020

040

060

080

0


Time

πc t

[e/M

Wh]

π+ t

[e/M

Wh]

π− t

[e/M

Wh]

Figure 4.7 Elspot prices followed by down-regulation and finally up-regulation prices.The red marks in the bottom of each plot indicate missing data points forthe respective variable. Only few data points are missing except from whentime changes from CET to CEST.

For the up-regulation case, it is seen that both prices and costs range from about0 to about 800 e/MWh. Even though most up-regulation costs are either zero(cases of a down-regulated system) or below 50 e/MWh, there are observationsof extremely high regulation costs of several hundred e/MWh. The smallestobserved up-regulation cost is -0.03 e/MWh. Negative regulation costs violate


the rules for trading on the regulation market presented in Equation (2.11).The problem could come from shifting currency values between gate closures atdifferent markets, or simply from bad data registration.

0 200 400 600 800

010

0020

0030

0040

00

Price [e/MWh]

Counts

of9216

(a) Up-regulation prices.

0 200 400 600 800

010

0020

0030

0040

0050

00

Cost [e/MWh]

Counts

of9216

(b) Up-regulation costs.

Figure 4.8 Histograms of up regulation prices and costs used for simulations.

In the histograms of down regulation prices and costs (Figure 4.9), the price isseen to span from around -200 to around 200 e/MWh. The both positive andnegative prices were expected as explained in Section 2.3. The regulation costsspan from -12.31 to about 200 e/MWh. 17 observations are of regulation pricessmaller than -1 e/MWh. Such a clear violation of Equation (2.11) can not becaused by currency changes. Why they occur is unknown, but if they are causedby faulty data handling, it is surely known not to have occurred in this work.When downloading the data from Energinet, tables can be made with all prices,with rows of all prices sharing time stamp. Similar observations have been donein e.g. the work on [11].

4.6 Time handling

As data is given with time stamps from different time zones, considerations mustbe done to make sure that they are matching when used.

Special care must be taken because some data is in a constant time zone (UTC)and other data is from a time zone with summer time, i.e. the time zone isshifting. Had they both been from either a time zone with summer time orfrom a constant one, only a constant would have to be added to or subtracted

4.7 Partitioning the data 51

−200 −100 0 100 200

010

0020

0030

0040

00

Price [e/MWh]

Counts

of9216

(a) Down regulation prices.

0 50 100 150 200

010

0020

0030

0040

00

Cost [e/MWh]

Counts

of9216

(b) Down regulation costs.

Figure 4.9 Histograms of down regulation prices and costs used for simulations.

from one time to match the other.

The as.POSIX* functions in R provide an easy way of handling this, since itcan both convert time stamps between time zones and calculate time differencesof time stamps of different time zones.

Since trading on Elspot which gate closures are in Danish time, the most handyis to create a table with time stamps in Danish time. First, a relative timecolumn is generated with time stamps in “hours since” 2006-01-01 00:00. Alltime stamps from both wind and price data are then converted into this unit,and with a loop, the table is generated line by line for each kind of data. This isnot a fast algorithm, but it eliminates the very confusing problem of comparingtime zones once and for all. Hereafter, R easily gives the time stamp in anytimezone needed. The results from the algorithm were checked on the originaldata structures at several points, both in summer and normal time.

4.7 Partitioning the data

When evaluating models with fixed parameters, data is usually split into anestimation and an evaluation part. This is not to “over-fit” data. When fittinga number of data points, a model will always give a fit that is at least as goodas any of its sub-models. In fact, data can always be perfectly fitted by use ofhigh order polynomials. This kind of models will however predict very poorly.


Different information criteria have been suggested in statistical modelling to in-clude a penalty for model complexity, among them Akaike’s Information Criteriaand the Bayesian Information Criteria. Another idea is to specifically evaluateprediction performance, which is done in the “leave one out” method, whereone point at a time is left out in the estimation procedure and then predicted.When having a lot of data, this method can be very time consuming. For timeseries analysis, the prediction testing has in stead been suggested to be carriedout by dividing data into an estimation set and an evaluation set.

When using adaptive models, this issue is a bit different. The estimation cannot be isolated to take part in only one part of the data but is always basedon recent measurements. However, an adaptive method can be dependent ona model structure, which is indeed the case for adaptive quantile regression.To decide on a quantile regression model, data is split up into a “decision set”and an evaluation set. In the decision set different models are tested, and theirperformances are compared. One will be chosen and used for the simulations.

How much data to use for the different parts depends on the amount of dataavailable. It is a trade-off between the wish of having a good base for a decisionand evaluation at the same time. Adaptive quantile regression needs some datafor initialisation where points are put into the bins, and the estimates becomingmore reliable. The first 1,000 data points were reserved for the initialisation.For some of the horizons (models), this resulted in all quantiles estimates equalto zero which has been noticed to be returned when too little data has beenrecorded. After 1,500 data points this behaviour is no longer seen for any of thetested models, and 1,500 data points were therefore used for the initialisation.This corresponds approximately to two months.

The decision set is then chosen to end on the 2006-08-31 at 10 Am. All pre-dictions needed for trading the production for 2006-09-01 on the Elspot marketthen belong to the simulation set. An overview of the data partition is shown inFigure 4.10. For an easy comparison of the performance on different markets,only full days with all predictions, measurements, and all prices available areused. In the figure, red spots indicate missing data points in a given time series,and the lower line summarises where all data is available and the simulationsare carried out.

In the simulation period, 384 days are without any missing data points. Thegeneration on these days is 2,228 p.u. which were worth 66,206 e/MWh. if ithad all been sold at Elspot without any deviations from contracts. The unit,e/MWh means e per installed MWh.

4.7 Partitioning the data 53

measurementsWPPT predictionsquantiles

regulation market pricesElspot pricesall

missing data pointsquantile regression decission


Time

Figure 4.10 Illustration of the availability of the different kinds of data. A red pointindicate missing data. The grey area is the period used for decision mak-ing on a quantile regression model, and hence simulations are only donein the white area. Notice that one red point leads to the discard of a wholeday of data. One red spot in the figure is however larger than a day onthe time line.


Chapter 5

Scenario generation

This chapter is an explanation of how scenarios of the electrical energy gener-ation are carried out. First, the whole procedure is outlined. Then the stepswhere choices by the user are necessary will be carried out. The procedureconsists of non-parametric estimation of probability density functions, transfor-mations of random variables by use of theory given in Chapter 3, and adaptiveestimation of covariance structures. The method is also described in [24], [20]and [13].

An illustration of the the overall procedure is shown in Figure 5.1. The proce-dure can be regarded in two major parts. First the estimation part leading toa representation of the observed process in a domain of realisations of variablesfollowing the standard normal distribution. From these, the autocorrelation ofthe transformed process can be estimated.

When having an estimate of the autocorrelation, realisations of the multivari-ate normal distribution with the estimated autocorrelation can be simulated.The inverse of the transformations done on data to obtain data that could beconsidered normally distributed, will now transform data back to the domain ofenergy generation.

In Figure 5.1, the colours of the components indicate if choices must be ex-plicitly made or not. The blue components are just function evaluations that

56 Scenario generation

Production data

Ft+k|t

F (·) ∼ Ûnif(0, 1) Φ−1(·) ∼ N(0, 1)

Σt+k|t

Simulations fromN(0, Σt+k|t)

Φ(·) ∼ Ûnif(0,1)are inter-dependent

F−1(·) areproductionscenarios.

Figure 5.1 Illustration of the process of simulation of power production scenarios.The blue boxes are applications of functions, while the green boxes are es-timations. Hence, the continuous lines indicate calculations that are donewithout decision making. Notice that Ûnif and N are used to denote trans-formations of data into the uniform and normal domains, respectively. I.e.they are not estimates of distributions, but the notation is used because thestrength of the assumptions are related to the accuracy of F .

are predefined either theoretically or by the modelling done in the green com-ponents. Two transformations are based on the non-parametrically estimatedcdf of the wind power production, and the generation of sequences followingan autocorrelated normal distribution is based on the estimation of, of course,the autocorrelation. Hence, the continuous lines represent deterministic trans-formations. They are all applications of Equations (3.5) which were proved inChapter 3. Simulations of realisations of the multivariate normal distributionwas described in Section 3.4

Once decisions have been made on the non-parametric pdf of the power gener-ation and on the autocorrelation, the procedure can be implemented. The restof this chapter is about the estimation of the pdf of the power generation andof the autocorrelation.

5.1 From quantile estimates to full distribution functions 57

5.1 From quantile estimates to full distribution

functions

The physical constrains that the nominal production is between zero and oneand that the distribution of the production is strictly increasing were formulatedin equations (5.14) and (5.15). These assumptions are checked to be fulfilledfor all considered sets of quantiles, and if necessary they are modified to fulfilthem. For each set of quantile estimates the following procedure is carried out.First, the quantiles are sorted. Then they are checked to lie within the interval[δx, 1 − δx], where δx is chosen to be δx := 10−2. The ones that are not in theinterval are substituted with values of difference δx starting from either δx ifsmaller than δx or ending at 1−δx if larger than this. The values are then againsorted, and then they are checked not to equal. If some of the quantiles equal,they are subsequently increased by δx and the quantiles are sorted again. Thisalgorithm ensures that both equations (5.14) and (5.15) are fulfilled.

The quantile estimates are then extended to an estimate of the full cdf. Asalready mentioned, the assumption that F (0) = 0, F (1) = 1 comes from the factthat normalised power is considered. Moreover as in [24], exponential functionsare used to model the tails between zero and the smallest estimated quantile,and between the largest available quantile estimate and one.

With N quantile estimates, the estimate of the cdf is extended to the followingpoints:

F (x) =

0, x = 0

ρ · exp(

q(α1)

x · ln(

α1

ρ

))

, x = i · δx, i ∈ 1, . . . , ⌊ q(α1)

δx⌋

αn, x = q(αn), n ∈ 1, . . . , N

1 − ρ · exp(

1−q(αN )

1−x · ln(

1−αN

ρ

))

, x = q(αN ) + j · δx,

j ∈ 1, . . . , ⌊ 1−q(αN )

δx⌋

1, x = 1

(5.1)where ρ ∈ R

+0 , and ⌊·⌋ is the floor operator. If the floor operations return zero

in the second or the third line it should be interpret as i or j belongs to theempty set, and the respective line disappears.

Between the points described in Equation (5.1), piecewise cubic Hermite inter-polation is used. Piecewise Hermite interpolation gives an interpolation schemeof pieces of third order polynomials. The first derivative is continuous, which thesecond is not necessarily. The applied method is described in [1] and is based onpiecewise Bi-cubic functions. It assures a monotonic interpolation scheme which


is needed for having a strictly increasing estimate of the cdf. It is implementedin the R-package called “Signal”. See [6] and [12] for spline interpolation.

Two examples of estimates of the full cumulative probability density functionare shown in Figure 5.2. Both the output from the adaptive quantile regressionalgorithm and the corrected quantiles are shown. In Figure 5.2(a), the quantilesestimates from adaptive quantile regression fulfils Equations (5.14) and (5.15)are shown. It shows the estimate of the distribution function for the producedpower the 25th of March 2006 at 10 am CET, where the estimate is based oninformation available one hour on beforehand. On the abscissa, bars indicatethe placement of the points defined by Equation (5.1).

The plot to the right in Figure 5.2 shows the estimation of the distributionfunction for the production on the 25th of February at 8 am CET, again the es-timation is based on information available one hour on beforehand. In this case,the quantile estimates from quantile regression give that negative production ispredicted to occur with positive probability. This is corrected by the describedalgorithm.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

F (p)F (p)

Power, p [p.u.]

P(P

<p)

(pt1 , put1)

(a) One hour prediction issued at 9 am onMarch the 25th, 2006.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

F (p)F (p)

Power, p [p.u.]

P(P

<p) (pt2 , p

ut2)

(b) One hour prediction issued at 10 p.m. onFebruary the 13th, 2007.

Figure 5.2 Illustration of linear interpolation and exponential extrapolation of esti-mated quantiles. The points marked by black circles are the points avail-able from the quantile estimation. The marks in the bottom indicate thepoints defined by Equation (5.1). With green, the transformation of theobservation into a realisation of a uniformly distributed random variableusing Equation (3.5) is illustrated.

