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High frequency data analysis for wind energy applications Escalante Soberanis, Mauricio Alberto 2015

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HIGH FREQUENCY DATA ANALYSIS FOR WIND ENERGY APPLICATIONS by  Mauricio Alberto Escalante Soberanis  Ingeniero Físico, Universidad Autónoma de Yucatán, 2007 Maestro en Ingeniería, Universidad Nacional Autónoma de México, 2008  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September, 2015  © Mauricio Alberto Escalante Soberanis, 2015 ii  Abstract  High frequency data (HFD) of three site studies in different geographic locations were analyzed to reproduce the power spectrum illustrated by Van der Hoven in 1957. His work represents the basis of wind energy standards such as averaging and variability in the frequency domain. The results presented in this thesis unveil discrepancies with Van der Hoven’s approach. A major eddy-energy peak is illustrated at a period of 2 days and a smaller eddy-energy peak contribution at frequencies higher than the region known as the spectrum gap. The variance in the microscale region was calculated by integrating the Power Spectral Density (PSD) over the corresponding range of frequencies. The economic value of this energy variance based on the turbulence kinetic energy of the wind data set is calculated. It is also concluded that, given the results of the study, HFD analysis in the frequency domain uncovers eddy-energy peaks that determine energy fluctuations in the short and long terms.  An algorithm was developed to detect delay times in the turbulence kinetic energy (TKE) and the energy dissipation rate ε on a continuous basis (thereby identifying the highest cross-correlation coefficients between them). The Kolmogorov turbulence order is applied to calculate the energy dissipation rate ε through the identification of the inertial subrange. The time scale in the variations of both parameters was successfully calculated and it is close to the time the air takes to circulate between the surface and the top of the atmosphere’s mixed layer. High correlation coefficients are found in the three site studies from 4am to 8am, and from 8pm to 12pm. The cross-correlation function also determines delay time scales in the range of 10-20 minutes and approximately 2 hours. The energy dissipation rate can be calculated to characterize wind variability in a particular iii  site that might affect the performance of a wind turbine. With these results, more information is generated that can be used in the wind turbine’s control system routines to improve its response under wind turbulence variations. iv  Preface  The literature review, data analysis, and design of the algorithms were done by M.A. Escalante Soberanis under the supervision of Dr. Walter Mérida. The initial and final drafts of all manuscripts were prepared by M.A. Escalante Soberanis with revisions edited and approved by Dr. Walter Mérida.  A version of Chapter 3 has been published. M.A. Escalante Soberanis and W. Mérida, “Regarding the influence of the Van der Hoven spectrum on wind energy applications in the meteorological mesoscale and microscale,” Renew. Energy, vol. 81, pp. 286-292, 2015.  A version of Appendix A has been published. M.A. Escalante Soberanis, A. Alnaggar, and W. Mérida, “The economic feasibility of renewable energy for off-grid mining deployment,” Extr. Ind. Soc., vol. 2, pp. 509-518, 2015.  The results illustrated in Appendix A were presented in the Nunavut Mining Symposium, in Iqaluit, Canada, April 2015.  A version of Chapter 4 has been submitted for publication. M.A. Escalante Soberanis and W. Mérida, “Analysis of Energy Dissipation and TKE Using HFD for Wind Energy Applications”.  Permission for reproduction of materials published in journals Renewable Energy and Extractive Industries and Society is covered under Elsevier B.V.’s author’s rights for scholarly purposes. v  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ...........................................................................................................................v List of Figures .................................................................................................................................x List of Symbols ........................................................................................................................... xiv List of Abbreviations ...................................................................................................................xv Acknowledgements .................................................................................................................... xvi Dedication ................................................................................................................................. xviii Chapter 1: Introduction ................................................................................................................1 1.1 Wind Energy in Numbers ............................................................................................... 1 1.2 Wind Potential Assessment............................................................................................. 2 1.3 Forecasting ...................................................................................................................... 5 1.4 Relevant Parameters of Wind Turbines .......................................................................... 6 1.4.1 Turbulence Intensity ................................................................................................... 6 1.4.2 Capacity Factor, Tip-Speed Ratio and Power Coefficient .......................................... 6 1.4.3 Pitch Angle................................................................................................................ 12 1.4.4 Yaw Alignment ......................................................................................................... 14 1.4.5 Grid Balance ............................................................................................................. 16 1.5 Motivation ..................................................................................................................... 16 1.6 Research Questions ....................................................................................................... 19 1.6.1 Is HFD Analysis Valuable? ...................................................................................... 19 vi  1.6.2 What is the Economic Significance of Finer Temporal Data Analysis? .................. 19 1.7 Literature Review and Key Contributions .................................................................... 20 1.7.1 Predicting Wind Conditions at the Turbine in Advance of Gusts ............................ 20 1.7.2 Extreme Wind Estimation and Structural Mitigation ............................................... 23 1.8 Objectives ..................................................................................................................... 25 1.9 Thesis Outline ............................................................................................................... 26 1.9.1 Wind Turbine Dynamic Response ............................................................................ 26 1.9.2 Eddy-Energy Contribution in Wind Fluctuation....................................................... 27 1.9.3 Wind Energy Dissipation and Turbulence Kinetic Energy....................................... 28 Chapter 2: Dynamic Response of Wind Turbines ....................................................................29 2.1 Wind Turbine Response ................................................................................................ 29 2.2 Turbine Response Models............................................................................................. 31 2.3 Contributions................................................................................................................. 34 2.4 Methodology ................................................................................................................. 34 2.4.1 Elements of Dynamic Response ............................................................................... 34 2.4.2 Calculations and Results ........................................................................................... 36 2.5 Significance and Repercussions .................................................................................... 49 2.6 Conclusions ................................................................................................................... 52 Chapter 3: Eddy-Energy Contributions in the Frequency Domain ........................................53 3.1 The Van der Hoven Spectrum....................................................................................... 53 3.2 Motivation ..................................................................................................................... 57 3.3 Statement of the Problem .............................................................................................. 58 3.4 Methodology ................................................................................................................. 58 vii  3.5 Results ........................................................................................................................... 62 3.6 Discussion ..................................................................................................................... 67 3.6.1 Financial Analysis ..................................................................................................... 69 3.7 Conclusions ................................................................................................................... 72 Chapter 4: Energy Dissipation Rate and TKE..........................................................................74 4.1 Wind Analysis in the Frequency Domain ..................................................................... 74 4.2 Objective ....................................................................................................................... 78 4.3 Methodology ................................................................................................................. 80 4.4 Results ........................................................................................................................... 84 4.5 Autocorrelation of TKE ................................................................................................ 94 4.6 Conclusions ................................................................................................................... 96 Chapter 5: Conclusions and Recommendations .......................................................................99 5.1 Conclusions ................................................................................................................... 99 5.2 Future Work and Recommendations .......................................................................... 100 Bibliography ...............................................................................................................................102 Appendices ..................................................................................................................................109 Appendix A Wind Energy in Off-Grid Mining Deployments in Northern Canada ............... 109 A.1 Introduction ............................................................................................................. 109 A.2 Case Study .............................................................................................................. 111 A.3 Methodology ........................................................................................................... 112 A.4 Results ..................................................................................................................... 113 A.5 Summary ................................................................................................................. 120 Appendix B Probabilistic Forecast of Wind Speed Peaks Using HFD .................................. 122 viii  B.1 Methodology ........................................................................................................... 122 B.2 Conclusions ............................................................................................................. 126  ix  List of Tables  Table 1.1 Summary of the literature review and expected contributions of the present thesis. ... 24 Table 2.1 Parameters required calculating settling times of a wind turbine under wind turbulence occurrence ..................................................................................................................................... 35 Table 2.2 Integration parameters in (2.8) for the six wind speed averages and maximum gust duration for the range of gust frequencies. The lambda ratios for the sensitivity analysis are also included. Significant figures are set taking the resolution of the physical resolution of the digitizing software ........................................................................................................................ 43 Table 3.1 Description of the data used to calculate E of the wind speed HFD ............................. 59 Table 3.2. Chosen time periods and energy spectrum values for the three sites of study ............ 69 Table 3.3. Estimated costs of wind energy as a function of wind speed [2] ................................. 70 Table 4.1 Description of the HFD, provided by Vestas-Canadian Wind Technology, used in this study. ............................................................................................................................................. 81 Table A.1 Solar and Wind resource assumed for the site study ................................................. 112 Table A.2 Diesel consumption and costs of different power generation deployments for the extraction phase. The amounts are based on a 10 year project lifetime at 6% interest rate per year..................................................................................................................................................... 114 Table B.1 Estimation examples for a 4 min period in a site study ............................................. 125  x  List of Figures  Figure 1.1 Global cumulative installed wind capacity 1996-2013 [5] ............................................ 1 Figure 1.2 Canadian cumulative installed wind capacity [5] .......................................................... 2 Figure 1.3 Ten minute averaged wind speeds and corresponding PDF  (exhibiting typical nominal speeds) from a weather station in the Canarian Archipelago [8] ...................................... 4 Figure 1.4 Example of a Cp-λ curve digitized from [14] .............................................................. 11 Figure 1.5 Pitch and attack angles at an airfoil in a HAWT ......................................................... 12 Figure 1.6 Theoretical power output curve of fictitious 300kW, 2MW, and 3MW wind turbines [16]–[18] ....................................................................................................................................... 14 Figure 1.7 Wind turbine signaling movement in the pitch angle and yaw ................................... 15 Figure 1.8 Fictitious time distribution of wind speed and risk of faults during 10 minutes ......... 17 Figure 2.1 Diagram of the extension to the model developed by Rauh and Peinke [27] ............. 31 Figure 2.2 Left: Idealized sinusoidal gust around the wind speed average. Right: Real gust measured at a height of 60m with a frequency of 1Hz ................................................................. 37 Figure 2.3 Reproduction of G1(f, v) from [37]. The curves were digitized for illustration purposes ........................................................................................................................................ 38 Figure 2.4 Reproduction of Cp0 used for the Vestas V25 in [38]. The original curve was digitized and reproduced via a second degree polynomial fit....................................................... 39 Figure 2.5 Reproduction of δ(v) used for the Vestas V25 in [37]. The original curve was digitized for illustration purposes ................................................................................................. 41 Figure 2.6 Response function g(t) of a Vestas V25 model at six wind speed averages .............. 44 xi  Figure 2.7 Settling times for a specific wind turbine model at six different wind speed averages and a relaxation function r0 = 0.2. Turbulence intensity Iu is set at 0.5. Settling times are also reported as Cpo was increased by 29% and G1 reduced by 50% ................................................. 45 Figure 2.8 Variation of settling times for the Vestas V25 wind turbine relative to the change in the relaxation function .................................................................................................................. 48 Figure 2.9 Wind speed measurements (1Hz) at 30m height in a site study in an interval of 10 minutes. Data provided by Vestas-Canadian Wind Technology, Inc. .......................................... 50 Figure 3.1 Reproduction of the Van der Hoven Spectrum at Brookhaven National Laboratory [51], edited for the purpose of this study ...................................................................................... 55 Figure 3.2 Left: Energy spectrum of Site 1 for a 6 day length of uninterrupted 1Hz wind speed data. Right: Energy spectrum of Site 1 for a 1 day uninterrupted 1Hz wind speed data .............. 63 Figure 3.3 Upper left: Energy spectrum of Site 2 for a 1 day length of 1Hz wind speed data in January. Upper right: Energy spectrum of Site 2 for a 5 day length 1Hz wind speed data in January. Lower left: Energy spectrum of Site 2 for a 1 day length of 1Hz wind speed data in March. Lower right: Energy spectrum of Site 2 for a 7 day length 1Hz wind speed data in March....................................................................................................................................................... 65 Figure 3.4 Left: Energy spectrum of Site 3 for a 2 day length of 5Hz wind speed data in September. Right: Energy spectrum of Site 3 for a 4 day length of 5Hz wind speed data in October .......................................................................................................................................... 67 Figure 3.5. Economic value and energy contained in the power spectrum for each site. The numeric labels indicate the wind speed average in ms-1 considered to make the calculations. The labels in the horizontal axis indicate the date considered for the calculation ............................... 72 Figure 4.1 Average wind speed and band of dispersion in a 20 minute wind speed time series .. 79 xii  Figure 4.2 Flow chart of the HFD analysis. .................................................................................. 80 Figure 4.3 Example of E(f) at a 10 minute time window for wind Site 1. Data was measured at a frequency of 1Hz........................................................................................................................... 85 Figure 4.4 Wind speed, EDR, and TKE for Site 1. Wind speed is illustrated in 10 minute averaged data, and both TKE and ε were calculated from those averages. Additionally, the cross-correlation coefficients between ε and TKE are illustrated in the same plot of wind speed ........ 86 Figure 4.5 Wind speed, EDR, and TKE for the two periods of Site 2. Wind speed is illustrated in 10 minute averaged data, and both TKE and ε were calculated from those averages. Additionally, the cross-correlation coefficients between ε and TKE are illustrated in the same plot of wind speed ............................................................................................................................................. 87 Figure 4.6 Wind speed, EDR, and TKE for the two periods of Site 3. Wind speed is illustrated in 10 minute averaged data, and both TKE and ε were calculated from those averages. Additionally, the cross-correlation coefficients between ε and TKE are illustrated in the same plot of wind speed ............................................................................................................................................. 88 Figure 4.7 Cross-correlation coefficients between ε and TKE from 0am to 4am on May 15th for the measurements at Site 1. The lag is expressed in minutes, considering 10 minute time periods....................................................................................................................................................... 90 Figure 4.8 Histograms of the cross-correlation coefficients with values equal or larger than 0.6 for the three site studies ................................................................................................................ 