IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2017

Online Regime Switching Vector Autoregression Incorporating Spatio-temporal Aspects for Short Term Wind Power Forecasting

SEAN GILLERAN

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING

Online Regime Switching Vector Autoregression Incorporating Spatio-temporal Aspects for

Short Term Wind Power Forecasting

Sean Gilleran

October 1st, 2017 School of Electrical Engineering Department of Electric Power and Energy Systems Nordic Master in Innovative Sustainable Energy Engineering – Energy Systems Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science. Examiner: Professor Lennart Söder Supervisors: Christer Modin, Elis Nycander

Abstract

This master thesis examines short term wind power forecasting time seriesmodels focusing on regimes conditioned to meteorological conditions and theincorporation of spatio-temporal aspects. Novel regime switching autoregres-sive and vector autoregressive models are proposed, implemented in a .NETframework, and evaluated. The vector autoregressive framework takes ad-vantage of cross-correlation between sites incorporating upstream online pro-duction information from all wind farms within a given region. The regimesare formed using K-means clustering based on forecast meteorological condi-tions. Each of the proposed models are fit to hourly historical data from allof 2015 for 24 wind farms located in Sweden and Finland. Forecasts are gen-erated for all of 2016 and are evaluated against historical data from that yearfor each of the 24 wind farms. The proposed models are successfully imple-mented into the .NET framework of Vitec Software’s Aiolos Forecast Studio,which is widely used in the Northern and Western Europe. Vitec’s Aioloswind power forecast model is based on an ensemble of numerical weatherprediction models and adaptive statistical machine learning algorithms. Theproposed models are found to have significantly lower mean absolute errorand root mean squared error compared to the Aiolos model and autoregres-sive model benchmarks. The improved short term wind power forecast willinform operation and trading decisions and translate to significant reduc-tions in balancing costs for Vitecs customers. The improvement is valued atas much as between 9.4 million Euros to 42.3 million Euros in reduced bal-ancing costs. Spatio-temporal aspects for wind power forecasting is shown tocontinue to be promising for improving current state-of-the-art wind powerforecasting.

Sammanfattning

I detta arbete undersoks och implementeras autoregressiva modeller forvindkraftprognoser for en kort tidshorisont. Metoden tar hansyn till sam-variationer i tid och rum mellan olika vindkraftanlaggningar och anvanderregimer som baseras pa vaderforhallanden for att forbattra prognoserna. Viforeslar nya autoregressiva regimer, implementerar modellerna i .NET ochutvarderar dem. Vektor autoregressiva modeller utnyttjar korrelationen mel-lan olika anlaggningar genom att ta med information i nartid fran andraanlaggningar i samma region i modellen och pa sa vis forbattra prognoserna.Regimerna skapas med en klustermetod for K-medelvarde som baseras pavaderforhallandena. Alla foreslagna modeller anpassas till historiska datafor 2015 for 24 vindkraftanlaggningar i Sverige och Finland. Prognoser ska-pas for 2016 och anvands for att utvardera modellerna for var och en av de24 anlaggningarna. De foreslagna modellerna har implementerats i .NET imiljon for Vitecs Aiolos Forecast Studio, vilket ar ett program som anvandsav manga operatorer i norra och vastra Europa for att gora vindkraftprog-noser. Aiolos modell baseras pa en rad olika numeriska vaderprognosmodelleroch adaptiva statistiska maskinlarningsalgoritmer. De foreslagna modellernavisar sig ha lagre fel jamfort med Aiolos modell och andra autoregressivamodeller som anvants som riktmarken. De forbattrade kortsiktiga vind-kraftsprognoserna kommer vara underlag for operativa och finansiella beslutfor Vitecs kunder och innebara betydande minskningar av balanskostnader.Forbattringen uppskattas kunna minska kostnaderna for Vitecs kunder medsa mycket som mellan 9.4 miljoner och 42.3 miljoner Euro. Att utnyttjakorrelationer mellan olika vindkraftanlaggningar visar sig ha fortsatt storbetydelse for att forbattra vindkraftprognoser.

Contents

1 Motivation 3

2 Introduction 4

2.1 Wind Power Forecasting Applications . . . . . . . . . . . . . . 42.2 State-of-the-art Wind Power Forecasting . . . . . . . . . . . . 72.3 Towards Improving State-of-the-art Wind Power Forecasting . 92.4 Forecast Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Spatio-temporal Aspects for Wind Power Forecasting . . . . . 12

3 Methodology 17

3.1 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.1 Input Data . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Forecast Models . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.1 Autoregression . . . . . . . . . . . . . . . . . . . . . . 193.2.2 NWP Residual Models . . . . . . . . . . . . . . . . . . 213.2.3 Vector Autoregression . . . . . . . . . . . . . . . . . . 223.2.4 Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.5 Two-step Regime Switching Model . . . . . . . . . . . 283.2.6 Seasonality . . . . . . . . . . . . . . . . . . . . . . . . 293.2.7 Parameter Estimation . . . . . . . . . . . . . . . . . . 30

3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.1 Software Development . . . . . . . . . . . . . . . . . . 323.3.2 Aiolos Forecast Model . . . . . . . . . . . . . . . . . . 33

4 Results and Analysis 36

4.1 Base Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Regime Models . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Diurnal Seasonality . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Two-step model benchmark . . . . . . . . . . . . . . . . . . . 41

5 Discussion 49

5.1 Contribution and Significance . . . . . . . . . . . . . . . . . . 495.2 Limitations and Future Research and Development . . . . . . 50

6 Conclusion 53

7 References 54

2

1. Motivation

Failure to mitigate climate change is one of the greatest risks currentlyfacing humanity. Catastrophic impacts to humanity and natural systems dueto climate change are increasing in severity with continued emissions of greenhouse gases and land use changes. Human systems and natural systems areinextricably linked and are both extremely vulnerable to climate change. [1]

There is a dramatic paradigm shift occurring in the world’s energy sys-tems driven by both the need to tackle climate change and technologicaldevelopment.This paradigm shift is characterised by decarbonisation, elec-trification and decentralisation. Wind power is playing an important role inthe decarbonisation of the energy system. The cumulative installed capacityof wind power was 487 GW at the end of 2016 [2]. The International EnergyAgency estimates that a share of 18% of global electricity will need to beprovided by wind power by 2050 to keep global warming below 2�C [3].

Wind power with its stochastic and variable nature presents challengesto integrating wind power into power systems and electricity markets. Windpower generally increases the total variability of the power system and in-creased flexibility of the power system is needed to maintain the balancebetween production and consumption of electricity [4]. The variability andlimited predictability of wind power make accurate wind power forecastsessential for reliable and economic power system operation. Errors in thewind power forecast increases required balancing services, namely regulationpower reserves. With increasing penetration of wind power in power systemsthese challenges become more critical. Reducing error in the forecasts forwind power reduces balancing costs and facilitates the integration of higherpenetration levels of wind power in a power system. [5]

3

2. Introduction

In this introduction, the applications and state-of-the-art of wind powerforecasting and introduced before discussing current e↵orts towards improv-ing the state of the art of wind power forecasting. Forecast evaluation isdiscussed in Section 2.4. A thorough discussion and literature review ofspatio-temporal aspects for wind power forecasting can be found in Section2.5.

The case study and input data is introduced in Section 3.1. The develop-ment and justification of each of the proposed models is given in Section 3.2.A discussion of the implementation, software development and Aiolos windpower forecast model are in Section 3.3.

The results and analysis of the proposed models evaluated with the casestudy follows in Section 4. A discussion of the contribution and significanceof this master thesis, as well as a discussion of the limitations and futureresearch and development is in Section 5. The final conclusions are in Section6.

2.1. Wind Power Forecasting Applications

Historically, almost all electricity production in power systems globallywas generated from the energy sources of coal, natural gas, other fossil-fuelderived products, biofuels and hydro. With the exception of electricity pro-duction from run-of-the-river hydro, electricity production from these sourcesis for the most part determined by the will of the owner.

The electricity production from wind power is inherently stochastic andits electricity production is determined not only by the will of the ownerbut by wind resource. As power systems integrate increasing amounts ofwind power, the operation of the power system needs to be more flexibleto compensate for the increased variability and uncertainty [4]. The totalvariability can be separated into two components, the variability of the windresource itself, and the variability of the wind power forecast error.

The fundamental characteristic for the reliable delivery of electricity ispower system stability. It is the ability of an electric power system to regain astate of operating equilibrium after being subjected to a physical disturbance[6]. Changes in load and production occur continuously and in the frame ofreference of power system stability, are defined as small disturbances to theequilibrium between load and production. These production disturbancesincrease with increasing wind power penetration and increased wind power

4

forecast error. Improved wind power forecasts with less error can thus reducethe pressure on the power system to adjust to these changing conditions andto maintain stability.

The e�cient operation of a power system requires forecasts of the demand,production and network conditions from seconds ahead to years ahead. Elec-tricity system participants with wind power in their respective power systemsuse wind power forecasts for their operations, trading, and planning. Theseparticipants include the network system operators, utilities, balance respon-sible parties, traders, and independent power producers. A survey of systemoperators regarding wind power integration found that 94% of the respon-dents indicated that the integration of a significant amount of wind powerwill ultimately depend on the accuracy of wind power forecasts [7].

Until the end of the twentieth century, power systems globally were or-ganised by state-owned, vertically-integrated utilities that were responsiblefor the generation, transmission, distribution, and retail. In contrast, in amodern power system activities are often separated. Competition in gen-eration and retail is often encouraged through markets. Transmission anddistribution as natural monopolies are regulated and are managed by systemoperators. Modern electricity markets often take the form of a centralisedmarket in which producers and consumers submit bids and o↵er functionsto a central market operator [8]. In some countries reserve capacity marketsexist in parallel to energy markets where participants are paid proportionalto the available capacity. There are also electricity markets for longer termfinancial contracts such as forwards, options, and derivatives. The electricitymarkets that are most relevant for short term wind power forecasting areday-ahead, intra-day, and balancing markets [5].

In day-ahead markets, also known as forward markets and spot markets,market participants submit bids to buy and sell electricity through price-quantity pairs for each block of time of the following day. The aggregate buyand sell curves are sorted by increasing and decreasing prices respectivelyby the market operator. The intersection between these two curves sets thesystem price by which the market is cleared. Sale bids with a price lowerthan the system price are accepted, and buy bids with a price higher thanthe sytem price are accepted. Market clearance that considers transmissionnetwork constraints in the market clearing procedure sets locational pricesfor each area or node of the power system rather than a single system price.[5]

J. Miettinen and H. Holttinen recently studied day-ahead hourly forecast

5

errors from Finland, Norway, Sweden and Denmark and the impacts of windpower plant dispersion on forecast errors in areas of di↵erent sizes [9]. Theyfound that mean absolute errors normalised to total energy production of thehourly resolution forecasts reduce from the value of 31.3% for a single regionto 9 % for the whole of the Nordic region. These errors would representsignificant balancing demands on the power system if it were not for beingable to later adjust day-ahead contract positions on intra-day markets. Windpower forecast errors reduce with shorter lead times and thus the originalpositions on the day-ahead market can be adjusted on the intra-day andbalancing markets [10]. Intra-day and balancing markets are seen as essentialto large scale integration of wind power [11].

Balancing markets, also known as real-time markets or regulation mar-kets, are the last chance for power system participants to participate in bal-ancing production and demand in the power system before the inherent sta-bility of the transmission and distribution network itself is relied upon. Thebalancing market handles imbalances due for forecast errors and is used toalleviate congestion problems during the operating hour by the activation ofregulating power from other market participants [5]. These balancing servicesinclude automatic frequency controlled reserves, load frequency reserves, andregulation power reserves. These services are also known as primary, sec-ondary and tertiary regulation respectively. Improved short term wind powerforecasts can reduce the required balancing needs [12]. The balancing needscan become critical if there is insu�cient flexible and competitive regulationcapacity available trough the balancing markets [5]. Wind power forecastswith reduced error thus not only reduces the associated balancing costs, butfacilitate higher penetration of wind power.

The latest summary report from the International Energy Agency WindTask 25, Design and Operation of Power Systems with Large Amounts ofWind Power, reports information on system balancing costs from studies onCanada, Denmark, Finland, Germany, Ireland, Norway, Sweden, the UK,and the US [4]. It is reported that the system balancing costs due to windpower forecast error is between 1 to 4.5 Euro /MWh for wind penetrationsof up to 20% of gross energy demand; this is approximately up to 10% ofthe wholesale value of the wind energy [4]. Lower balancing costs can beachieved through improved wind power forecast performance, the allowanceof interconnection capacity to be used for balancing purposes, aggregatingwind farms over larger geographical regions, and scheduling closer to thedelivery time [4].

6

2.2. State-of-the-art Wind Power ForecastingCurrent state-of-the-art wind power forecasts are based on a combination

of physical and statistical models. Giebel et al. comprehensively reviewedthe state of the art in the short term prediction of wind power; for furtherreading the author directs the reader to this work which reviews more than380 journal and conference papers [10]. Models with an emphasis on a sta-tistical approaches and online measured power data outperform models withemphasis on physical approaches for lead times up to 3 to 6 hours ahead. Fi-nal wind power forecast models thus seek to optimally combine wind powerforecast models depending on the lead time of interest.