Based on the points given in Equation (5.1) the expectation of the productionis calculated using a middle sum approximation to the integral (for continuousrandom variables, the expectation is defined as the integral over the value of the

5.2 Evaluation of probabilistic forecasting 59

variable times the probability density in that point, see any textbook on basicprobability theory). This will be referred to as EF (Ew)

5.2 Evaluation of probabilistic forecasting

The precision of point forecast models is expressed by functions of their predic-tion errors and is easily interpreted as e.g. the average distance between theforecast and the realization value. When dealing with probabilistic forecasts,evaluation of strength becomes more complicated.

Proposals for evaluation of probabilistic forecasts are given in [23]. Since onlyone realization is known for each estimated distribution, performance measuresof quantile regression models must express the performance of the whole setof quantile estimates. The evaluation is based on three properties: reliability,sharpness, and resolution.

The characteristic function for a set, A, is given by (in accordance with [3])

χA : Rn → 0, 1, χA(x) =

0 x 6∈ A

1 x ∈ A(5.2)

where n is a positive integer.

The fraction of outcomes smaller than or equal to a quantile estimate, q(α), ofnominal proportion, α, and prediction horizon, k, is then given by

aαk =

1

N

N∑

t=1

χ]−∞,q

(α)

t+k|t](pt+k) (5.3)

For a perfectly calibrated set of quantiles, this equals α.

The deviation from α is the bias, b(α)k , of a set of quantile estimates [23]:

b(α)k = α − a

(α)k (5.4)

As the bias depends on both the horizon and the nominal proportion, threedifferent averages are defined. First, the prediction horizon is eliminated, then


the nominal proportion, then both:

bk(α)

=1

max (K)

∑

k∈K

b(α)k (5.5)

b(α)k =

1

Nα

Nα∑

i=1

b(αi)k (5.6)

b(α)k =

1

max (K)

1

Nα

∑

k∈K

Nα∑

i=1

b(α)k (5.7)

where K denotes the set of considered prediction horizons, and Nα is the numberof estimated quantiles.

The problem with bias for quantile regression is the same as for point predictions.The model can have an arbitrarily large standard error and a bias of zero atthe same time. Take for simplicity the example of a quantile regression modelforecasting only two quantiles forming a central prediction interval. As long asthe estimated interval is central, it will have a mean bias of zero (Equation (5.6)).Sharpness is the ability to narrow down the probability mass. The measure ofinterest is the widths of the central prediction intervals:

δ(β)t,k = q

1−β/2t+k|t − q

β/2t+k|t (5.8)

For each prediction time, this gives as many sharpness measures as estimatedprediction intervals. Averaging can be done, first over time, then over horizons,and finally over prediction intervals:

δ(β)

k =1

N

N∑

t=1

δ(β)t,k (5.9)

δ(β)

=1

max(K)

∑

k∈K

1

N

N∑

t=1

δ(β)t,k (5.10)

δ =1

Nβ

Nβ∑

i=1

1

max(K)

∑

k∈K

1

N

N∑

t=1

δ(β)t,k (5.11)

Resolution is the ability to differentiate the predictions depending on the forecastconditions, which is in this case limited to be the point forecast. The resolutioncan be checked by analysing δ as function of the point forecast.

In [23], a simple and unique “skill score” is suggested to express both bias,

5.2 Evaluation of probabilistic forecasting 61

sharpness, and resolution. It is given by

Sc(

f , p)

=

n∑

i=1

(

ξ(αi) − αi

)(

p − ˆq(αi))

(5.12)

This skill score is an increasing measure, with 0 representing perfect quantiles.Again, since several horizons are modelled, it is useful to introduce an averagedversion of the measure. Averaging over the horizons writes:

Sc =1

Nk

∑

k∈K

Sc(f , p) (5.13)

where Nk denotes the number of elements in K.

5.2.0.1 Checking physical constrains

For using Equation (3.6), the assumption of a strictly increasing cdf defined inall the range of the random variable must be met. As quantiles only representa number of points (x, F (x)), assumptions must be made to have an estimateF of the cdf, F , as defined in Equation (3.2). An estimated set of quantileswill be denoted as the vector, qα consisting of as many elements, q(α), as in thevector α of the nominal proportions. The number of elements is n.

The physical constrain that Ew ∈ [0, 1] gives that P (Ew ≤ 0) = 0 and P (Ew ≤ 1) =1. This is physical knowledge that is added to the estimates from a statisticalmethod. This means that a reasonable collection of quantiles satisfies

q(αi) ∈]0, 1[, ∀αi ∈ α (5.14)

And for describing a strictly increasing cdf:

q(α1) < q(α2), ∀α1, α2 ∈ α, α1 < α2, α ∈]0, 1[n (5.15)

The smallest set that includes all acceptable quantile estimates is now defined:Q is defined as the set of quantile estimates that satisfy Equations (5.14) and(5.15).

Two different kinds of probabilistic forecasts will now be compared. One isbased on adaptive re-sampling. The other method is the adaptive quantileregression described in Section 3.2. In [10], an optimised C++ implementationof this method was developed, and this implementation has been used to do theestimations.


None of these methods necessarily yield quantile estimates that fulfil the require-ments of being increasing and within [0, 1] p.u. as given in Equations (5.14) and(5.15). As will be explained in Section 5.1, a procedure to ensure these proper-ties is applied before the quantiles are used. Therefore, it has also been decidedto apply this procedure on the estimates before evaluating their performances.However, it is interesting how well the methods meet these requirements. Ameasure for this is introduced. For a given collection of estimates, it equals thefraction that fulfils the requirements and thus by definition are in the set Q andis denoted γ:

γk(T ) :=1

#T

∑

t∈T

χQqαt,k (5.16)

where #T denotes the number of elements in T . Again, a mean value over thehorizons is defined.

γK,T :=1

NK

∑

k∈K

γk(T ) (5.17)

5.3 Using the Adaptive Quantile Regression al-

gorithm

In this section, some experiences with the Linux program by Christian VillerHansen provided by Enfor doing adaptive quantile regression will be explained.Even though the methods used for construction of the code is explained in [10],the program it self lacks documentation. Analysing the output, it was foundthat some precautions must be taken when using it. The explanatory variableused in the model is the WPPT predictions. The spline base functions that canbe chosen are natural cubic splines and periodic splines. Natural cubic splineshave continuity of C2 meaning that the second derivative is continuous, withsecond derivative equalling zero at the endpoints of the interval of interpolation.This option has been used. For information on splines, see i.e. [12].

5.3.1 Handling missing point predictions

In [10], handling of missing values in data is discussed. Elements assigned to“NaN” (IEEE standard “Not a Number”) are stated not to influence the algo-rithm. The program returns quantiles forming narrow distributions around zerowhen given a NaN as input. In the case of a missing explanatory variable datapoint, three approaches are obvious. One is to return NaN’s, another is to return

5.4 Comparing performance of different models 63

the marginal distribution, and the third would be to not to update the estimate.The result returned does not correspond to any of these suggestions. A pitfallwould be to interpret these as the distribution minus the mean value. When theexplanatory variable is missing, such a distribution can not be estimated. Sincethis behaviour is not documented in [10], it was found surprising, and not torun the risk that it would influence later estimations, these were decided to beleft out. The algorithm does not use time as an explanatory variable, and hencethe elements of missing data points can be removed, and when the estimationhas been carried out, the resulting data structure can be extended with NA’s(“Not Assigned”) at the places where the input data were removed. In practice,this is done in R by registering the positions and removing the NaN’s, creatingan empty data structure to contain all the estimates, present or not. Hereafterthe results from adaptive quantile regression are inserted in the data structureat the elements that do not match the registered elements from the NaN re-moval processes. This way, R will leave the estimates when point predictionsare missing as “NA”.

5.3.2 Initialisation period

First, 1000 hours (approximately 42 days) were used to initialise the adaptivequantiles algorithm. It was then seen to return zeros for all nominal proportionsthe first 500 hours for some of the estimates. Therefore, the initialisation periodwas changed to 1500 hours. The algorithm then succeeded to return non-zeroestimates for all prediction times.

5.4 Comparing performance of different models

In this section, performances of several different setups of adaptive quantileregression, and the provided probabilistic forecasting based on adaptive re-sampling will be compared. Since the latter does not include an estimate ofthe median, only the rest of the estimates of the 5th, 10th,... up to the 95thquantiles will be used in the comparison. Moreover, only models for horizons of1 to 37 hours will be used. This is because only they are of interest in the usedmarket models.

As explained in Section 3.2, adaptive quantile regression models the quantilesas functions of the explanatory variables with splines. Knots must thereforebe chosen, and bins can be used to ensure that the spline approximation relieson data in all of the data range. In [19], the best results are obtained with 5


bins and the knots placed on the bin barriers. The number of bins and theirplacement are however not well argued. Here, models with two to seven binswill be evaluated. In [10] models are built with knots placed in the middle ofthe bins. Both placing the knots in the middle and on boundaries of the binswere tried.

The placement of the bins has been done in two different ways. One is a naiveapproach where they are spread uniformly over the range of the explanatoryvariable. In the other approach, they are spread uniformly over the distributionof the explanatory variable, so that equally many observations will fall in eachbin. Take for instance the example of having four bins. Bins with boundaries atthe estimated quartiles then approximately contain one fourth of the data each.The distribution function of the explanatory variable was fixed at zero and oneat the boundaries of its range. The placement of the bins is based only on thedecision data.

The number of observations in each bin also has to be chosen. Bin sizes of300, 500, 1000, and 1500 observations were tried. These numbers are based onexperiences made in [19] and [11].

The C++ implementation moreover has a module called “Bin penalty” to obtaina more uniform distribution of the observations in each bin. This should decreasethe dependency on the placement of the bins. The influence of this module wasalso investigated.

The available options and general observations on their influence on the per-formance are listed in Table 5.3. All in all it gives a grid of models spanningover 6 · 2 · 2 · 4 · 2 = 192 model setups. An algorithm was written to performit, and the results for the best performing model for each number of bins willhere be given. The setups and summaries of the results are listed and plottedin Appendix B.

The setups of the best of these models for each number of bins are shown inTable 5.1 and measures of their performances are shown in the upper part ofTable 5.2. It is seen that for all numbers of bins, the placement of bins byquantiles of the explanatory variable performs better. As shown with plots inAppendix B, this effect is large.

The general picture is that the skill score is increasing and the absolute valueof the bias is decreasing with the number of bins. While the sharpness does notseem to be that depending on the number of bins, the ability to fulfil Equa-tions (5.14) and (5.15) certainly is. Even though five out of six of the bestmodels with different number of bins, this module did not prove to be bene-ficial. The mean of skill scores were slightly smaller with the module enabled


Table 5.1 Setup of the best fitted models with knots placed on either bin barriers orbin centres. Measures of their performances are given in Table 5.2

Numberof bins

Bin placementmethod

”Bin penalty”enabled

Knot place-ment

2 quantiles TRUE barriers3 quantiles FALSE middle4 quantiles TRUE barriers5 quantiles TRUE barriers6 quantiles TRUE barriers7 quantiles TRUE barriers

than without. Bin sizes of 1000 or 1500 have yielded the best results.

It was also tried to combine the two different placements by having one knotin the centre of each bin and one at each minimum and maximum of the pointpredictions, i.e. zero and one. In the setup of these models, knowledge from thefirst models were used. Since the skill score was generally increasing with thenumber of knots, only four to six bins were used in this second run. Moreoverall bins were placed by quantiles of the point predictions. Bin sizes were set to1000 and 1500 observations, and the “Bin Penalty” module was not used. Thesesetups and their performances are summarised in the middle part of Table 5.2.The last line in the table contains the evaluation of the forecasts based onadaptive re-sampling.

Even though the best results have been obtained with largest bin sizes tried,experiments of further increasing the bin sizes are worthless with this data. Asonly around 5000 data points are used for each model, larger bin sizes wouldjust result in bins that were not full, and therefore exactly the same results.

Using the skill score in Equation (5.12), the best tested model is found using 7bins placed using quantiles of the explanatory variable, with 1500 observationsin each bin, knots placed in the middle of the bins and at 0, 1, and “Bin penalty”disabled.

The information provided in Table 5.2 is narrowed down to averages over modelsof different horizons and their different quantile estimates. A lot of informationmay be lost in this, and therefore the quantities are also analysed as functions oftime or prediction horizon. Since independent models are used for each horizon,they could actually be chosen independently.