91 Figure 4.9 Left: Histogram of frequencies for the lags between ε and TKE. Right: Histogram of frequencies for the time of the day when the highest correlation occurs ...................................... 93 Figure 4.10 Autocorrelation coefficients for the three site studies over the previously mentioned periods ........................................................................................................................................... 95 xiii  Figure A.1 Left: Costs involved in a power generation plant that includes both wind energy and conventional sources. Right: Diesel consumption for the same power plant relative to the number of wind turbines included in it, and the wind farm participation in the power generation ......... 116 Figure A.2 Left: NPC variation with different values of the interest rate. The installation of ten wind turbines exhibits the highest rate of change in the NPC at all interest rates. Right: Savings per kW installed of wind generated power ................................................................................. 117 Figure A.3 Tornado chart of five parameters’ impact on the COE for the extraction phase. COE of base case was calculated as $0.674 USD ............................................................................... 120 Figure B.1 Cumulative frequency histograms of the time when the peaks occur every 30s in a site study. As the number of peaks increases, the histograms show where the majority of peaks occurs .......................................................................................................................................... 124 Figure B.2 Cumulative frequency histogram of the peaks are highly dependent on the average and the TI. ................................................................................................................................... 124  xiv  List of Symbols 𝑌 Predicted time series. 𝑓(𝑣) Reference time series. 𝑣0 Upstream wind speed. 𝐶𝑝,𝑚𝑎𝑥 Maximum power coefficient. 𝑣2 Downstream wind speed. 𝑣𝑟  Wind speed at the rotor. ε Forecast residual. Energy dissipation rate. 𝐹 Force. Fourier transform. 𝐶𝐹  Capacity Factor. 𝐸𝑇 Energy produced in a time period. 𝑇 Time period. 𝑃𝑅 Rated power of the wind turbine. 𝜔𝑟 Rotational speed of the blade. 𝜌 Air density 𝐴 Cross-sectional area. ?̇?𝑟 Mass flow rate. 𝑝 Pressure. 𝑃 Power output. 𝐶𝑝,𝑚𝑎𝑥 Maximum power coefficient. 𝑣𝑟𝑒𝑙  Relative wind speed. 𝛾 Angle of relative wind speed. α Angle of attack. φ Pitch angle.    Standard deviation of wind speed.  1T  Gust duration.  1( , )G f v  Correction function of the wind turbine.  f  Gust frequency.  1 2/   Lambda ratio.  sP  Stationary power output.    Angular frequency of wind speed.  0pC  Power coefficient.  0r  Relaxation function.  v  Wind speed.  uI  Turbulence intensity.  v  Average wind speed. ¨00 , ppC C  First and second derivatives of the power coefficient relative to the wind speed.  ( )v  Delta function. ( )g t  Turbine response function. 𝑆(𝑓) Spectral energy density. 𝐸(𝑓) Discrete spectral energy. ∆𝑛 Difference between harmonic indices. 𝑚 Mass of air. 𝑅 Correlation coefficient. 𝑁 Total number of data. xv  List of Abbreviations  GPD  Generalized Pareto distribution GRF  Gust Response Factor HAWT Horizontal axis wind turbines IEC  International Electrotechnical Commission O&M  Operation and maintenance PDF  Probability Density Function PSD  Power Spectral Density TI  Turbulence intensity TKE  Turbulence kinetic energy VAWT Vertical axis wind turbine ABL  Atmospheric Boundary Layer COE  Cost of Electricity DAQ  Data Acquisition  LLC  Limited Liability Company NPC  Net Present Cost PV  Photovoltaic UBC  University of British Columbia  US  United States xvi  Acknowledgements This thesis has been fulfilled thanks to the support of many people. I want to thank my supervisor Dr. Walter Mérida, for his guidance during this project and the opportunity to work with him during this past few years. I also want to thank Dr. Roland Stull for the time he spent with me guiding and sharpening my ideas to deliver the best of me in this thesis. Special thanks to Dr. Ryozo Nagamune and Dr. Ziad Shawwash for taking the time to read my thesis as supervisory committee members, making comments to improve the document and my expertise in the field.  I want to thank Dr. Shankar Raman Dhanushkodi for his time revising my work, for his support, and for sharing his knowledge with me. I want to thank Devin Todd and Luke Damron for reading my thesis and giving me valuable insights. I want to thank my research group, especially the people I started with: Maximilian Schwager, Devin Todd, Robert Alink, and Sarah Flick, for sharing both knowledge and friendship. I want to thank the members that joined the research group along the way: Jochen Schoenweiss, Sebastian Prass, Pradeep Kumar, and Peter Kalisvaart.  I want to thank the Consejo Nacional de Ciencia y Tecnología (CONACYT, Mexico) for the financial support during my doctoral studies. Special thanks to the Secretaría de Educación Pública (SEP, Mexico) for the complimentary scholarship granted to me. I want to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for the financial support to the projects associated with this thesis.  My most sincere gratitude to my parents, Elga and Benedicto, who have been my first support from the day I decided to be a scientist. For their endless effort to give me the best education I xvii  could ever receive, most importantly, the education they gave me. I want to thank my brothers and sisters, for their moral support, their company in distance, and for increasing the number of members in our big family.  Finally and most importantly, I want to thank my wife, Brisa, for her support, patience, and enormous sacrifice she made joining me in this journey… and in the journeys to come. xviii  Dedication           To Isabel and Samuel, my inspiration and motivation1  Chapter 1: Introduction  1.1 Wind Energy in Numbers Wind energy has had an important impact in the energy market. It had an average annual growth of 30% globally from year 1998 to 2008. As illustrated in Figure 1.1, the average annual growth from 2008 to 2012 was about 20%, surpassing the annual global electricity market increase, which was over 6% in the same period [1]–[4]. The latest report of total wind capacity installed around the world indicates approximately 318.8 GW at the end of 2013, being more than 12% growth compared to the end of 2012 [5].  6 8 10 1417 2431 39485974941211591982382833183701996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014050100150200250300350400Capacity (GW)Year Figure 1.1 Global cumulative installed wind capacity 1996-2013 [5]  The installed capacity in Canada increased from 137 MW in 2000 to 7803 MW in the beginning of 2013, becoming the 12th largest nation in terms of total wind installed capacity. In the middle of 2014, the total Canadian wind capacity has reached 8500 MW, being the 9th country with the largest total wind energy installed capacity [3], [4]. The evolution of the Canadian wind installed 2  capacity is illustrated in Figure 1.2 [5]. As it can be seen, the Canadian increase in wind installed capacity was of about 25% in 2013, surpassing the global growth for in the same period.  0.20 0.24 0.32 0.440.681.461.852.373.324.015.276.207.809.692001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 20140246810Capacity (GW)Year Figure 1.2 Canadian cumulative installed wind capacity [5]  Wind energy can bring significant savings when the economic feasibility is calculated accurately, considering all the economic and technical constraints. These savings also depend on the location of the wind site and the purpose of the deployment (e.g. industrial or residential). Power generation through wind has been considered, for instance, by mining developers to supply their processing plants’ energy demand. An example of this type of analysis is presented in Appendix A.  1.2 Wind Potential Assessment The optimal installation and operation of a wind farm can be fulfilled if careful assessment of the resource is carried out. The wind resource assessment provides information to select the best wind turbines for the site and their optimal configuration. This is typically carried out through statistical analysis of wind speed and direction data based on 10 minute averages at different heights of 3  measurement (10-200m) [6]. Those averages are calculated from HFD, normally collected at 1 Hz and measured at meteorological towers. Such towers can be located several kilometers apart to assess the wind potential of regions rather than small site studies [7]. These grid dimensions are taken into account when larger extensions for more than one wind farm are targeted. According to Jain, new models using finer grid measurements (wind speeds, mechanical loads, etc.) are considered a priority to ensure future presence of the wind industry in the energy market [2].  The most common statistical tool to study wind resource is the Weibull distribution. This function is normally used with 10 minute averages and is determined by the mode, median and average wind speed of a specific site. Figure 1.3 illustrates an example of a frequency distribution for wind speed, with the Weibull trend line [8]. Capturing the highest gust with minimum yaw in this range increases exponentially the power output of a wind turbine. Those speeds may vary according to the wind turbine model. Speeds below 4 m s-1 (cut-in speed) have low probability to occur, while the speeds between 4 m s-1 and 13 m s-1 (cut-in and rated speeds) have the greatest probability to occur. The probability of obtaining wind speeds larger than 13 m s-1 is also lower than that of the region between cut-in and rated speeds. However, those wind speeds still occur and the wind turbine’s control system must be able to maintain a stable output under such conditions.  4  0 2 4 6 8 10 12 14 16 18 20 22 24024681012Frequency (%)Wind Speed (m/s) Frequency Weibull TrendCut-in speedRated speedCut-out speed Figure 1.3 Ten-minute averaged wind speeds and corresponding PDF  (exhibiting typical nominal speeds) from a weather station in the Canarian Archipelago [8]  The calculation of Weibull PDF is essential to describe statistically the predominant wind speed and direction in a site study [2], [9], [10]. The PDF is calculated from at least one year of measurements in a site study. The PDF describes the general distribution of wind speeds at a site, but it does not give information on its temporal or frequency behavior.  Monahan performed a study on the sea surface wind speeds to corroborate the validity of the Weibull PDF in the frequency distribution [11], [12]. The author calculated the mean, standard deviation, skewness, and kurtosis fields of the wind components’ resultant and found that the two-parameter Weibull distributions provides a good approximation to the probability density function of the resultant.  5  1.3 Forecasting Considering a statistical prediction model, the generic prediction method can be expressed as    ( )Y f X     (1.1)  where Y is the predicted time series for onsite wind data, 𝑋 is the reference time series, 𝑓(𝑥) is a transfer function, and ε is the residual. The computation of the transfer function is the main objective of a generic prediction model. The smaller value of ε, the better the prediction is [2].  Typical models to estimate wind speed are mesoscale and microscale models. They use wind data acquired previously and generate estimations for desired regions, containing wind speed and direction, at multiple heights [2]. The scale applicability of microscale models is of the order of 10 km. Mesoscale models describe weather phenomena with a spatial resolution of 20-2,000 km and a temporal resolution of minutes to days.  According to Abiven [13], wind energy operators agree that wind resource studies must go beyond the determination of baseline quantities such as annual average wind speeds or the annual energy production. The recommendation aims to optimize wind turbine class selection, operation, and wind farm design. The improvements to models based on average quantities lead to reduced occurrence of malfunctions and increased lifetime. This can be achieved by generating finer information on wind variability to avoid structural mitigation and power output unbalances caused by intense wind events.  6  1.4 Relevant Parameters of Wind Turbines 1.4.1 Turbulence Intensity An increase in turbulence leads to reduced energy production and greater structural loading of the turbines. High turbulence causes energy output to diminish and affects the loading, durability, and operation of a turbine [2]. Turbulence intensity (TI) is a measure of wind variability; it is defined as the ratio of the standard deviation of wind speed and the wind speed average, as expressed in (1.2). The time window used to calculate TI depends on the purpose of the calculation, but it is typically calculated in 10 minute averages.   TIv   (1.2)  1.4.2 Capacity Factor, Tip-Speed Ratio and Power Coefficient The capacity factor of any power plant is the ratio of the actual energy output over a period of time and its potential output if it had operated at full rated capacity during the same period [10]. It can be expressed as   TFRECTP   (1.3)  where 𝐸𝑇 is the energy actually produced by the system, 𝑇 is the time period, and 𝑃𝑅 is the rated power of the wind turbine. The typical capacity factor of wind farms around the world is in the range of 0.25-0.35, with turbine height from 30 to 100 m, respectively [10]. Small increases in the capacity factor lead to higher energy outputs and revenue, but it can also lead to unbalances in the 7  grid, in the case of grid-connected turbines [2]. Such unbalances are normally generated by mismatching the supply of wind turbines with the conventional grid demand.  The ratio between the velocity of the rotor tip and the wind velocity is best known as the tip-speed ratio of a wind turbine, and can be expressed as   rrv    (1.4)  where 𝜆 is the tip-speed ratio, 𝜔𝑟 is the rotational speed of the blade, 𝑟 is the radius of the rotor, and 𝑣 is the wind speed. A turbine cannot extract the total power contained in the wind; part of wind’s kinetic energy is transferred to the rotor and the rest passes through the turbine in the form of wake [10]. The power coefficient is the ratio of the actual power developed by the rotor and the total power of the wind’s kinetic energy. It is mathematically expressed as   32 TpTPCA v   (1.5)  where 𝑃𝑇 is the power developed by the rotor, 𝜌 is the air density, and 𝐴𝑇 is the area swept by the rotor. In 1919, German physicist Albert Betz published his work on determining the maximum amount of energy that can be extracted from the wind. He proved that the limit for the power coefficient is approximately 59% from its total kinetic energy. This limit is today known as the Betz limit and its mathematical proof can be found elsewhere [2], [9], [10]. The Betz limit is derived next, as it is in [2]. 8   Applying conservation of mass to cross-sectional areas 𝐴0, 𝐴𝑟, and 𝐴2 with constant density:   0 0 2 2r rA v A v A v    (1.6)  where 𝐴0, 𝐴𝑟, and 𝐴2 are upstream, rotor, and downstream cross-sectional areas. Also, 𝑣𝑥 is the average wind speed at 𝐴𝑥. From Newton’s second law, the force 𝐹 applied on the rotor by wind is   0 2 0 2( ) ( )r r rF m v v A v v v      (1.7)  where 𝜌 is the density of the air at standard conditions and 𝑚𝑟̇  is the mass flow rate. The force is also due to the pressure difference across the rotor, expressed as   0 2 0 2( ) ( )r r r r rF A p p A v v v      (1.8)  Bernoulli’s law can be applied in: a) Flow along stream-lines from 𝐴0 to the front face of the rotor; and b) flow from the back surface of rotor to 𝐴2.   2 20 0 01 12 2r rp v p v      (1.9)  2 22 0 21 12 2r rp v p v      (1.10)  9  Subtracting (1.9) from (1.10)   2 20 2 0 21( )2r rp p v v     (1.11)  Equating (1.8) and (1.11):   2 20 2 0 2 0 2( ) ( )2r r rrFp p v v v v vA        (1.12)  0 2( )2rv vv   (1.13)  which implies that 𝑣𝑟, the wind speed at the rotor, is average of the free-stream wind speed and the wind speed in the wake. The power 𝑃 delivered to the rotor by the wind is:   2 20 2 0 2 0 2 0 21 1( ) ( ) ( )( )2 2r r r r r r r r rP Fv p p A v A v v v A v v v v v           (1.14)  Combining (1.13) and (1.14):   2 20 2 0( ) 2 ( )r r r r rP A v v v A v v v       (1.15)  Maximum power is reached when:  10   200 2 3r rrPv v vv    (1.16)  Then   023rv v   (1.17)  2 013v v   (1.18)  2 30 082 ( ) ( )27r r r rP A v v v A v      (1.19)  ,max30160.5931 272prPCA v     (1.20)  𝐶𝑝 − 𝜆 curves are normally used in wind turbine design as they characterize the rotor performance independent of the rotor size and the site parameters [10]. Their use can be found in literature for diverse applications, such as the optimization of the rotor’s power output through adaptive control strategies [14]. An example of these curves is illustrated in Figure 1.4.  11  0 1 2 3 4 5 6 7 8 9 100.000.050.100.150.200.250.300.350.40CprMaximum Cp= 0.38Optimum  = 7.19 Figure 1.4 Example of a 𝐂𝐩 − 𝛌 curve digitized from [14]  The power coefficient increases initially with the tip-speed ratio until the former reaches a maximum at certain value of the latter. The power coefficient subsequently decreases with further increase in the tip-speed ratio [15]. The decreasing trend of the power coefficient after its maximum can be inferred since the energy taken from the wind remains constant after the rated output has been reached, and the wind speed contains larger amounts of energy. Figure 1.4 illustrates a power coefficient curve for a wind turbine [14], which exhibits power coefficients close to 0.4. Horizontal axis wind turbines (HAWT) have power coefficients close to 0.5 at lower tip-speed ratios than the ones illustrated in Figure 1.4. This is why HAWT are much more popular and preferred in the big scale of the wind energy market.  12  1.4.3 Pitch Angle The relative wind speed 𝑣𝑟𝑒𝑙 is the resultant of the tangential speed of the turbine’s airfoil 𝜔𝑟 at the tip and the horizontal wind speed 𝑣, holding an angle of 𝛾. The angle of attack α is that at which wind strikes a blade from a reference point at the blade. The optimal angle of attack will yield high lift and low drag forces at the blade. The pitch angle φ is the one between the airfoil’s chord and the direction of motion. In a HAWT, motion has a vertical direction and the angle of attack changes along the length of the blade [2]. The relationship of these concepts is illustrated in Figure 1.5 and mathematically expressed in (1.21)-(1.23).  Figure 1.5 Pitch and attack angles at an airfoil in a HAWT    22relv v r    (1.21)    1tanrv    (1.22)    90        (1.23)  13  The pitch mechanism controls the angle of the blades with respect to the plane of rotation. Pitch control allows a turbine to capture energy at low wind speed and to capture constant amount of energy at wind speed above the rated speed [2].  Modern wind turbines start generating power at a typical cut-in speed of 3.5 𝑚 𝑠−1 and the rated speed varies between 12 − 15 𝑚 𝑠−1, depending on the design, as illustrated in Figure 1.6. In the same way, cut-out wind speeds are located around 25 ms-1, also depending on the wind turbine’s design. Between the cut-in and rated wind speeds, the pitch mechanism adjusts the blade pitch to the optimum angle of attack, so the turbine operates at its maximum efficiency. Between the rated and cut-out speeds, the control mechanism change the blade pitch, subsequently changing the angle of attack, to diminish the aerodynamic efficiency of the blades and dissipate the excessive rotor power [10]. The pitch control keeps the wind turbine delivering the rated power output, even at wind speeds higher than the rated value. The control system has to be very sensitive to respond fast under variations in wind velocity and avoid sudden changes in the rotor speed.  Figure 1.6 illustrates the power output of three different turbines, extracted from the manufacturer’s brochures (Vestas, Siemens, and General Electric). The pitch angle is placed at its optimal position between the cut-in speed and the rated speed to maximize the energy capture. One important challenge is to control the pitch when the wind speed surpasses the rated speed because the rotor has to work at the rated speed. This has the purpose to prevent the turbine of generating power beyond the nominal limit. Moreover, the power system must ideally react prior to wind speeds beyond the cut-out speed, to avoid damage that can put the turbine permanently down.  14  0 5 10 15 20 25 30 350500100015002000250030003500Power Output (kW)Wind Speed (m/s) 3.0MW Vestas 2.3MW Siemens 1.5MW GEModel Vestas V112 Siemens SWT-2.3-93 GE 1.5-77Rater power output 3.0MW 2.3MW 1.5MWDiameter 112m 90m 77mHub height 85-119m 80m 65-80mRated wind speed 12m/s 13-14m/s 12m/sCut-in speedRated speed Cut-out speed Figure 1.6 Theoretical power output curve of fictitious 300kW, 2MW, and 3MW wind turbines [16]–[18]  The wind turbine’s control system is constantly analyzing information to assess every variation in the rotor’s orientation to have stable and continuous power production. Optimization in the production is important because even small increases (of approximately 1%) in energy output may lead to additional annual revenue per turbine of $50,000 to $100,000 USD [2].  