Models with emphasis on physical approaches are based on outputs frommesoscale and microscale Numerical Weather Prediction (NWP) models,namely wind speed and wind direction. Physical laws are used in NWPmodels to forecast the weather. Predicting a future state of the atmosphereis achieved fundamentally by starting from the assimilation of the observedweather and integrating the governing partial di↵erential equations account-ing for dynamic, thermodynamic, radiative and chemical processes. State-of-the-art mesoscale NWP providers forecast down to a geospatial scale of1 to 2 kilometres, with a temporal scale of 30 minutes to 1 hour resolution,and which are updated every 1 to 3 hours [13].

Statistical treatments are also used at multiple stages of a modelling pro-cess that has emphasis on physical aspects. Model output statistics andensembles are used to reduce bias and error in the outputs from the NWP.Wind speed and direction are then downscaled to the specific location andhub height of the turbine from an optimised selection of the surroundingNWP grid points and elevation data. This downscaled wind speed and di-rection is then converted to power with a power curve. The power curveis estimated from the historical statistical relation between forecasted windspeed and direction and measured power. If historical power measurementsare not available, the manufacturers power curve can be used, although al-most always with poorer performance [10]. It is at the next step that theline between models with an emphasis on physical approaches and statisticalapproaches starts to blur, as both model approaches utilise online measuredpower data to improve model output statistics using adaptive recursive meth-ods.

For shorter forecast horizons and where the recently measured onlinepower data is available the measured data is used as the key input, tak-ing the form of an explicit statistical model using advanced time-series ap-

7

proaches of classical time-series analysis or less frequently, artificial neuralnetworks [10]. While artificial neural networks show similar results to clas-sical time-series approaches, they do not permit a clear interpretation of theunderlying processes being modelled [14]. Gallego et al. (2011) [15] foundthat making generalised autoregressive models conditional to local wind di-rection captured the impact of the wind direction on the wind variability.Additionally, they found making the models conditional to wind speed cap-tured the non-linear behaviour related to the power transformation process .Trombe and Pinson (2012) [16] examined the predictive performance of themost recent approaches in the wind power forecasting literature for very shortlead times. They considered combinations of linear auto regressive modelswith various regime switching models, and o↵site predictors. The approacheswere extended to a probabilistic framework using the generalised logit normaldistribution. Trombe and Pinson (2012) found that hidden Markov-Regimeswitching models had superior predictive performance over threshold regimeswitching models based on observable lagged values, confirming the findingsof Pinson et al. (2008) [17]. The hidden Markov-Regime switching models arebased on an unobservable process that represents a hidden weather regime.Trombe and Pinson (2012) suggest that integrating higher spatio-temporalresolution information could further improve the predicative performance[16].

When upscaling from a single turbine to a wind farm the wake losses canbe accounted for using statistical and semi-empirical methods [10]. Upscal-ing from a single wind farm to a group of wind farms in a point forecastframework can be achieved using simple summation. Alternative approachesinclude using representative farms as input in upscaling models.

Probabilistic forecasts have been shown to provide additional value overpoint power forecasts [18] [16]. Given the stochastic nature of wind power,there exists an intrinsic uncertainty in the wind power generation process andits prediction. A probabilistic framework results in more optimal decisions,both in terms of power system management and electricity trading [19] [12].

In a probabilistic framework, upscaling from a single wind farms marginaldistribution to form the probabilistic forecast for a group of windfarms can-not be achieved with simple summation. Marginal distributions can onlybe summed if they are independent. The aggregated error of a group farmsis less than the error of a single farm. This feature is explained by twoe↵ects; the aggregated production is smoother due to the partially uncorre-lated series making its prediction easier, and secondly the forecast errors are

8

partially uncorrelated [9]. As wind power production is correlated in spaceand in time, probabilistic forecasts which provide information about the un-certainty of wind power forecasts for various locations and lead times, as wellas about their spatio-temporal dependencies, can improve both power sys-tem management and trading decision-making processes that rely on windpower forecasts. This can be achieved for example through random fieldbased statistical methods [20] [21].

In a recent survey of wind power forecast end users conducted as a part ofInternational Energy Agency Wind Task 36 on Forecasting, less than 10% ofend users currently make use of probabilistic forecasts [22]. The main reasonattributed to not transitioning to probabilistic forecasts despite their clearbenefits is di�culty interpreting the forecasts and a lack of transparency inregulations.

2.3. Towards Improving State-of-the-art Wind Power Forecasting

There are many research areas with potential to improve the currentstate of the art in wind power forecasting, many of which are identified inthe International Energy Agency Wind Task 36 Wind Power Forecasting[22]. Improved forecast skill of Numerical Weather Prediction (NWP) modeloutputs has significant potential to improve wind power forecasting. Po-tential developments include: better ensemble variability calibration, morerapid update cycle, improved model physics, and finer spatial and temporalresolution [22]. The key challenge areas for NWP advances in prediction per-formance can come from scientific and technological innovation in computing,development of the representation of physical processes in parameterizations,integrating advanced data assimilation algorithms, and improved character-isation of uncertainties through ensemble methods [13]. Other issues suchas the prediction of icing [23] and ramp events [24] are important. Anotherresearch area regards capturing the interaction between wind farms wake ef-fects as wind farms continue to increase in size and the wakes form one farmcan interact with another. Additionally, wind farm wake e↵ects could be in-cluded in NWP models; as wind farms increase in size, particularly o↵shorewind farms, they are becoming large enough to have a noticeable e↵ect onlocal weather [22].

International Energy Agency Wind Task 36 Wind Power Forecastingidentifies spatio-temporal forecasting as an area to improve prediction per-formance; and indicates that while several methods have been researched,they have yet to be implemented in operational models and further methods

9

should be investigated [22]. Further, the works have only been verified forhigh wind regions where the driving wind comes from the ocean such as Den-mark, Portugal, and Southern Australia, and thus case studies are requiredfor other regions. This online upstream information can be sourced fromanemometer measurements, radar, lidar, or as studied in this master thesis,the wind farm power production data.

2.4. Forecast Evaluation

For point forecasts, mean absolute error (MAE) and root mean squarederror (RMSE) are the most common performance measures [16]. MAE givesthe same weight to all errors, while the RMSE penalizes variance as it giveserrors with larger absolute values more weight than errors with smaller ab-solute values. The two measures together provide information for expectedbalancing costs and some insight of the distribution of errors [25]. Madsenet al. (2005) [26] propose a common set of performance measures includingbias, MAE, RMSE as well as to include the error distribution as a histogram.While this thesis focuses on point forecasts, it is worth noting that probabilis-tic forecasts are inherently more di�cult to evaluate; Pinson et al. (2007) [27]propose a framework for the evaluation of probabilistic forecasts to provideinformation on the reliability, sharpness, resolution and skill.

The performance measures of wind power forecasts are often made nor-malised to the installed capacity both in the market place and in literature[28]. Normalising measures to the installed capacity makes wind power plantswith lower capacity factors to appear to have better performance scores whencompared high capacity factor wind power plants. Further, the capacity fac-tors of the wind power plant are often not presented together with the eval-uation making interpretation of the forecast errors for power system studiesdi�cult [10]. Normalising performance to the capacity overcomes the issuefaced when normalising to the production in the case where the production is0. Normalising the performance to the cumulative production over the periodof evaluation similarly solves this problem, and is increasingly done both inresearch and the market place in the evaluation of solar power forecasts [29].Madsen et al. (2005) [26] recommend normalising to the installed capacityand not the production arguing that for new wind farms the installed capac-ity information is easy to assess, while the mean production is hard to knowwith su�cient accuracy. This master thesis presents performance measuresof MAE and RMSE which are normalised to the installed capacity for ease ofcomparison to other research for the reader. The capacity factor information

10

is also provided. For the case study of this thesis, the installed capacity is329 MW and the capacity factor is 0.298. Normalising the MAE and RMSEto the mean production would lead to error scores given by 1 / 0.298 = 3.36times higher than scores normalised by the installed capacity.

The end user of the forecast defines what constitutes a good forecast[9]. Electricity market participants seek to maximize their profit by min-imizing the balancing costs. This leads to a mean absolute error measurebeing more important evaluated over the temporal resolutions of the blocksin which electricity is traded [5]. Further, overestimation or underestimationin production can be incentivised if the balancing costs are expected to bea di↵erence between up-regulation and down-regulation prices [19]. Powersystem operators seek to operate the power system reliably by avoiding sig-nificant forecast errors while also minimizing the total errors and the use ofregulating power [5]. Among the most important features to forecast froma power system perspective are sudden and pronounced changes and rampevents, for example due to a passing storm front, which has been a growingarea of research [10].

The interpretation of the predictive performance of wind power forecastmodels requires care. The variability of the underlying wind power produc-tion is strongly correlated with the di�culty in forecasting the power pro-duction [30]. A non-exhaustive list factors a↵ecting both variability of theunderlying wind power production includes: the design of the wind farmsconsidered; the number of wind farms considered and the total power ca-pacity together with the geographic spread and cross correlation of the windfarms; the capacity factors of the wind farm, the topology of the surround-ings and whether the farms are onshore, near-shore, or o↵shore; the climate,weather, and season; the lead time considered; and the temporal resolutionof the forecast [30] [9]. Forecast error decreases with increasing geographicspread due to aggregated production being smoother due to the partiallyuncorrelated series making its prediction easier, and secondly the forecasterrors are also partially uncorrelated [9]. Increasing temporal resolution alsosmoothens the group wind power production and forecast error just as itdoes with increasing spatial domains. Wind power production averaged overshorter temporal domain of 1 to 5 minutes will show higher variability thanwind power production averaged over an hour [10]. The characteristics ofthe power curve including cut in speed, output speed, and cut-out speed aswell as the various gradients along the power curve influence the di�cultyof forecasting wind power. Steeper power curves with higher cut in speeds

11

are typically more di�cult to forecast because small changes in the windspeed result in larger changes in power produced. Certain weather and asso-ciated seasons are more di�cult to forecast than others such as major stormsand high instability [10]. Wind power forecasting models evaluated for windpower plants in regions with more challenging meteorological conditions willshow poorer model performance.

Given the di�culty in obtaining good information on all of potential fac-tors influencing predictability when comparing models that have been eval-uated on di↵erent sets of wind turbines, caution should be exercised. Thepractice of comparing one model performance to another model or bench-mark models with the same data set is a useful tool for the comparison ofpredictive skill. Simple persistence forecasts for the very short term windpower forecasts or simple first order auto regressive forecasts can be usedbenchmark models for short term wind power forecasting [26] [10] [31]. Itis worth noting however that persistence and auto regressive forecasts arenot a static benchmarks, wind farms with higher autocorrelation will havebetter performing persistence and auto regressive forecasts. Further, persis-tence and auto regressive models vary depending on the temporal resolutionon which they are based. Persistence and autoregressive models are used asbenchmarks herein.

2.5. Spatio-temporal Aspects for Wind Power Forecasting

The current state-of-the-art short term wind power forecasts has beenshown to be improved by accounting for the spatial temporal propagationof either wind power or wind power forecast errors between wind farms [32][31].

For lead times of 3 to 6 hours ahead current state-of-the-art short termwind power forecasts rely heavily on wind speed forecasts issued by Numer-ical Weather Prediction (NWP) [10]. NWP models are typically only issuedevery two to three hours [13] due to high computational expense of dataassimilation and physical model processes utilising millions of inputs frommany di↵erent sources including ground weather stations, weather balloons,and satellites. The time from initial observation are recorded to when thedata assimilation and NWP model processes are completed can be up severalhours on high-performance computers despite that the highest-performancecomputers employed in NWP place in the top 20 of the most powerful sys-tems in the world [13]. Weather is chaotic, chaos theory even has roots ine↵orts to quantify atmospheric predictability [33]. The more time between

12

observation and forecast periods, the more chaos can enter the system; andthis is part of the reason that weather forecast skill beyond a few days aheaddeteriorates rapidly. The challenges of this chaotic nature exists already forforecasting several hours ahead. The first minutes of the newly issued mete-orological forecast can be based on observations that are several hours old,which can extend up to being 6 hours old before the next NWP forecast up-date is issued. This weakness can be computationally e�ciently addressed byutilising spatial temporal aspects, taking upstream online information that isas little as several seconds old and incorporating that directly into the windpower forecast. This allows more recent spatio-temporal information thatwas previously only available through NWP outputs for longer time horizonsto be available to very short term forecasting.