In the bias of the models from Table 5.2 are plotted as functions of nominal


Table 5.2 Performance measures for the different model setups.

Number of bins Bin size γ δ · 103 b · 103 Sc

AQR, Knots at bin centers/barriers2 1500 0.45 91.05 −122.43 −0.3663 1500 0.59 101.16 −2.42 −0.3564 1000 0.69 96.44 −38.56 −0.3375 1000 0.75 104.50 30.93 −0.3436 1500 0.79 100.65 7.01 −0.3377 1000 0.81 98.74 −21.37 −0.334

AQR, Knots at bin center and 0, 14 1500 0.72 98.78 51.03 −0.3435 1500 0.76 98.17 −27.58 −0.3336 1500 0.80 101.59 18.48 −0.3347 1500 0.82 99.20 −5.20 −0.331

Adaptive resampling- 0.70 98.79 −2.42 −0.340

proportion. Only the model with two bins is left out, because the bias of thismodel is so large that it made the plot worthless for comparing the rest of themodels. It is seen that especially three models perform well throughout theproportions, namely the model with 6 bins and the knots placed on the binbarriers, the one with 7 bins and knots placed in bin centres and at 0 and 1,and finally the model based on adapted re-sampling. It should be noticed thatcommon for the two models based on adaptive quantile regression is that theyhave bin sizes of 1500 observations. The best of the three models does notexceed a bias of ±3% at any nominal proportion.

In Figure 5.4 the bias of the same models are plotted against horizon. Thesame three models perform well again. Especially the one based on adaptivere-sampling lies constantly around zero.

As a last plot of performance measures, Figure 5.5 shows the average skill scoreversus horizon. The best models are hard to distinguish, but they perform betterthan the model based on adapted re-sampling throughout the considered hori-zons. The better of the curves are decreasing throughout the horizons withoutany sudden jumps. This is as expected.

The model based on adaptive quantile regression with 7 bins, bin sizes of 1500


0.2 0.4 0.6 0.8−0.

06−

0.04

−0.

020.

000.

020.

040.

060.

08

3 bins4 bins5 bins6 bins7 bins1st run2nd runARS

Nominal proportion

Bia

s,b(

α) k

Figure 5.3 Bias versus nominal proportion for the different probabilistic forecastingmodels.

0 5 10 15 20 25 30 35

−0.

15−

0.10

−0.

050.

000.

050.

10


Horizon [hours]

Bia

s,b k

(α)

Figure 5.4 Bias versus horizon for the different probabilistic forecasting models.

observations has yielded the highest average skill score and the highest abilityto fulfil requirements formulated for the sets of quantiles. No specific problemswith bias or skill score had been noticed at some horizons or at some nominal


0 5 10 15 20 25 30 35

−0.

45−

0.40

−0.

35−

0.30

−0.

25


Horizon [hours]

Sc

Figure 5.5 Skill score versus horizon for the different probabilistic forecasting models.

proportions, and hence this model is chosen to be used.

From now on, when referring to the estimated cdf, FEw(x), of the production,estimates from this model will be meant.

Table 5.3 Options varied in the configuration of the adaptive quantile regression mod-els and observations on their influence on the performance.

Option observation remark

Number of bins 2,3,4,5,6,7 More bins increase Sc

and decrease b.Bin size 300,500,1000,1500,2000 1000, 1500, and 2000

perform well.Bin placement Uniformly and by quan-

tilesBy quantiles much bet-ter.

Knots placement At bin barriers, in themiddle of bins, and inthe middle of the binsplus at 0, 1

At bin centres and at0, 1 better.

Bin penalty module enabled, disabled Not important.

5.5 Adaptive estimation of the autocorrelation 69

5.5 Adaptive estimation of the autocorrelation

The algorithm used for estimation of the autocorrelation of the random vari-ables transformed into the normal domain is developed in [24]. The procedureconverges to the exponential smoothing algorithm known from[17]. Let X(t)

denote the vector of all observations of electrical energy generation up to time t

transformed into the normal domain. The unbiased estimate of the covariancematrix is

Σt =

(

t − 2

t − 1

)

Σt−1 +

(

1

t − 1

)

X(t)X(t)T(5.18)

An adaptive estimate of the covariance matrix is then given by the exponentialsmoothing algorithm:

Σt = λ · Σt−1 + (1 − λ)X(t)X(t)T(5.19)

where λ is the forgetting factor. It is chosen to 0.95 and has not been optimised.

Since the distribution of X depends on the estimate of the distribution of thepower production, it is not perfectly the standard normal distribution (i.e. theinverse of the estimated function of data is not perfectly uniformly distributed).Therefore, the used estimate, Σt, is re-calibrated by element-wise division withthe outer product of the diagonal elements.

Because of this recursive formula, the generation of the scenarios is an algorithmthat is started at the beginning of the decision set and continued throughoutthe decision set and the test set.


Chapter 6

Example on simulations and

optimisation

In this chapter the focus is on one specific period, March the 22th 2007 from6 am to noon. This is done to illustrate the methods used in the optimisationby a case study. The period have been chosen by searching for what could beconsidered a “typical” period on the modelled six hour market. The criteriawas that mean and variance of the production in the period should be close tothe average of the means and variances of all blocks the trading periods on themodelled 6 hour market. The chosen six hour block was the only one with arelative difference of both mean and variance of less than 5% from the overallones.

6.1 The example period

First, the data from the period will be presented, and the storage devices willbe exemplified with two different contracts. One made up specifically for illus-tration, the other is the WPPT model.

In Figure 6.1, probabilistic forecasts are shown along with power measurements,point forecasts given by the WPPT model, and the estimate of the expectation

72 Example on simulations and optimisation

calculated from the quantile regression. Even though the considered marketscenario only focuses on the trading decision in the 2 to 7 hours horizons, the 1hour horizon is also of interest in the simulations because of the interdependencestructure in the power variables. It is seen that the WPPT model overestimatesthe production throughout the 7 considered hours. The mean of the predictedprobability functions are very close to the WPPT predictions, and it also over-estimates the production throughout the period.

0.1

0.2

0.3

0.4

0.5

0.6

10%20%30%40%50%60%70%80%90%WPPT

meas.

06:00 07:00 08:00 09:00 10:00 11:00 12:00

Time

Pow

er[p

.u.]

E(Pt)

Figure 6.1 Probabilistic forecasts issued on 2007-10-19 at 11 Pm.

First, an example with a contract constructed to show the differences betweenthe storage devices is shown. The contract and the actual production are bothshown in the plots in Figure 6.2. Moreover the difference, E∼, of the two isshown (see Equation (2.26)). −E∼

t is the demand for Est to balance out the

production and the contract at time t. In Figure 6.2(a) the storage levels areshown versus time while Figure 6.2(b) shows the deviations between the contractand the deliveries for the three different systems.

All storage devices start with the storage level 0.25 p.u. The first two hours showthe same limitations of the three devices. Both the ideal EES and EES1 balanceout the production lack compared to the contract, but with the difference thatthe storage level of EES1 drops more than the one for IEES. This is because ofthe conversion loss in EES1 that IEES does not have. In the same period, EES2fails to balance out the contract and the delivery. EES2 has a moreover than theconversion loss of EES1 the limitation that it can at maximum deliver 0.05 p.u.within an hour. This is what EES2 does in the first two hours. Mathematically,IEES and EES1 are described by the third line in Equation (2.27), while EES2

6.1 The example period 73

is described by line 1.0.

00.

20.

40.

60.

8

EwEcL, IEESL, EES1L, EES2E~

06:00 07:00 08:00 09:00 10:00 11:00 12:00

Horizon

Ener

gy

[p.u

.]

(a) Storage levels.

0.0

0.2

0.4

0.6

0.8

EwEcEd, IEESEd, EES1Ed, EES2E~

06:00 07:00 08:00 09:00 10:00 11:00 12:00

HorizonE

ner

gy

[p.u

.]

(b) Deviance between contract and deliveredamounts of energy.

Figure 6.2 Simulations of the systems with different EES given the shown contract.

From 8:00 to 9:00 Am, the situation is the same for IEES, since this EES becauseof its perfect conversion efficiency still has sufficient storage to balance out thecontract and the delivery. EES1 runs empty, whereas EES2 manages to supplythe needed energy to balance out contract and the contract. In other words it isduring this hour and advantage of EES2 compared to EES1 that it was not fastenough during the first two hours. It is a little hard to see from the small plot,but the slopes of the storage level curves for IEES and EES2 differ according totheir different discharging efficiencies. IEES is again described by the third lineof Equation (2.27), while EES1 is described by line 2, and EES2 by line 3.

From 9:00 to 11:00 Am the situation is opposite to the situation seen during thefirst two considered hours. There is a large need for charging, and EES2 fails toabsorb all of the production surplus. The difference in the slopes of the storagelevel of IEES and EES1 shows the difference in their charging efficiencies. IEESand EES1 are described by line 3 in Equation (2.27), while EES2 is describedby line 5.

During the last considered hour, the need for charging is more than 0.4 p.u.(E∼). IEES charges all of it, while EES1 runs full, since it has a capacity limitof 0.5 p.u. EES2 does not run full but is too slow. IEES is described by line 3in Equation (2.27) while EES1 is described by line 4, and EES2 is described byline 5.

In other words IEES succeeds at keeping Ed at zero throughout the period. This


is equivalent to the fact that it is described by line 3 in Equation (2.27). From8:00 to 9:00 Am, ESS1 is in a small power shortage, and it fails to absorb theproduction surplus in the last considered hour resulting in a positive Ed. EES2only succeeds at keeping Ed at zero between 8:00 and 9:00 Am. The exampledemonstrates the very different behaviour of the three devices, and it also showsthat the different limitations are correctly implemented.

In stead of the quite naive contract shown in Figure 6.2 the WPPT predictionswill now be used as contracted energy. With this contract, the storage levelsand the energy flows in the systems with the three defined storage devices areshown in Figure 6.3.

−0.

10.

00.

10.

20.

30.

4

EwEcL, IEESL, EES1L, EES2E~

06:00 07:00 08:00 09:00 10:00 11:00 12:00

Time

Ener

gy

[p.u

.]

(a) Storage levels.

−0.

10.

00.

10.

20.

30.

4EwEcEd, IEESEd, EES1Ed, EES2E~

06:00 07:00 08:00 09:00 10:00 11:00 12:00

Time

Ener

gy

[p.u

.]

(b) Contract and deviations from this.

Figure 6.3 Simulations of the systems with different EES with WPPT predictions ascontracts.

First of all, the deviations between the contract and the electricity are generallysmaller with this contract. Only at from 8:00 to 9:00 Am the deviation is larger.On the other hand the contract is consequently too large causing all three storagedevices to empty.

As described in Section 2.3.1, the revenue obtained from a given series of con-tracts and system operation, depends on both Elspot prices, and on up anddown regulation costs. For the considered period, these prices are shown inFigure 6.4. The Elspot price is between 24,40 e/MWh and 25,97 e/MWh. Itis increased until 9 Am, from where it slightly decreases for the rest of the pe-riod. The up and down regulation prices reflect that the system balance changeswithin the period. From 7 to 10 Am, the up regulation price is higher than theElspot price. Between 10 and 11 Am, they equal all three of them, and at from11 Am to noon, the down regulation price is lower than the Elspot price. This

6.1 The example period 75

2025

3035

07:00 08:00 09:00 10:00 11:00 12:00

Time

e/M

Wh

πc

π−

π+

π−

π+

Figure 6.4 Elspot and regulation prices and predictions for the chosen 6 hour period.

reflects that the physical system has been up regulated from 7 to 10 Am, andproducers that deliver less than contracted must pay a higher regulation pricethan the Elspot price for the imbalance that they impose. Between 10 and 11Am, the system was in balance, and between 11 Am and noon the system wasdown regulated, and the price payed for the production delivered in surplus tocontracts was lower than the Elspot price. Between 8 and 9 Am the biggest devi-ation between Elspot and regulation prices is observed; this is of 10,42 e/MWhcorresponding to a deviance of 24% from the Elspot price. The predictions ofthe regulation prices are also shown.