1.4.4 Yaw Alignment Yaw 𝛽 is defined as the direction change of the rotor around a vertical axis [2], [9], [10]. The yaw mechanism allows the turbine to align the plane of rotation to be perpendicular to the direction of wind. Active yaw mechanisms include electromechanical components and a control system which can be found in large generation wind turbines [2]. An illustration of a wind turbine describing the mechanisms previously mentioned is presented in Figure 1.7. Yaw control is used to move the rotor either close or away from the wind direction, depending on the desired characteristics of the 15  power output. If higher wind speeds occur, the yaw mechanism will regulate the power output by misaligning the rotor axis and the wind horizontal vector [10].   Figure 1.7 Wind turbine signaling movement in the pitch angle and yaw  The change in the yaw of the hub may be as fast as 2° s-1 [19], depending on the size of the turbine, making a high frequency model applicable for a turbine to react before a significant wind speed variation is estimated a few minutes prior to its occurrence. However, the costs of power consumption during the alignment have to be considered. Nowadays, the stepper motors involved in the yaw are operated at high efficiency and low cost. A technical report written for the company Catch the Wind presented data collected from an operating wind turbine pertaining to yaw misalignment [20]. They provided evidence that there is at least a 𝑐𝑜𝑠3𝛽 dependency between power loss and yaw misalignments between -20o and +20o. This yields a power loss output of approximately 20% when the yaw is 20o. Furthermore, they reported the turbine spent significant 16  time out of alignment with the wind, due to the lack of higher resolution prediction techniques. With accurate remote measurements of the inflow on very short (greater than one Hz) time scales, the parameters in the yaw control algorithms can be optimized over a large range of values, including the possibility of a continuously yawing turbine [20].  1.4.5 Grid Balance Grid balance is an important requirement for the adequate operation of wind turbines connected to the grid and to provide continuous and stable power supply. Unbalances are due to variations of both wind speed and of voltage and frequency on the side of the conventional generation. Coupling wind with conventional power production requires voltage errors no greater than +5 V and frequency errors of no greater than 0.3 Hz [15]. According to BC Hydro requirements, nominal operation during disturbances of synchronous generation rated 1 MVA is maintained within a frequency range of 59.5 − 60.5 𝐻𝑧 and 90 − 106% of nominal voltage [21]. Ten-minute averaged wind data based estimation models do not consider transient effects of wind speed, such as gusts. Thus sudden gusts occurring within ten-minute intervals can unbalance the frequency and voltage coupling, leading to turbine stoppage and increased costs associated with grid re-coupling.  1.5 Motivation The data used for current estimation and assessment models are mainly 10 minute and 1 hour averages. Figure 1.8 illustrates a fictitious time distribution of high frequency wind speed at a wind site, over 10 minutes. When the wind speed’s value is above the rated speed, the pitch angle needs to be continuously adjusted so the rotor speed is maintained at a constant value. One main reason 17  for disturbances in the grid-connected wind turbines is due to faults in the pitch angle control that is unable to respond properly to sudden changes in the wind speed [22].   Figure 1.8 Fictitious time distribution of wind speed and risk of faults during 10 minutes  For the majority of blade designs in a wind turbine, the angle of attack that holds a maximum lift configuration is 15° for wind speeds below the rated speed [2], [9]. This angle is fixed to maximize the energy capture at low wind speeds. As the wind speed approaches the rated speed, the angle of attack starts to decrease to maintain a constant rotor speed and the power output constant. For variable rotational speed operation the turbine’s blade must be oriented in such way that the tip-speed ratio remains constant and the power coefficient is maximized. This action is taken while the rotational speed is adjusted and then the pitch angle should return to the point of highest efficiency. For wind speed above the rated speed, pitch control is necessary to maintain the rotational speed and the power output constant [9], [23]. For wind speeds above the cut-out speed, the pitch angle is set at 90°, also known as feather position. This position will contribute to stop 18  the rotor and reduce the energy capture to zero. The objective of the pitch angle manipulation is to increase the energy capture and, therefore, the capacity factor. For further information about blade pitch and the angles involved in the energy capture, the reader is advised to refer to [9], [23], Section 4 in both references. For gusts higher than the rated wind speed, the turbine needs to have a fast response to avoid faults that yield to grid disturbances and ultimately turbine stoppage. If a gust at the cut-out speed is found, the turbine must immediately stop to avoid damage. If the turbine could also prevent damage to its structure by anticipating the gusts over the cut-out speed, then the costs of operation and maintenance could decrease with the consequent increase in its lifetime. These are currently open questions, and to achieve these improvements, the turbine must have finer predictive capability to act prior a wind event. Such information can be also useful either to balance the power production with that of the conventional grid, or to increase the power output from the wind to satisfy the demand. This thesis addresses the identification of wind patterns in the frequency domain to estimate the significance of capturing the identified wind fluctuations.  10 minute averaged data do not allow visualization of transient effects in the wind, including turbulence and wind gusts. The averages are typically calculated from 600 wind speeds measured every second during the 10 minute interval and the HFD are discarded after the calculation. There are no standards to handle HFD (1 Hz or higher) or incorporating them into wind variation models, or power scheduling strategies. The present thesis addresses this technology gap.  19  1.6 Research Questions 1.6.1 Is HFD Analysis Valuable? When wind events occurring in within ranges of 10 minutes are neglected, the turbine may or may not capture the energy of gusts from different directions and at different speeds. The present research project was aimed to mitigating the uncertainty of the conventional average predictions. The importance relates to the possibility of increasing the power output or reducing the operation and maintenance costs of a wind farm. It is also valuable to know if the wind gust may cause any structural damage to the turbine or induce any unbalance in the electrical grid.  1.6.2 What is the Economic Significance of Finer Temporal Data Analysis? Lifetime of the wind turbine is extended when structural damage is prevented. Operation and maintenance (O&M) costs are also reduced when fatigue is diminished. It is valuable to know the quantity of resources which may be saved as part of assessing the economic significance of this research project. Jain reported [2] an annual average cost of 45-60 USD/kW in O&M for wind turbines, but no available references in literature estimates the economic impact of prevention measures. This assessment will be carried to identify the economic advantages.  When using the extra energy from gusts, the possible unbalances in the grid must be taken into account. If the power output of the turbine exceeds the demand of the grid, the use of energy storage systems becomes a factor that aids balance between the energy demand and supply, involving greater costs and technology deployment. If the grid experiences unbalances, additional costs are involved due to turbine stoppage and electrical power outages.  20  The use of finer predictions provides information of wind speed and direction in a shorter range of time. This yields capture of longer and stable wind events; therefore, greater and continuous grid-balanced power output can be reached. In this way, the capacity factor of a wind facility is increased. As illustrated in the previous section, small increases in the capacity factor lead to attractive revenues for the suppliers. This thesis aims to establish the relationship between the energy capture and the increase in revenue of the wind industry, through the HFD analysis.  1.7 Literature Review and Key Contributions 1.7.1 Predicting Wind Conditions at the Turbine in Advance of Gusts The random variations in wind generated power affect a power system's stability if the percentage generation of wind-converted power is substantial and if the power system's inertia is low enough to absorb random disturbances [24]. At this time, DC-AC converters in the turbine are fast enough to handle the frequency changes due to wind speed variations. Since the electrical disturbances are practically solved from the side of the wind turbine, current research is focused on the unbalances from the grid due to the connection with renewable and conventional sources. Efforts in error analysis have been carried out and possible solutions are emerging to have a grid fed with both conventional and renewable energy [22], [25]. It is necessary to study the fault history in the grid when connected to renewable systems. This is not only important with wind energy, but also with the rest of the renewable sources (e.g. solar), which present variability in its nature.  Nannayakkara et al. [24] proposed a pitch control which maintains the rotor’s pitch angle at constant as long as the output power is less than the rated power of the wind turbine. Active pitch regulation is started when the output power exceeds the rated power. Accurate prediction of the 21  effective wind speed is necessary to have an adequate pitch control. This prediction can be achieved by statistical tools that anticipate, for example, turbulence variability and eddy-energy peaks. This gap in the statistical analysis are addressed in this thesis.  Short term distribution will be the main goal, considering the relationship between statistics and the deterministic nature as well. Statistical methods include inherently the physical laws in the behavior of wind. Bierbooms and Cheng [26] developed a probabilistic method to determine the long term distribution of the extreme response of wind turbines. It relies on the quick determination of the response to a single gust, with given amplitude, by means of the constrained simulation. They stated that the integrity of a wind turbine structure involves analysis of fatigue loading as well as extreme loading arising from the wind climate. They made a contradistinction between the considerations of IEC (International Electrotechnical Commission) standards for extreme wind events, which are formulated as coherent gusts of inherently deterministic character, with the gusts experienced in real situations. Such gusts are of stochastic nature with a limited spatial extension.  Rauh and Peinke [27] developed a phenomenological model to predict the average power output of a wind turbine, based on the longitudinal wind speed fluctuations. The model is particularly suitable to examine the dynamic effect of turbulent wind fields which can be described by standard models of boundary layer meteorology.  Abiven et al. [13] studied the one Hz measurements and computational simulations to assess wind turbine siting, as a part of a resource evaluation project. This is one of the first attempts to deal with HFD. The author’s results indicated that the analysis of HFD lead to different occurrences for 22  specific wind directions than the simulations. This was a result of the complex orography of the terrain studied, which the computational model didn’t take into account with full detail.  Gust prediction has been widely studied since meteorologists tried to determine weather conditions. Sallis et al. [28] developed an algorithm for wind gust prediction based on neural networks. Cadenas et al. [29] demonstrated that the single exponential smoothing method is a good alternative for the wind forecasting. These studies were developed with 10 minute average data and they are examples of prediction methods using statistical approaches. No HFD were involved in the studies.  The economic significance of HFD analysis must be assessed before the estimation model is developed. The question in this research is not only about the reliability of the forecast, but also on its impact in the overall revenue. The analysis can improve the competitiveness and the penetration of the wind industry in the energy market, and that point is where the importance of HFD relies. This gap not covered by previous work is addressed in this thesis. As illustrated in Appendix B, due to the random nature of turbulence, HFD used for wind speed prediction in the very short term (few seconds ahead) is not better than the persistence forecast. It is illustrated in further sections of the present work that a variability analysis in the frequency domain must be carried out. An example of statistical prediction is illustrated in Appendix B, here only 60% of success in the predictions is observed.  23  1.7.2 Extreme Wind Estimation and Structural Mitigation According to Cheng and Yeung [30], US wind engineers are using extreme gust as the basic wind speed to quantify the destructive capacity of extreme winds since 1996. In order to understand better these destructive wind forces, it is important to know the appropriate representations of these extreme gust wind speeds. Cheng and Yeung [30] used a generalized Pareto distribution to analyze the extreme gust wind speeds. However, stations affected by hurricanes and tornadoes were excluded from the study. Finally, 143 weather stations in the US were selected for the analysis.  There are also studies on fatigue loads which are performed with stochastic wind field simulations. However, the analysis of extreme gust loads is carried out with deterministic wind signals. Cheng and Bierbooms [31] investigated extreme gust loads based on a rational approach and also developed a gust model that takes into account the stochastic property of a wind gust. A simulation of typical downburst of a storm was developed by Chay et al. [32], using a technique with a parametric analytical model of the downburst's non-turbulent winds, and amplitude modulated Gaussian stochastic process for the turbulent component. They used 20 minute duration storms, but also considered the effect of varying such durations. A sine wave was also considered as the waveform for the relation between wind speeds and time. A Gumbel fit was used for stationary events and Reverse-Weibull fit was used for non-stationary events, in the probability distributions. Piccardo and Solari [33] studied the effect of wind gusts on vertical structures, based on the calculation of the Gust Response Factor. These kind of studies are useful to develop strategies to mitigate structure loading of the wind turbine’s tower due to strong winds. Table 1.1 summarizes the relevant literature in the field along with the corresponding research questions addressed in the present thesis. 24  Topic Strategy Gap filled by the present thesis Interconnection with the grid and grid balance  Series grid-side converter to balance the stator voltage under grid unbalances [25]  Turbine’s shutdowns are mostly due to grid faults [22]  Voltage and frequency difference of less than 5 V and 0.3 Hz [15] HFD analysis to describe wind energy fluctuations and contribute to a continuous balance with the electrical grid Wind speed and direction prediction  Single Exponential Smoothing, MAE=0.5 ms-1 [29]  Artificial Neural Networks, MAE=0.6 ms-1 [29]  MAE=1.8 ms-1 [3]  Machine-learning algorithm with 88.45% of accuracy in gust prediction [28]  Kalman filtering, 1.7 of absolute bias [34]  SKIRON modelling, 2.7 of absolute bias [34] Complement estimation models with variability analysis in the frequency domain and quantify contributions to wind variance Power forecasting  Kalman filtering, NMAE=11% of nominal power [34]  SKIRON modelling, NMAE=11% [34] Analyze wind variability to estimate the energy availability in the wind spectrum Yaw alignment  Establish a cosine-cube for power and yaw in yaw values between +20° [20]  Proposal of active yaw mechanism [19] Identification of eddy-energy peaks in the frequency spectrum for a better control in the turbine’s orientation Turbulence  Proposal of pitch angle controller during turbulent wind [24] Complement the calculation of TI with the energy fluctuations in the frequency domain Extreme wind speed and fatigue  Gumbel extreme value function as the best model for annual extreme gusts [30]  Stochastic gust model proposed to calculate gust load [31] Identify larger eddy-energy contributions that may cause structural damage in the rotor and tower Table 1.1 Summary of the literature review and expected contributions of the present thesis.  There are published studies in wind energy using data measured in developing wind farms under adverse seasonal climate phenomena, such as storms and hurricanes  [9], [10]. Experimental data 25  during the occurrence of storms can lead to statistical tools to develop better understanding of these phenomena. When extreme weather is experienced in a wind farm, structural loading of the tower’s turbine becomes an important issue, implying higher costs in maintenance, fixing, and the early turbine’s substitution. These costs can be reduced if preventive measures are carried out. HFD analysis for extreme speed estimation can be a tool to mitigate structural loading. A study on wind variability in the frequency domain is useful to characterize the eddy contributions in the energy spectrum. Through this analysis the power producer can have more information on the turbulence variation to anticipate strong variance that might affect the turbine tower’s structure.  1.8 Objectives The most common output of existing models is related to wind speed estimation in the time domain, rather than variability estimation. The main objective of the proposed research is to increase the resolution of the current wind variability estimation models using a statistical approach, and generalize it for any onshore wind farm. This objective is achieved by analyzing HFD in the frequency domain to calculate the variance contribution to the energy spectrum.  Hypothesis: The analysis of HFD (e.g. 1 Hz) contains important information that must be included in the wind potential assessment studies. This information must also accompany estimation models to complement their input with the analysis of wind fluctuations in the frequency domain.  Objectives: 1. Validate and quantify the economic value in HFD analysis, predicted by an industrial partner. 26  2. Improve the temporal resolution (within ten min) of current wind speed spectrum studies. 3. Provide a method to identify the main eddy-energy contributions in the frequency domain that can be used to improve the performance of wind turbines. 4. Identify larger fluctuations (eddy-energy size, wind turbulent kinetic energy variations) whose estimates can prevent structural damage of wind turbines and increase their productivity.  1.9 Thesis Outline 1.9.1 Wind Turbine Dynamic Response Chapter 2 of the present thesis consists of determining the time response of a wind turbine, taking into account the dynamic characteristics of the turbine model. The analysis includes a variation in the dynamic characteristics to describe how the response changes under such variation.  The time required by the rotor to reach the steady state under wind variations was calculated and it is known as settling time. Wind variations of a specific level of TI under seven different wind averages are considered. However, settling time also depends on the duration of the gust and the dynamic characteristics of the wind turbine under study. The time to reach the steady state support the argument that finer analysis is needed to aid a more stable performance.  For this first section an analysis of the turbine’s response and gust occurrence in the frequency domain was carried out. This approach is important because it complements current studies that consider wind estimation in the time domain.  