Wind power production data from wind power plants is an attractive pre-dictor of power output because there is no conversion or scaling required [34].In contrast, wind speeds recorded by small aenometers are more susceptibleto turbulence from nearby structures and micro currents in the wind flow.Attaining quality live wind speed observations at hub height is challenging.Radar and Lidar development is reducing this challenge [35] [36]. Aenome-ters on the turbines su↵er from turbulence and disturbance of the wind flowfrom the blades and nacelle. Aenometers at multiple heights on dedicatedtowers can be costly and aenometers at ground level require extrapolationof the wind speed profile [37]. Despite variance of hub heights and topo-logical features, real-time measurements of wind power production data arereasonable reflections of the wind characteristics from which the propagationof meteorological conditions through space and time to another wind powerplant can be inferred [32] [31] [21].

The first approaches to capture spatio-temporal aspects showed signifi-cant improvement over persistence benchmarks but often relied upon expertknowledge of local meteorological conditions to choose the predictors fromgeographically dispersed sites. Larson and Westrick (2006) [38] and Gneit-ing et al. (2006) [39] examine forecasting a potential site located at the exitto the Columbia River Gorge using upstream meteorological observationsfrom the entrance of the Gorge. Damousis et al. (2004) [40] utilize informa-tion available upstream under prevailing wind conditions for the Thessalonikiarea. Hering and Genton (2010) [41] continued the previous studios near theColumbia River Gorge proposing trigonometric direction diurnal model andbivariate skew-t model statistical models. For these approaches to be ex-tended to areas with more complex topology or to larger areas would require

13

more generalised models without requiring significant expertise for identify-ing suitable model structures and estimation of the parameters.

Wind power and wind power forecast error fields propagate in space and intime depending on meteorological and geographic aspects, timing, and windfarm characteristics. The wind power and wind power forecast errors of windfarms in a given region is cross-correlated at significant values up to severalhours and several hundred kilometres apart [42] [43]. Von Bremen et al.(2010) as reported in [10] found the forecast error cross-correlation diminishesfaster over land than over sea due to the di↵erence between o↵shore andonshore wind conditions. Tastu et al.(2011) [42] propose a generalized modeland showed that the historical spatial and temporal propagation of windpower forecast error fields over a set of wind farms can significantly improveforecast skill. The recent wind power or forecast errors as they evolve in realtime for each wind farm together with meteorological forecast conditions areinputs into the model which can continuously form an updated forecast asnew data becomes available.

Recent research approaches to capture spatio-temporal aspects utilise arange of machine learning approaches including artificial neural networks,gradient boosting, random fields and classical time series approaches. Artifi-cial neural networks have thus far only received attention for spatio-temporalwind speed forecasting [44] [45] and for solar power forecasting [46]. It isnot yet clear if artificial neural networks bring additional value to time se-ries problems over classical approaches as significant expertise is required tocompare the state-of-the-art of both approaches. Bessa et al. (2015) [47]examine component-wise gradient boosting within a vector autoregressiveframework combining observations of solar generation through smart metersand distribution transformer controllers. Kou et al. (2013) [48] examinegenerate probabilistic wind power forecasts from an online sparse Bayesianmodel based on a warped Gaussian process including measurements fromnearby wind farms and NWP data. Wytock and Kolter (2013) [49] proposea sparse Gaussian conditional random field fitted with a second-order activeset method. The main weakness of this work is a significant computationalexpense of 160 minutes for a case study using seven wind farms. Lenzi etal. (2017) [21] recently examined Gaussian random fields to capture spatio-temporal aspects for probabilistic individual and aggregated forecasts. Lenziet al. show that capturing spatio-temporal dependency is required to gener-ate aggregated probabilistic forecasts that are calibrated, as is also shown byTastu et al. (2015) [20], who propose a Gaussian copula function to capture

14

the underlying spatio-temporal dependencies. To reduce the computationalexpense, Tastu et al. (2015) utilise sparse precision matrices, and Lenzi et al.(2017) translate the data using knot-based linear combinations from a reso-lution of 15 minutes to 3 hours. Lenzi et al. (2017) do not indicate the loss inpredictive skill from the use of knots, if any. Despite these e↵orts to reducecomputational expense, both approaches are computationally expensive.

Classical time series analysis has been the most popular approach in re-cent research of spatio-temporal aspects for wind power forecasting, whichcan be attributed to the clear interpretation of the approach. Tastu et al.(2010) [34] expand on the autoregressive model proposed in [42] to a vec-tor autoregressive model, which is extended to a probabilistic frameworkunder a parametric approach employing a truncated multivariate Normaldistribution. The vector autoregressive model shows significant predictiveperformance gains when made conditional to the average wind direction andaverage wind speed. Tastu and Pinson (2014) [50] further examine spatio-temporal aspects for conditional vector autoregressive models extended toa probabilistic framework. Both parametric and non-parametric approachesare examined and a non-parametric approach using adaptive quantile regres-sion with the conditional vector autoregressive model as input had superiorpredictive performance compared to a state-of-the-art benchmark based onlocal information only.

He et al. (2014) [51] propose a model using Markov chains that includethe ramp trend information and spatio-temporal dependencies to generateaggregated point and probabilistic forecasts. He et al. (2015) [52] proposea vector autoregressive model conditional to wind direction and speed fitwith sparsity-constrained maximum likelihood. In both papers, the authorsexamine the aggregate forecasts for a single 300.5 MW wind farm for a leadtime of 10 minutes and with an input data temporal resolution of 10 minutes.A significant limitation of this study is that the spatio-temporal dependencieson a relatively small spatial scale (5 x 5 km) of a single wind farm is unlikelyto be well captured by 10 minute averaged power data, leading to a mismatchbetween spatial and temporal resolutions. If one considers on this scale thatthe meteorological processes propagate through space at the wind speed,then for wind speeds over 30 km/h the spatio-temporal dependencies willonly partially be captured with the given temporal resolution. A smallertemporal resolution input would likely improve the results significantly.

Dowell and Pinson (2016) [32] propose a logit-normal distribution witha mean estimated by a vector autoregressive model and a variance given

15

by exponential smoothing for each site independently. They achieve fittingcomputation times of approximately 20 minutes with the proposed sparsityhalving the required time to fit the model. The model is tested for 22 windfarms spanning some 1800 km across South Australia, Victoria, Tasmaniaand the Australian Capital Territory. The methods of Dowell and Pinson(2016) su↵er from mismatch of spatial and temporal scales as in He et al.[51] [52], but in this case the spatial resolution is severely excessive for the5 minute temporal resolution of the data. This can mismatch is seen in theoverly sparse parameter matrix with only wind farms within approximately40 km of each other showing significant cross-correlation. With an order of3, only the last 15 minutes of recently observed wind power data is used toforecast the next lead time. Meteorological systems and related wind con-ditions cannot propagate 1800 km in 20 minutes. Significant computationperformance gains would be achieved if wind-farms outside a certain thresh-old distance were simply separated into separate formulations.

Cavalcante et al. (2017) [31] propose various sparse structures for a vectorautoregressive model using a least absolute shrinkage and selection operatorframework. The alternating direction method of multipliers is applied tofit the various least absolute shrinkage and selection operator models takingadvantage of parallel computing and rapid convergence. They achieve goodcomputational performance with time required to fit the model on the orderof seconds for the case study with hourly data for 66 wind farms located inthe same control area. Their model outperformed autoregressive and vectorautoregressive model benchmarks, as well as the sparse vector autoregressivemodel from Dowell and Pinson [32]. Cavalcante et al. (2017) stress that theirproposed approach can be extended to include features of other methods inthe research such as using the model for spatiotemporal correction of forecasterrors, making the model conditional to weather conditions, or generatingprobabilistic forecasts based on the logit-normal distribution.

16

3. Methodology

The methodology introduced here builds from wind power forecastingmodels based on local information only to a spatio-temporal model. Aspectsof the final proposed model are inspired by elements of the spatio-temporalmodel proposed by Tastu et al. (2010) [34]. The proposed model aimsto incorporate upstream online production information from all wind farmswithin a given region conditional to the meteorological conditions to improvethe forecasts. The introduced models made conditional to the meteorologicalconditions through regimes. The regimes are based on k-means clustering ofmeteorological variables from NWP models aiming to characterize distinctweather regimes under which the spatio-temporal aspects, as introduced inSection 3.2.4.

To the best of the authors knowledge such regime switching modes char-acterizing distinct weather regimes for wind power forecasting together withspatio-temporal time series models has not yet been explored. Very recentlya similar spatio-temporal forecast framework for very-short-term wind speedforecasting was published by Browell et al. (2017) [53]. They propose regimesbased on an atmospheric classification of wind and pressure fields at the sur-face level, and the geopotential height field at the 500 hPa level extractedfrom the MERRA-2 reanalysis dataset with an hourly temporal resolutionover the UK. To be implemented and run operationally, their method wouldneed to be adapted to determine the atmospheric mode from the analysis offorecasts provided by a Numerical Weather Prediction (NWP) model ratherthan from a historical reanalysis data set. Their paper eloquently introducesthe framework for spatio-temporal forecasting with clear notation and servesas inspiration to the layout and notation of the methodology introduced here.

All models introduced here were implemented into an existing wind powerforecasting model developed by Vitec Software AB, called Aiolos. The Aio-los wind power forecasting model is widely used in Northern and WesternEurope, and is based on Numerical Weather Prediction (NWP) models andadaptive statistical machine learning algorithms. Further information onAiolos Forecast Model is in Section 3.3.2. The forecasting models were de-veloped in Microsoft’s Visual Studio Enterprise 2016 and implemented inthe .NET framework in the programming languages C# and VB.NET. Theimplementation is further discussed in Section 3.3.1. The presented modelswere fit using ordinary least squares as discussed in Section 3.2.7. A casestudy was run as discussed in Section 3.1.

17

3.1. Case Study

3.1.1. Input DataThe wind power forecasting models proposed herein are tested on hourly

wind power measurements and hourly historical numerical weather prediction(NWP) model meteorological outputs data for 24 wind farms, 23 of whichare in Sweden and 1 of which is in Finland. The approximate location of the24 wind farms is shown below in Figure 1. The 22 more southern wind farmsof the 24 span an area of approximately 300 km x 300 km. This geographicspread matched with hourly resolution data is deemed by the author as asuitable spatio-temporal scale match; however, shorter resolution data mayincrease the benefits of the consideration of spatio-temporal information forthe wind farms that are in closer proximity. The total installed capacity ofthe 24 wind farms 329 MW and the capacity factor is 0.298. The hourly windpower measurements are the power averaged over each hour, with the unitsof MWh per hour. The utilised NWP model meteorological outputs includepressure at sea level, temperature, wind speed at at height of 80 meters, andwind direction. The NWP model meteorological outputs were taken fromthe historical predictions from SMHI’s HARMONIE-AROME for each of the24 wind farm locations.

The wind power forecasting models proposed herein are fit to histori-cal data from January 1st to December 31st 2015 for the 24 wind farms.Forecasts were then generated for each of the proposed models from 1 hourahead to 36 hours ahead for January 1st through to December 31st 2016and evaluated against historical data from that year. The author emphasisesthe importance of splitting the data set into separate fitting and evaluationsets. All of the proposed models are then compared, including the selectionof orders and regime combinations, through the generation and evaluation ofa total 12.1 billion out-of-sample point forecasts.

Evaluation of the forecast has been conducted using mean absolute error(MAE) and root mean squared error (RMSE) as they are the most commonperformance measures for point forecasts [16]. MAE gives the same weightto all errors, while the RMSE penalizes variance as it gives errors with largerabsolute values more weight than errors with smaller absolute values. Thetwo measures together provide information for expected balancing costs andsome insight of the distribution of errors [25]. The total installed capacity ofthe 24 wind farms is 329 MW and the capacity factor is 0.298. Normalisingthe MAE and RMSE to the mean production would lead to error scores given

18

Figure 1: Location of the 24 Wind Farms Examined in the Case Study

by 1 / 0.298 = 3.36 times higher than scores normalised by the installedcapacity.

3.2. Forecast Models3.2.1. Autoregression

First consider the problem of predicting the wind power for a single windfarm based on local information. The wind power measured at time t iscontained in the time series Y = {y1, y2, . . . , yT}. The wind power forecastingframework that follows aims to calculate at time t the predicted wind poweryt+⌧ , where ⌧ is the lead time, by solving for some function f⌧ (·) which mapsa vector of explanatory variables onto yt+⌧ , expressed as,

yt+⌧ |t = f⌧ (xt) . (1)

The function is solved by minimising some function of the forecast error,which is given by

19

et+⌧ = yt+⌧ |t � yt+⌧ . (2)

Wind power time series are serially correlated in time. The autocorrela-tions of wind power from the 24 wind farms that are examined in the casestudy are shown below in Figure 2.

Figure 2: Autocorrelation of wind power series from 24 wind farms.

It is thus reasonable for the recent past wind power values in Y to be inthe vector of explanatory variables,

yt+⌧ = f⌧ (y1, y2, . . . , yT ) , (3)

and for the function f⌧ (·) to be the weighted sum of p past values plus aconstant c⌧ , forming the familiar autoregressive model of order p,

yt+⌧ = c⌧ +p�1X

i=0

ai,⌧yt�i . (4)

The selection of the order p of the model and the estimation of the parametersc⌧ , ai,⌧ for i = 0, . . . , p� 1 will be introduced shortly in Section 3.2.7.