With these prices, the expected revenues using the different storage devices iscalculated and shown in Figure 6.5. In Figure 6.5(a), the true regulation pricesare used, while the predicted regulation prices are used in Figure 6.5(b). Thedifference between the two is surprisingly small at a first glance. To understandthis, one must return to Figure 6.3(b) and 6.4. First of all, the deviances fromthe contracts are rather small and only different from zero from after 9am forEES2 and after 10 Am for IEES and EES1. Moreover, the up regulation pricehappens to be quite well predicted between 9 Am and 10 Am, so in this intervalthe regulation revenue for EES2 will be similar in the two cases.

76 Example on simulations and optimisation−

20

24

68

10

RcR, IEESR, EES1R, EES2Rd, IEESRd, EES1Rd, EES2

06:00 07:00 08:00 09:00 10:00 11:00 12:00

Time

Rev

enue/

p.u

.[e

/M

Wh]

(a) Regulation prices known.

−2

02

46

810

RcR, IEESR, EES1R, EES2Rd, IEESRd, EES1Rd, EES2

06:00 07:00 08:00 09:00 10:00 11:00 12:00

TimeR

even

ue/

p.u

.[e

/M

Wh]

(b) Regulation prices predicted.

Figure 6.5 Expected revenues using different EES using WPPT predictions as con-tracts. A small difference between the two cases is seen for IEES EES1from 10 Am to noon.

6.2 Simulations

For generating scenarios of the production, an estimate of the correlation of theestimated correlation matrix for the normally distributed random variables isneeded. This estimate is plotted in Figure 6.6. As expected, the autocorrelationis generally decreasing with the prediction horizon but everywhere positive. Itis noticed that the structure of the autocorrelation at a quick glance looks likethe one of an AR(1) process. Such an autocorrelation function is exponentiallydecaying with the horizon.

Together with the estimated probability distributions this interdependence struc-ture is now used to simulate 10,000 scenarios for this period. 50 of the simulatedscenarios are shown in Figure 6.7. It is seen that most of the production sce-narios are larger than the actual production. This is in accordance with themarginal distributions plotted in Figure 6.1. The correlation structure are alsoreflected in the simulations. Once a scenario is relatively large (or small) it hasa tendency to stay at a relatively high (low, respectively) level. In other wordsthe distribution of the future production conditioned on the present productionis positively correlated with this. It does therefore not equal the marginal dis-tribution of the future production. This is the modelling of the inertia in thewind power.

A plot of the scenarios as the one in Figure 6.7 is very illustrative and easily

6.2 Simulations 77

1

2

3

4

5

6

7

1 2 3 4 5 6 7

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Hori

zon

[hours

]

Horizon [hours]

Figure 6.6 Image of the correlation matrix for the normally distributed variables usedto simulate power production for the first 6 hours on March the 22nd 2007.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

predictionmeasurementsimulations

0.1

0.2

0.3

0.4

0.5

0.6

0.7

predictionmeasurementsimulations

06:00 07:00 08:00 09:00 10:00 11:00 12:00

Horizon [hour]Horizon [hour]

Pow

er[p

.u.]

Pow

er[p

.u.]

Figure 6.7 50 scenarios of the production in the considered 6 hour period. The sim-ulated production at 6:00 Am is also shown because of the correlation be-tween this and future production.

understandable at a first glance. The overview of the density of the simulationsis however fuzzy. In stead, histograms of the production level versus time are


shown in Figure 6.8. This representation, on the other hand, has the disadvan-tage that it does not reflect the interdependence of production. Since it onlyreflects marginal distributions, it should be very similar to Figure 6.1

1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

0

1

2

3

4

5

6

7

Horizon [hour]

Pow

er[p

.u.]

Figure 6.8 Tracking the probability density function of the production through time.

Decisions will be based on these scenarios, and it is therefore crucial that theyare reliable. They are as explained based on the estimated distributions of theproduction, which were concluded to be sufficiently reliable in Section 5.4. Now,it must be checked that the scenarios are in accordance with the predicted distri-butions. This is done by observing the scenarios transformed via the estimatedcdf[24]. These should be uniformly distributed between 0 and 1. For all of the7 horizons together, this is done in Figure 6.9. The distribution indeed lookslike realizations of uniformly distributed variables.

The only thing that has not been investigated is now in the check of the reliabilityof the scenarios is the interdependence structure. This will not be done in thiswork.

Since an EES is a summation over the prediction errors under certain limitations,a simulated sum of the these are of interest. Using the WPPT predictions,the prediction errors with these 10,000 production scenarios are summed. Theempirical distribution based on these 10,000 simulations is shown in Figure 6.10.This corresponds to using an ideal storage starting at storage level zero butwithout limits on storage level. A physical interpretation of the point zero is astarting point which is sufficiently high for the storage never to empty. Since alldeviations are equalled out by the storage, the revenue will in this case equal

6.2 Simulations 79

0.0 0.2 0.4 0.6 0.8 1.0

050

010

0015

0020

0025

0030

0035

00

Power [p.u.]

Counts

(of70,0

00)

Figure 6.9 A histogram of the production scenarios transformed by the estimated cdf

of the production. These should be close to uniformly distributed.

the sum of the spot prices times the contracted energy.

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0

010

020

030

040

050

060

0

End storage level [p.u.]

Counts

(of10,0

00)

Figure 6.10 Histogram of the sum of errors using the WPPT predictions on 10,000simulated production scenarios. Compare with Figure 6.7.

One may have a good idea about the mean of such an empirical distribution only


from looking at the underlying marginal distributions shown in Figure 6.1. Butthe length of the tails are heavily influenced by the interdependence structureshown in Figure 6.6. The correlations between the production at different hori-zons contribute to the need of storage capacity, since they correspond to rigidityin the production over time. This is seen from Equation (3.13). Because of theinterdependence of the errors, the variances of the sums of the errors are largerthan the sums of the variances of the errors. Moreover, the distribution is pos-itively skewed. This is because the predicted distribution has a mean smallerthan 0.5 p.u. The extreme realizations larger than expected can then deviatemore from the expectation than the extremes smaller than the expectation.

6.3 Bussiness as usual simulations

Now, the system is operated with storage devices using the WPPT model forfixing the contracts. Figure 6.11 is a plot of the pdf of the storage levels versustime using the three example storage devices. Since IEES has unlimited capacity,the range of this plot has been decided by the extremes of the storage levels.The bin sizes in the evaluation of the histograms have been fixed at 0.02 p.u.,since a variation in this would make the densities in the first plot incomparableto the two others.

The density of the storage level concentrates strongly at zero. Because of theinterdependence structure of the errors, scenarios of negative errors (productionlower than estimated expectation) result in storage levels that are trapped atzero. On the other hand, scenarios with positive prediction errors lead to a largespread on the storage level.

For EES1, the density spreads rapidly to span from minimum to maximumcapacity, and the interdependence structure of the errors make the storage levelstrap at both zero and half a p.u. It is seen that more density concentrates atzero than at maximum capacity level.

The WPPT prediction is seen from Figure 6.1 to be very close to the mean ofthe estimate of the pdf obtained with adaptive quantile regression on whichthe scenarios are based. The production scenarios therefore also have a meanvery close to the WPPT predictions. Assume the predictions were unbiased, i.e.E (E∼

τ ) = 0 for all τ in the prediction interval. The probability that EES1 isempty at time t + k given that it is at L0 at time t is given by the definition in

6.3 Bussiness as usual simulations 81

2 3 4 5 6 7

0.0

0.5

1.0

1.5

2.0

0

2

4

6

8

10

2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0

2

4

6

8

10

12

2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0

2

4

6

8

10

12

Horizon [hours]

Sto

rage

level

[p.u

.]Sto

rage

level

[p.u

.]Sto

rage

level

[p.u

.]

Figure 6.11 The pdf of the storage levels versus time using WPPT predictions ascontracts for each of the three EES. From above, the histograms are forIEES, EES1, and EES2. Notice that the storage level axis in the firstexample goes much higher than in the two latter. This is because IEEShas unlimited capacity.


Equation (2.14) as

P (Lt+k = 0|Lt) = P

Lt +∑

τ∈1,...,k

∇Lt+τ = 0

(6.1)

But as F∇Ltis a complicated function of Lt−1 this probability is hard to calcu-

late. For the purpose of understanding, let EES1 be approximated by EES1∼

similar to EES1 except that it has no limits on the capacity, meaning thatLEES1∼

min is infinitely small and LEES1∼

max is infinitely large. The greater the in-terdependence structure of the errors, the better probability that the storagelevel in EES1∼ is smaller than or equal to zero is to the probability that EES1is empty. This approximation is much more easily calculated, since the level isjust a sum of the contract errors multiplied with the corresponding conversionefficiency (see Figure 2.5 and Equation (2.16) if in doubt):

P(

LEES1t+k = 0|LEES1

t

)

≈ P(

LEES1∼

t+k ≤ 0|LEES1∼

t

)

= P

(

LEES1∼

t + ηEES1ch ·

∑

τ∈1,...,k:E∼t+τ >0

E∼t+τ

+1

ηEES1dch

·∑

τ∈1,...,k:E∼t+τ <0

E∼t+τ ≤ 0

)

As E∼ was assumed to be distributed with mean zero and not skew, there willbe equally many positive and negative terms. But because ηEES1

dch is smaller thanone the negative terms will be numerically larger than the positives one. For anyk ≥ 1, the expectation of the sum is smaller than LEES1∼

t , and the expectationof the sum will go to minus infinity as k goes to infinity. So, even though thesum of the prediction errors is a martingale, the storage level in ESS1∼ is not.Even though EES1 is certainly different from EES1∼, they share the propertythat a positive E∼ = a leads to a numerically smaller change in the storagelevel than E∼ = −a.

The distribution of the errors is positively skewed (see Figure 6.10.). This meansthat the median is smaller than the mean, and more scenarios are therefore ofnegative prediction errors than positive. This also leads to a higher concentra-tion of storage levels at Lmin

ees than at Lmaxees .

Two reasons for the skew distribution of the storage levels in EES1 have beengiven. To test the effect of them independently, one could simply pass theerrors through an EES like EES1 but with lossless conversions. This will notbe carried out here, since the most important observation is that the losses onboth charging and discharging generally tend to empty the storage.

6.4 Optimisation 83

The distributions of the storage level in EES2 (last plot in Figure 6.11) differsfrom the others by the extremes diverging as staircases. This is because of thelimited charging and discharging speeds. Again, probability mass concentratesat the extremes but less than for EES1. This is because it now takes severalsucceeding prediction errors of equal sign to drive the system to an extremewhile for EES1 just one prediction error can completely fill or empty the storage.This is the aspect that was also discussed in relation to the example shown inFigure 6.3 where the limitation to balance out contracted and delivered energyduring one hour had the advantage in the succeeding hours that the storage wasnot emptied. This could of course also be when having need for storing. Theadvantage would then be that a few prediction errors would not fill the storage.The regulation costs are then spread to more delivery hours, decreasing the riskof having large regulation costs concentrated in one delivery hour.

The “staircase” behaviour in the evolution of the pdf of the storage level inEES2 also reflects that the energy balance depends on the flow direction. Eventhough or exactly because E

s,minEES2 = −E

s,maxEES2 , the storage level discharges faster

than it charges.

These examples of using EES2 pose the question if, when using the passivestorage strategy, one could actually gain of a physical limitation on the charg-ing/discharging rates. This is a heavy argument against the passive operationstrategy and calls for control strategies of the storage level.

6.4 Optimisation

The contracts are now optimised using the optimisation criteria given in Sec-tion 2.6. The optimised contracts are plotted in Figure 6.12. The predicteddistributions of the storage levels with the optimised contracts are seen in Fig-ure 6.13 for the case of maximising the expected revenue and in Figure 6.14 forthe case of minimising the expected regulation costs.

For carrying out the optimisation, the function called nlminb (Nonlinear Mini-mization subject to Box Constraints) from the stats package in R is used. Itsupports the needed constraints and converged faster in test runs than otheroptimisation algorithms in R, such as optim.

Using the IESS, basically energy should be stored if a higher price is known tobe offered later. In Figure 6.4 the highest spot price was seen to be available forenergy contracted to be delivered between 8:00 and 9:00 Am. This is why noenergy is contracted to be delivered before this period. And during this period

84 Example on simulations and optimisation0.

00.

20.

40.

60.

8

WPPTIEESEES1EES2max revenuemin reg. costs

07:00 08:00 09:00 10:00 11:00 12:00

Time

Ec

[p.u

.]