27  1.9.2 Eddy-Energy Contribution in Wind Fluctuation Chapter 3 of this thesis demonstrates the use of HFD to reproduce the power spectrum calculated by Van der Hoven in 1957. His work represents the basis of wind energy standards such as averaging and variability in the frequency domain. The results presented here unveil discrepancies with Van der Hoven’s approach, which can be related to constraints in the computing capabilities in the 1950’s. A major eddy-energy peak at a period of 2 days and a smaller eddy-energy peak contribution were unveiled at frequencies higher than the region known as the spectrum gap. The variance calculated by the area under the curve indicated that the spectral energy is mainly due to the PSD values located in the microscale region. The economic value of this energy was calculated within the range of frequencies where the dynamic characteristics of a wind turbine can take advantage of these gusts. It is also concluded that, given the results from the analysis, HFD analysis in the frequency domain uncover eddy-energy peaks that determine energy fluctuations in the short and long terms. This information is missed every time data are erased from the existing monitoring systems or kept in a storage device without further analysis.  The energy fluctuation was used to approximate the amount of energy that is not utilized, and these figures were converted to a simple economic estimate based on the industrial partner’s revenue models. Savings in O&M costs can be achieved through detailed studies on the effect of eddy-energy contributions regarding structural loading. Using the results here unveiled to modify the turbine’s orientation and energy capture can prevent excessive structural fatigue and increase the lifetime of the wind turbine.  28  1.9.3 Wind Energy Dissipation and Turbulence Kinetic Energy Chapter 4 of this thesis consisted of applying the Kolmogorov theory through the PSD in HFD to identify the range where the inertial subrange is predominant. The inertial subrange is characterized by the energy transfer from large to small size eddies and it is used it to calculate the energy dissipation rate. The energy dissipation rate in this range is mainly on the small eddies and it has to balance the energy production in the large scale. When this parameter was compared to the TKE, the time scale of the dissipation rate and TKE was found, implying a delay in both parameters due to a change in the variability of wind speed. The cross-correlation function was used to calculate the lag between these two parameters, obtaining a main lag variation in the range of 1.5-2.5 hours. This lag variation depends on other parameters, such as temperature and pressure changes in the lower atmosphere and such variation can be calculated for each site study. This is the first attempt to work with HFD (1Hz) for wind energy applications, as they are typically discarded from the resource assessment. It is also demonstrated that high frequency turbulence is one of the sources of pitch errors, which cause turbine failures and losses. A more detailed description of turbulence and eddy formation is required to minimize pitch errors and ultimately increase the capacity factor. 29  Chapter 2: Dynamic Response of Wind Turbines  2.1 Wind Turbine Response With the increasing penetration of wind energy in the market [2], [35], [36], more stable and reliable power production is required to enable optimal interaction with the grid. Since the current wind estimation methods are based on averages of 5, 10 and 60 minutes, finer fluctuations in wind speed and direction are being ignored. Under the current control systems, the wind turbine has to modify its orientation and pitch angle concurrently with wind speed changes. Due to this limitation, the wind turbine may be unable to anticipate intense fluctuations in wind speed that could produce an unbalance between the turbine’s output and the grid.  A turbine response study is desired before discussing the time resolution of wind fluctuation models. The wind turbine settling times are also needed to determine the optimum interaction between wind events and the turbine control system. The settling time is the time required for the turbine’s output to reach and remain within a given error band following a change in wind speed. After settling time, steady state is said to be reached. The stable error band is considered as 10% of the turbine function stimulus, as explained in [27].  It is worthy to differentiate between the concepts of wind estimation and wind fluctuation models. Wind estimation is referred in this work to the models that aim to deliver wind speed forecast at a certain time window. Wind fluctuation is referred to the models that analyze the changes in the wind variance, mainly in the frequency domain. In other words, the latter are studies on the eddy-energy contributions to the wind variance. 30  Current models for wind speed and direction estimation are based on 10 minute averages and modern data acquisition systems capable of measuring in a time window of less than a second. If the response time of the wind turbine is considered to be in the range of a few seconds, then the turbine’s control system could be missing changes in wind resource profiles with shorter duration. However, the wind speed estimation becomes more challenging when the time window of the model is reduced. In this case, the PSD can be a useful tool to describe eddy-energy contributions in the wind spectra.  Response analysis can contribute to the development of novel techniques that increase the resolution of current estimation models. It can also provide more accurate description on wind conditions in advance of a strong variation (from a few seconds to nearly a minute). The research gap in this field and the need for HFD analysis to optimize the energy capture in a wind farm have been identified elsewhere [2], [20], [27], [37].  In this study, the dynamic response of a horizontal axis wind turbine was analyzed considering the model proposed by Rosen and Sheinman [37], [38] and later applied by Rauh and Peinke [27]. The dynamic response equations were solved using Matlab to calculate the turbine settling time under average wind speeds not considered by Rauh and Peinke. This work presents the results of a dynamic model applied to settling times of a wind turbine from a moderate wind fluctuations approach. Although wind turbine response has been extensively studied, there is no report on the settling time of the turbine when a wind disturbance has occurred. There is no information available on the use of the dynamic response to increase the time resolution of current estimation models. 31  The objective of wind fluctuation models in the frequency domain is to identify periodic fluctuations in wind and to capture gusts with longer duration than the settling time of a particular model of wind turbine. It also depends on the characteristics of a site with a specific annual mean wind speed. In this sense, long and slow gusts are preferred over short gusts with high speeds.  2.2 Turbine Response Models The characteristic response of a wind turbine model under different wind speed averages and turbulence intensities allows the turbine’s control system to manipulate pitch and yaw settings accordingly. Gust duration and amplitude are two main parameters that govern the turbine’s behavior. The turbine function illustrates the response amplitude of the wind turbine in the time domain. Moreover, the consideration of the dynamic characteristics of both the wind and the wind turbine, rather than the static parameters, reduces false estimation of the turbine’s energy output. The results of [27], [37], [38] were reproduced and a sensitivity analysis was performed, generalizing the methodology, as illustrated in  Figure 2.1.   Figure 2.1 Diagram of the extension to the model developed by Rauh and Peinke [27] 32  Different studies reporting fast response (0.05-0.15 s) for grid-connected wind turbines considered only the grid-side electronic converters [22], [25], [39]. To have a more complete response time of the wind turbine, the dynamic characteristics need to be considered, rather than the static ones.  In [40] a control strategy is proposed to minimize the impact of wind speed disturbances on the power quality and stability. The authors simulated a pitch control malfunction represented by a severe change in the pitch angle variation. The proposed control strategy led to a voltage drop of 8% of the nominal voltage and a recovery time of approximately half a second. However, they did not consider severe changes in the wind speed.  The dynamic response of turbine power during changes in the reference speed of the generator was analyzed in [35] as a part of a strategy to maximize power output and reduce mechanical stress. The authors improved the response of the mechanical variables when a ramp signal from 4-12 ms-1 was used. They developed an algorithm that achieves a smoother response of the conversion system than previous conventional methods. This represents savings in maintenance and failure occurrence, although those were not quantified. The dynamic response of the turbine power and rotor speed to changes in the reference speed of the rotor led to stabilization times around 0.1 s. The mechanic response was not discussed by the authors, which is a gap that is filled in this chapter. A short stabilization time of the conversion system is required, as it is strongly related to the control mechanisms that deliver stable power output. This time is significantly shorter than the turbine’s mechanical response, which is expected to be a fraction of a minute.  33  Rosen and Sheinman [38] developed three prediction models of the average output power of the wind turbine Vestas V25 taking into account: (1) the average static power output for the first model, (2) the dynamic characteristics of the wind, and (3) the dynamic characteristics of both the wind and the turbine. Wind variations of high frequencies (>1 Hz) causes the turbine to fail in responding to the fast variations in the wind speed, maintaining its power output constant [38].  Rauh and Peinke [27] developed a model of the dynamic response of a wind turbine considering the atmospheric boundary layer theory. They developed a turbine function that calculates the time a Vestas V25 turbine takes to stabilize after a change in the wind speed has occurred. They used a wind speed average of 8 ms-1 and found a settling time of 26 s. Their data was provided in a study by Rosen and Sheinman [37], [38] for the 200kW turbine located in the Israeli settlement Beit-Yatir in 1994. These authors also considered the blade root moments specified in [9] to estimate the relaxation time under pitch angle variations. Although the model was developed for a discontinued wind turbine, the theory can be applied to modern turbines with improved control systems. Real data of the power output from a wind turbine are not always available due to confidentiality and proprietary rights considerations. This is an important constraint in the development of the dynamic response models.  A simulation study of an offshore wind turbine response is reported in [41], regarding a wide range of wind speeds (3-25 ms-1). The authors took into account the blade root moments when wind fluctuations occurred as it was considered in [3] and [7], along with the tower base moment. The main objective was to derive the long term load prediction for a 5 MW wind turbine.  34  2.3 Contributions i) A general procedure to calculate settling times for different wind turbine models was designed. As a secondary outcome, possible changes in the resolution of estimation models can be based on the results of the dynamic response. The addressed research question is stated as follows:  What is the significance of the dynamic response in wind turbines to determine the best resolution of forecasting models?  ii) The settling time of a Vestas V25 during wind speed fluctuations are reported using the dynamic characteristics of the wind turbine response. The results for six different wind speed averages within the range 7-12 ms-1 are presented.  iii) A sensitivity analysis of the results reported for the V25 is carried out, increasing the power coefficient by 20% and decreasing the correction function by 50%. The values of the power coefficient were chosen taking into account contemporary wind turbine characteristics.  2.4 Methodology 2.4.1 Elements of Dynamic Response The procedure illustrated in the following section can be adapted to modern wind turbines if the elements of dynamic response are well identified and experimentally obtained. These elements are listed in Table 2.1 as a guide to generalize the calculations. These parameters were taken from 35  previous references to reproduce and extend the results shown in [27]. The solution of those parameters for different wind turbine models can be accomplished by means of both simulation and experiments. It will describe the power extracted from wind under a wide range of turbulence intensities.  Parameter Symbol Definition Power coefficient 𝐶𝑝0 The fraction of energy captured from the wind for static conditions Correction function 𝐺1(𝑓, ?̅?) Ratio of the power produced under dynamic and steady conditions Delta function 𝛿(?̅?) A function of the static characteristics of the wind turbine Relaxation function 𝑟0  Mechanical response of the blades during pitch control Table 2.1 Parameters required calculating settling times of a wind turbine under wind turbulence occurrence  In the model, the information required to determine the settling times include: a. The coefficient of performance curve from the manufacturer. b. The function 𝛿(?̅?) of the static characteristics of the wind turbine. c. The correction function 𝐺1(𝑓, ?̅?) that accounts for the dynamic characteristics of the wind turbine. This can be obtained by the design of specific experiments. d. The relaxation function 𝑟(𝑡) depending both on the wind speed and its first derivative. To simplify the problem one can assume 𝑟(𝑡) = 𝑟0. This constant can be obtained by measurements of blade root bending moment for any particular turbine under changes in the pitch angle.  36  The general approach is based on the calculation of the dynamic parameters of any modern turbine. Measurement of the turbine’s output under wind speed changes in wind tunnel experiments or in a real wind system make this calculation possible. Simulation of the power output of the turbine under constant and variable wind speeds can deliver the dynamic characteristics as well. The combination of these parameters yields a general expression of the turbine function. The settling time results from the numerical integration of the turbine function. The influence of each parameter on the model and the method for its calculation is explained in the next section.  2.4.2 Calculations and Results Gust duration is related to the frequency of the wind gust in (2.1) as follows:   112Tf   (2.1)  where 𝑇1 is the gust duration: the time between two consecutive points where the wind speed is equal to the average, and f is the frequency of the gust. Frequency and gust duration are illustrated in Figure 2.2, considering a sinusoidal gust in an idealized case. Left side is shown only for illustrative purposes and does not represent the real behavior of wind. Shinozuka and Jan used a discrete representation to describe a realistic gust using the PSD [38]. The dynamic response of modern wind turbines has to be considered before increasing the resolution of the estimation models. The first parameter considered in [37] to calculate the response function of the Vestas V25 in [27] is the function 𝛿(?̅?), where ?̅?  is the average wind speed. This is a function of the static characteristics of the wind turbine and is illustrated in Figure 1 of [37]. 37     Wind speedaverageWind speed [ms-1]Time [s]Gust duration T11 2 3 4 5 6 7 8 9 10 11 1213.514.014.515.015.5  Time [s]Wind speed [ms-1]   Figure 2.2 Left: Idealized sinusoidal gust around the wind speed average. Right: Real gust measured at a height of 60m with a frequency of 1Hz  The second parameter is the correction function 𝐺1(𝑓, ?̅?). This is a function of the frequency and the average wind speed. It modifies the TI taking into account the dynamic characteristics of the wind turbine. For practical turbulence intensities the dependence of 𝐺1(𝑓, ?̅?) on the gust amplitude is very small and thus can be neglected. 𝐺1(𝑓, ?̅?) was calculated in [37] for wind speed averages from 7 𝑚 𝑠−1 to 12 𝑚 𝑠−1 using a detailed dynamic model of the Vestas V25, including the characteristics of the electric network at the site. Different design details of the turbine were estimated based on its inspection [37]. Correction functions 𝐺1(𝑓, ?̅?) are plotted in Figure 2.3, reproduced from the original in [37]. The frequency is in the range 0.051 ≤ 𝑓 ≤ 0.632 approximately. Otherwise, the correction function is considered as 1 because the turbine cannot respond at high frequencies and is not able to extract the energy from short gusts (less than 1 𝑠).  38  0.01 0.1 10123456787 ms-111 ms-18 ms-19 ms-1 10 ms-1 G(f,V)  f (Hz)12 ms-1 Figure 2.3 Reproduction of 𝐆𝟏(𝐟, ?̅?) from [37]. The curves were digitized for illustration purposes  The static power coefficient 𝐶𝑝0 is the fraction of energy captured from the wind by the turbine and is a function of wind speed 𝑣. It modifies the power which is produced by a wind turbine in case of a constant wind speed. This parameter has also dynamic contributions when the turbine operates in a real environment, but these can be neglected for simplification, as explained in [38]. It is given by the following equation:   30s pP KC v   (2.2)  𝐾 is a constant equal to 0.5 ∗ 𝜌 ∗ 𝐴𝑟, where 𝜌 ins the air density at standard conditions and 𝐴𝑟 is the area covered by the rotor blades. 𝑃𝑠 is the stationary power production and 𝐶𝑝0 is the power coefficient curve which is a function of the average wind speed. The curve used for the V25 model is illustrated in Figure 2.4, reproduced from the original in [38]. 39    6 7 8 9 10 11 120.320.340.360.380.400.42Cp0ms-1) Figure 2.4 Reproduction of 𝑪𝒑𝟎 used for the Vestas V25 in [38]. The original curve was digitized and reproduced via a second degree polynomial fit  Rauh and Peinke introduced the function 𝐹1, which is related to 𝐺1 as follows:   211 12 22 0( ) ( )1G Fr     (2.3)  where 𝜆1 and 𝜆2 are the first and second derivatives (with respect to the mean wind speed) of the stationary power curve, respectively, according to Rauh and Peinke [27]. 𝜔 is the angular frequency (2𝜋𝑓) of the wind gust and 𝑟0 is the relaxation function of the turbine regarding its response to changes in wind speed. The angular frequency is considered in the range of 0.3188 <𝜔 < 3.97 𝑟𝑎𝑑 𝑠−1. This range is set according to the range of frequencies where frequency f is used. In this case a constant value of 0.2 𝑠−1 is assigned to the relaxation function, according to a 40  particular case illustrated in Figure 3.76 of [9]. This value depends on the TI 𝐼𝑢 [27] which is a measure of the variability in the wind speed and is given by:   uIv   (2.4)  where 𝜎 is the standard deviation of the average wind speed in a 10 minute period of time. This value is considered to be in the range of  0 ≤ 𝐼𝑢 ≤ 0.5 for this work.  When a second order fitting is applied to 𝐶𝑝0 and introduced in (2.2), a linearization can be made in the range of 6.75 ≤ ?̅? ≤ 12 𝑚 𝑠−1, where the slope is considered equal to 𝜆1. The value of the slope is approximately 87𝐾 𝑚2 𝑠−2. The parameter 𝜆2 is related to ?̅? as follows:    2 03 pv vKC    (2.5)  where    2) )16( )' ( '' (( )p pp pC v C vvvC v vvC      (2.6)  is a function of the static characteristics of the wind turbine. It was first developed in [42] and then calculated for the Beit-Yatir Vestas V25 wind turbine in [37], where its plot is illustrated in Figure 2.5. 41  7 8 9 10 11 12-1.0-0.50.00.51.01.5  () (ms-1)  Figure 2.5 Reproduction of 𝜹(?̅?) used for the Vestas V25 in [37]. The original curve was digitized for illustration purposes  Rearranging (2.5):   2 212 0873 ( ) pm svv C    (2.7)  The ratio in (2.7) depends on the wind speed average over a 10 minute period. Rauh and Peinke [27] took Figure 2 in [37] and calculated the correction function 𝐺1(𝑓, ?̅?) by a logarithmic approximation. Then they used (2.3) to obtain 𝐹1(𝜔) and numerically solved the next integral to obtain the turbine’s response function:       210 00 1 2 20cos sin(ωt)4 ( )exp r t r tg t r F dr       (2.8) 42   They performed this procedure for an average wind speed of 8 𝑚 𝑠−1. In this chapter the calculations for different wind speed averages and report the settling time of the particular wind turbine model are produced. The response function 𝑔(𝑡) generally depends on 𝑣 ̅and 𝐼𝑢 [27]. The result of the numerical evaluation is plotted in Figure 2 of [27] for a wind speed average of 8 𝑚 𝑠−1. An extension of that calculation for the remaining wind speed averages considered in [38] is illustrated in Figure 2.6. This requires the incorporation of data that was not considered in [27] and allows the delivery of new and relevant results for the sake of wind turbine response.  