20

3.2.2. NWP Residual ModelsTwo additional models based on local information only are described.

Forecast error of the Aiolos wind power forecast model, as discussed in Sec-tion 3.3.2, based only on NWP information are also serially correlated intime, but to a weaker extent. The mean of the autocorrelation function ofwind power and NWP only Aiolos model forecast error from the 24 windfarms are shown below in Figure 3.

Figure 3: Autocorrelation function (ACF) of wind power series and forecast error seriesaveraged over 24 wind farms.

As discussed earlier, wind power forecast models with NWP wind speedinputs are outperformed by purely statistical based forecasts in the firstfew hours ahead [10]. Despite expected weaker predictive performance, itis still reasonable to propose an autoregressive model of the p recent pastNWP based model forecast error values contained in the time series W ={w1, w2, . . . , wT} as explanatory variables, forming an autoregressive modelof order p,

wt+⌧ = c⌧ +p�1X

i=0

ai,⌧wt�i . (5)

The predicted wind power yt+⌧ can then calculated by subtracting the pre-dicted forecast error wt+h from the original forecast qt+h, written

21

yt+⌧ = qt+⌧ � wt+⌧ . (6)

The Aiolos wind power forecast model with real time information is basedon a similar model that is a linearly fading moving average of the p recentpast NWP based model forecast error values contained in the time seriesW = {w1, w2, . . . , wT}. The selection of the order p of the Aiolos modeldetermines the parameters ai,⌧ to linearly fade from 1 to 0 for each orderi = 0, . . . , p� 1.

3.2.3. Vector AutoregressionThe interdependency among lagged wind power generation for spatially

dispersed sites can be captured by extending the autoregressive time seriesmodels to vector autoregressive models, as discussed in detail earlier in Sec-tion 2.5.

The cross-correlation of wind power at wind farm 5 with each of theother wind farms in the examined set of wind farms in the case study isshown below in Figure 4. A positive cross-correlation with a given series ata negative lag indicates that the respective series contains information thatcan be used to improve the forecasts. The cross-correlation function of aseries with itself is its autocorrelation function, and is included in Figure 4as the red dotted curve. It is interesting to note that the cross-correlationbetween a lag of 4 and 12 hours with several of the wind farms is strongerthan the auto-correlation information at those respective lags. The weakestcross-correlation curves are for the two wind farm sites that are much furtheraway, with one in Western Finland and the other in Northern Sweden asshown earlier in Figure 1.

Wind power measurements made at time t and N wind farms are embed-ded in the vector yt 2 RN . The vector-valued time seriesY = {y1,y2, . . . ,yT}enables a simple representation of a vector autoregressive process of order p,expressed as

yt+⌧ = C⌧ +p�1X

i=0

Ai,⌧yt�i , (7)

where Ai,⌧ 2 RN⇥N are matrices of parameters and C⌧ 2 RN are vectorsof constants. Unique parameter matrices are fit for each lead time ⌧ . Theparameter matrix is displayed for clarity as

22

Figure 4: Cross-correlation function (CCF) of wind power series between wind farm 5 and23 other wind farms.

Ai,⌧ =

2

6664

a11i,⌧ a12i,⌧ . . . a1Ni,⌧a21i,⌧ a22i,⌧ . . . a2Ni,⌧...

.... . .

...aN1i,⌧ aN2

i,⌧ . . . aNNi,⌧

3

7775. (8)

The parameters on the diagonal of Ai,⌧ give the autocorrelation e↵ects andthe o↵-diagonal parameters give cross-correlation between each of the windfarms. In other words, each wind farm power series not only now has pre-dictors given by the lagged values of its own series, but the lagged valuesof all of the other wind farms in the set. The input vector of wind powermeasurements of size N is given by

yt�i =

2

6664

y1t�i

y2t�i...

yNt�i

3

7775. (9)

The target vector of wind power measurements of size N is given by

23

yt+⌧ =

2

6664

y1t+⌧

y2t+⌧...

yNt+⌧

3

7775. (10)

The vector of constants of size N is given by

C⌧ =

2

6664

c1⌧c2⌧...cN⌧

3

7775. (11)

3.2.4. RegimesSince wind power is a result of meteorological conditions, it is desirable to

include variables in the models thus far introduced that are able to charac-terise the meteorological conditions. Tastu et. al (2010) [34] were able to im-prove the wind power predictive performance of a spatio-temporal model bymaking it conditional to the meteorological conditions, reducing the RMSEby between 4.08% to 18.46% at one hour ahead for groups of wind farms inDenmark over Wind Power Prediction Tool, a state-of-the-art wind powerforecast model. This was achieved through conditional parametric modelswith a linear structure, which are similar to vector autoregressive models butfor which the matrices of parameters Ai,⌧ 2 RN⇥N are replaced by smoothfunctions. These functions were conditioned on average wind speed and di-rection. Their key finding of note here is that the propagation of the forecasterrors was better explained by accounting for the meteorological conditions.

The meteorological conditions are to be captured here using regimes basedon clustering of meteorological variables. The meteorological variables usedto form the regimes include P pressure at sea level, T temperature, WSwind speed at at height of 80 meters, and WD wind direction. The datawas taken from the historical predictions of these variables from SMHI’snumerical weather prediction model HARMONIE-AROME for each of the24 wind farm locations in the case study.

Two types of regimes are proposed. The first type includes individualregime modes, which are based on meteorological conditions specific to each

24

site. The second type includes field regime modes, which are based on themean of each of the meteorological variables across all of the sites. For theautoregressive models the regime allocation can be made for each wind farmbased on the value of the meteorological variables at that location. However,for the vector autoregressive models, the regime allocation in its proposedform needs to be made based on the mean of each of the meteorologicalvariables across all sites.

The individual regime mode at time t is denoted bymt 2 s = {1, 2, . . . , k}and similarly, the field regime mode at time t is denoted by Mt 2 S ={1, 2, . . . , K}. Consider now auto regression and vector autoregressive modelsthat are mode specific, and which switch regimes by the allocated regimemode for a given data point. The auto regressive model in Equation (4)becomes

yt+⌧ = c⌧mt +p�1X

i=0

ai,⌧mtyt�i , (12)

and the vector autoregressive model in Equation (7) becomes

yt+⌧ = C⌧Mt +p�1X

i=0

Ai,⌧Mtyt�i . (13)

The individual regime mode mt and field regime mode Mt are discreteand result in separate auto regressive and vector autoregressive models re-spectively to be fit for each regime. The parameters are estimated using onlythe subset of the available training data which corresponds to the respectivemode.

In this spatio-temporal framework characterised by many parameters, itis beneficial to capture the propagation of these meteorological systems us-ing a limited number of regimes despite the complex nature of the systems.The desire for a limited number of regimes is that more regimes decreasethe number of samples available for parameters estimation, and insu�cienttraining data can result in poor parameter estimates. In the atmosphericclassification for regime switching vector autoregressive model for the pre-diction of wind speeds, Browell at al. found that 3 regimes were optimal outof the original 21 possible atmospheric regimes identified in the reanalysisdata [53].

25

It is intended that the regime switching models developed here be generalso that the models can be expanded to other countries or regions with outlocal meteorological expertise. K-means clustering was used to form regimesby allocating the meteorological variables, P , T , WS, and WD into clus-ters. K-means clustering is a type of unsupervised learning used to find andallocate groups in a given set of data. The forecast performance for each fore-cast model was optimised by searching for the optimal number of regimes,given by k for individual regimes and K for field regimes, reflecting here thenumber of clusters to be considered. Each of the considered meteorologicalvariables was normalised using Gaussian normalization. The K-means algo-rithm is initialised with allocation to a randomly selected cluster. In the firststep, the means of the clusters, or centroids, are calculated and updated. Inthe next step, each data point qt is assigned to its nearest centroid mi, bythe squared Euclidean distance, given by

argminmi2M

vuutnX

i=1

(qt �mi)2.

The algorithm is then looped through these two steps until there is no changein the clustering or the maximum number of iterations constraint is met. Theclustering results of the K-means algorithm may be a local optimum and nota global optimum.

The scatter plots of each of these meteorological variable forecasts isshown below with respect to the measured wind power for a given windfarm in Figure 5. The relation between the wind speed forecast at a givenlocation and the measured wind power reflects the familiar wind power curve,and the large variability of the measured wind power for a given wind speedforecast is a reminder of the challenging and stochastic nature of wind powerforecasting. While the relations between wind power and the remaining threemeteorological variable forecasts of pressure, temperature and wind directionare weaker, each of the meteorological variables has potential to provide fur-ther information to inform the allocation of the regime, and to better describethe spatio-temporal propagation of the meteorological conditions.

The result of one of the K-means clustering algorithm runs showing theclustered groups with field wind speed and field temperature as the selectedmeteorological variables is shown below for illustration in Figure 6.

All possible combinations of the 4 meteorological variables are consideredand tested as potential sets of explanatory variables for allocation of the

26

Figure 5: Plots of measured wind power at one wind farm over a year against forecastsfor wind speed, wind direction, temperature and pressure.

meteorological regime. The intuitive logic behind testing all possible combi-nations of the meteorological variables is that pressure, temperature, windspeed and wind direction can each provide additional information to bettercharacterize a given regime. It can be conjectured that low pressure eventswill on average be associated with certain wind processes. If pressure is thenconsidered together with wind speed and wind direction information can fur-ther characterise the low pressure event. If pressure, wind speed and winddirection are then considered together with temperature it may better cap-ture potential convection processes. Ultimately the underlying processes thatdefine the meteorological regimes do not need to be known, the hypothesisis merely that clustering these meteorological variables together can bettercharacterize the meteorological processes and their evolution than consideringthem individually. The optimal combination of meteorological variables andthe optimal number of clusters are selected for each model proposed hereinby examining the improvements in predictive skill of each of the respective

27

Figure 6: K-means Clustered Groups with Field Wind Speed and Field Temperature asSelected Meteorological Variables

models.

3.2.5. Two-step Regime Switching ModelRecall that two types of regimes are proposed, the first type includes

the individual regime modes and the second type includes the field regimemodes. It is conjectured that it is beneficial to consider both types in asingle model. The individual regime modes capture meteorological processesspecific to each location, and the field regime modes capture the propagationof meteorological processes from one site to another. For this reason, a twostep model is proposed. In the first step a regime switching autoregressivemodel as proposed in Equation (12), with regimes based on location specificmeteorological variables, is used to generate forecasts. In the second step, aregime switching spatio-temporal vector autoregressive model is proposed tocapture the spatio-temporal propagation of the forecast errors of the regimeswitching autoregressive model from the first step, with regimes based on themean of each of the meteorological variables across all sites.

Consider the p recent past forecast error series R = {r1, r2, . . . , rT} ofwind power forecasts made with the proposed regime switching autoregres-sive model, as given in Equation (12). These forecast errors can be set asexplanatory variables, forming a vector autoregressive model of order p,

28

rt+⌧ = C⌧ +p�1X

i=0

Ai,⌧Mtrt�i , (14)

Similar to Equation (6), the final predicted wind power vectors, for clarity,denoted Ft+h, can then formed by adding the predicted forecast error vectorsrt+h to the vector of original regime switching autoregressive model forecastvectors yt+h. Forming a model expressed as

Ft+⌧ = rt+⌧ + yt+⌧ . (15)

This approach allows for the information from the meteorological regimes ateach individual location to corrected independently under individual regimesbefore considering the spatio-temporal dependencies under a regime charac-terized by the mean meteorological conditions across all locations.

3.2.6. SeasonalityWind power time series exhibit exhibit diurnal seasonality due to the

underlying diurnal seasonality of wind speeds and meteorological conditions.This can be observed by the slight increase in autocorrelation and cross-correlation values at a lag of 24 hours in Figure 2 and Figure 4 respectively. Asimple way of characterising any additional remaining part of the underlyingmeteorological conditions not already captured in the previously introducedregimes is therefore is to include the time of day exogenously. This can beachieved by including a boolean function dh(t) of a set of variables h 2 Hwhere H is determined by the temporal resolution and is the number ofdiscrete measurements in a day. For the case of hourly data, h is the time ofthe day, H = {0, 1, . . . , 23], th is the time of day of the associated value inthe time series at time t, and

dh(t) =

(1 for th = h

0 for th 6= h. (16)

The intercept c⌧ of the regime switching auto regressive model given in Equa-tion (12) is then replaced by bh,⌧ reflecting time dependent intercepts, and

29

the regime switching autoregressive model with exogenous diurnal variables,is given by

yt+⌧ =p�1X

i=0

ai,⌧mtyt�i +X

h2H

bh,⌧mtdh(t+ ⌧) . (17)

A similar extension can be made for the other vector autoregressive mod-els, take the regime switching vector autoregressive model as an example,as presented in Equation (13), where the vector of constants C⌧ 2 RN isthen replaced by Bh,⌧ 2 RN reflecting a vector of time dependent intercepts,and the set of diurnal variables is replaced by a vector of diurnal variablesBh,⌧ 2 RN . The regime switching vector autoregressive model with exoge-nous diurnal variables, can then be expressed as

yt+⌧ =p�1X

i=0

Ai,⌧Mtyt�i +X

h2H

Bh,⌧Mtdh(t+ ⌧) , (18)

where Ai,⌧ 2 RN⇥N are matrices of parameters.