Figure 6.12 WPPT and the optimized contracts.

the discharge of the storage device is a balance of selling as much as possible atthis highest price and not expecting to pay too much regulation costs.

Minimising regulation costs with IEES does not make any sense since contractsof zero energy to be delivered is a global minimum. The EES would then juststore all the production.

The maximisations of the expected revenues for EES1 and EES2 yield quitesimilar results. The contracts for EES2 are a little greater than those for EES1until 10:00 Am, and from 11:00 Am to noon, the contract for EES2 is a littlegreater then the one for EES1.

The minimisation of the regulation costs yield contracts following the WPPTprediction more closely. The optimisation strategy of minimising regulationcosts is about issuing contracts that can with high certainty be met. The waythat this is accomplished can be seen in Figure 6.15(a). For EES2, the expectedstorage level is constantly at 0.25 where is starts. For EES1 it is done by makingtoo large a contract for the first hour and then gradually fill up the storage.But the charging and discharging efficiencies are, on the other hand, expectedto equal this out to some extend. In stead, the reason is likely to be found inthe large spread of the production scenarios in the last part of the consideredperiod. Since the predicted down-regulation cost is higher than the predictedup-regulation, the storage should be able to absorb surplus production later.And in the beginning the spread is much smaller, so the expected up regulation

6.4 Optimisation 85

cost to be paid for planning on decreasing the storage level is small.

2 3 4 5 6 7

0.0

0.5

1.0

1.5

0

10

20

30

40

2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0

10

20

30

40

2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0

10

20

30

Horizon

Sto

rage

level

[p.u

.]Sto

rage

level

[p.u

.]Sto

rage

level

[p.u

.]

Figure 6.13 The pdf of the storage levels versus time in the example period for thethree example storage devices where the contracts have been optimised bymaximising the expected revenue for the period.

This is what is seen in the second plot in Figure 6.14. Even though around 20%


(the pdf takes an average value of approximately 10 in an interval of width 0.02p.u) of the realisations result in a storage level lower than 0.02 and likely an upregulation cost, it makes it possible to avoid down regulation later in the periodwhere the probability masses concentrate at 0 and 1.

2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

2

4

6

8

10

12

2 3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0

2

4

6

8

10

12

Horizon

Sto

rage

level

[p.u

.]Sto

rage

level

[p.u

.]

Figure 6.14 The predicted pdf of the storage level versus time in the example periodfor EES1 and EES2 where the contracts have been optimized by minimis-ing the expected regulation costs.

6.5 Realization 87

0.1

0.2

0.3

0.4

0.5

0.6

IEESEES1EES2max revenuemin reg. costs

07:00 08:00 09:00 10:00 11:00 12:00

Time

E(L

|Ft c−

1)

[p.u

.]

(a) Expected L.0

510

1520


07:00 08:00 09:00 10:00 11:00 12:00

Time

Rev

enue

[e/p.u

.](b) Expected R.

Figure 6.15 Results for the six hour example with different contract strategies andEES.

6.5 Realization

The outcomes of the different trading strategies are now evaluated. All resultsare listed in Table 6.1. The table consists of two columns. The first columnexpresses the expected revenue estimated from the scenarios based on informa-tion available one hour before gate closure, i.e. at 6:00 Am. In the calculationof these the naive prediction of the regulation costs is used. The other columnis the revenue calculated based on the true production of electrical energy andthe true regulation costs for the given period.

Moreover, the “Perfect knowledge” without storage setups need some comments.The first one, just called “Perfect knowledge” is the value of the truly generatedenergy, first estimated from the generated scenarios, then evaluated with thetrue production. The “Realised” revenue in the perfect knowledge setup istherefore just the Spot value of the true production.

The row called “Perfect knowledge, ref” is a more correct reference for thecomparing with the storage setups. The storage devices are half full, i.e. at thelevel of 0.25 p.u. at the beginning at the period, so maximising the revenue forthese setups allow the possibility of selling this energy as well. This setup wasimplemented by defining a storage device with perfect discharging efficiency,and with infinitely large maximum discharge speed but without the ability tocharge (Es,min

ees = 0). Then, the contract was optimised, maximising the revenue.It should be noticed that this setup is neither a perfect reference, since EES1


and EES2 have imperfect discharging efficiencies. This means that the startingstorage levels of 0.25 p.u. in these setups only correspond to the starting storagelevel times the discharging efficiency, i.e. 0.25p.u. · 0.85 = 0.2125p.u.

0.0

0.1

0.2

0.3

0.4

0.5

0.6 IEES

EES1EES2max revenuemin reg. costs

07:00 08:00 09:00 10:00 11:00 12:00

Time

E(L

|Ft c−

1)

[p.u

.]

(a) Storage levels.

05

1015

20


07:00 08:00 09:00 10:00 11:00 12:00

Time

Rev

enue

[e/p.u

.]

(b) Revenues.

Figure 6.16 Realised results for the six hour example with different contract strategiesand EES.

The general picture is that the realised revenues are lower than expected. Thisis because the power predictions for the period are considerably higher thanthe actual production. The realised revenues are plotted versus time in Fig-ure 6.16(b).

Since the example has been analysed throughout the steps in the optimization,the sums of the revenues will not be thoroughly discussed here. But an impor-tant observation is that there is no clear tendency that EES1 performs betterthan EES2.

6.5 Realization 89

Table 6.1 Results of different setups in the example 6 hour period. The unit is e perhour per installed MWh.

Setup Expected Realised

No storagePerfect knowledge 8.11 6.00Perfect knowledge, ref 9.00 5.78WPPT as Ec 8.43 5.78EFEw

(Ew) 8.43 5.88

IESSWPPT as Ec 7.45 7.07E (R) maximised 7.56 6.83

EES1WPPT as Ec 8.15 6.91E (R) maximised 8.69 6.67E (C) minimised 7.98 6.82

EES2WPPT as Ec 8.23 6.86E (R) maximised 8.75 6.50E (C) minimised 8.22 6.86


Chapter 7

Results

This chapter is a presentation of the simulations over the full simulation periodfor both 6 and 24 hour runs. For an overview of the period that is analyzed,return to Figure 4.10. The period consists of 384 full days.

In Table 7.1 total revenues are shown when no optimization is performed onthe contracts issued. “Perfect knowledge” refers to the case where perfect pointpredictions of the production are available, and the contracts perfectly matchthe production, ie. the Elspot price is obtained for all of the production.

Table 7.2 presents an overview of the total revenues for all of the optimizedmodels. EF (R) refers to the case where Ec has been optimized for each delivery

period as in Equation (2.31). EF (C) refers to the case where Ec has beenoptimized as in Equation (2.36).

The results are divided into two times two culomns. The first two columnsare the mean revenue obtained with the optimised contracts with 1000 newsimulated production scenarios. The last two columns are the revenues when thesystem is run with the real production data. The expected revenues are shownfor controlling that the optimisation has worked and to see if the scenarios havebeen representative for the real production process.

92 Results

Table 7.1 Revenues without optimisation. The revenues are normalised to by the rev-enue of selling all of the production at Elspot and not facing any regulationcosts.

Contracted energy 6 hours 24 hours

No storage

Perfect knowledge 1.000 1.000WPPT 0.964 0.936E

F(Ew) 0.965 0.936

IEES

WPPT 0.969 0.957E

F(Ew) 0.979 0.957

EES1

WPPT 0.965 0.936E

F(Ew) 0.966 0.936

EES2

WPPT 0.964 0.937E

F(Ew) 0.965 0.937

Table 7.2 Results of different setups throughout the test period. The revenues arenormalised to by the revenue of selling all of the production at Elspot andnot facing any regulation costs. “R.p.” is an abbreviation for “regulationprices”.

Setup Expected Realised

6 hour 24 hour 6 hour 24 hour

IEES

EF(R) maximised, r.p. known 1.214 1.205 1.028 1.177

EF(R) maximised, r.p. pred. 1.179 1.162 1.017 1.140

EES1



EF(C) minimised 0.939 0.936 0.949 0.934

EES2



EF(C) minimised 0.983 0.943 0.969 0.938

Chapter 8

Discussion

In this chapter, the results presented in Chapter 7 and aspects about the usedmethods will be discussed.

All simulations with storage devices have been started with storage levels at 0.25p.u. This accounts for some extra value that in these simulations. On the otherhand, they can also end up at a positive storage level that could have been soldto increase the revenue. However, for EES1 and EES2 this can at maximumaccount for 0.5 p.u. Sold at the highest observed Elspot price in the period, thevalue of this is approximately 100 e per MWh installed. As the results are allrevenues of more than 60,000 e per MWh installed, this will simply be ignoredfor EES1 and EES2.

8.1 Simulations without optimisation

The reference results contain information about how much of the revenue that islost due to regulation costs when using a state of the art prediction model of theproduction. This revenue is only 3.6% less than when using contracts exactlymatching the production. First of all it must be remembered that the WPPTmodel has an MAE of approximately 4.2% for 6 hours periods, and approxi-mately 6.0% for 24 hours periods (see Table 4.1), although these are calculated

94 Discussion

for all of the data and not just the simulation period. Within the simulation pe-riod, the distribution of the regulation state on the regulation market is shownin Figure 8.1. The regulation state should be given by Equation (2.11) butas discussed in Section 4.5, these rules are not perfectly reflected in the data.In stead, the cases of down regulation have been identified as the when thedown-regulation price is more than 0.1 e smaller than the Elspot price, the up-regulation cases as when the up regulation price is at least 0.1 e greater thanthe Elspot price, and the “system in balance” cases as the rest.

down regulation system balance up regulation

050

010

0015

0020

0025

0030

00C

ounts

of9216

Figure 8.1 The distribution of the regulation state on the regulation market within thesimulation period.

It is seen that the most common situation is that no regulation cost is present,and up-regulation is a little more common than down regulation. The probabil-ity of paying regulation costs is the sum of the probabilities of the coincidencesof the producers imbalance and the system imbalance colliding:

P(

Cd > 0)

= P(

Ed > 0)

· P (C- > 0) + P(

Ed < 0)

· P(

C+ > 0)

(8.1)

The expectation of the regulation costs can, if independence is assumed betweensystem balance and producer’s delivery balance, be estimated by decomposinginto the two situations of positive regulation costs:

E(

Cd)

= P(

Ed < 0)

· E(

C+)

+ P(

Ed > 0)

· E (C-) ≈ 8e/MWh (8.2)

The mean Elspot price for the simulation period is 32.1 e/MWh. Combinedwith the MAE of only 4.2% or 6.0% this gives a very little expected regulationcost loss. In plus, the bias of the WPPT predictions is positive, giving that the

8.2 Simulations with optimisation 95

producer delivery balance collide with the system balance in less than 1/3 of thetime (again, the system is more frequently up-regulated than down regulated).Because of this, it is not necessarily better to have a better prediction model,which the interpolation of the estimated distribution function could represent.This is actually the problem leading to the optimal contracts without an EESfrom equation (2.35).

Using an EES with the WPPT predictions as contracts sell seems to have littleeffect. Only IEES increases the revenues a little. For this EES it must be takeninto account that the positive bias in the point predictions leads to a storagelevel in the end of the simulation period of 68.2 p.u. and 73.0 p.u. for the 6and the 24 hours periods, respectively. If these were traded at the mean Elspotprice within the period that would lead to additionally 3.3% and 3.5% of therevenue obtained when no regulation costs are faced, respectively. Then thetotal revenue is very close to the one using perfect contracts. It is noticed thatEES1 and EES2 perform about equally well. Actually EES2 yields a little higherincome on the 24 market, which may come as some surprise, since EES2 is asEES1 but further limited.

8.2 Simulations with optimisation

The overall pictures of the results are quite different for the two different opti-misation strategies.

8.2.1 Maximisation of expected revenues

When using the true regulation prices in the maximisation of the expectedrevenue, the use of IEES results in an expected increase of revenue of 21.4%and 20.5% for the two markets. This mainly reflects variations in the Elspotprice, since IEES provides a lossless way to store the generated electrical energywhenever a higher price is known to come later in the delivery period. Thisbehaviour was illustrated in the case study in Section 6.4. For the 6 hoursmarket, it realises to a gain of only 2.8%. But compared to using the WPPTmodel as contracts, it is still a gain of 7.4%. For the 24 hour market, the realisedrevenue turns out to be closer to the expected. Here the gain is 23.0%.