The settling time is the time required for the response curve to reach and stay within a range of 10% of the difference between the instantaneous input and steady values. The settling time of the turbine response is 26 𝑠 after a change in wind speed has occurred, for an 8 𝑚 𝑠−1 average wind speed. This is a key indicator of the need to include wind estimations with finer resolution than the current 10 minute averages. At this point it is not possible to assure that the turbine will capture the energy from the gust; however, it certainly takes less than half a minute to reach the steady state.  For the numerical integration of (2.8) different angular frequency limits were chosen for each wind speed average. These limits are illustrated in Table 2.2, along with the values of the maximum gust durations and lambda ratios in (7). The frequency limits were chosen according to the correction function 𝐺1 calculated in [37] and later partially used in [27]. The behavior of the correction functions in the frequencies below the lower limit was omitted for simplicity, considering 𝐺1 = 1. The upper limit of the angular frequencies is set when 𝐺1 and the dynamic characteristics of the 43  turbine fail to respond to the fast variations in the wind. These conditions are analogous to equation (48) in [27].  Wind Speed Average [m s-1] Angular Freq. Limits [rad s-1] Max. Gust Duration [s] 𝝀𝟏/𝝀𝟐 Vestas V25 [m s-1] 𝝀𝟏/𝝀𝟐 𝑪𝒑𝟎 20% higher [m s-1] 7 0.29-1.44 10.8 8.9 9.2 8 0.32-3.97 9.8 10.8 11.5 9 0.24-4.12 13.1 21.8 26.2 10 0.31-4.61 10.1 -30.3 -22.6 11 0.30-3.84 10.5 -6.7 -6.0 12 0.28-4.61 11.2 -3.2 -3.0 Table 2.2 Integration parameters in (2.8) for the six wind speed averages and maximum gust duration for the range of gust frequencies. The lambda ratios for the sensitivity analysis are also included. Significant figures are set taking the resolution of the physical resolution of the digitizing software  The numerical solution of (2.8) for the six wind speed averages is illustrated in Figure 2.6 and the plot of the settling times is reported in Figure 2.7. In despite of using the same boundary conditions, different results between the calculations reported here and those reported in [27] are observed in the turbine function for the average wind speed of  8 𝑚 𝑠−1 in the range of 0 − 5 𝑠. The rest of them are new results and there is no reference to compare with. The reference reported positive values while the results in this work exhibit negative values over the same period of time.  The results illustrated at the bottom of Figure 2.6 exhibit a shorter response time for the wind speed average of 8 𝑚 𝑠−1. Although the rest of the wind speed averages remain within the same response time, more than 50% decrease in the turbine function is observed for the six average wind speeds. 44  This decrease indicates that the correction function is a determinant factor in the turbine’s balance under turbulent conditions. Modern turbines with lower correction values are less prone to be affected by turbulent winds and remain in the steady state when wind fluctuations are experienced.   Figure 2.6 Response function 𝒈(𝒕) of a Vestas V25 model at six wind speed averages  The turbine function was also calculated for a 20% increase in 𝐶𝑝0 from the original values of the V25 model. The results are not illustrated in this analysis as they were very similar to those illustrated in Figure 2.6. Changes in 𝐶𝑝0 and 𝐺1 did not have a significant impact on the turbine’s response and changes in 𝐺1 were directly proportional to changes in the turbine function. The study of wind fluctuations in the frequency domain must be addressed to minimize the effect of 45  turbulence on the wind turbine’s response. The values of 𝐺1 can be manipulated in the design of both the wind turbine and the control system, considering the fluctuation winds at the installation site.  7 8 9 10 11 120510152025303540455055Time [s]Wind Speed Average [ms-1] Vestas V25 1.2Cp0 & 0.5G1 Figure 2.7 Settling times for a specific wind turbine model at six different wind speed averages and a relaxation function 𝐫𝟎 = 𝟎. 𝟐. TI is set at 0.5. Settling times are also reported as 𝑪𝒑𝒐 was increased by 29% and 𝑮𝟏 reduced by 50%  The results of the numerical integration of (2.8) exhibited a settling time of approximately 13 s, half of the 26 𝑠 calculated in [27] for the average wind speed of  8 𝑚 𝑠−1. There is no reference to compare the other five averages, illustrated in Figure 2.7. When wind speed approaches the rated speed of the Vestas V25 settling times in the range of 12.5-14 s are observed. This behavior may 46  be caused by the blade pitch control actions at the rated speed, which also affects the blade moment, as explained in  [41]. There is a need for experimental data that allows the investigation of the turbine’s settling time at higher wind speed averages. This would also contribute to validate the present model experimentally. Data for this application is rare, and when they exist, the access to them is often restricted by proprietary rights.  The time that the Vestas V25 wind turbine requires to reach the steady state under wind speed variations is no longer than 17 s, except for the wind speed average of  7 𝑚 𝑠−1. At the specified wind speed average the settling time is set approximately at more than 50 s, much longer than the rest of the results. This singularity may be due to a small fluctuation of the turbine’s response function slightly larger than the 10% of the original overshoot value or an artifact of the integral’s numerical solution. However, a corrected settling time might be considered to be close to the rest of the values in Figure 2.7 with significant confidence. When the average wind speed is approaching the rated speed, the settling time decreases. The minimum settling time is set at  10 𝑚 𝑠−1 which is approximate equal to the maximum gust duration in its integration limits. This is one main reason to think that 10 minute based estimation models are not fine enough to detect wind variability in the response range of the wind turbine. Normally the turbine is oriented at a maximum capture position below the rated speed, while the pitch angle varies at speeds above the rated speed. In this way the power generation remains constant at the nominal value, discarding the energy surplus in the wind speeds above the rated speed. Pitch control becomes critical at this point, as it must prevent turbine damage caused by the rotation at higher speeds than the nominal one.  47  If the time the turbine spends in the steady state is increased, then a more reliable and stable performance can be reached. This may lead to an increase in the overall operating time of the turbine and less unbalances when the system is connected to the grid. When the estimation is made accurately, the abrupt change in the wind can be anticipated by the turbine’s control system to modify the pitch angle and minimize the yaw.  Energy capture from gusts with the longest duration increases the overall power production of the farm without compromising the grid balance. This is explained by the weak influence of the dynamic characteristics when low wind gust frequency (or long wind gust duration) is predominant, as stated in [38]. The smaller the dynamic influence, the more steady behavior is delivered. In the case of the wind turbine at Beit-Yatir, wind averages between  10 − 12 𝑚 𝑠−1 with gusts above ~ 15 s are the most beneficial for the power production. These characteristics in the wind provide the turbine with enough time to reach the steady state. When this happens, the power produced will be also stable and continuous, reducing losses in regulation, conversion, and transmission to the conventional grid. When the relaxation function 𝑟0 varies for each wind speed average, the settling times also change. Relaxation time depends on the mechanical response of the turbine to sudden changes in wind speed, relative to the average. The variation of settling times when relaxation time is changed for the six wind speed averages is illustrated in Figure 2.8.  48  0.0 0.2 0.4 0.6 0.8 1.00102030405060708090100    r0 [s-1]Time [s] 10m/s 11m/s 12m/s 7m/s 8m/s 9m/s10 5 2 11/r0 [s] Figure 2.8 Variation of settling times for the Vestas V25 wind turbine relative to the change in the relaxation function  There are local maxima in the settling time for all averages when the relaxation function is in the range of 0.4 − 0.6 𝑠−1. The overall settling times for the larger wind speed averages are shorter than those from the smaller averages. This can be interpreted as a more stable behavior of the wind turbine when the wind speed average approaches the rated speed. The shorter settling times are found in the vicinity of  𝑟0  equal to 0.2. This is the relaxation function that was used in [27], where the authors stated that  𝑟0  depends only on the mean wind speed. However, the variations in Figure 2.8 indirectly represent changes in the mean wind speed and mechanical response to changes in the blade root bending moment and the inertia of the total system. These changes follow the variations in the pitch angle to adapt the airfoil to the variations in wind speed. The response function also depends on design and manufacture conditions of the wind blade. Modern wind turbine designs (later than the V25 model) have a reliable mechanical response of the airfoil to gust occurrence and they are expected to have short relaxation times. Realistic values of the relaxation time  1/𝑟0  are in the range of 4-10 s because they only represent the moment 49  fluctuations in the blade’s root attached to the rotor. The wind fluctuations affecting the blade’s root are transmitted to the rest of the turbine’s system, and a general method to calculate the time to overcome these fluctuations is here presented. These results can be applied only for the Vestas V25 model under the particular conditions of its connection to the grid of Beit-Yatir. However, the analogous procedure can be applied to modified values of the power coefficient and correction function, where similar behavior can be expected.  2.5 Significance and Repercussions Current estimation models use 10 minute wind speed and direction averages to predict future averages. When continuous pitch and yaw control is desirable, higher resolution fluctuation analysis could improve the accuracy of the control system when the energy of a gust is captured.  Figure 2.9 illustrates 10 minute measurements of HFD at a confidential site study collected in 2005 (property of Vestas-Canadian Wind Technology, Inc.). If the turbine model at Beit-Yatir operated during 10 minutes at the wind speed illustrated in Figure 2.9, the turbine’s system might not respond properly to variations neglected in a single calculated average. The wind speed average in Figure 2.9 is 16.35 𝑚 𝑠−1, which is above the rated speed of the V25 (13.8 𝑚 𝑠−1). TI during that interval is 0.17, which is considered moderate turbulence [2] and it is found in the range of the response model studied here.  50   Figure 2.9 Wind speed measurements (1Hz) at 30m height in a site study in an interval of 10 minutes. Data provided by Vestas-Canadian Wind Technology, Inc.  Wind speed varies from values below and above the rated speed, which means that pitch has to be varied from maximum to limited energy capture positions. When a conventional estimation is made the turbine can fail to respond to the short term variations of wind speed when they occur. Due to the inaccuracy of estimating wind speed in the very short term, as illustrated in Appendix B  this application is presented as an opportunity to avoid overestimation of the energy capture and to maintain the production at the rated power, when wind speed exceeds the rated speed.  Gust analysis is a complex task, since the wind speed estimation becomes unfeasible in the very short term. Rosen and Sheinman [37], [38] suggested a function to model a real gust and it was originally proposed by Shinozuka [43]. It consists of a wind speed that is a function of the frequency and the PSD. This approach has been applied previously to make data like those in 51  Figure 2.9 feasible to study them in the frequency domain and convert the random process in a function with periodic behavior. This approach will be studied in the next chapters.  The importance of new wind fluctuation models with higher resolution relies on the capture of gusts with a minimum yaw when they approach the rotor of the turbine. These gusts must ideally be longer than the settling time of each specific turbine in the case of the average of current 10 minute interval. With the capture of these gusts the turbine may reach the steady state for a longer time, minimizing losses in regulation and conversion. This could be also important for siting and matching turbine designs with the resource. The overall capacity factor of the turbine and the revenue of the supplier could also be higher.  Different techniques for gust analysis and prediction have been applied to improve the forecast of wind condition for energy applications [3], [24], [26], [28], [29], [34]. There is no evidence at this point of an agreement on how to use HFD analysis in those techniques. Improvements in wind forecasting involve minimization of errors in the estimation models and in the mechanical control of wind turbines. The performance of wind turbines involves measures to reduce the fluctuations in the response of the mechanical and electric systems such as frequency and voltage variations. Optimization of yaw and pitch angle is a measure to prevent structural damage and increase the power output and its stability. These can be taken when HFD is available and analyzed for wind fluctuation analysis.  52  2.6 Conclusions The work reported by Rosen and Sheinman (1992 and 1994), and later used by Rauh and Peinke (2004) was applied to the dynamic response of a wind turbine. The resulting model was partially applied to investigate the settling time of a Vestas V25 wind turbine located at Beit-Yatir. The methodology was repeated for a power coefficient curve with values 10% higher than the V25 model and a correction function with values 50% lower than the V25 model. The results indicate that no significant contribution to the turbine function is observed when the power coefficient is varied. On the other hand, the main contribution to the turbine function variation is the change in the correction function, whose changes are directly proportional to those in the turbine function (one to one). Higher resolution fluctuation approaches are required in the control systems of turbines connected to the grid. This strategy will shorten the ratio between the TI due to the dynamic fluctuations in the wind and the statistic TI.  The calculated settling times also confirm the need for wind fluctuation analysis with higher resolution. The relevant gusts last for a few seconds, typically. Hence, fluctuation analysis in the frequency domain at shorter time periods requires the analysis of high frequency wind speed data. Such analysis can improve the performance of current control systems for grid-connected turbines. Further studies at the meteorological mesoscale and microscale need to be taken into account to capture wind events that contribute the better response of wind turbines. The economic benefit of the link between the wind turbine’s dynamic response and the meteorological microscale will be illustrated in the next chapter. These approaches must look towards the delivery of more stable and reliable power from the wind turbine.  53  Chapter 3: Eddy-Energy Contributions in the Frequency Domain  3.1 The Van der Hoven Spectrum While wind power is becoming increasingly significant in the electricity market, the challenges regarding the intermittency of the wind resource influence the power delivery from wind turbines [44]. This influence causes higher maintenance costs due to frequent changes in the dispatch since the wind turbine has to respond to wind variations. Several forecasting techniques have been developed to estimate the wind speed in the range of a few hours to several days and even weeks [29], [45]–[47]. These methods generally offer results in the time domain and their main inputs are 10 minute wind speed averages. The increased randomness of wind in finer temporal windows makes forecasting unfeasible, and the study of wind speed in the frequency domain can fill this gap. An example of statistical wind speed forecast is illustrated in Appendix B of the present work. Moreover, the wind power spectrum can be used to plan reliable schedules through fluctuation forecasts [48] and the identification of periodic deterministic components of a time series [49].  Power spectrum analysis for wind data has been recognized as a very useful tool to provide a more detailed description of wind speed variability [44], [48], [49]. It enhances both the study of identification of variation patterns and the distribution of turbulent energy over the frequency domain [50]. In 1957, Isaac Van der Hoven gave the first recognized step to analyze the wind spectrum in a wide range of frequencies [51]. The study was carried out with wind speed data measured during approximately 10 months and contained different average readings, from 2 seconds to 5 days. One of the main uses of the Van der Hoven spectrum in the wind industry was to set the 10 minute averages as the main input to characterize important parameters such as the 54  TI and the Weibull parameters. The 10 minute period is found in the middle of the so-called Van der Hoven gap, where the contribution to wind energy fluctuations is considered negligible [44], [48], [51]–[54]. More recent studies of wind power spectra have been carried out under special circumstances such as the study of Lixiao et al [55], where the authors took wind data from a typhoon to model wind loads over tall structures. The authors developed their study focusing mainly on the inertial subrange of the spectra, where the energy transfer from larger to smaller eddy is predominant.  A study developed by Vincent et al [56] recalled the discrepancy that several authors exhibit with the spectral gap in the Van der Hoven spectrum. One would expect that in such portion of the spectrum little wind variations may be experienced, but this statement has been questioned. Furthermore, the authors aim for a spectral gap that is not always well-defined and, in some cases, does not exist. The authors also separated the spectral behavior into different atmospheric conditions using the Hilbert-Huang transform.  A reproduction of the Van der Hoven spectrum is illustrated in Figure 3.1, including the differentiation between the meteorological mesoscale and the microscale. The first one corresponds to phenomena with space scales larger than 3km and with time scales longer than about 1hr, while the latter covers the smaller values of space and time [57]. The illustrated region in the frequency domain is where the dynamic characteristics of a wind turbine model [27], [37], [38] must be considered to model its power production accurately. Outside such frequency range the influence of the wind’s dynamic effects becomes negligible [38]. In the low frequency region, the correction function illustrated in Figure 2.3 is slightly affected by wind events long periods 55  (20 − 100 𝑠). In the higher frequency region, the correction function also experiences small variations with periods shorter than 1 𝑠 approximately. It is also noticeable that the units illustrated in the ordinate in Figure 3.1 are equivalent to the power of the signal divided by its corresponding frequency. In this way, when the area under the curve on this semi-log graph is calculated, the result is the total energy variance in the frequency domain. Although the units of the abscissa are not represented in standard units, they were kept to easily compare the results with the ones illustrated in Figure 3.1.  1E-3 0.01 0.1 1 10 100 10000.00.51.01.52.02.53.03.54.04.55.05.56.06.5Our main regionof interestEnergy Spectrum GapEnergy Spectral Estimates (m2/s2)Frequency (cycles/h)Horizontal Wind SpectrumBrookhaven 91, 108, and 125mMesoscale MicroscaleDynamic responseof the wind turbine Figure 3.1 Reproduction of the Van der Hoven Spectrum at Brookhaven National Laboratory [51], edited for the purpose of this study  Some authors have used the Van der Hoven spectrum as a reference for their wind data analysis in the frequency domain [48], [52], [54]. To the knowledge of the author, there is no evidence in the literature that aims to update that study, although the processing and computational capabilities 56  used for the analysis are nowadays more accessible and advanced. This study discusses the completeness of the spectrum curve presented in 1957 emphasizing the energy gap. The hypothesis of the present study is that the actual eddy-energy peak distribution over the frequency domain can be different from the original reference and its description must be reconsidered on a case by case basis. Van der Hoven calculated the spectral estimates using the computing tools available at the time for a single geographic location with its own meteorological characteristics (Brookhaven National Laboratory). There is no recent study that validates the spectrum gap with finer data nor using current computer tools available for the analysis.  