3.2.7. Parameter EstimationThe model parameters �⌧,s =

⇥A0,⌧,s . . . Ap�1,⌧,s B0,⌧,s . . . B23,⌧,s

⇤

can be estimated by minimising some function of the forecast errors givena historical dataset. A unique set of parameters is fit for each lead timeindependently.

For 1 t T � ⌧ of wind power measurements with regime mode s andassociated diurnal variables the matrix of input data of size (pN + 24)⇥ T ,is given by a horizontal concatenation of explanatory variables, expressed as

X⌧,s =

2

666666666666664

. . . y1t . . .

. . . y2t . . ....

. . . yNt . . ....

. . . yNt�p�1 . . .

. . . d0(t+ ⌧) . . ....

. . . d23(t+ ⌧) . . .

3

777777777777775

. (19)

30

The corresponding target matrix of wind power measurements for p + ⌧ t T of size N ⇥ (T � p � ⌧) is given by horizontal concatenation of thetarget vectors

Y⌧,s =

2

6664

. . . y1t . . .

. . . y2t . . ....

. . . yNt . . .

3

7775. (20)

The above parameter matrices, input matrices and target matrices are de-scribed for the full regime switching vector autoregressive model with di-urnal seasonality. The respective matrices for the other proposed modelsare formed by removing the non-relevant elements, for example the regimeswitching vector autoregressive model without diurnal seasonality would notcontain the related diurnal variables and parameters.

The sum of the squared error is chosen here as the cost function for all ofthe proposed models, known as ordinary least squares, with the parametersestimates given by the solution to

argmin�i,⌧

k E⌧,s k22= argmin�i,⌧

k Y⌧,s � �⌧,sX⌧,s k22 . (21)

The ordinary least squares estimator may be eloquently expressed as

�⌧,s = (X0

⌧,sX⌧,s)�1X

0

⌧,sY⌧,s . (22)

This cost function can be extended by introducing regularisation to con-strain resulting solution potentially enabling better generalisation [54]. Reg-ularsation induces sparsity into the parameter matrix, mitigating the poten-tial weakness of ordinary least sqaures to overfit problems in high parameterspaces [55]. The least absolute shrinkage and selection operator or ridge re-gression are common penalty parameters for regularisation methods [54]. Themodel can also be extended to utilise parallism and more rapid convergencecan by solving the problem with alternating direction method of multipliersas proposed by Calvalcante et al. (2017) [31]. It is conjectured regularisationand parallelism are not required for the relatively small case study of hourlyresolution for 24 wind farms and thus are not studied here, they are prudent

31

future extensions for the consideration of significantly larger data sets withsmaller time resolution and more wind farms as the fitting problem growswith the square of the number of wind farms considered, increasing di�cultyof fitting the problem and computation expense. This conclusion is supportedby the recent works of Browell et al. (2017) [53] which produced promisingwind speed forecast improvement for 23 sites using a regime switching vectorautoregressive model without employing regularisation.

3.3. Implementation

3.3.1. Software DevelopmentThe proposed wind power forecasting models were developed Microsoft’s

Visual Studio Enterprise 2016 and implemented in Vitec’s .NET frameworkin the programming languages C# and VB.NET. Microsoft Visual Studiois an integrated development environment and its source code editor, buildautomation tools and debugger were used to build the proposed models.There are several reasons for this integrated development environment choiceover developing the model in another other environments or language such asR or MATLAB. It was intended from the beginning that this thesis make asignificant contribution to the currently implemented wind power forecastingin industry. Aiolos forecasting studio is implemented in the .NET frameworkand so implementing the new model in the .NET framework allows the modelto be implemented with ease into the existing Aiolos Forecast Studio.

As discussed in Section 2.3, incorporating spatio-temporal aspects canbe computationally expensive, and writing and developing the model in thechosen level of abstraction language allows for good computational perfor-mance to be obtained. The implementation of the model in terms of itscomputational e�ciency is an important focus as it is closely linked to theoverall e↵ectiveness of the model. In the case one or more measured windpower data series are not usable, due to the data being unavailable or dueto planned downtime, it can not be included in the spatio-temporal modeland the model becomes incomplete. It is practical to give the end user theability to rapidly refit the model excluding the wind farm series with issues,and to generate spatio-temporal forecasts for the other wind farms withoutissue. The wind farm series with issues can then have forecasts generatedindependently of the other wind farms. As such issues happen regularly inpractice, it is conjectured that the proposed regime switching vector autore-gressive model will be fit multiple times per week even even though the model

32

parameters will only drift slowly with time if fit to a reasonable amount ofhistorical data.

Minor preprocessing of data is achieved through a script that skips missingdata and records the counts. The data for all wind farms have less than 1%of missing data for the presented case study and can be considered of verygood quality. As discussed earlier, scripts were developed for future caseswhere series may need to removed from the spatio-temporal models due tomissing data and allocated for forecasts based on local information only. Asimple post processing script checks that all forecasts were within the boundsof 0 and the installed capacity of the wind farm.

All development and testing was performed in Microsoft’s Visual StudioEnterprise 2016 on a laptop running 64 bit Windows 10 Enterprise withan Intel Core i7-4600U CPU @ 2.1 GHz processor, and 8.00 GB of RAM.For the presented case study, all models proposed herein achieved readingof data and fitting times below 30 seconds apart from the two step modelwhich took up to 1 minute. The two step model involved generating forecastsbased on locational information before making spatio-temporal correctionsas presented in Section 3.2.5. A year of point forecasts can be generatedin less than 2 seconds, and evaluated for the purpose of this thesis also inless than 2 seconds. Better model performance can be achieved with smallextensions to the proposed models by introducing regularisation and by usingfitting algorithms that can take advantage of parallelism and more rapidconvergence such as in the models proposed by Calvalcante et al. (2017)[31].

3.3.2. Aiolos Forecast ModelThis Master thesis has been written in collaboration with the Energy

group of Vitec Software AB. Vitec is a leading developer and supplier ofEnergy software on the Northern European market with approximately 170customers, providing software for companies including E.ON, Fingrid, Sven-ska Kraftnat, Fortum, Helsinki Energy, and Vattenfall. The Aiolos windpower forecasting model is part of Vitec’s software-as-a-service Aiolos Fore-casting Studio which provides forecasts of electricity load, heat load, run ofriver hydro production, solar power production, and wind power production.Users of Aiolos Forecasting Studio include transmission system operators,distribution system operators, power producers, retailers, and traders.

An introduction to the Aiolos forecasting model follows. Aiolos windpower forecasting model utilises an ensemble of numerical weather predictions

33

(NWP) from the weather forecast providers they cooperate with which variesdepending on the region under consideration. Most of the NWP providersincludes weather parameters for the next 72 hours at half hourly or hourlyresolutions. Longer resolution NWP parameter predictions are typical oftime horizons beyond three days and up to ten days into the future. The keyweather parameters from NWP include wind speed, wind direction, pressure,temperature, and pressure all of which are provided at various heights from10 m to 800 m above sea level. The latitude and longitude of each wind powerplant are used to interpolate NWP meso-scale forecast to better capture thesite specific wind conditions. The benefit of this interpolation is that windfarms near the edge of grid boundaries can incorporate forecasts from theneighbouring boundary. The vertical wind profile is estimated based on bothNWP wind speeds at various heights, pressure, and temperature. The windspeed is vertically interpolated to the hub height of each wind power plant. Amodel based on a trimmed non-linear regression between the historical powerproduction and weather forecasts is then fit. The model is made conditionalto the wind direction using regimes. The number of wind direction regimes isoptimised based on an evaluation of performance against historical data. Ifhistorical data is not available, adjusted power curves are used. Wind powerforecasts are then estimated for each available NWP provider and variousensembles of those NWP models to form preliminary production forecasts.Depending on performance of each preliminary production forecast for eachwind farm, an optimised weighting of available NWP ensembles is formed forthe final forecast. The optimal ensemble can also change with the predictionhorizon. Maintenance functionality in Aiolos Forecast Studio allows mainte-nance planning and and to account for scheduled maintenance in the forecast.Real-time production data is utilised to individually correct the very shortterm forecasts taking advantage of auto correlation of forecast errors at windpower plants. The Aiolos final wind power forecast is displayed in tables andgraphs in Aiolos Forecast Studio and are available for export in a numberof formats. Aiolos forecast studio allows the import of external wind powerforecasts from other wind power providers which can also be weighted withthe final Aiolos forecast.

The historical Aiolos model forecasts, production, and meteorologicaldata were exported from Vitec’s Aiolos Forecast Studio. Aiolos ForecastStudio export functionality to comma separated value (.csv) file types wasutilised to allow the model to be run without fully integrating the model toAiolos Forecast Studio. The full integration of the model requires integrating

34

the functionality with the user interface which was outside of the scope of thisthesis. This work will later be completed by the user interface developers atVitec. The Aiolos model forecasts, production, and meteorological data wasread from the .csv files types. Vitec has developed a custom data type withintheir .NET dynamic library for Aiolos Forecast Studio to allow large timeseries data types to be e�ciently stored with other important characteristicsof the underlying asset and time information allowing for ease of access tothe data. This data type was used to store the read data from the .csv fileand allows for the settings of the model to be both easily adjusted as wellas assisting with the full implementation of the model into Aiolos ForecastStudio.

35

4. Results and Analysis

The results and analysis from the case study follow. All the wind powerforecasting models proposed in Section 3 are tested on hourly wind powermeasurements and hourly historical numerical weather prediction (NWP)model meteorological outputs data for the 24 wind farms in the case study.Additionally persistence forecasts are generated for benchmarking purposes,where the forecast at some future time is given by the most recently measuredwind power. The performance measures of mean absolute error normalisedto the installed capacity (MAE) and root mean squared error normalisedto the installed capacity (RMSE) are utilised here. For the most part allperformance measures reported are the average of the individual errors, butare simply referred to at MAE and RMSE. Note it must be stressed theperformance measures are not evaluated on the aggregation of the forecasterrors. Aggregation of forecast errors significantly reduces the total forecasterror scores [30]. The total installed capacity of the 24 wind farms is 329MW and the capacity factor is 0.298. Normalising the MAE and RMSE tothe mean production would lead to error scores given by 1 / 0.298 = 3.36times higher than scores normalised by the installed capacity. For furtherdetail of the case study see earlier discussion in Section 3.1.

Up to an order of 12 is considered for all of the proposed autoregressiveand vector autoregressive models. For all models an order of 2 is foundto be the most appropriate selection, with higher orders giving negligibleimprovement in predictive performance.

4.1. Base Models

The models defined as base models here are those without diurnal season-ality and without regimes conditioned by meteorological conditions. Each ofthe base models outperforms the persistence benchmark for MAE and RMSEscores as seen in Figure 7 and Figure 8. The poorest performing base modelsare those based on the errors of the Aiolos wind power forecasting modelbased on NWP. The Aiolos wind power forecast model with real time mea-surements included is based on the linearly fading moving average of therecent past NWP based model error values, denoted Fade Residual NWP. Itis outperformed by the autoregressive model of the recent past NWP basedmodel forecast error values, denoted AR Residual NWP, and given earlier inEquation 6. The poorer performance reflects excessive weight give to NWPbased information which worsens the potential predictive skill for short term

36

wind power forecasting. This result supports the common finding in the re-search that models with emphasis on statistical methods outperform thosewith emphasis on physical methods for times horizons up to 3 to 6 hoursahead [10]. Figure 3 shown earlier further explains this result, with the au-tocorrelation of NWP based observed forecast errors declining much fasterwith time compared to the autocorrelation of measure wind power.

Figure 7: Mean Absolute Error (MAE) Normalised to Installed Capacity and Root MeanSquared Error (RMSE) Normalised to Installed Capacity Performance of Base Models

The base autoregressive model based on measured wind power measure-ments, denoted AR and given earlier in Equation (4), outperforms the basevector autoregressive model also based on measured wind power measure-ments beyond 2 hours ahead, denoted VAR and given earlier in Equation(7). While the VAR model outperforms AR for 1 hour ahead as expected,the result of weaker performance for 2 hours ahead and beyond was not ex-pected as previous research on similar models indicates the consideration ofspatio-temporal information to improve the prediction skill compared to toforecasts based on local information alone. A potential explanation for thisresult is poor fitting of the model using ordinary least squares without regu-larisation to induce sparsity to the parameter matrix. He et al. (2015) [52],Dowell and Pinson (2016) [32] and Cavalcante et al. (2017) [31] all utiliseregularisation in the fitting of their respective models. However, He et al.(2015) Dowell and Pinson (2016) only examine the predictive performance of

37

the first lead time. While Browell et al. (2017) [53] do not utilise regularisa-tion, they only benchmark their vector autoregressive model to persistenceand it is thus possible that it does not outperform an autoregressive model.