When predicting the regulation prices, the picture is the same, just with lowervalues. The expected revenue is larger on the six hour market, but turns outto be higher at the 24 hours market. Compared to using WPPT predictions

96 Discussion

as contracts, this still accounts for an increase in revenues of 5.4% and 19.1%,respectively.

It seems that the possibility of storing generated electricity for higher contractprices is more stable, the longer the horizon. It is surprising to see results thatis that are 15% and 14% lower than expected on the six hours market, usingthe perfect and the naive regulation prices predictions.

Using EES1 and EES2 the pictures are a little different. First of all, all revenues- expected and realised - are lower than when using IEES. And for EES2 theyare all lower than when using EES1. When using EES1, the mean realisedrevenue of these four setups is 9.7% lower than when using IEES. For EES2 thisnumber is 10.6%. So for the used values of limitations, the main effect is seenon limiting the storage capacity and using losses on the conversions, and not onthe limitations on the conversion speeds. However, since these limitations haveonly been tried with single values, a sensitivity analysis cannot be performed.

In one setup EES1 and EES2 behave differently than IEES. The realised rev-enues with predicted regulation prices are largest on the six hour market. Sinceit is only the case when predicting the regulation prices, it certainly calls forbetter prediction models for the regulation prices.

With EES1 and EES2, The expected gains on the 6 hours markets comparedto using WPPT predictions are 16.3 and 12.3%, respectively. On the 24 hoursmarket they are 5.6 and 4.1% respectively. The general result is that the ex-pected increases in revenues using the maximisation of the expected revenuestrategy is that the more limitations on the EES, the less the gain.

8.2.2 Minimisation of expected regulation costs

This optimisation strategy yields lower expected revenues than maximisationof the expected revenue for all of the four cases. The mean of the expectedrevenues are in average 8.8% lower than those where the expected revenue hasbeen maximised, and regulation costs predicted.

The realised revenues are generally also smaller using this strategy. Though,using EES2 it is slightly larger on the six hour market, and the difference betweenthe revenues are smaller for the two strategies on the 24 hours market.

This strategy is, as also discussed in Chapter 6, very conservative, and the differ-ences between expected and realised revenues are smaller than when maximisingexpected revenue. The obtained results are very close to using

8.2 Simulations with optimisation 97

8.2.3 General issues in the simulations

It is seen that the expected revenues are generally higher than the realised ones.This must be because the scenarios on which the decisions have been made can-not be considered reliable. As explained in Chapter 5, the scenario generationis build upon two main assumptions: that the estimated distributions describesthe data sufficiently well, and that the estimation of the autocorrelation is good.The latter was done with the assumption that a forgetting factor of λ = 0.95would be appropriate. Although this has neither been optimised nor tested, itis not believed to be the main reason for these disappointing results.

In the identification of an appropriate model for estimation of the pdf of thepower generation, it was found that the higher the number of bins, and the largerthe bin sizes, the better a fit was obtained. This was done mainly by believingthe average skill score defined in Equation 5.13 calculated on a decision set wouldlead to the best model. However, it is now the question if this has led to an overfitting of the decision set in stead of to the model with the best ability to predictfuture distributions. With a bin size of 1500 and 7 bins it takes at least 10,500 tofill up the bins, from where observations will excluded as new observations enterthe bins. This will lead to a very low degree of adaption to change of behaviourin data. In Figure 4.5 it was seen that the bias of the WPPT predictions, whichform the explanatory variable for the adaptive quantile algorithm, changes frombeing small and fluctuating, gets larger, first positive, then negative, and is thenlarge and positive through the last three months. The first months with a smallbias actually form the decision period. This could be why a small degree ofadaption has a positive influence on the bias and on the overall skill score inthis period.

The evaluation of the predicted distributions has not been carried out for thefull period. However, Figure 8.2 shows the cumulated error of using the expec-tation of the predicted distributions as point forecasts. The bias actually seemto be larger than for the WPPT predictions. This certainly reflects that thedistributions are biased.

The practical problem in the modelling process has been that models were testedon only the first month, and when they were run on the whole test set, it tookabout a week to finish all simulations. This was very unfortunately done toolate to change the prediction model.

98 Discussion0

2040

60


Time, t

∑

τ<

tǫ τ

Figure 8.2 Cummulated sum of errors for the estimated expectation based on the pre-dicted distributions.

8.3 Discussion of the strategies

Even though the expectations from the simulations and the realisations revealsproblems with the underlying modelling of the data, the expectations themselvescontain information about how well the optimisation strategies could increasethe revenues, given that the distribution of the production was well fitted.

The expected increases in revenues are larger for all the optimisations using bothknown and predicted regulation prices. When the regulation prices are know,this comes as no surprise, since the optimisation procedure would otherwisehad stopped at the WPPT predictions. When using the naive predictions ofregulation, this is not an evident result, since the regulation prices forecastingcould be so poor that wrong decisions would made.

The resulting charging and discharging of the storage devices from the differentstrategies are of interest, since they express how well the system is prepared tocope with different prediction errors. Figure 8.3 show the distributions of thestorage levels in EES1 when predicting the regulation costs and using the twodifferent optimisation strategies.

It is clearly seen how maximisation of the expected revenues tend to empty thestorage. This behaviour was already expected when it was formulated, and it

8.3 Discussion of the strategies 99

0.0 0.1 0.2 0.3 0.4 0.5

020

0040

0060

00

Storage level [p.u.]

Counts

(of9212)

(a) Expected revenue maximised

0.0 0.1 0.2 0.3 0.4 0.5

050

010

0015

0020

0025

00

Storage level [p.u.]

Counts

(of9212)

(b) Expected r.c. minimised

Figure 8.3 Distributions of the storage level in EES1 throughout the test period usingdifferent optimisation strategies, predicting regulation costs.

was the reason for also trying the other optimisation strategy. Minimisationof the expected regulation costs also tend to empty the storage but much less.The fact that it does must be explained partly by the larger predicted down-regulation costs than up-regulation costs, and partly by the generally positivebias in the predicted distributions learned from Figure 8.2.

Even though the results show that the potential of using the maximisation of theexpected revenue is the greater of the two, this behaviour of emptying the storagecannot be considered optimal. The probability of facing up-regulation costs istoo large. One way to deal with this would be to use a risk aversion strategy.Returning to the risk function given in Equation (3.15), the only loss function,Λ, that have been used so far is the random variables them selves, i.e. revenueand regulation costs. One could change this loss function into for instance thesquared error of the estimate which would make the risk function the meansquared error (mse) of the estimate, or one could try other linear combinationsof the expectation and the variance of the estimate. This would probably resultin a method that combines the “aggressive” behaviour of the maximisation ofexpected revenue with the “conservative” behaviour of the minimisation of theexpected regulation costs strategy.

Another possibility would be to use a prediction model for the spot price, andthen ascribe this value - or a function of it - times the discharge efficiency tothe energy stored. Maybe this method could even be combined with the riskaversion approach.

100 Discussion

8.4 Discussion of the passive operation strategy

Throughout this thesis, EES have only been used to smooth out predictionerrors, and without considering future prediction errors. This has, from themodelling point of view, the big advantage that it reduces the objective func-tion in the optimisation problem to only have one free variable at each time step,namely the trading contract. This approach have not been justified from a theo-retical point of view and could be just as big a problem as an easy solution. Theresults that EES2 in some cases perform better than EES1 certainly indicatesthat a control strategy for the charging/discharging could be beneficial.

One could try to turn around the problem and let the contracts be determinedby a point forecast model and then optimise the operation of the EES. Thiswould give the possibility to update the optimisation problem hour-by-hourwhile trading on the Elspot market can only be done for whole days at a time.Maybe, the regulation prices could be forecasted with good precision if thehorizon is only one hour.

A more direct way to deal with the problem would be to optimise both contractsand storage operation before gate closure. Then hour-by-hour optimise thestorage operation using the information that a certain contract has been issued.The solution to the optimisation problem of deciding on the contract, would haveto take into account the preceding contract, because that is the contract thatdeals with the delivery period within which the new contract must be issued.In the time before the delivery period starts, the solution to optimisation of thestorage operation then depends on both the contracts for the present time, andthe contract that was just issued for the delivery period coming. When the newdelivery period is entered - and before next gate closure - the optimisation ofthe storage operation only depends on one contract, namely the contract forthe present. At for instance the 24 hours market modelled in this work, onlyone contract is fixed from midnight to noon, but from noon to midnight thereis both a contract for that very day, and a contract for the following day.

If this much more complex operation strategy is implemented, studies wouldstill be needed to be done to decide on an appropriate objective function.

8.5 General aspects about storage

In this work, no considerations have taken into account of costs of EES. Thedata that have been used for the modelling comes from a large area in northern

8.5 General aspects about storage 101

Jutland. Half an hour of full production from a large wind farm, can accountfor so much energy that it could not possibly be stored. The methodology israther feasible for smaller production units, or the EES should be much smaller.

The main problem leading to the need for storage is that energy prices andregulation costs are fluctuating. This reflects variations in, on one side, thedemand, and on the other, the costs of supplying. Introducing storage is amethod for balance out the costs of supplying in a production environment ofstochastic behaviour. Another possibility is to emphasis on the demand. If thedemand could be smoothed out or made more flexible, there would be less needfor storing. This could be done with electric cars that are likely to be morecommon in the future. In stead of buying a certain power when connecting adevice to a traditional socket, other sockets could be equipped with interfaceswhere a certain amount of energy could be bought to delivery within a certaintime at a lower price. Similar systems could be used for some heating and coolingpurposes, where the user do not have any benefits from having the delivery atan exact time. In such a system, the optimisation problem would be on thedistribution side. But it would share some fundamental properties with theproblem analysed in this work.

102 Discussion

Chapter 9

Conclusions

A methodology has been developed for using an electrical energy storage (EES)to smooth out deviances between delivery contracts and power production in aproduction environment of random behaviour. The methodology has been testedusing data from wind power generation in the Northern part of Jutland, spotprices from Elspot and regulation prices from western Denmark. The testinghas been carried out through all in all 384 days. It has been applied on marketmodels of delivery periods of 6 and 24 hours, with gate closures immediatelybefore the delivery periods.

Optimisation of trading has been accomplished with Monte Carlo simulationbased on non-linear prediction of the distribution of power production and itscovariance structure. For the prediction of the distributions, adaptive quantileregression was applied. The optimisation has been done using two differentobjective functions. One approach was to maximise the expected revenue fromthe contracts, the other was to minimise the expected regulation costs. Theoptimisation procedures were limited only to take into account one deliveryperiod at a time on the considered markets.

Without using an EES, the regulation costs within the period used for simula-tions have only constituted a loss of 3.6% of the revenue that would have beenobtained if the production was sold on Elspot and no regulation costs were faced.Using an electrical energy storage for avoiding regulation costs have therefore

104 Conclusions

been an approach with little possible benefits. This is because of relatively smallregulation prices and because regulation prices are only met when contributingto the system imbalance. If it is possible to predict the sign of the systemimbalance, this would therefore be a simpler way to avoid regulation costs.

Three different examples of EES were modelled. One with only a lower boundon the storage level, without limits on charge and discharge rates and withoutlosses associated with charging and discharging. The second EES was with bothupper and lower limits on the possible storage level and losses associated withenergy conversion but without limits on the conversion rates. The third EEShad both limits on maximum and minimum storage level, losses were associatedwith conversion, and the conversion speeds were also limited.

Only for the lossless EES, the methods were seen to increase the revenues signifi-cantly in the considered period. With a very simple forecast model for regulationprices, the revenue was increased with 5.4% on the six hours market and 19.1%on the 24 hours market using when maximising the expected revenue on before-hand, compared to using a state of the art point prediction model as tradingcontracts.

When limiting the maximum storage level to half of an hour production capac-ity and lossy conversions, the gain of using this strategy disappeared, and theincome was exactly the one obtained using the point forecasts as contracts. Forthe 24 hours market, this gain was at 2.3%.

When moreover limiting the energy that can be delivered to lie within 0.05 of anhours production capacity, the gains of maximising the expected revenue were0 and 0.6% for the six hours and the 24 hours markets respectively.

The revenues observed were seen to be lower when minimising the regulationcosts. The gains of this strategy were all found to lie within -1.7% and 0.4%compared to using the point forecasts as contracts.