Wind resource analysis of many site studies are widely reported in the literature and carried out using basically the same techniques [7]. However, more attention must be focused on the study of wind variability in the frequency domain. Moreover, for most of the studies reported in literature, wind averaged data is typically used to describe wind conditions in both the time and frequency domains. Such averages are normally calculated from HFD, which is measured in meteorological stations and then discarded by the data acquisition system (DAQ). This study’s approach includes the exclusive use of HFD in the calculation, allowing one to run the same computing routine with one data set divided in several time periods. Evidence of HFD use in wind spectra analysis can be found in literature. Larsén [58] et al developed a study using HFD and ten-minute average wind speeds and temperature to model their mesoscale spectra. The authors calculated the spatial variability of wind and temperature in a range up to a few hundred kilometers paying special emphasis in the presence of gravity waves.  57  3.2 Motivation Abderrazzaq and Aloquili developed an experimental study which concluded that 30% of the faults (turbine stoppage) and energy losses in the wind turbine’s system are due to pitch errors [22]. Blade root bending moments have a response of a few seconds to sudden changes of the blade pitch angle in modern turbines [9]. During this response time, the electronic components of the wind turbine’s system have to respond to such variations as well. As a conclusion, better control of the pitch angle will bring a smoother response of the wind turbine forced with wind speed variations. Wind turbines have response times corresponding to different wind variation periods [27]. The study of such variations in the frequency domain must be carried out to identify patterns in the wind speed [44] and reduce faults caused by pitch misalignments. This topic is also addressed in the present study.  The frequency spectra analysis of wind corresponds to its dynamic variations, while the resource assessment studies the static characteristics from a statistical approach. It was first stated by Healey [59] that the excess kinetic energy in the fluctuations of wind speed, above the hourly average, may be important, depending on the TI and response time of the wind turbine. This same approach was continued in the work of Rosen and Sheinman [37], [38], concluding that ignoring wind’s dynamic nature due to turbulence results in over predictions of more than 10% of the power output. In the present work, the wind spectra were studied to identify the frequency of the main contributions to the variability and increase the information available regarding the wind variability.  58  3.3 Statement of the Problem The Van der Hoven spectrum is characterized by a gap in the energy contribution from a period of 2 hours to approximately 5 minutes. This means that the eddy-energy variation doesn’t have any significant contributions at the corresponding frequencies. This information is nowadays the basis to the ten-minute average calculations in the wind energy industry. This method includes the calculation of the wind speed averages, TI, and the Weibull parameters, neglecting the dynamic characteristics of wind and their effects on the wind turbine performance. However, the kinetic energy contained in the variance spectra during this period is not detected by traditional wind resource methods, and it is not clear if the wind turbine is able to capture it.  3.4  Methodology HFD (1 𝐻𝑧 and 5 𝐻𝑧) were used to calculate the spectral energy 𝐸 of the wind speed at three different wind sites. The frequency range of interest is that located within the wind turbine’s dynamic response range in Figure 3.1. The description of the data is illustrated in Table 3.1, along with the time of the year when it was collected and the length of the samples. The source files used in this study contain a larger data volume, but only representative segments were taken for the study. The data was collected and provided by Vestas – Canadian Wind Technology, Inc. The HFD is proprietary information and the locations of the site studies are not to be disclosed, which does not affect the effectiveness of the method. The numerical calculations were carried out in Matlab as the main data and code processor.   59  Site Frequency of Measurements [𝑯𝒛] Height of Measurements [𝒎] Period of Measurements Average Wind Speed [𝒎𝒔−𝟏] TI Site 1 1 60  May 24 May 14-20 15.41 5.57 0.19 0.84 Site 2 1 60  January 16 January 16-20 March 18 March 12-18 6.59 4.77 3.31 3.65 0.36 0.60 0.44 0.41 Site 3 5 Unknown  September 20 October 22 8.51 9.74 0.73 0.22 Table 3.1 Description of the data used to calculate 𝑬 of the wind speed HFD  Period lengths in which the data is divided are defined as follows:   , 1,2, ,kTT kkN     (3.1)  1kkfT   (3.2)  where 𝑇𝑘 is the length of each 𝑘-period, measured in seconds, in which the total data length 𝑇 is divided, 𝑘 is the number of periods from which 𝐸 is calculated, and 𝑓𝑘 is the corresponding frequency to the 𝑘-period. In the present study, time periods ranged from approximately 20 𝑠 to 2 days were considered to calculate the 𝐸 contribution. In this way, 𝑁 is a counter which can have values from one up to 20,000, depending on the data length and the frequency of measurement. Then the data is analyzed in matrices of 𝑖 rows and 𝑗 columns to calculate and average spectral energy. The index 𝑖 refers to the position of a wind speed datum in a given period, and its value 60  goes from one to 𝑇𝑘. The index 𝑗 refers to the number of 𝑘 period where certain wind speed is allocated and goes from one to 𝑘. Thus the data series will be taken as the matrix    ( 1)*ij ki j Tv v     (3.3)  where 𝑣 is a wind speed value. Each time period considered to calculate the Fourier transform is the inverse value of the frequency; for instance, if a 12 hour period is targeted, the equivalent frequency will be 0.083 cycles per hour, using the units of Van der Hoven. In some cases, periods of a few minutes were the shortest ones because no change in the trend of the spectrum was observed with shorter periods. For this reason the calculation was truncated when 𝐸 for higher frequencies remained more or less constant.  The spectral energy is calculated as explained by Stull [57] and his methodology was adapted to the conditions of this study. The author approximated the spectral energy density by    ( )fE fSn  (3.4)  where 𝐸(𝑓) is the discrete spectral energy and ∆𝑛 is the difference between neighbouring harmonic indices. In this study, that difference was set at 1, as no datum in the series was skipped. Stull also associates the energy spectrum with the concept of eddy motion and velocity fluctuations, calling the spectrum velocity the ‘discrete energy spectrum’. This name is used sometimes for all variance 61  spectra. If ∆𝑛 is substituted by the difference between neighbouring frequencies ∆𝑓, the parameter calculated will be the PSD and such difference will not be equal to one anymore.  The discrete energy spectrum can be approximated by the double product of the Fourier transform’s magnitude squared [57], as   2( ) 2 ( )E f F f   (3.5)  12 /0( )( )Ni fk Nkv kF f eN       (3.6)  where 𝐹 is the Fourier transform. The first element’s norm in the Fourier transform series represents the average of the data series. For this reason, the first element of the Fourier transforms is typically discarded to calculate the energy spectrum 𝐸(𝑓) [57]. This action is performed by taking the elements of the Fourier transform from the second element to the last one. Furthermore, due to the random nature of turbulence, the main contribution to the variance is quantified by the second element of the Fourier transform, which corresponds to the fundamental frequency. Thus only the second element was used to calculate 𝐸 and quantify the variance contribution at a specific frequency, as shown in (3.7).      2,( ) jE f E f   (3.7)  The plotted values of 𝐸 observed in the results section are calculated by averaging the second elements of the energy spectrum at over all the 𝑘-periods evaluated. The spectrum for each site 62  was calculated taking different time periods from the data and averaging the energy spectrum of each one of them. The first two points in the spectrum are the result of 𝐸 for the entire day and the average of the two halves of the same day. If one day is divided into three periods of 8 hours, the energy spectrum illustrated in the results will be the average of the three energy densities. The measurement uncertainties were not available, but the ultrasonic anemometers used in the meteorological towers are not a significant source of error (approximately 2%) [60]. Thus, the sensor uncertainty was neglected in the data.  3.5 Results The calculated results for all the sites were plotted using a logarithmic abscissa with units in 𝑐𝑦𝑐/ℎ𝑟 to make an easy comparison with Figure 3.1, and a linear ordinate with units in 𝑚2 𝑠−2. The spectra were calculated in groups of 500 sub-periods to decrease the computing time, until the full spectra is reached, which was mostly truncated at a frequency in the range of 10 −100 𝑐𝑦𝑐 ℎ𝑟−1.  The results for Site 1 are illustrated Figure 3.2, where a smoother decrease in the spectrum contribution is observed, opposed to the well defined peaks in the Van der Hoven spectrum. It is also evident that the major eddy-energy peak is located approximately 2 days, opposite to the conclusion in reference [51] that that identifies a major peak located at a 4 day period. However, it concurs with the results illustrated in [54], where a similar trend is observed at a site study in Bushland, Texas. It can be observed that working with 1 day data implies a stronger influence of the highest frequencies in the power spectrum. Another peak is observed between a 3 and 5 hour 63  time period, which likely follow changes in the air temperature profile during the day and the consequent buoyancy effects.  0.1 1 10 100 10000.00.51.01.52.02.53.03.5May 24E(m2 s-2)f(cyc h-1) 0.01 0.1 1 10 100 1000012345678910E(m2 s-2)f(cyc h-1)May 14-205% from the energypeak at the mesoscale Figure 3.2 Left: Energy spectrum of Site 1 for a 6 day length of uninterrupted 1Hz wind speed data. Right: Energy spectrum of Site 1 for a 1 day uninterrupted 1Hz wind speed data  From a wind turbine’s power generation perspective, the fluctuation in the kinetic energy is significant between those ranges of time. This fluctuation is directly associated with the turbulent component of wind speed and, at the same time, with the dimensions of the dominant eddies. Taylor’s hypothesis illustrates a simple method to calculate the diameter of eddies by multiplying the average wind speed by the time period for it to pass [57]. However, this calculation may not be evident when looking at a time series, and the wind spectrum offers an accurate way to look at different eddy sizes in the frequency domain. Eddy sizes are, consequently, a measure of the variation in the kinetic energy in a range of time that depends on the frequency and the mean wind speed. In this way, the localization of the meteorological scales depends on the mean wind speed, which indicates the rate of the eddy motion. 64   The higher frequency peak illustrated by Van der Hoven located in the frequency range of 20 −300 cycles per hour becomes significant only compared with the 𝐸 values in the vicinity of 0.1 cycles per hour. The eddy-energy contribution of the microscale is only 5% of the 𝐸 value at the mesoscale eddy-energy peak. It suggests a weak eddy-energy contribution to the variance in the microscale and a combination of multiple eddies that increase the randomness in that period. The 𝐸 in the microscale plays a major role when the dynamic response of the wind turbine is considered. The 𝐸 illustrated in Figure 3.2 represents average densities calculated over 6 days for the left side plot and a day for the right side plot. Averaging in those periods can result in different turbulence intensities, which may vary the spectrum density as well. The differences in the 𝐸 averages at, for instance, a frequency of 0.1 cycles per hour indicate a larger average variance in May 24th than the one observed from May 14th to the 20th.  Figure 3.3 illustrates the corresponding results for the study in Site 2. It is observed that the eddy-energy contribution at the highest frequency is slightly larger when a single day is studied compared to the average calculated over five days. Upper right of Figure 3.3 illustrates agreement with [54] for the distribution of the eddy-energy peaks, exhibiting a maximum at a 0.02 cycles per hour frequency or 2 day time period. The 𝐸 in lower right of Figure 3.3 also illustrates a significantly lower contribution compared to that reported in January, which may be due to lower wind speed average and TI. The major eddy-energy peaks are located at an approximate time period of 15 hours, with a smooth decrease in the spectral density when the frequency is approaching one cycle per hour.  65    0.1 1 10 100 10000.000.150.300.450.600.750.901.051.201.351.50January 16E(m2 s-2)f(cyc h-1) 0.01 0.1 1 10 1000.00.51.01.52.02.53.03.54.04.5January 16-20E(m2 s-2)f(cyc h-1) 0.01 0.1 1 10 100 10000.00.10.20.30.40.50.60.7March 18E(m2 s-2)f(cyc h-1)0.01 0.1 1 10 1000.00.10.20.30.40.50.60.70.8March 12-18E(m2 s-2)f (cyc h-1) Figure 3.3 Upper left: Energy spectrum of Site 2 for a 1 day length of 1Hz wind speed data in January. Upper right: Energy spectrum of Site 2 for a 5 day length 1Hz wind speed data in January. Lower left: Energy spectrum of Site 2 for a 1 day length of 1Hz wind speed data in March. Lower right: Energy spectrum of Site 2 for a 7 day length 1Hz wind speed data in March  It is observed that Site 2 exhibits a better agreement with the results of Van der Hoven, in terms of the eddy-energy peak located in the microscale region. The influence of the microscale region 66  in the total variance is more evident than that of the other site studies. This influence can be explained by lower average wind speeds with a more frequent incidence of gusts in the vicinity of 100 cycles per hour. Even when the eddy-energy peaks in the microscale region is significantly lower than the mesoscale ones, the total energy variance defined by the area under the curve is predominantly found in the former. This phenomenon is due to the dissipation of the larger eddies into smaller ones, creating regions with increased wind shear [57].  Figure 3.4 illustrates the result for Site 3 with 5 Hz measurements. These plots also exhibit eddy-energy peaks at the longest period (1 day) with a growing trend towards a period of 2 days. In this site study, there is no other significant contribution from the rest of the frequencies if they were compared to the major peaks in the mesoscale. The left plot indicates that larger masses of air are being slowly transported in the atmosphere, also bringing changes in the ambient temperature. It also indicates a large difference in the average wind speeds between two consecutive days, which was observed in the data series. This is also explained by the strong TI illustrated in Table 3.1. While the first half of September 20th showed mild gusts, the second half of the data series showed wind speed around 30 𝑚 𝑠−1. The energy spectrum was not calculated for the next days due to anomalies in the measurement, which indicated not reliable data. The same type of anomalies was observed around October 22nd, when the sensors indicated invalid values of wind speeds.  67  0.1 1 10 100 100005101520253035September 20E(m2 s-2)f (cyc h-1) 0.1 1 10 100 10000.000.250.500.751.001.251.501.752.002.25October 22E(m2 s-2)f (cyc h-1) Figure 3.4 Left: Energy spectrum of Site 3 for a 2 day length of 5Hz wind speed data in September. Right: Energy spectrum of Site 3 for a 4 day length of 5Hz wind speed data in October  3.6 Discussion Although the largest eddies are usually the most intense compared to the high frequency ones, the energy density content of the latter is significantly larger. This high frequency region is the one that affects the wind turbine’s dynamic response to turbulence.  Sheinman and Rosen [37] reported a function that gives a ratio of the average TI and the TI that accounts for the dynamic characteristics of the wind turbine Vestas V25, illustrated in Figure 2.3 of the present thesis. The peak of the correction function for different wind speed averages are very close to the one located at the meteorological microscale in the Van der Hoven spectrum. This is explained by the response characteristics of the wind turbine, and the variability’s influence on such response in the frequency domain. The frequency range in which the dynamic characteristics of wind have an impact on the turbine’s response is within 0.01 − 1 𝐻𝑧 or 36-3600 cycles per hour. Gusts occurring at a frequency in the vicinity of 0.07 Hz or 250 cycles per hour 68  cause the strongest impact in the delivery of power from the wind turbine. This impact is represented by the peak observed in the correction function, which corrects the statistical TI with a dynamic one by a maximum factor of 8. It is at this frequency value where 𝐸 has a major contribution, as illustrated in Figure 3.2 through Figure 3.4.  The objective of Sheinman and Rosen was to improve the power generation forecasting, by correcting the link between wind and power variation. Considering the frequency range of the dynamic response illustrated in Figure 2.3, the characterization of the wind variance contribution within the same range must be assessed on a continuous basis. This analysis unveils the relationship between the energy variation and the behavior of the wind turbine’s dynamic response. For instance, Sheinman and Rosen illustrate the direct relationship between the PSD and the correction function to calculate a modified TI as   1/21,mod( ) ( , )ni iivPSD f G f vIv     (3.8)  where 𝐼𝑣,𝑚𝑜𝑑 is the modified TI, 𝐺 is the correction function, 𝑓𝑖 is the 𝑖 − 𝑡ℎ frequency in the spectrum, and ?̅? is the wind speed average. A modified TI of 2.10 can be approximated from a PSD as the one expressed in Figure 3.4 left, with a TI of 0.22 (Table 3.1). This calculation was made using the correction function illustrated in Figure 2.5 for the wind turbine Vestas V25 and the corresponding PSD for Site 3. The range of frequencies considered for the calculation was 0.01 − 1 𝐻𝑧, as illustrated in the plot of the correction function. Calculating numerically (3.8) can be performed with the availability of finer wind speed data, since typical 10 minute averages 69  are equivalent to a frequency of approximately 0.0033 Hz. For this reason, storing HFD for further analysis must be considered rather than using them to exclusively calculate wind speed averages.  3.6.1 Financial Analysis A financial analysis was performed using the energy peaks described in Figure 3.2 to Figure 3.4, where the ordinate axis have units of energy per unit mass. These peaks indicate the energy available in the variance at their corresponding frequency. In this way, a dimensional analysis can be performed to obtain the revenue content of the eddy-energy peak. Table 3.2 illustrates the energy spectrum of the chosen days to calculate the revenue from the variance contributions.   Energy Spectrum [m2 s-2] Frequency [cyc h-1] Period Site 1 0.5 30 May 24 Site 2 2 20 January 16 Site 3 0.2 30 October 22 Table 3.2. Chosen time periods and energy spectrum values for the three sites of study  Such revenue can be expressed as   sEEm   (3.9)  Where 𝐸 is the total available average energy at the peak’s frequency, 𝐸𝑠 is the energy in the spectrum per unit mass, and 𝑚 is the air mass that passes through the rotor at during a time 1/𝑓 and is expressed as  70   2r vmf   (3.10)  where ?̅? is the average wind speed of the period time of study, 𝑓 is the frequency of the eddy-energy peak, 𝜌 is the air density at standard conditions, and 𝑟 is the rotor radii. Three single days were analyzed, one for each site, and then generalized the results for yearly values. One important assumption is that the spectral energy peak will have the same value during the entire day. In this way, the time 1/𝑓 repeated during one full day becomes 24 hours. Thus (3.9) can be expressed as   2sE T r vE   (3.11)  where 𝑇 is the time period equivalent to 24 hours, and the total energy 𝐸 will be expressed in MWh per day.  