The weakness of not utilising regularisation in the fitting process for theVAR model is mitigated through the Two-step Model proposed in Section3.2.5. This can be seen as the two-step model, here without regimes or diurnalcomponents, denoted VAR Residual AR, is seen to outperform the AR model.This e↵ectively allows all autoregressive information to be removed from theVAR fitting procedure, reducing the di�culty to achieve a good fit for themodel [54].

The percentage improvement of each of the base models predictive skillcompared to persistence is shown below in Figure 8. Comparing RMSE andMAE measures for 1 hour ahead, it is interesting to note that only VARand VAR Residual AR marginally outperform persistence for the first leadin terms of MAE. In contrast, in terms of RMSE all of the base modelsoutperform persistence for the first lead time. This indicates that each of thebase models is able to reduce the larger errors more so than persistence asRMSE penalises outliers.

Figure 8: Mean Absolute Error (MAE) Normalised to Installed Capacity and Root MeanSquared Error (RMSE) Normalised to Installed Capacity Percentage Improvement of BaseModels Over Persistence

The improvement shown for the AR, VAR, and VAR Residual AR base

38

models compared to the Fade residual NWP model is already of significantvalue for Vitec’s Aiolos wind power forecasting model which currently hasits real time corrections based on the Fade Residual NWP model.

4.2. Regime Models

Making the autoregressive models conditional to the meteorological con-ditions through the proposed regimes significantly improves the predictiveskill. Recall from Section 3.2.4, the meteorological conditions are capturedhere using regimes based on K-means clustering of the P pressure at sealevel, T temperature, WS wind speed at at height of 80 meters, and WDwind direction. The optimal number of regimes specified for the K-meansalgorithm ranges from 3 to 16 depending on the model and meteorologicalvariables considered. The consideration of more than 2 meteorological vari-ables as inputs to form the regimes is found to have negligible impact on thepredictive performance of all the benchmarked models. This indicates thatthe consideration of 3 or more of the considered meteorological variables doesnot provide additional information that is beneficial. This may be conjec-tured to be due to the noisy and interdependent nature of these variables,as can be seen in the earlier displayed plots in Figure 5. The confoundednature of these variables also can explain why each of the combinations ofthese variables resulted in similar improvements in forecast skill.

The various combinations of regime inputs to form the regime switch-ing autoregressive models show similar improvements compared to the baseautoregressive model as shown below in Figure 9 and Figure 10. These im-provements in forecast skill align with previous research on regime switchingautoregressive models from Gallego et al. (2011) [15] and from Trombe andPinson (2012) [16].

The regimes that included forecast wind speed as an input to the K-meansclustering show good performance and are only outperformed combinationof pressure and temperature as inputs to the K-means clustering. The com-bination of temperature and pressure predictions as inputs to the K-meansclustering with 7 groups was found to have the best average performanceover the first 6 hours, with percentage improvement in MAE over the baseautoregressive model of 3.9%, 11.1%, 18.0%, 23.7%, 28.5%, 31.9%, for eachof the first 6 respective lead times. It is interesting that the combination oftemperature and pressure outperforms regimes based on wind direction andwind speed. Making wind power foreacasting models conditional to winddirection and wind speed is not only the most common in research [10], they

39

Figure 9: Mean Absolute Error (MAE) Normalised to Installed Capacity and Root MeanSquared Error (RMSE) Normalised to Installed Capacity Performance of Regime SwitchingAutoregressive Models

are the most logical. It is shown that pressure and temperature deserve fur-ther consideration for very short term wind power forecasting models whichare made conditional to meteorological conditions.

The percentage improvement compared to the base autoregressive modelincreases significantly with the time horizon. Notably, each of the regimeswitching autoregressive models under performs the base autoregressive modelby between 0 to 1% in terms of RMSE for the first hour ahead, while showing3.6% to 4.0% improvement in terms of MAE, as shown in Figure 10

It can be asserted that making the autoregressive model conditional tothe forecast meteorological conditions allows the model to better capture thelocal underlying meteorological processes that determine wind power gener-ation, namely wind speed and how it evolves through time.

4.3. Diurnal Seasonality

The consideration of the diurnal seasonality extensions, as presented inSection 3.2.6, showed some improvement in predictive skill over the basemodels. However the addition of diurnal seasonality extensions to the regimeswitching autoregressive and vector autoregressive models showed negligibleimprovements. Browell et al. (2017) [53] reports a similar finding that thediurnal seasonality provides negligible additional benefits after the spatio-temporal model is made conditional to the meteorological conditions. It is

40

Figure 10: Mean Absolute Error (MAE) Normalised to Installed Capacity and RootMean Squared Error (RMSE) Normalised to Installed Capacity Percentage Improvementof Regime Switching Autoregressive Models Over Base Autoregressive Model

conjectured that the diurnal seasonality is captured by the regimes based onmeteorological conditions. This can be explained as the diurnal seasonalityexhibited by wind power is a function of the diurnal seasonality of the under-lying meteorological conditions, for example, wind speeds may be higher onaverage during afternoons at coastal locations due to local convection causinglocal sea-breezes.

4.4. Two-step model benchmark

The two-step regime switching model, as proposed in Section 3.2.5 achievedthe best performance of all the proposed models. Figure 11 shows the MAEand RMSE performance of the two-step regime switching model denotedTwo-step Model, the best performing regime switching autoregressive modeldenoted AR Reg(P,T), the base autoregressive model denoted AR, the FadeResidual NWP as currently used in the Aiolos wind power forecast model,and persistence. All of the proposed models show significant improvementover persistence and the Fade Residual NWP model of Vitec’s Aiolos windpower forecasting model, especially for 2 hours head onwards. The percentagereduction in MAE of the two-step model over the regime switching autore-gressive model benchmark is 4.7% for 1 hour ahead and increases to 34.1%at 6 hours ahead. The reductions in forecast error from the hour ahead to sixhours equates to subtantial monetary value in terms of reduced balanacing

41

costs for Vitec’s customers [5]. Recall that the users of Aiolos ForecastingStudio include transmission system operators, distribution system operators,power producers, retailers, and traders. For transmission and distributionsystem operators with increasing penetration of wind power, the improvedforecasts are important to the reliable and economic power system operation.The larger reduction of forecast error at several hours can be utilised by thecustomers for more e�cient and economic optimisation, both for internal en-ergy dispatch and externally for customers active on intra-day markets whocan trade their position [5].

Figure 11: Mean Absolute Error (MAE) Normalised to Installed Capacity and Root MeanSquared Error (RMSE) Normalised to Installed Capacity Performance

Recall the first step of the two-step model is the generation of prelimi-nary forecasts using a regime switching autoregressive model with regimesbased on location specific meteorological variables. In the second step, theregime switching spatio-temporal vector autoregressive model captures thespatio-temporal propagation of the forecast errors of the regime switchingautoregressive model from the first step, with regimes based on the mean ofeach of the meteorological variables across all sites.

The best performing two-step model is based on the errors of the bestperforming regime switching autoregressive model, with pressure and tem-perature as inputs to to the K-means clustering to form the individual regimemodes.

All of the two-step models based on the various combinations of regime

42

switching autoregressive models was found to improve the predictive per-formance of the underlying autoregressive model. Consideration of highernumbers of regime modes is found to reduce the predictive performance ofthe two-step model.

The additional consideration of field regime modes based on the mean ofeach of the meteorological variables across all of the sites for the vector au-toregressive in the two-step model is found to provide negligible improvementif the individual regime mode is included in the underlying autoregressivemodel. For the case of the base autoregressive model without regimes as theunderlying model, the inclusion field regime modes for the vector autoregres-sive in the two-step model is found to provide a modest improvement.

As discussed in detail earlier in Section 2.5, making vector autoregressivemodels conditional to the meteorological conditions was shown by Tastu etal. (2010) [34] to improve the predictive performance. Tastu et al. (2010)benchmarked the additional percentage reduction in RMSE to WPPT, andreports that the making vector autoregressive models conditional to the windspeed and wind direction reduced the RMSE for the hour ahead by a further0.72% to 3.5%. As Tastu et al. (2010) did not test an autoregressive modelmade conditional to the wind direction and wind speed, it is not clear if theirreported improvements for making the model conditional to wind directionand wind speed would also be shown for an autoregressive model.

It is conjectured that consideration of the proposed field regime modesshould further improve the predictive performance of the two-step model.However, this result is not found here, and may be attributed to splitting thedata set into many sets which reduces the quality of the fit of the model dueto insu�cient number of data points. Regularisation may mitigate this issueenabling better fitting of the model given the higher parameter to datapointratios involed with splitting the data into many discrete sets [55].

The MAE of the two-step model is 5.51% for 1 hour ahead and increasesto 6.53% at 6 hours ahead. The RMSE of the two-step model is 11.13% for1 hour ahead and increases to 12.67% at 6 hours ahead. The percentageimprovement in MAE and RMSE of the two-step model over each of theother respective models is shown below in Figure 12. The deterioration ofpredictive performance is extremely slow and significantly outperforms thestate-of-the-art in research at several hours ahead [10].

There is some dispersion in forecast performance for the individual windfarms. The individual MAE of the two-step model ranges from 3.13% to7.44% for 1 hour ahead and increases to values from 3.68% to 8.80% at 6

43

hours ahead. The individual RMSE of the two-step model ranges from 5.99%to 13.72% for 1 hour ahead and increases to values from 7.84% to 14.10% at6 hours ahead. The variance in individual performance is quite consistantaccross all of the proposed forecasting models, reflecting the variance is at-tributable to the variability and related di�culty in forecasting of the windfarms [30]. This shows the importance of benchmarking model performanceto other models rather than comparing MAE and RMSE measures directlyfrom wind farm to another, even for those within the same case study, asdiscussed in detail earlier in Section 2.4.

The percentage reduction in MAE of the two-step model over the baseautoregressive model benchmark is 4.6% for 1 hour ahead and increases to34.1% at 6 hours ahead. The percentage reduction in RMSE of the two-step model over the base autoregressive model benchmark is 0.6% for 1 hourahead and increases to 21.8% at 6 hours ahead. The smaller percentageimprovement in RMSE compared to MAE for the first hour ahead reflectsthe existence of small numbers of large forecast errors.

In comparison, Calvalcante et al. (2017) [31] reported a percentage re-duction in RMSE over an autoregressive model benchmark of 5.7% at 1 hourahead, which increases to 7% at 3 hours head, and then fades to 4% at 6hours ahead. The slower deterioration of forecast skill at longer time hori-zons reflects that the spatio-temporal information persists over these timehorizons. The quality of the spatio temporal information is reflected in Fig-ure 4 shown earlier showing very strong cross-correlation at lags up to 12hours. The cross-correlation information shown in Calvalcante et al. (2017)are weaker and fade faster after 3 hours, which is reflected in the deteriora-tion of their forecast skill with time. The greater improvement in RMSE ofCalvalcante et al. (2017) for the first hour is conjectured to be attributed tothe superior fit of their model by the use of regularisation achieved with theleast absolute shrinkage and selection operator framework. It is interestingto note that Calvalcante et al. (2017) reported a reduction in MAE over anautoregressive model benchmark of 7.54% at 1 hour ahead compared to the4.6% reduction at 1 hour ahead for the propose two-step model which is notsubstantially di↵erent.

The percentage reduction in MAE of the two-step model over the regimeswitching autoregressive model benchmark is 0.8% for 1 hour ahead andincreases to 2.4% at 6 hours ahead. The percentage reduction in RMSE of thetwo-step model over the regime switching autoregressive model benchmarkis 1.3% for 1 hour ahead and increases to 3.4% at 6 hours ahead. While

44

Figure 12: Mean Absolute Error (MAE) Normalised to Installed Capacity and Root MeanSquared Error (RMSE) Normalised to Installed Capacity Percentage Improvement of Two-Step Model Over Each Other Respective Model

these improvements are modest, the regime switching autoregressive modelcan be considered among the state-of-the-art approaches for point forecasts[15] [16], and even modest reductions in forecast error at the hour ahead canequate to subtantial monetary value in terms of reduced balanacing costs [5].

Each of the proposed models and associated features that lead to keyimprovements in the forecasts are reflected in Figure 11 and Figure 12. Theconsideration of the measured wind power directly into the autoregressivemodel showed significant improvements compared to the consideration ofobserved forecast error based on a NWP model. Making the autoregressivemodels conditional to the weather conditions through the regimes producedanother substantial improvement in forecast skill. Finally, the two-step modelcapturing spatio-temporal aspects through vector autoregression improvedthe performance of the regime switching autoregressive models.

45

The 1 hour ahead forecast errors of the best performing regime switchingtwo-step model for one of the wind farms over the whole of 2016, and ahistogram showing the distribution of these forecast errors are shown belowin Figure 13 and Figure 14 respectively. The wind farm chosen has a themedian of forecast performance out of the set of wind farms. The distributionof the forecast errors is approximately Gaussian. A short summary of thedistribution of forecast errors follows: 59% of the forecast errors are within+/- 5%, 79% of the forecast errors are within +/- 10%, 93.8% of the forecasterrors are within +/- 20%, and 99.5% of the forecast errors are within +/-40%.