The results were not as good as expected. This was explained by a problemin the prediction of the distributions of the production. Based on simulations,the method is estimated have a potential of increasing the revenue with up toabout 10% for the EES with losses on conversions, depending on market model.Trading on the six hours market yielded results that were about ten percentbetter than on the 24 hours market.

The optimisation strategies have been discussed. Risk aversion strategies havebeen suggested to improve the objective function in the optimisation problemthat already has been solved. Furthermore it has been discussed how the opti-misation problem can be extended to improve the revenue. Here, one suggestion

9.1 Future work 105

were to include a term expressing an expected value of the stored energy. An-other suggestion was to optimise with respect to the storage operation in steadof the trading, i.e. to leave behind the idea of only using the EES for smoothingout deviances between contracts and production. This way, the strategy couldadapt to changes in regulation prices.

The last, and definitely the most ambitious, suggestion was to optimise bothcontracts and EES operation. In this formulation, the EES operation can beoptimised hour-by-hour taking into account the contracts that must be issuedbefore gate closures.

9.1 Future work

First of all, it is suggested that the decision on a model for prediction of quantilesis reviewed and the optimisations re-run. It is expected that the results givenin this work can be improved significantly.

Implementation of different loss functions in a risk aversion approach is sug-gested to see if the performance of the “aggressive” maximisation of the expectedrevenue can be combined with the stability of the risk-averse strategy of min-imising the expected regulation costs. If the spot prices can be predicted withsufficient precision, it may be better to include a term expressing the expectedvalue of the stored energy in the objective function.

It should be tried to change the optimisation problem into an optimisation ofthe operation of the EES. The problem could then be further extended to coverboth trading and storage operation.

106 Conclusions

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List of Figures

2.1 Illustration of the energy balance in the considered system. . . . 10

2.2 Sketch of a power curve for a wind turbine. . . . . . . . . . . . . 11

2.3 Illustration of up-regulation by the TSO. The flows between theactors are energy in MWh. . . . . . . . . . . . . . . . . . . . . . 15

2.4 Illustration of trading and delivery periods at two different markets. 16

2.5 Illustration of the energy conversion between the storage deviceand the power net and the limitations hereof. . . . . . . . . . . . 22

2.6 Illustration of three examples of the limitations of the chargingand discharging of a storage device. . . . . . . . . . . . . . . . . . 23

2.7 Illustration of the first two lines of the solution in Equation (2.27).These are the cases that result in negative Ed

t . . . . . . . . . . . 26

2.8 Illustration of the third line of the solution in Equation (2.27). . 26

2.9 Illustrations of line 4 and 5 in Equation (2.27). . . . . . . . . . . 27

2.10 Total revenues for a fixed production versus deviation, Edt , from

contract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

112 LIST OF FIGURES

3.1 Illustration of the cost function in the quantile regression estima-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Western Denmark divided into five regions. Data come from pro-duction in region 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 The production measurements normalised to the available capacity. 43

4.3 A histogram of the production measurements recorded hourlybetween 2006-01-01, 2:00 am and 2007-10-26, 7:00 pm, both inUTC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Bias versus horizons. . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 BIAS versus time. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6 BIAS versus power. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.7 Elspot prices followed by down-regulation and finally up-regulationprices. The whole dataset. . . . . . . . . . . . . . . . . . . . . . . 49

4.8 Histograms of up regulation prices and costs used for simulations. 50

4.9 Histograms of down regulation prices and costs used for simulations. 51

4.10 Illustration of the availability of the different kinds of data. . . . 53

5.1 Illustration of the process of simulation of power production sce-narios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Illustration of linear interpolation and exponential extrapolationof estimated quantiles. . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 Bias versus nominal proportion for the different probabilistic fore-casting models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Bias versus horizon for the different probabilistic forecasting models. 67

5.5 Skill score versus horizon for the different probabilistic forecastingmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

LIST OF FIGURES 113

6.1 Probabilistic forecasts issued on 2007-10-19 at 11 Pm. . . . . . . 72

6.2 Simulations of the systems with different EES given the showncontract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3 Simulations of the systems with different EES with WPPT pre-dictions as contracts. . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4 Elspot and regulation prices and predictions for the chosen 6 hourperiod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.5 Expected revenues using different EES using WPPT predictionsas contracts. A small difference between the two cases is seen forIEES EES1 from 10 Am to noon. . . . . . . . . . . . . . . . . . . 76

6.6 Image of the correlation matrix for the normally distributed vari-ables used to simulate power production for the first 6 hours onMarch the 22nd 2007. . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.7 50 scenarios of the production in the considered 6 hour period.The simulated production at 6:00 Am is also shown because ofthe correlation between this and future production. . . . . . . . . 77

6.8 Tracking the probability density function of the production throughtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.9 A histogram of the production scenarios transformed by the esti-mated cdf of the production. These should be close to uniformlydistributed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.10 Histogram of the sum of errors using the WPPT predictions on10,000 simulated production scenarios. Compare with Figure 6.7. 79

6.11 The pdf of the storage levels versus time using WPPT predictionsas contracts for each of the three EES. . . . . . . . . . . . . . . . 81

6.12 WPPT and the optimized contracts. . . . . . . . . . . . . . . . . 84

6.13 The pdf of the storage levels versus time in the example periodfor the three example storage devices where the contracts havebeen optimised by maximising the expected revenue for the period. 85

114 LIST OF FIGURES

6.14 The predicted pdf of the storage level versus time in the exam-ple period for EES1 and EES2 where the contracts have beenoptimized by minimising the expected regulation costs. . . . . . . 86

6.15 Results for the six hour example with different contract strategiesand EES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.16 Realised results for the six hour example with different contractstrategies and EES. . . . . . . . . . . . . . . . . . . . . . . . . . . 88

8.1 The distribution of the regulation state on the regulation marketwithin the simulation period. . . . . . . . . . . . . . . . . . . . . 94

8.2 Cummulated sum of errors for the estimated expectation basedon the predicted distributions. . . . . . . . . . . . . . . . . . . . . 98

8.3 Distributions of the storage level in EES1 throughout the test pe-riod using different optimisation strategies, predicting regulationcosts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.1 Skill scores and average bias for the estimated models. . . . . . . 122

List of Tables

2.1 Summary of the revenues for the producers in the up-regulationexample illustrated in Figure 2.3. . . . . . . . . . . . . . . . . . . 15

2.2 The storage devices, IEES, EES1, and EES2, to be used in theanalysis. ∗ The maximum charge and discharge rates are limitedby Lmin

ees and Lmaxees . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 Evaluation properties for the point forecasts from WPPT. . . . . 48

5.1 Setup of the best fitted models with knots placed on either binbarriers or bin centres. Measures of their performances are givenin Table 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Performance measures for the different model setups. . . . . . . . 66

5.3 Options varied in the configuration of the adaptive quantile re-gression models and observations on their influence on the per-formance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1 Results of different setups in the example 6 hour period. Theunit is e per hour per installed MWh. . . . . . . . . . . . . . . . 89

116 LIST OF TABLES

7.1 Revenues without optimisation. The revenues are normalised toby the revenue of selling all of the production at Elspot and notfacing any regulation costs. . . . . . . . . . . . . . . . . . . . . . 92

7.2 Results of different setups throughout the test period. The rev-enues are normalised to by the revenue of selling all of the pro-duction at Elspot and not facing any regulation costs. “R.p.” isan abbreviation for “regulation prices”. . . . . . . . . . . . . . . 92

Appendix A

Derivation of Taylor

expansions

Let X be an unbiased estimate of X . f is a mapping from and into the realnumbers and is continous and two times differentiale in X. Then, a taylorexpansion of second order gives

E

(

f (X)∣

∣

∣X)

= E

(

f(

E

(

X |X)

+(

X − E

(

X |X))))

≈ E

(

f(

E

(

X |X))

+ f ′(

E

(

X |X))

·(

X − E

(

X |X))

+1

2· f ′′

(

E

(

X |X))

·(

X − E

(

X |X))2

∣

∣

∣

∣

X

)

Since the estimate X is assumed to be unbiased, E

(

X |X)

= X and

E

(

X − E

(

X |X)

|X)

= 0, the second of the terms disappears:

E

(

f (X)∣

∣

∣X)

≈ E

(

f(

X)

+1

2· f ′′

(

X)

·(

X − X)2)

= f(

X)

+1

2· f ′′

(

X)

· V

(

X |X)

(2.4)

118 Derivation of Taylor expansions

For the variance, only a first order Taylor expansion is used.

V (f (X)) = V

(

f(

E

(

X |X)

+(

X − E

(

X |X)))

|X)

≈ V

(

f(

E

(

X |X))

+ f ′(

E

(

X |X))

·(

X − E

(

X |X))∣

∣

∣X)

= V

(

f(

E

(

X |X))∣

∣

∣X)

+ V

(

f ′(

E

(

X |X))

· X∣

∣

∣X)

+ V

(

f ′(

E

(

X |X))

· E(

X |X)∣

∣

∣X)

= f ′(

E

(

X |X))2

· V (X)

Again, because X is unbiased:

V (f (X)) ≈ f ′(

X)2

· V (X) (2.5)

Appendix B

Results of adaptive quantile

regression analysis

In the following table, abbreviations are used for some of the headings. “nb”means “number of bins”, “bsize” stands for “bin size”, “binpen” for “Bin penaltyenabled”, “kp.method” for “knot placement method”, and “bp.method” for “bin

placement method”. γ, δ, b, and Sc are all averages over the horizons 1 to 37,and only the decision period is used.

mn nb bsize binpen kp.method bp.method γ δ · 103 b · 103 Sc

1 2 300 TRUE middle even 0.01 437.4 235.08 −2.5252 2 300 FALSE middle even 0.01 437.4 235.08 −2.5253 2 500 TRUE middle even 0.01 437.4 235.08 −2.5254 2 500 FALSE middle even 0.01 437.4 235.08 −2.5255 2 1000 TRUE middle even 0.01 437.4 235.08 −2.5256 2 1000 FALSE middle even 0.01 437.4 235.08 −2.5257 2 1500 TRUE middle even 0.01 437.4 235.08 −2.5258 2 1500 FALSE middle even 0.01 437.4 235.08 −2.5259 2 300 TRUE barriers even 0.01 411.6 74.64 −1.398

10 2 300 FALSE barriers even 0.01 411.6 74.64 −1.39811 2 500 TRUE barriers even 0.01 411.6 74.64 −1.39812 2 500 FALSE barriers even 0.01 411.6 74.64 −1.39813 2 1000 TRUE barriers even 0.01 411.6 74.64 −1.39814 2 1000 FALSE barriers even 0.01 411.6 74.64 −1.39815 2 1500 TRUE barriers even 0.01 411.6 74.64 −1.39816 2 1500 FALSE barriers even 0.01 411.6 74.64 −1.39817 2 300 TRUE middle quantiles 0.36 127.6 45.19 −0.51918 2 300 FALSE middle quantiles 0.37 118.1 8.41 −0.47919 2 500 TRUE middle quantiles 0.39 117.3 35.66 −0.46620 2 500 FALSE middle quantiles 0.38 112.9 32.77 −0.54621 2 1000 TRUE middle quantiles 0.42 109.4 32.99 −0.43622 2 1000 FALSE middle quantiles 0.41 110.5 −36.50 −0.38323 2 1500 TRUE middle quantiles 0.43 106.5 13.82 −0.41924 2 1500 FALSE middle quantiles 0.42 108.9 −31.10 −0.37125 2 300 TRUE barriers quantiles 0.37 89.3 −36.73 −0.420