Wind Speed Average [ms-1] Cost [USD per MWh] 7 90 8 70 9 60 10 50 11 50 12 50 13 50 Table 3.3. Estimated costs of wind energy as a function of wind speed [2]  The approximation of the monetary value of that energy was calculated using the wind speed averages addressed in Chapter 2 and an additional theoretical rated speed of 13 𝑚𝑠−1. The 71  considered cost per MWh generated by a wind turbine in 2011 was illustrated by to Jain [2], from 6 𝑚𝑠−1 to 10 𝑚𝑠−1. As the wind speed averages considered in this analysis are in the range of 7 −13 𝑚𝑠−1, the average cost of the energy generated by a wind turbine for the wind speeds not considered by Jain was set at the same cost of 10 𝑚𝑠−1. These values are illustrated in  Wind Speed Average [ms-1] Cost [USD per MWh] 7 90 8 70 9 60 10 50 11 50 12 50 13 50 Table 3.3.   The energy and monetary values for the three site studies are illustrated in Figure 3.5. A wind turbine radius of 50 m and air density at standard conditions were considered. The economic value per turbine per year fluctuates between $10,000 and $70,000 USD at the rated speed of most wind turbines (13 𝑚 𝑠−1). This amount becomes more significant when a wind farm is considered rather than a single wind turbine. The monetary values illustrated in Figure 3.5 were calculated using a power coefficient of 0.5 as the maximum fraction of energy that can be extracted from the wind. The total energy 𝐸 was calculated in MWh per year, considering that the spectral energy will have the same value in each site during the entire year. To calculate the daily potential cost savings per turbine, the total amounts in Figure 3.5 would have to be divided by 365. The energy fraction extracted by each particular wind turbine model (not considered in this study) needs to be assessed for a more accurate approximation. Thus the results illustrated in this study represent the total 72  average energy that can be extracted from the wind variance in the specified time periods. One case study of a wind farm’s overall revenue is illustrated in Appendix A of the present work.  1 2 3015000300004500060000750009000001883755637509381125September 20thMay 24th7 8 9 10 11 12 137 8 9 10 11 12 13MWh per yearEconomic Value ($ USD)Site7 8 9 10 11 12 13January 16th Figure 3.5. Economic value and energy contained in the power spectrum for each site. The numeric labels indicate the wind speed average in 𝐦𝐬−𝟏 considered to make the calculations. The labels in the horizontal axis indicate the date considered for the calculation  3.7 Conclusions The results indicate that major eddy-energy peaks are observed at periods of 2 days for the sites investigated, opposite to Van der Hoven’s main outcome of 4 days. This means that the spectrum peaks in the macro meteorological scale indicate that major fluctuations occur every other day in the three site studies. These results also indicate that the contribution in the higher frequency end of his study is actually negligible. Although the values of the spectra in the Van der Hoven gap are in these results not zero, their contribution is very small compared to the major peaks in the macro 73  meteorological scale. The fluctuations in the kinetic energy indicate that wind energy models can result in over prediction of a wind turbine’s energy production if only wind speed averages and turbulence intensities are taken into account. Smaller but significant eddy-energy peaks are observed in the range of 3-5 hours, also indicating that average prediction must be adjusted to kinetic energy fluctuations in the short term. Although some of the energy peaks are in agreement with Van der Hoven’s baseline, significant differences in the frequency of other peaks are observed (up to 160 cycles per hour). Furthermore, it is also concluded here that the spectral gap must not be generalized, as explained by Vincent et al. [56]. Furthermore, individual assessments must be carried out to find the characteristic spectrum of specific site studies.  The TI at the frequencies where the dynamic response of general wind turbines is located (between 0.01 𝐻𝑧 and 1 𝐻𝑧) is significantly higher when the modified value is calculated. This indicated that the wind variance has a stronger impact when the turbine’s dynamic characteristics are considered. The impact of the energy density is more apparent when it is converted into a monetary value. The results indicate a theoretical average annual extra revenue between $10,000.00 USD and $70,000 USD, considering a single 50m radius wind turbine at a rated wind speed average of 13 𝑚𝑠−1 in the three site studies. This implies that the energy spectrum in the frequency domain must be studied on a case by case basis to establish the degree of variation. This might be different from what is normally stated in the literature. Each installed wind farm can have its own characterized spectrum, where the frequency of the major eddy-energy contributions can vary according to its particular topographic and weather characteristics. 74  Chapter 4: Energy Dissipation Rate and TKE  4.1 Wind Analysis in the Frequency Domain Wind data history is typically used for two main purposes: wind energy assessment and wind speed forecasts. The Weibull distribution is the most common tool to assess the available wind energy of a potential wind site. However, this tool is limited in that it only describes long term averages of wind speed and direction at a wind turbine location and fails to capture phenomena, such as turbulence, occurring at shorter time scales. There are several techniques of weather and wind estimation used in literature, such as numerical weather prediction, artificial neural networks, and auto-regressive moving average models [45]. All of these models intend to estimate average values of wind speed in time periods ranging from a few minutes to several days by processing periodic samples of wind speed data.  Wind data are available in several forms, depending on the frequency of measurement and length of time averaging. Wind speed and direction are typically sampled at frequencies of 1Hz or higher and the resulting HFD are processed to calculate averages over time windows ranging from 10 minutes to 1 year in duration. HFD are usually discarded from the computer memory of the measurement system after these averages are calculated. These average wind speed data are the main input to the Weibull distribution and most estimation models. Herein, HFD of three different wind sites located at confidential locations are presented. The HFD was provided by Vestas, Canadian Wind Technology Inc., an industrial partner in this research thesis. The company and the authors are concerned that valuable information is lost by discarding HFD. In spite of the additional information contained in HFD, the author is not aware of any current agreements or 75  standard for HFD analysis. From the different options in HFD study, the focus was put on a first step to analyze wind variability in the frequency domain.  Wind speed estimation is typically reported in the time domain, while the characteristics of wind variability are reported in the frequency domain, through spectral analysis [44]. The frequency domain approach for wind energy applications separates wind speed into mean and turbulent components and describes the amplitude and duration of gusts. As the temporal window of wind estimation becomes shorter (e.g. less than a minute), the estimation becomes less feasible, as the randomness in the wind increases. Furthermore, predictable changes in the time domain are dominated by the synoptic and mesoscale processes in the atmospheric boundary layer, rather than the turbulent portion of the wind distribution [57]. However, HFD can be valuable when used to describe wind variability in the frequency domain. This approach can be used to characterize wind data and even find patterns that could be used for more effective wind turbine operation [44]. According to current literature findings, turbulence plays a major role in the wind potential assessment in the time domain [7], and wind pattern characterization in the frequency domain [27], [61].  Fourier analysis has been performed on wind data because it is an accessible and reliable tool to study time series in the frequency domain. It describes wind variability throughout the year and can be used to characterize patterns for a specific site study [44], [48], [53], [54]. Spectral analysis also relates the contribution to variance over the frequency components of a time series. As turbulence is also related to variability, it has been shown that the power spectrum analysis is highly correlated with the TI [57]. TI is defined as the ratio of the standard deviation over the 76  average of the wind speed. Thus, the Fourier transform can be useful to complement wind resource potential assessment. Strong TI is a contemporary problem in wind turbines, as it can lead to pitch errors and faulty behavior in the power generation.  There is some evidence in the literature of HFD analysis to complement wind potential assessment or turbulence estimation, such as the studies carried out by Monahan, and McBean and Elliot [62], [63]. In this chapter HFD is analyzed as a complement to current assessment tools to provide improved descriptions of the wind’s turbulent component in the frequency domain. Two main concepts used in this study are the TKE and the energy dissipation rate ε. TKE is defined as the kinetic energy’s component of the turbulence, while ε represent the energy loss rate in the small scale turbulence, characterized by the cascade of energy from larger length scale to small scale eddies [57]. In this study the degree of correlation between TKE and ε was calculated to identify the time of the day when that correlation reaches a maximum.  The Kolmogorov turbulence order is a parameter that relates the energy dissipation in the turbulence by assuming that the smallest eddies see only turbulent energy cascading down the spectrum at the rate of the dissipation. It has been widely used when the wind frequency spectrum is analyzed to identify the characteristic range of frequencies of the inertial subrange [44], [48], [53], [57], [61]. It is this region of the spectra where the energy is transferred from the larger eddies to the smaller ones in an inertial process [57]. One way to calculate dissipation is through the energy spectrum using the Kolmogorov turbulence theory. In the present work this theory is used to determine the dissipation rate as it varies throughout the day.  77  The relationship between ε and TKE has been previously studied elsewhere, highlighting the importance of the turbulence kinetic energy in describing the atmospheric boundary layer and its dynamics. Li et al. [64] developed a method to estimate the TI from ε and validated their model under both normal wind and typhoon conditions. Kantha and Hocking developed a method [65] to calculate the energy dissipation rate from radar measurements in the free atmosphere. Their study was focused on a better understanding of the spatio-temporal variability of turbulence. In [23], Kalapureddy et al. studied the diurnal and seasonal variations of the energy dissipation rate ε through measurements made by a lower atmospheric wind profiler. They calculated specific values of ε for a tropical region at different altitudes, focusing specifically on the convective boundary layer. It is in this region where the turbulence generates a well-mixed region and also where the authors observed that ε has its largest values. The aforementioned methods calculate the values of dissipation from the TKE, but there is not enough analysis of the time delay between those parameters. The existence of a time delay indicates the time in which a correlation between two parameters is still valid. In the present chapter this gap is addressed.  Different empirical and theoretical relations between TKE and ε for calculating their instantaneous values can be found elsewhere [23], [64], [65]. However, no information on the cross-correlation between those parameters have been found, although it may be useful to calculate their time delay. For example, Jensen and Busch [66] have mentioned that the dissipation may occur in regions with intermittent turbulence, meaning stationary and non-stationary periods. It can be concluded that this phenomenon could involve a difference between the dissipation rate and the production of turbulent eddies:  78  “One can think of this as if the local dissipation rate runs faster than the local cascading process, such that the average supply of small enough eddies is slower than the average rate at which they are dissipated.”  Moreover, Panofsky and Dutton [67] stated that ε can be modelled as   TKE   (4.1)  where 𝜏 is the time scale of those parameters. The time scale indicates when significant changes in the parameters start to occur. Both literature sources imply that there is a time scale in which the ε and the TKE experience significant changes due to diurnal and seasonal variations. It is also evident that both parameters decay together and the changes also depend on the major eddies size and wind speed average. Hereby a method is presented to evaluate the degree of correlation that yields to an estimation of turbulent wind variations. An algorithm that calculates the lagged cross-correlation coefficients between the TKE and ε was designed to estimate variations in the wind turbulence. This measure can contribute to improve the turbine’s control and response to wind variability.  4.2 Objective As the time window of a wind forecast becomes smaller, a wind speed forecast model becomes unfeasible due to the influence of turbulence and increasing random variations in the wind speed time series. For this reason, HFD may not be used to estimate wind speed in the very short term 79  (0 − 30 𝑠). However, estimation of turbulence variability can be a feasible alternative to exploit the information contained in HFD.  The objective of this chapter is to provide a method to calculate the level of correlation between the energy dissipation rate and the TKE using HFD with a sampling rate between 1 Hz and 5 Hz as the main input. Using the cross-correlation coefficients, the TKE fluctuation can be estimated from the variations of ε at a specific time. The TKE is directly proportional to the square of the standard deviation σ or variance of any wind speed time series. The fluctuations in the TKE influence the width of the band between the dashed lines, illustrated in Figure 4.1. This band can be narrow or broad depending on the dispersion of the data series.  10121416182022Wind Speed (m s-1)Time (s)Can we predictthis change?0 200 400 600 800 1000 12000 200 400 600 800 1000 120010121416182022 Wind Speed (m s-1)Average Figure 4.1 Average wind speed and band of dispersion in a 20 minute wind speed time series  80  4.3 Methodology Figure 4.2 shows a flow chart of the methodology used in this study. The first part of the flow chart indicates that the HFD is the main input of the algorithm that calculates 𝜀 and TKE, and the cross-correlation coefficient of the two parameters. In this study the cross-correlation coefficients higher than 0.6 were captured to record the time in which those are observed and its corresponding dominant lag. Sample periods of four hours were taken and the cross-correlation coefficients between the TKE and 𝜀 were calculated up to a 2 hour lag, where the function was truncated.   Figure 4.2 Flow chart of the HFD analysis.  The dataset used in this study was provided by the industrial partner, Vestas-Canadian Wind Technology, and the location of the three site studies will remain confidential. Relevant parameters of the site studies are illustrated in Table 4.1. Uninterrupted portions of data were selected to report accurate and reliable results, without the use of interpolation for any blackout periods. 81   Site Frequency of Measurements [Hz] Height of Measurements [M] Date Range of Data Average Wind Speed [ms-1] TI Site 1 1 60  May 15-19 5.17 0.79 Site 2 1 60  January 11-20 March 12-18  4.28 3.65 0.59 0.41 Site 3 5 Unknown  September 20-21 October 21-24 12.37 13.67 0.61 0.59 Table 4.1 Description of the HFD, provided by Vestas-Canadian Wind Technology, used in this study.  The energy spectrum function 𝐸(𝑓) is the double the square norm of the Fourier transform of the time series, as illustrated in Equation (3.5). The Fast Fourier Transform was applied to the data in different time windows and then calculated the square norm to generate 𝐸(𝑓). The inertial subrange, which is characterized by the cascade of energy from large to small eddies, was identified within each 𝐸(𝑓) calculation of the three sites. This phenomenon occurs in the atmosphere on a continuous basis, and the identification of the inertial subrange is not sufficient information to be able to anticipate these variations. By plotting the spectral density, the least-squares best fit line of the logarithmic plot was computed. According to the Kolmogorov turbulence theory, the energy spectrum function in the inertial subrange is:     2/3 5/3f fE C    (4.2)  82  where 𝐶 is the Kolmogorov constant in the range of 1.53 and 1.68, 𝜀 is the energy dissipation rate and 𝑓 is the frequency [57]. When the equation is represented in its logarithmic form, the exponent of 𝑓 in (4.2) is the slope of the best fit line, while the y-intercept can be considered as a measure of the dissipation rate, as shown in:     2/35log3logE f lf ogC     (4.3)  2/3A logC   (4.4)  3/210AC      (4.5)  where 𝐴 is the y-intercept of the best fit line. The wind speed time series was divided into 10 minute periods for the three sites and calculated 𝐸(𝑓) for all periods comprising each dataset. The ten-minute periods were chosen according to the wind energy standard, based on the Van der Hoven spectrum gap, illustrated in Chapter 3. An algorithm that scans the 𝐸(𝑓) series and locates the region where the slope is closest to the -5/3 value described by Kolmogorov’s theory was also developed. The next step of the algorithm was to calculate the range of the dissipation rate. A Kolmogorov constant equal to 1.53 was selected, as its selection does not affect the results, since the variations of the dissipation rate in time were the main focus.  The dissipation rate can also be related to the TKE, which is a measure of the energy contained in the turbulent portion of the wind speed. This is also a widely used parameter of wind variance in the energy industry and meteorology [57]. The TKE is typically expressed as  83   21 ( ' )2TKEvm   (4.6)  'v v v    (4.7)  where 𝑚 is the mass of the air, 𝑣 is the actual wind speed and ?̅? is the average wind speed of the period time of study. Periods of 10 minutes were analyzed for the three sites. Eastward, northward, and upward components of the wind speed are normally used for the calculation of the TKE, but the vector norm magnitude of the three components was used for simplicity.  The lagged cross-correlation coefficients between the dissipation rate and the TKE were calculated to quantitatively describe the similarity in their waveforms. The equation for the lagged cross-correlation function follows:         10, 1/2 1/21 10 0N jk jk k k jkTKEN j N jk k k kk kTKE TKER LTKE TKE                    (4.8)  where 𝑅𝜀,𝑇𝐾𝐸 is the cross-correlation coefficient of ε and TKE, and L is the lag between the same parameters, also equivalent to 𝑗∆𝑡 [57]. The portions of data are defined by the subscripts 𝑗 and 𝑘, which increase with the lag, and also define the averages as   101 N jk kkN j     (4.9)  1101 N jk k jkTKE TKEN j    (4.10) 84  In other words, 𝑘 counts the elements of the series from the zero element to 𝑁 − 𝑗 − 1, which represents the boundary element of the corresponding lag. If the target cross-correlation coefficient is 1 hour, 𝑘 will have values from 0 to 5, given that the coefficient is calculated in ten-minute averages. As 𝑁 = 24 at all times, because of the 4 hour groups of data, 𝑗 = 18 for the 1 hour lag.  The standard error of the cross-correlation function is shown in [68] as follows:   1SEN k  (4.11)  For this study, the TI was also used to measure the evolution of the wind variability in the three site studies. TI is a measure of the variability of wind speed and is normally calculated at a standard time window of 10 minutes for wind energy applications, as follows   TIv   (4.12)  where 𝜎 is the standard deviation and ?̅? is the average wind speed.  