Figure 13: One Hour Ahead Forecast Errors of the Regime Switching Two-step Model forOne Wind Farm for Every Hour of 2016

46

Figure 14: Histogram of One Hour Ahead Forecast Errors of the Regime Switching Two-step Model for One Wind Farm for All Hours of 2016

The regime switching two-step model hour ahead forecasts and the mea-sured wind power are shown for illustrative purposes for the same wind farmfor a randomly chosen week in 2016 below in Figure 15. The wind farmvalues are normalised to the installed capacity for data privacy.

Figure 15: Measured Wind Power and Regime Switching Two-step Model 1 Hour AheadForecast Normalised to Installed Capacity for a Randomly Chosen Week in 2016.

While short term wind power forecasting with emphasis on statisticalmethods is usually only evaluated for the first 6 hours ahead, it can be

47

interesting to examine the performance over longer time horizons. The per-formance of the two-step model from 1 hour ahead to 36 hours ahead interms of MAE and RMSE is shown below in Figure 16. The model showsquite strong performance from 2 hours ahead up to 12 hours ahead beforea wind power forecasting model with more emphasis on NWP inputs wouldtake over with better predictive skill [10]. This improvement in forecast skillover the 2 hours ahead to 12 hours ahead can translate to more e�cient andeconomic intra-day dispatch operations and planning.

Figure 16: Mean Absolute Error (MAE) Normalised to Installed Capacity and Root MeanSquared Error (RMSE) Normalised to Installed Capacity of Regime Switching Two-stepModel

48

5. Discussion

5.1. Contribution and Significance

This master thesis contributes to the limited research on spatio-temporalaspects for short term wind power forecasting, examining time series modelsfocusing on incorporating spatio-temporal aspects conditional to meteoro-logical conditions. It has been shown that the inclusion of spatio-temporalaspects consistently improves the state-of-the-art in wind power forecast-ing. The promising performance encourages further examination of spatio-temporal aspects.

Novel regime switching autoregressive and vector autoregressive modelsare proposed. The best performing two-step model utilises an underlyingregime switching autoregressive model with a vector autoregressive modelto take advantage of cross-correlation between sites incorporating upstreamonline wind power measurement information from all wind farms within agiven region. The proposed novel regimes are formed using K-means clus-tering based on forecast meteorological conditions and show significant im-provements in forecast skill for all of the time-series models.

The case-study thoroughly examines the performance of each of the pro-posed models. The strong performance of the models from 2 hours aheadup to 12 hours ahead reflects the quality of the proposed regimes as wellas the quality of the underlying spatio-temporal information. This showsthe importance of matching the spatial and temporal scales to capture thespatio-temporal aspects. The hourly resolution here is seen to be appropiatefor the majority of the wind farms within a 300 km x 300km area. Shortertemporal resolution would likely improve the forecast performance of windfarms within a 50 km x 50 km of each other for very short time horizons upto 1 hour ahead. This consideration of matching spatial and temporal scalesis one that has lacked in previous research, such as in he et al. (2014) [51],He et al. (2015)[52] and Dowell and Pinson (2016) [32], as discussed earlierin Section 2.5.

The model was implemented into the .NET framework of Vitec Software’sAiolos Forecast Studio, which is widely used in Northern and Western Eu-rope. All of the proposed models were found to have significantly lower meanabsolute error and root mean squared error compared to the Aiolos modeland autoregressive model benchmarks. The proposed regime switching au-toregressive model is ready to be implemented in Aiolos Forecast Studio withlittle work. The proposed two-step model is di↵erent to current Aiolos Fore-

49

cast Studio design and would require additional changes to the user interfacebefore being able to be implemented. It is hoped the promising results of thevector autoregressive model are to be further investigated by Vitec, namelythe inclusion of regularisation in the fitting process and the consideration ofshorter time resolution case studies.

The improved short term wind power forecast available to Vitec will in-form operation and trading decisions and translate to significant reductionsin balancing costs for Vitecs Aiolos Forecasting Studio customer’s consistingof transmission system operators, distribution system operators, power pro-ducers, retailers, and traders. An estimate of this value is di�cult to estimateas much of the data required is not publicly available. A simple estimate ofthe value of the improved wind power forecasts for Vitec’s customers is esti-mated here to give an idea of the order of magnitude of the benefits. Viteccurrently produces forecasts for approximately 12 GW of installed capacityof wind power with an estimated capacity factor of 0.3. The prosed regimeswitching vector autoregressive model reduces the mean absolute foreast er-ror at 1 hour ahead by 4.7% and at 2 hours ahead by 29.8% compared to thecurrent Aiolos wind power forecast with real time corrections. The systembalancing costs due to wind power forecast error is between 1 to 4.5 Euro/ MWh for wind penetrations of up to 20% of gross energy demand; this isapproximately up to 10% of the wholesale value of the wind energy [4]. Ifthe consideration of wind power forecasts for system operation and planningis conducted within the hour ahead or within two hours a head, the improve-ment is valued at as much as between 9.4 million Euros to 42.3 million Eurosin reduced balancing costs. As similar improvements have been shown forthe consideration of spatio-temporal aspects down to 5 minutes ahead, thesereductions in balancing costs hold for shorter time horizons [32]. Hodge etal. (2015) [56] and Xie et al. (2014) [57] present more detailed evaluationsof significant value of more accurate forecasting.

5.2. Limitations and Future Research and Development

A limitation of the proposed vector autoregressive models is the use ofordinary least squares as the fitting method without regularisation. It isconjectured that the generally weaker performance of the direct base vectorautoregressive models compared to base autoregressive models is explainedby poor fitting of the model using ordinary least squares without regulari-sation. Regularsation induces sparsity into the parameter matrix enablingbetter generalisation and mitigating the potential weakness of ordinary least

50

sqaures to overfit problems in high parameter spaces [55]. The least absoluteshrinkage and selection operator or ridge regression are potential penalty pa-rameters to extend the fitting method to include regularisation [54]. He etal. (2015) [52], Dowell and Pinson (2016) [32] and Cavalcante et al. (2017)[31] all utilise regularisation in the fitting of their respective models. Thislimitation was partially overcome by use of the two-step model reducing thedi�culty of fitting the vector autoregressive problem by e↵ectively remov-ing the strong autoregressive components. However, it is likely poor fittingmethodology did not allow the field regime modes in the two-step model tocontribute significantly to better capturing the spatio-temporal propagationof the forecast errors from the regime switching autoregressive model.

Reasonable computational performance, is achieved for the proposed mod-els, with fitting times all less than 1 minute. Parallelism and e�cient fittingalgorithms are not required for the relatively small case study of hourly res-olution for 24 wind farms and thus is not studied here, it is a prudent futureextensions for the consideration of significantly larger data sets with smallertime resolution and more wind farms as the fitting problem grows with thesquare of the number of wind farms considered, increasing di�culty of fittingthe problem and computation expense. The model can be extended to utiliseparallism and more rapid convergence by solving the problem with alternat-ing direction method of multipliers as proposed by Calvalcante et al. (2017)[31]. Calvalcante et al. (2017) achieve good computational performance withtime required to fit the model on the order of seconds for the case study withhourly data for 66 wind farms located in the same control area.

The match of spatial and temporal scales is proposed here to be veryimportant in the consideration of spatio-temporal aspects for improving pre-diction performance. This requires further investigation to establish whichtemporal resolutions are best suited to which spatial scales and geographicdispersion for optimal prediction perforance. Few studies explicitly tacklethe problem of how spatio-temporal correlation changes with the averagingperiod as most studies consider a single temporal resolution when investi-gated spatio-temporal correlations. Louie et al. (2014) [58] and St. Martinet al.(2015) [59]report correlation decreases with the averaging period and de-creases exponentially with separation distance. All averaging periods under38 hours exhibit a distance at which wind power becomes uncorrelated thatis proportional to the averaging period. Shorter averaging periods becomeuncorrelated for shorter distances.

It is interesting to note in one study for solar power, Perez et al. (2012)

51

examined the distance at which fluctuations of the clear sky index becameuncorrelated for various temporal resolutions, and found a resolution of 20seconds, 1 minute, 5 minutes, and 15 minutes became uncorrelated at dis-tances of 500 m, 1 km, 4 km and 10 km respectively. These distances reflectthe limits the consideration of spatio-temporal information can improve so-lar power forecasts. While solar power is clearly a function of very di↵erentprocesses to wind power, these spatio-temporal aspects are related mostlyto the propagation of clouds, which in turn are a function of wind speed, sosome parallels may be conjectured.

Regarding the inclusion of spatio-temporal aspects for time series mod-els, erroneous measured data from one wind farm or changes in the farmsproduction that are not related to the meteorological conditions have poten-tial to harm the forecast quality for other wind farms and can be seen as asignificant limitation. For implementation of spatial temporal forecasts, thepreprocessing of incoming production data is very important to the reliabil-ity of the forecasts and preprocessing standards should be developed and nosuch standards currently exist [28].

Making the models conditional to the meteorological conditions throughthe proposed regimes encourages continued research and development of al-ternative model approaches, as also called for by Tastu et al.(2010) [34],Dowell and Pinson (2016) [32] and Cavalcante et al. (2017) [31]. The pathweather systems follow as they propagate through space and time is due topressure di↵erentials and is not in the same direction as the wind directionnor is the speed the same as the wind speed. Wind direction and windspeed alone can not always capture the evolution of the meteorological pro-cesses and can potentially explain why the regimes based only on pressureand temperature has showed marginally superior performance to those basedon wind speed and wind direction. Topological features such as mountains,valleys and trees can funnel wind flow and can result in large changes tothe wind direction at the surface compared to the geostrophic wind direc-tion and wind speed. Thus regimes based on geostrophic wind speeds anddirections together with atmospheric stability would be interesting to inves-tigate. Hidden Markov-Regime switching models based on an unobservableprocess have showed superior performance over threshold regime switchingmodels based on observable lagged values [17] [16]. It would be interesting forMarkov-Regime switching models to be compared to the proposed K-meansclustering regimes in future research.

52

6. Conclusion

The proposed novel models are successfully implemented into the .NETframework of Vitec Software’s Aiolos Forecast Studio, which is widely usedin Northern and Western Europe. The proposed short term wind power fore-casting time series models made conditional to meteorological conditions andwhich incorporate spatio-temporal aspects show significant improvements inforecast skill. This is reflected by significant reductions in mean absoluteerror and root mean squared error compared to the Aiolos model and au-toregressive model benchmarks. The vector autoregressive framework in theproposed two-step model is able to take advantage of cross-correlation be-tween sites incorporating upstream online production information from allwind farms within a given region. The regimes are formed using K-meansclustering based on forecast meteorological conditions, with regimes basedon wind speed, and the combination of pressure and temperature showingthe best ability to characterize the regimes to improve the predictive perfor-mance. Appropriate matching of spatial and temporal scale is important toachieving full performance of spatio-temporal wind power forecasting models.The selected case study shows a good match between spatial and temporalscales as reflected by the high cross correlations between the sites and strongperformance of the two-step model up to a lead time of 12 hours ahead.

The improved short term wind power forecasts will inform operation andtrading decisions and translate to significant reductions in balancing costsfor Vitecs customers. The improvement is valued at as much as between9.4 million Euros to 42.3 million Euros in reduced balancing costs. Spatio-temporal aspects conditioned to meteorological conditions for wind powerforecasting shows to be promising for improving current state-of-the-art windpower forecasting, reducing balancing costs, and ultimately facilitating thecontinued integration of wind power.

53

7. References

[1] C.B. Field, V.R. Barros, D.J. Dokken, K.J. Mach, M.D. Mastran-drea, M. Chatterjee T.E. Bilir, Y.O. Estrada K.L. Ebi, R.C. Genova,B. Girma, E.S. Kissel, A.N. Levy, S. MacCracken, P.R. (eds.) Mastran-drea, and L.L. White. IPCC, 2014: Summary for Policymakers. In:Climate Change 2014: Impacts, Adaptation and Vulnerability - Con-tributions of the Working Group II to the Fifth Assessment Report.Technical report, IPCC Intergovernmental Panel on Climate Change,2014.

[2] GWEC. Global Wind Statistics 2016. Technical report, Gobal WindEnergy Council, 2017.

[3] Wind Energy - Technology roadmap -. Technical report, InternationalEnergy Agency, 2013.

[4] Hannele Holttinen, John Olav Tande, Ana Estanqueiro, Emilio Gomez-lazaro, Bernhard Lange, Michael Milligan, Charles Smith, Mark O Mal-ley, and Jody Dillon. Summary of experiences and studies for WindIntegration IEA Wind Task 25. Proceedings of WIW2013, pages 22–24,2013.