120 Results of adaptive quantile regression analysis


26 2 300 FALSE barriers quantiles 0.37 92.9 −18.85 −0.42027 2 500 TRUE barriers quantiles 0.41 91.8 −21.86 −0.41528 2 500 FALSE barriers quantiles 0.40 91.0 −104.31 −0.37729 2 1000 TRUE barriers quantiles 0.44 92.8 −71.31 −0.37530 2 1000 FALSE barriers quantiles 0.42 83.6 −177.88 −0.39231 2 1500 TRUE barriers quantiles 0.45 91.1 −122.43 −0.36632 2 1500 FALSE barriers quantiles 0.43 86.2 −164.58 −0.38533 3 300 TRUE middle even 0.11 363.0 126.00 −1.84034 3 300 FALSE middle even 0.11 350.9 83.78 −1.59435 3 500 TRUE middle even 0.11 351.8 91.40 −1.51536 3 500 FALSE middle even 0.11 329.5 75.10 −1.36837 3 1000 TRUE middle even 0.12 322.5 61.24 −1.20438 3 1000 FALSE middle even 0.12 322.5 61.24 −1.20439 3 1500 TRUE middle even 0.12 322.5 61.24 −1.20440 3 1500 FALSE middle even 0.12 322.5 61.24 −1.20441 3 300 TRUE barriers even 0.12 230.2 −20.02 −0.73342 3 300 FALSE barriers even 0.11 223.6 −15.19 −0.70543 3 500 TRUE barriers even 0.12 218.6 −3.91 −0.64044 3 500 FALSE barriers even 0.12 182.3 −49.65 −0.59245 3 1000 TRUE barriers even 0.12 179.4 −51.08 −0.59046 3 1000 FALSE barriers even 0.12 179.4 −51.08 −0.59047 3 1500 TRUE barriers even 0.12 179.4 −51.08 −0.59048 3 1500 FALSE barriers even 0.12 179.4 −51.08 −0.59049 3 300 TRUE middle quantiles 0.56 108.7 15.69 −0.39650 3 300 FALSE middle quantiles 0.56 110.0 13.47 −0.40751 3 500 TRUE middle quantiles 0.58 105.2 12.25 −0.39552 3 500 FALSE middle quantiles 0.58 107.5 24.13 −0.38553 3 1000 TRUE middle quantiles 0.59 102.4 2.58 −0.36154 3 1000 FALSE middle quantiles 0.59 103.7 20.45 −0.36755 3 1500 TRUE middle quantiles 0.60 102.5 11.61 −0.35856 3 1500 FALSE middle quantiles 0.59 101.2 −2.42 −0.35657 3 300 TRUE barriers quantiles 0.57 97.0 15.44 −0.36758 3 300 FALSE barriers quantiles 0.57 96.1 39.21 −0.37759 3 500 TRUE barriers quantiles 0.58 96.9 58.85 −0.37260 3 500 FALSE barriers quantiles 0.59 100.1 98.00 −0.41061 3 1000 TRUE barriers quantiles 0.61 98.7 94.12 −0.37862 3 1000 FALSE barriers quantiles 0.60 100.9 106.16 −0.39863 3 1500 TRUE barriers quantiles 0.61 98.8 104.00 −0.38864 3 1500 FALSE barriers quantiles 0.61 99.2 102.71 −0.38665 4 300 TRUE middle even 0.20 282.9 240.58 −2.83566 4 300 FALSE middle even 0.20 292.1 254.34 −2.86067 4 500 TRUE middle even 0.20 293.3 253.00 −2.76168 4 500 FALSE middle even 0.20 310.1 245.45 −2.57169 4 1000 TRUE middle even 0.21 314.8 240.11 −2.45570 4 1000 FALSE middle even 0.21 311.9 236.64 −2.41171 4 1500 TRUE middle even 0.21 315.6 239.31 −2.42372 4 1500 FALSE middle even 0.21 315.6 239.31 −2.42373 4 300 TRUE barriers even 0.20 170.0 −25.98 −0.93474 4 300 FALSE barriers even 0.20 171.8 −14.86 −0.87075 4 500 TRUE barriers even 0.21 154.7 −28.94 −0.72176 4 500 FALSE barriers even 0.21 152.4 −34.51 −0.65377 4 1000 TRUE barriers even 0.22 149.2 −16.47 −0.54378 4 1000 FALSE barriers even 0.22 150.1 −17.14 −0.51679 4 1500 TRUE barriers even 0.22 150.3 −14.64 −0.51780 4 1500 FALSE barriers even 0.22 150.3 −14.64 −0.51781 4 300 TRUE middle quantiles 0.67 115.0 36.72 −0.40982 4 300 FALSE middle quantiles 0.67 113.0 33.74 −0.40383 4 500 TRUE middle quantiles 0.68 111.1 30.57 −0.38484 4 500 FALSE middle quantiles 0.68 112.8 3.30 −0.36585 4 1000 TRUE middle quantiles 0.69 113.9 7.24 −0.35686 4 1000 FALSE middle quantiles 0.68 114.9 10.45 −0.35787 4 1500 TRUE middle quantiles 0.69 113.9 11.75 −0.35588 4 1500 FALSE middle quantiles 0.69 114.0 12.04 −0.35589 4 300 TRUE barriers quantiles 0.67 96.4 4.25 −0.34990 4 300 FALSE barriers quantiles 0.67 96.3 −6.11 −0.34491 4 500 TRUE barriers quantiles 0.68 96.0 −15.65 −0.34292 4 500 FALSE barriers quantiles 0.68 95.5 −34.18 −0.34093 4 1000 TRUE barriers quantiles 0.69 96.4 −38.56 −0.33794 4 1000 FALSE barriers quantiles 0.69 96.5 −45.37 −0.33895 4 1500 TRUE barriers quantiles 0.70 96.5 −48.97 −0.33796 4 1500 FALSE barriers quantiles 0.69 96.5 −50.70 −0.33797 5 300 TRUE middle even 0.3498 5 300 FALSE middle even 0.3499 5 500 TRUE middle even 0.34

100 5 500 FALSE middle even 0.34101 5 1000 TRUE middle even 0.35102 5 1000 FALSE middle even 0.35103 5 1500 TRUE middle even 0.35104 5 1500 FALSE middle even 0.35105 5 300 TRUE barriers even 0.28 216.3 177.93 −1.086106 5 300 FALSE barriers even 0.28 221.5 190.35 −1.179107 5 500 TRUE barriers even 0.29 187.8 186.77 −0.992108 5 500 FALSE barriers even 0.29 193.4 194.35 −0.956109 5 1000 TRUE barriers even 0.30 160.9 197.94 −0.844

121


110 5 1000 FALSE barriers even 0.30 167.2 206.93 −0.891111 5 1500 TRUE barriers even 0.30 158.9 198.98 −0.836112 5 1500 FALSE barriers even 0.30 158.9 198.98 −0.836113 5 300 TRUE middle quantiles 0.73 108.3 32.83 −0.381114 5 300 FALSE middle quantiles 0.73 107.2 30.97 −0.379115 5 500 TRUE middle quantiles 0.74 108.3 45.48 −0.378116 5 500 FALSE middle quantiles 0.74 107.6 45.20 −0.379117 5 1000 TRUE middle quantiles 0.74 107.5 42.43 −0.375118 5 1000 FALSE middle quantiles 0.74 107.6 44.09 −0.375119 5 1500 TRUE middle quantiles 0.75 108.0 46.77 −0.378120 5 1500 FALSE middle quantiles 0.75 108.0 46.77 −0.378121 5 300 TRUE barriers quantiles 0.74 100.1 30.79 −0.356122 5 300 FALSE barriers quantiles 0.74 101.3 26.81 −0.352123 5 500 TRUE barriers quantiles 0.75 102.4 29.42 −0.348124 5 500 FALSE barriers quantiles 0.75 102.3 26.41 −0.344125 5 1000 TRUE barriers quantiles 0.75 104.5 30.93 −0.343126 5 1000 FALSE barriers quantiles 0.75 105.0 34.46 −0.344127 5 1500 TRUE barriers quantiles 0.76 105.2 36.16 −0.344128 5 1500 FALSE barriers quantiles 0.76 105.2 36.16 −0.344129 6 300 TRUE middle even 0.49130 6 300 FALSE middle even 0.49131 6 500 TRUE middle even 0.50132 6 500 FALSE middle even 0.49133 6 1000 TRUE middle even 0.50134 6 1000 FALSE middle even 0.50135 6 1500 TRUE middle even 0.50136 6 1500 FALSE middle even 0.50137 6 300 TRUE barriers even 0.35 166.9 83.83 −0.709138 6 300 FALSE barriers even 0.35 167.9 76.97 −0.632139 6 500 TRUE barriers even 0.36 147.4 50.39 −0.536140 6 500 FALSE barriers even 0.36 145.7 43.82 −0.525141 6 1000 TRUE barriers even 0.37 136.2 10.08 −0.462142 6 1000 FALSE barriers even 0.37 134.0 14.44 −0.480143 6 1500 TRUE barriers even 0.37 134.8 14.59 −0.466144 6 1500 FALSE barriers even 0.37 134.9 14.80 −0.466145 6 300 TRUE middle quantiles 0.78 102.4 −7.88 −0.356146 6 300 FALSE middle quantiles 0.78 104.0 −7.97 −0.356147 6 500 TRUE middle quantiles 0.78 103.7 −13.92 −0.356148 6 500 FALSE middle quantiles 0.78 105.1 −12.66 −0.357149 6 1000 TRUE middle quantiles 0.78 105.2 −17.39 −0.354150 6 1000 FALSE middle quantiles 0.78 105.3 −17.65 −0.354151 6 1500 TRUE middle quantiles 0.78 105.2 −17.73 −0.354152 6 1500 FALSE middle quantiles 0.78 105.2 −17.73 −0.354153 6 300 TRUE barriers quantiles 0.78 97.4 7.64 −0.344154 6 300 FALSE barriers quantiles 0.78 97.9 4.74 −0.339155 6 500 TRUE barriers quantiles 0.78 99.1 12.12 −0.339156 6 500 FALSE barriers quantiles 0.78 99.4 4.47 −0.337157 6 1000 TRUE barriers quantiles 0.79 100.6 6.18 −0.337158 6 1000 FALSE barriers quantiles 0.79 100.7 7.26 −0.337159 6 1500 TRUE barriers quantiles 0.79 100.7 7.01 −0.337160 6 1500 FALSE barriers quantiles 0.79 100.7 7.01 −0.337161 7 300 TRUE middle even 0.55162 7 300 FALSE middle even 0.55163 7 500 TRUE middle even 0.56164 7 500 FALSE middle even 0.56165 7 1000 TRUE middle even 0.56166 7 1000 FALSE middle even 0.56167 7 1500 TRUE middle even 0.56168 7 1500 FALSE middle even 0.56169 7 300 TRUE barriers even 0.47170 7 300 FALSE barriers even 0.47171 7 500 TRUE barriers even 0.48172 7 500 FALSE barriers even 0.48173 7 1000 TRUE barriers even 0.49174 7 1000 FALSE barriers even 0.49175 7 1500 TRUE barriers even 0.50176 7 1500 FALSE barriers even 0.50177 7 300 TRUE middle quantiles 0.81 101.1 5.83 −0.354178 7 300 FALSE middle quantiles 0.81 101.6 4.86 −0.354179 7 500 TRUE middle quantiles 0.81 101.0 −4.63 −0.345180 7 500 FALSE middle quantiles 0.81 102.1 −1.36 −0.345181 7 1000 TRUE middle quantiles 0.81 101.7 −8.69 −0.343182 7 1000 FALSE middle quantiles 0.81 101.7 −8.73 −0.343183 7 1500 TRUE middle quantiles 0.81 101.7 −8.73 −0.343184 7 1500 FALSE middle quantiles 0.81 101.7 −8.73 −0.343185 7 300 TRUE barriers quantiles 0.81 96.0 −7.03 −0.339186 7 300 FALSE barriers quantiles 0.81 96.8 −9.31 −0.338187 7 500 TRUE barriers quantiles 0.81 97.5 −17.18 −0.335188 7 500 FALSE barriers quantiles 0.81 98.2 −15.77 −0.335189 7 1000 TRUE barriers quantiles 0.81 98.7 −21.37 −0.334190 7 1000 FALSE barriers quantiles 0.81 98.7 −21.36 −0.334191 7 1500 TRUE barriers quantiles 0.81 98.7 −21.37 −0.334192 7 1500 FALSE barriers quantiles 0.81 98.7 −21.37 −0.334

122 Results of adaptive quantile regression analysis

0 50 100 150

−2.

5−

2.0

−1.

5−

1.0

−0.

5

filled: bins quantile basedempty: bins uniformly placedsmall: knots in bin centerslarge: knots on barriersbin sizes:30050010001500

0 50 100 150

−0.

10.

00.

10.

2

filled: bins quantile basedempty: bins uniformly placedsmall: knots in bin centerslarge: knots on barriersbin sizes:30050010001500

Model Number

Bia

sS

c

Figure B.1 Skill scores and average bias for the estimated models.

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