[5] Juan M. Morales, Antonio J. Conejo, Henrik Madsen, Pierre Pinson, andMarco Zugno. Integrating Renewables in Electricity Markets, volume 205of International Series in Operations Research & Management Science.Springer US, Boston, MA, 2014.

[6] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose,C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. VanCutsem, and V. Vittal. Definition and Classification of Power SystemStability. 2004.

[7] L. Jones and C. Clark. Workshop, International Integration, Large-scale Power, Wind Systems, Power Networks, Transmission Wind, O↵-shore Plants, Power Germany, Langen. In 10th Internationbal WorkshopLarge-Scale Integration of Wind Power Into Power Systems, 2011.

[8] Mohammad Reza Hesamzadeh and Darryl R. Biggar. The Economicsof Electricity Markets. Wiley-IEEE Press.

54

[9] Jari Johannes Miettinen and Hannele Holttinen. Characteristics of day-ahead wind power forecast errors in Nordic countries and bene fi ts ofaggregation. pages 959–972, 2017.

[10] Caroline Giebel, Gregor; Brownsword, Richard; Kariniotakis, George;Denhard, Michael; Draxl. The State-Of-The-Art in Short-Term Predic-tion of Wind Power: A Literature Overview, 2nd edition. 2011.

[11] Christoph Weber. Adequate intraday market design to enable the inte-gration of wind energy into the European power systems. Energy Policy,38(7):3155–3163, jul 2010.

[12] Audun Botterud, Zhi Zhou, Jianhui Wang, Jean Sumaili, Hrvoje Keko,Joana Mendes, Ricardo J. Bessa, and Vladimiro Miranda. Demand Dis-patch and Probabilistic Wind Power Forecasting in Unit Commitmentand Economic Dispatch: A Case Study of Illinois. IEEE Transactionson Sustainable Energy, 4(1):250–261, jan 2013.

[13] Peter Bauer, Alan Thorpe, and Gilbert Brunet. The quiet revolution ofnumerical weather prediction. Nature, 525(7567):47–55, 2015.

[14] G.W. Chang, H.J. Lu, Y.R. Chang, and Y.D. Lee. An improved neuralnetwork-based approach for short-term wind speed and power forecast.Renewable Energy, 105:301–311, 2017.

[15] C. Gallego, P. Pinson, H. Madsen, A. Costa, and A. Cuerva. Influence oflocal wind speed and direction on wind power dynamics ? Application too↵shore very short-term forecasting. Applied Energy, 88(11):4087–4096,nov 2011.

[16] Pierre-julien Trombe and Pierre Pinson. High-resolution forecasting ofwind power generation with regime- switching models and o↵-site ob-servations. pages 1–24, 2012.

[17] P. Pinson, L.E.A. Christensen, H. Madsen, P.E. Sorensen, M.H. Dono-van, and L.E. Jensen. Regime-switching modelling of the fluctuations ofo↵shore wind generation. Journal of Wind Engineering and IndustrialAerodynamics, 96(12):2327–2347, dec 2008.

55

[18] Tilmann Gneiting. Editorial: Probabilistic forecasting. Journal of theRoyal Statistical Society: Series A (Statistics in Society), 171(2):319–321, apr 2008.

[19] Pierre Pinson, Christophe Chevallier, and George N. Kariniotakis. Trad-ing Wind Generation From Short-Term Probabilistic Forecasts of WindPower. IEEE Transactions on Power Systems, 22(3):1148–1156, aug2007.

[20] Julija Tastu, Pierre Pinson, and Henrik Madsen. Space-Time Trajecto-ries of Wind Power Generation: Parametrized Precision Matrices Undera Gaussian Copula Approach. In Modelling and Stochastic Learning forForecasting In High Dimensions. Springer. Lecture Notes in Statistics,volume 217, pages 267–296. 2015.

[21] Amanda Lenzi, Ingelin Steinsland, and Pierre Pinson. Benefits of spatio-temporal modelling for short term wind power forecasting at both indi-vidual and aggregated levels. 2017.

[22] G Giebel, J Cline, H Frank, W Shaw, P Pinson, B-M Hodge, G Karin-iotakis, J Madsen, and C M?hrlen. Wind power forecasting: IEA WindTask 36: Future Research Issues. Journal of Physics: Conference Series,753(3):032042, sep 2016.

[23] Rolv Erlend Bredesen, Norway Rene Cattin, Niels-Erik Clausen, Den-mark Neil Davis, Pieter Jan Jordaens, Zouhair Khadiri-Yazami, RebeckaKlintstrom, Andreas Krenn, Ville Lehtomaki, Goran Ronsten, MatthewWadham-Gagnon, and Meventus Wickman. IEA Wind TCP Recom-mended Practice 13 Wind Energy in Cold Climates, 2nd Edition. pages1–43, 2017.

[24] Lei Yang, Miao He, Junshan Zhang, and Vijay Vittal. Support-Vector-Machine-Enhanced Markov Model for Short-TermWind Power Forecast.IEEE Transactions on Sustainable Energy, 6(3):791–799, jul 2015.

[25] T. Chai and R. R. Draxler. Root mean square error (RMSE) or meanabsolute error (MAE)? Geoscientific Model Development, 7(3):1247–1250, 2014.

56

[26] Henrik Madsen, Pierre Pinson, George Kariniotakis, Henrik Aa. Nielsen,and Torben Nielsen. Standardizing the Performance Evaluation of Short-TermWind Power Prediction Models. Wind Engineering, 29(6):475–489,dec 2005.

[27] Pierre Pinson, Henrik Aa. Nielsen, Jan K. M?ller, Henrik Madsen, andGeorge N. Kariniotakis. Non-parametric probabilistic forecasts of windpower: required properties and evaluation. Wind Energy, 10(6):497–516,nov 2007.

[28] G Giebel, J Cline, H Frank, W Shaw, P Pinson, B-M Hodge, G Karin-iotakis, J Madsen, and C Mohrlen. Wind Power Forecasting: IEA WindTask 36 and Future Research Issues. Journal of Physics ConferenceSeries, 753, 2016.

[29] Richard Perez, James Schlemmer, Karl Hemker, Sergey Kivalov, AdamKankiewicz, and John Dise. Solar energy forecast validation for ex-tended areas & economic impact of forecast accuracy. Conference Recordof the IEEE Photovoltaic Specialists Conference, 2016-Novem(1):1119–1124, 2016.

[30] Jan Dobschinski. How good is my forecast? Comparability of windpower forecast erros. 13th International Workshop on Large Scale Inte-gration of Wind Power into Power Systems as well as on TransmissionNetworks for O↵shore Wind Farms, pages 1–5, 2014.

[31] Laura Cavalcante, Ricardo J. Bessa, Marisa Reis, and Jethro Browell.LASSO vector autoregression structures for very short-term wind powerforecasting. Wind Energy, 20(4):657–675, apr 2016.

[32] Jethro Dowell and Pierre Pinson. Very-Short-Term Probabilistic WindPower Forecasts by Sparse Vector Autoregression. IEEE Transactionson Smart Grid, 7(2):763–770, 2016.

[33] Edward N. Lorenz. Deterministic Nonperiodic Flow, 1963.

[34] Julia Tastu, Pierre Pinson, and Henrik Madsen. Multivariate Condi-tional Parametric Models for a spatiotemporal analysis of short-termwind power forecast errors. . . . of the European Wind Energy . . . , 2010.

57

[35] L Valldecabres, W Friedrichs, L von Bremen, and M Kuhn. Spatial-temporal analysis of coherent o↵shore wind field structures measuredby scanning Doppler-lidar. Journal of Physics: Conference Series,753:072028, 2016.

[36] Pierre-julien Trombe, Pierre Pinson, and Henrik Madsen. AutomaticClassification of O↵shore Wind Regimes With Weather Radar Observa-tions. IEEE Journal of selected topics in applied earth observations andremote sensing, 7(1):116–125, 2014.

[37] Michael C. Brower. Overview of a Wind Resource Assessment Cam-paign. In Wind Resource Assessment, pages 13–22. John Wiley & Sons,Inc., Hoboken, NJ, USA, may 2012.

[38] Kristin A. Larson and Kenneth Westrick. Short-term wind forecastingusing o↵-site observations. Wind Energy, 9(1-2):55–62, jan 2006.

[39] Tilmann Gneiting, Kristin Larson, Kenneth Westrick, Marc G Genton,and Eric Aldrich. Calibrated Probabilistic Forecasting at the StatelineWind Energy Center. Journal of the American Statistical Association,101(475):968–979, sep 2006.

[40] I.G. Damousis, M.C. Alexiadis, J.B. Theocharis, and P.S. Dokopoulos.A Fuzzy Model for Wind Speed Prediction and Power Generation inWind Parks Using Spatial Correlation. IEEE Transactions on EnergyConversion, 19(2):352–361, jun 2004.

[41] Amanda S. Hering and Marc G. Genton. Powering Up With Space-TimeWind Forecasting. Journal of the American Statistical Association,105(489):92–104, mar 2010.

[42] Julija Tastu, Pierre Pinson, Ewelina Kotwa, Henrik Madsen, and Hen-rik Aa. Nielsen. Spatio-temporal analysis and modeling of short-termwind power forecast errors. Wind Energy, 14(1):43–60, jan 2011.

[43] Robin Girard and Denis Allard. Spatio-temporal propagation of windpower prediction errors. Wind Energy, 16(7):999–1012, oct 2013.

[44] Younes Noorollahi, Mohammad Ali Jokar, and Ahmad Kalhor. Usingartificial neural networks for temporal and spatial wind speed forecastingin Iran. Energy Conversion and Management, 115:17–25, may 2016.

58

[45] Ummuhan Baaran Filik and Tansu Filik. Wind Speed Prediction UsingArtificial Neural Networks Based on Multiple Local Measurements inEskisehir. Energy Procedia, 107:264–269, 2017.

[46] A.G.R. Vaz, B. Elsinga, W.G.J.H.M. van Sark, and M.C. Brito. An ar-tificial neural network to assess the impact of neighbouring photovoltaicsystems in power forecasting in Utrecht, the Netherlands. RenewableEnergy, 85:631–641, jan 2016.

[47] Ricardo J. Bessa, Artur Trindade, and Vladimiro Miranda. Spatial-Temporal Solar Power Forecasting for Smart Grids. IEEE Transactionson Industrial Informatics, 11(1):232–241, feb 2015.

[48] Peng Kou, Feng Gao, and Xiaohong Guan. Sparse online warped Gaus-sian process for wind power probabilistic forecasting. Applied Energy,108:410–428, aug 2013.

[49] Matt Wytock and J. Zico Kolter. Large-scale probabilistic forecasting inenergy systems using sparse Gaussian conditional random fields. In 52ndIEEE Conference on Decision and Control, pages 1019–1024. IEEE, dec2013.

[50] Julija Tastu, Pierre Pinson, Pierre Julien Trombe, and Henrik Mad-sen. Probabilistic forecasts of wind power generation accounting for ge-ographically dispersed information. IEEE Transactions on Smart Grid,5(1):480–489, 2014.

[51] Miao He, Lei Yang, Junshan Zhang, and Vijay Vittal. A Spatio-Temporal Analysis Approach for Short-Term Forecast of Wind FarmGeneration. IEEE Transactions on Power Systems, 29(4):1611–1622,jul 2014.

[52] Miao He, Vijay Vittal, and Junshan Zhang. A sparsified vector au-toregressive model for short-term wind farm power forecasting. In 2015IEEE Power & Energy Society General Meeting, pages 1–5. IEEE, jul2015.

[53] Jethro Browell, Daniel R. Drew, and Kostas Philippopoulos. ImprovedVery-short-term Wind Forecasting using Atmospheric Classification.pages 1–20, 2017.

59

[54] Yanfei Wang, Changchun Yang, and Anatoly G. Yagola. Optimizationand Regularization for Computational Inverse Problems and Applica-tions. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011.

[55] D. Donoho and Victoria Stodden. Breakdown Point of Model SelectionWhen the Number of Variables Exceeds the Number of Observations.In The 2006 IEEE International Joint Conference on Neural NetworkProceedings, pages 1916–1921. IEEE, 2006.

[56] B.-M Hodge, A Florita, J Sharp, M Margulis, D Mcreavy, and LockheedMartin. The Value of Improved Short- Term Wind Power Forecasting.Technical report, National Renewable Energy Laboratory, 2015.

[57] Le Xie, Yingzhong Gu, Xinxin Zhu, and Marc G. Genton. Short-TermSpatio-Temporal Wind Power Forecast in Robust Look-ahead PowerSystem Dispatch. IEEE Transactions on Smart Grid, 5(1):511–520, jan2014.

[58] Henry Louie. Correlation and statistical characteristics of aggregatewind power in large transcontinental systems. Wind Energy, 17(6):793–810, jun 2014.

[59] Clara M St. Martin, Julie K Lundquist, and Mark A Handschy. Variabil-ity of interconnected wind plants: correlation length and its dependenceon variability time scale. Environmental Research Letters, 10(4):044004,apr 2015.

60

TRITA TRITA-EE 2017:144ISSN 1653-5146

www.kth.se