4.4 Results A sample of the calculations is illustrated in Figure 4.3 to illustrate the behavior of 𝐸(𝑓) for the first 10 minutes of measurements. This procedure was repeated for 10 minute windows through the duration of all the data sets given in Table 4.1. The frequency region where the slope of a least square best fit line most closely matches to that of Kolmogorov’s inertial subrange value of -5/3 was identified. The rest of the 𝐸(𝑓) series were calculated from the three site studies, starting at 85  midnight of the first day. The dissipation rate 𝜀 can be calculated from the y-intercept of the best fit lines in Figure 4.3, according to (4.3) through (4.5). The y-intercept is found at the frequency of 1Hz, given that the spectral density is represented in a logarithmic plot. The inertial subrange was identified in the 10 minute energy spectrum plots, typically at frequencies between 0.01 Hz and 0.1 Hz, and the y-intercepts were used to calculate the coefficient of dissipation. It seems that extending that range of frequency seems to fit in the trend line. However, the extension of such range would actually increase the difference between the Kolmogorov slope and the best fit line’s slope.   1E-3 0.01 0.11E-81E-71E-61E-51E-41E-30.01E(m2 s-2)f (Hz)Intercept = -6.06636Slope = -1.69711 Figure 4.3 Example of 𝐄(𝐟) at a 10 minute time window for wind Site 1. Data was measured at a frequency of 1Hz  Figure 4.4 through Figure 4.6 illustrate stack plots of the three sites, divided into wind speed, TKE per unit mass, and energy dissipation rate ε. In the same plots, the cross-correlation coefficients 86  with absolute values higher than 0.6 are illustrated. Strong correlation between ε and the TKE is observed at Site 2, where lower values of wind speeds were measured. Furthermore, the lower wind speed distribution observed in a period of 7 days exhibits periodic patterns in the TKE distribution. From this perspective, one can also observe that as the wind speed increases, ε and TKE also increases. Dissipation rate is also inversely related to the size of the characteristic eddy length scale. This is, the larger the dissipation is, the smaller the dominant eddy sizes are [57]. In the case of the plot for Site 1, in Figure 4.4, higher correlation coefficients are not regular and the overall TI is high.  0:00 0:12 1:00 1:12 2:00 2:12 3:00 3:12 4:00 4:12 5:001E-121E-101E-81E-60.00.61.21.80:00 0:12 1:00 1:12 2:00 2:12 3:00 3:12 4:00 4:12 5:0005101520 (m2s-3)Day:Time EDRSite 1May 15-19  TKE/m (m2s-2) TKE/m Wind Speed  (ms-1)0.60.81.0 CrosscorrelationCrosscorrelation Figure 4.4 Wind speed, EDR, and TKE for Site 1. Wind speed is illustrated in 10 minute averaged data, and both TKE and ε were calculated from those averages. Additionally, the cross-correlation coefficients between ε and TKE are illustrated in the same plot of wind speed  87  0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:001E-121E-101E-81E-60240 1 2 3 4 5 6 7 8 9 10036912 (m2s-3)Day:Time EDRSite 2Jan 11-20  TKE/m (m2s-2) TKE/m  (ms-1) Wind Speed-1.0-0.50.00.51.0 CrosscorrelationCrosscorrelation 0:00 0:12 1:00 1:12 2:00 2:12 3:00 3:12 4:00 4:12 5:00 5:12 6:00 6:12 7:001E-121E-101E-81E-60.00.51.01.50:00 0:12 1:00 1:12 2:00 2:12 3:00 3:12 4:00 4:12 5:00 5:12 6:00 6:12 7:0002468 (m2s-3)Day:Time EDRSite 2Mar 12-18 TKE/m (m2s-2) TKE/m  (ms-1) Wind Speed-1.0-0.50.00.51.0 CrosscorrelationCrosscorrelation Figure 4.5 Wind speed, EDR, and TKE for the two periods of Site 2. Wind speed is illustrated in 10 minute averaged data, and both TKE and ε were calculated from those averages. Additionally, the cross-correlation coefficients between ε and TKE are illustrated in the same plot of wind speed   88  0:00 0:06 0:12 0:18 1:00 1:06 1:12 1:18 2:001E-101E-81E-61E-40510150:00 0:06 0:12 0:18 1:00 1:06 1:12 1:18 2:0006121824 (m2s-3)Day:Time EDRSite 3Sep 20-21  TKE/m (m2s-2) TKE/m  (ms-1) Wind Speed-1.0-0.50.00.51.0 CrosscorrelationCrosscorrelation 0:00 0:12 1:00 1:12 2:00 2:12 3:00 3:12 4:001E-131E-111E-91E-70369120:00 0:12 1:00 1:12 2:00 2:12 3:00 3:12 4:0008162432 (m2s-3)Day:Time EDRSite 3Oct 21-24  TKE/m (m2s-2) TKE/m  (ms-1) Wind Speed-1.0-0.50.00.51.0 CrosscorrelationCrosscorrelation Figure 4.6 Wind speed, EDR, and TKE for the two periods of Site 3. Wind speed is illustrated in 10 minute averaged data, and both TKE and ε were calculated from those averages. Additionally, the cross-correlation coefficients between ε and TKE are illustrated in the same plot of wind speed  The correlation between the changes of ε and the TKE might be visually difficult to inspect in Figure 4.4 through Figure 4.6 at some points. Thus a correlation function between the two 89  parameters needs to be calculated to illustrate how is the parameters vary throughout the day. The cross-correlation function was applied to determine the lags at which the changes between the two parameters are observed, and a persistence forecast is no longer valid. Groups of 24 values of ε and TKE in the 10 minute time periods were used, being a total of 4 hours of measurements and up to 2 hours of lag. The cross-correlation for the first 4 hours of Site 1 is illustrated in Figure 4.7 as an example of how this parameter looks like in the calculations. In this particular period, cross-correlation coefficients of approximately 0.6 and 0.7 are observed for the lags of 50 and 70 minutes, respectively. The standard error was calculated from the theory of Box and Jenkins, illustrated in (4.11), using the lags and the number of data pairs as the inputs. The error is directly proportional to the lag, supporting the fact that increasing lags becomes less reliable, and the cross-correlation function needs to be continuously recalculated to ensure accuracy.  Figure 4.7 illustrates the cross-correlation coefficients for the first 4 hours of measurements at Site 1, but the same function was calculated for the remaining data sets. The cross-correlation coefficients indicate a lag between the TKE and ε, and it exhibits positive and negative values. A positive lag corresponds to the periods of time in which a time series is in phase with itself, or have a phase shift shorter than one quarter of a cycle, while a negative lag indicates a phase shift larger than on quarter of a cycle. It is explained in [57] that ε and TKE are typically greater during the day, when buoyancy effects are more predominant due to greater thermal gradients. In the same reference, the author explains that the increase in the TI is followed by an increase in the dissipation rate under the same weather conditions. Typically the highest cross-correlation coefficient between ε and the TKE happen at a zero lag, indicating that the changes in the two parameters occur 90  simultaneously. There are some cases observed in the illustrated plots, where negative cross-correlation coefficients were calculated caused by the phase shift in the averaging periods.   0 20 40 60 80 100 120-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0CrosscorrelationLag (min) Figure 4.7 Cross-correlation coefficients between ε and TKE from 0am to 4am on May 15th for the measurements at Site 1. The lag is expressed in minutes, considering 10 minute time periods  The previous plots illustrate evidence of decaying dissipation during night time, which may be an additional reason why wind turbines are most likely to produce less power in that time of the day. The main reason for less generation during night time are lower wind speeds, due to a more stable thermal stratification near the ground. A cross-correlation coefficient is accepted given that it is higher than 0.6, and the time in which such value is calculated was captured. As all the cross-correlation functions are calculated in 4 hour ranges, the coefficients were calculated in lags of up to a truncation of 120 minutes. 91  20 40 60 80 100 120 1400123  CountLag (min)Site 1May 15-19 0 20 40 60 80 100 120 140024681012141618Site 2Jan 11-20  CountLag (min) 0 20 40 60 80 100 120 1400246810121416182022Site 2Mar 12-18  CountLag (min) 0 20 40 60 80 100 120 1400123456Site 3Sep 20-21  CountLag (min) 0 20 40 60 80 100 12001234567Site 3Oct 21-24  CountLag (min) Figure 4.8 Histograms of the cross-correlation coefficients with values equal or larger than 0.6 for the three site studies 92  Figure 4.8 illustrates histograms of the five available data sets in which the lags with cross-correlation coefficients higher than 0.6 were captured. The largest number of accepted coefficients is observed in Site 2, where moderate wind speeds were measured. There is a dominant lag observed between 110 and 120 minutes, followed by the range between 10 and 30 minutes. This trend is observed in the three sites, although the 2 hour lag is more evident in site 1.  In the case of Site 1, the accepted correlation coefficients are not as frequent as in the other cases. Site 1 experienced a higher TI during the measurement period, which can represent a higher degree of variability in the wind speed. In the case of the other sites, the dominant lag corresponds to zero minutes, indicating a simultaneous change in the ε and TKE. Other lags were observed between those parameters, but they must be analyzed carefully, as very long lags are rejected. The reason for the rejection is that mass of air passing over a point does not affect the masses that are coming from far away the same point. A lag of 120 minutes, or 2 hours, is observed in the three sites, but there is no deterministic theory that can support the influence of a mass of air over such a long time, thus the time lag is rejected. Moreover, when the cross-correlation level decreases, there is no basis to indicate a further increase. The main dominant lag is located between 0 and 10 minutes, which indicates that a decrease or increase in the TKE and in ε occur simultaneously, given that a positive correlation is calculated. It is interesting to notice that the decrease in the cross-correlation coefficients is comparable to the time delay between those parameters. Considering the decrease in the cross-correlation coefficients, a time delay between 10 and 20 minutes can be estimated. It is also interesting to capture the time where the highest correlation coefficients were measured to associate them with the estimation of the overall atmospheric conditions that may cause the delay. This time delay is very similar to the convective time scale, which is in the order of 10-20 minutes, 93  and represents the time period for air to circulate between the surface and the top of the atmosphere’s mixed layer [57], but this might be a coincidence.  Figure 4.9 illustrates the histogram of the lags for all three sites, with the purpose of generalizing the results obtained in the calculations, and the time of the day where those coefficients are dominant. The highest frequency corresponds to a zero lag, and the second highest corresponds to a lag of 10 minutes. A lag located at zero minutes means the variations in one parameters occur within the 10 minute block of the variations in the second parameter. The 2 hour lag exhibits a local maximum at all sites, and this trend is observed in Figure 4.9 left. However, this cannot conclude that there is a relationship between TKE variations at such a long lag. Moreover, it may result from a folding effect of the cross-correlation function.   0 20 40 60 80 100 120 14001020304050  CountLag (min)12 AM 04 AM 08 AM 12 PM 04 PM 08 PM 12 AM051015202530354045  CountTime of the Day Figure 4.9 Left: Histogram of frequencies for the lags between ε and TKE. Right: Histogram of frequencies for the time of the day when the highest correlation occurs  94  The calculation of non-zero lags gives the opportunity to estimate a change in the TKE when a change in 𝜀 is observed. As the correlation coefficients were calculated in periods of 4 hours, the day was divided in 6 bins to show the dominant hours of the day. It is observed that the early morning (between 4 am and 8 am) and the late night (between 8 pm and midnight) have higher correlation values between the two parameters under study. As illustrated by Stull [57], TKE experiences a rise typically after midnight, when the major cause of turbulence is shear, and ε increases as well, but this variation is highly dependent on the weather conditions. However, most of the lags between the parameters indicate a phase shift in the averaging periods considered to calculate the particular cross-correlation coefficient of TKE and ε. The designed algorithm is able to run on a continuous basis to facilitate the detection of a variation in any of the two parameters at the times with highest cross-correlation occurrence. In this way, a decision can be made in terms of wind turbine control and reaction to wind turbulence. This methodology should be applied for every site study and the results are not to be generalized, as the geographic and weather characteristics will affect its atmospheric conditions.  4.5 Autocorrelation of TKE The autocorrelation function is very useful to investigate how well correlated is a signal with itself under different lags. The use of the autocorrelation function has been used previously in wind energy related research [8], [47], [50], [52], [55], [69]–[72]. Pérez et al [69] used the autocorrelation function and fitting analysis to determine the importance of daily and yearly cycles in wind speed and temperature profiles. They found that wind cycles in wind are observed on a daily basis, which is more evident at a height of 200 m. The temperature profile exhibited both daily and yearly cycles at 40 and 200 m. The autocorrelation coefficients of the TKE were 95  calculated to identify the characteristic variations of the three sites. Equation (4.8) was applied to the same variable (TKE) to calculate its autocorrelation coefficients and the dissipation rate ε was substituted by the TKE as well. Figure 4.10 illustrates the results of these calculations and one can see that the curves behave differently for every location and time of the year. The diurnal effects are better observed in Site 2 (March dataset), where the TKE is highly correlated with itself every 1 day approximately. In the other cases the peaks are rather attenuated, indicating a lower and more lagged correlation. One can see in general that the autocorrelation starts at one at zero lag and then quickly decreases. This type of calculation allows to identify for how long we can use the persistence forecast in the TKE and how strong the further variability patterns are in each particular site.  00:00 01:00 02:00 03:00 04:00 05:00-0.630.000.63-0.630.000.63-0.630.000.63-0.630.000.6300:00 01:00 02:00 03:00 04:00 05:00-0.630.000.63 Site 1, May 15-19   Site 2, Jan 11-20  Site 2,Mar 12-18   Site 3, Sep 20-21RTKE RTKERTKERTKE RTKETime (DD:HH) Site 3, Oct 21-24 Figure 4.10 Autocorrelation coefficients for the three site studies over the previously mentioned periods  96  For example, Site 3 for the data measured in September, exhibits a shorter lag (about 2 hours) with an autocorrelation coefficient above 0.60. Considering an exponential decay for all cases, the time scales values are 13, 10, 4, 4, and 11 hours, respectively, from top to bottom of Figure 4.10. This time scale represents the time in which significant changes occur in the TKE.  A representation of the TKE budget in the frequency domain was introduced by McBean and Elliott [63]. The authors showed found that the dissipation of TKE is approximately equal to that of the local production. It was also found that atmospheric instability is directly proportional to the vertical flux of the TKE’s vertical component increases with instability. Divergence of TKE’s vertical flux is equal to the buoyancy production, with negative sign. Their work also extends the knowledge on how the TKE and dissipation are occurring at the same time, but not at the same scales [57], [63]. While production is feeding large eddies, dissipation is predominant in smaller scale eddies. But most importantly, the convergence point between buoyant and shear production with dissipation is characteristic of the inertial energy transfer from larger to smaller eddies. Thus McBean and Elliott’s work represents an alternative to illustrate the inertial subrange.  4.6 Conclusions A methodology to use HFD for wind variability studies was developed, using accessible mathematical tools to determine energy dissipation and identify the inertial subrange in a specific time period. The cross-correlation function was used to study the temporal lag between ε and the TKE. 𝐸(𝑓) and energy dissipation analysis indicated that frequencies between 0.01-0.1Hz are within the inertial range. When ε is calculated, significant changes are observed to follow diurnal and nocturnal patterns. This effect produces a lag between the variations of ε and the TKE during 97  different times of the day. This lag also follows the thermal variations that comes with the daily solar cycle, when periodic heating and cooling of the ground occurs. In general, high cross-correlation for a lag in the range of 0-10 minutes was found for the site studies, which implies that the zero lag is predominant and the variations are simultaneous. A second lag of about 2 hours was found, which can be interpreted as the cycle length of the two parameters, which keep varying simultaneously, rather than a 2 hour lag between TKE and ε variations. However, further analysis must be made to discard the possibility of a folding effect in the cross-correlation function. The predominant times for high cross-correlation coefficients are between 4 am and 8 am, and 8 pm and 12 pm, which indicates a strong influence of the daily solar cycle on the wind energy dissipation rate and the TKE. This analysis also determined the times when dissipative turbulence wind is predominant. This contribution may be combined with a prediction model to forecast not only the wind speed average, but the level of dissipation and the degree of variation under different atmospheric conditions. A time delay of 10-20 minutes was found for the averaged cross-correlation functions, which is in agreement with the convective time scale that communicates changes in the surface heat flux to the atmospheric mixed layer.  Variations in the TKE and ε occur necessarily at the same time and the cyclic variation can be estimated by the presented algorithm that calculates and detect lagged cross-correlation coefficients between them. The characterization of energy dissipation rate and TKE provided in this study are promising tools for integration into pitch control algorithms. Although the illustrated methodology was applied to three sites; it can be generalized for other wind farms around the world. The temporal lag observed between TKE and ε contributes to have a broader scope of the diurnal and nocturnal variations of the wind speed. 98   The autocorrelation function was used to determine variability patterns in the wind, depending on the site topology and weather characteristics. The diurnal cycle is not always evident in an autocorrelation plot and the exponential time scale can vary from site to site. Combining analysis on turbulence’s time delay with current wind speed estimation models can lead to modify the turbine’s control routine to respond to significant changes in wind variability. 99  Chapter 5: Conclusions and Recommendations  5.1 Conclusions Novel approaches to implement the use of HFD in the study of wind energy were presented. The argument expressed in Chapter 1 is based on the response characteristics of a wind turbine. It is observed that its response time window is in the range of 2 to 20 seconds, approximately, where the gusts of duration within such range have the strongest impact on the turbine behavior. For this reason, special emphasis should be put on the study of HFD, for each specific wind farm or potential wind site. The determination of a wind turbine’s settling time under different variability conditions must be carried out when matching a specific turbine model with the corresponding weather characteristics of a wind site.  The Van der Hoven spectrum has been defined as one of the basis of wind turbine standards. One of the most significant contributions to the field is the settling of a 10 minute average basis to describe wind conditions in a time series. This approach was re-addressed to evaluate the accuracy of his conclusions. One main discrepancy with his results is that the major eddy-energy peak is located at a time period of 2 days, being of high importance when a wind study foresees wind variations in the range of the mesoscale. These results are not to be generalized and the methodology must be applied on a case by case basis. It is suggested that Van der Hoven’s results need to be updated and the calculations and analysis carried out in this research represent the first attempt to do so.  100  A combination of this theory and the dynamic characteristics of a wind turbine model allowed the estimation of the modified TI in the model. This approach can also be generalized to different wind turbine models, when the correction function is available. The monetary value of TKE budget highlighting the extra revenue that the HFD analysis can bring was also calculated.  One main reason to calculate the cross-correlation coefficients between the TKE and ε is that their variations do not occur necessarily at the same time. Thus a method to anticipate variations in the TKE budget or wind speed dispersion was developed. The calculation of the energy dissipation rate proved to be a useful tool to anticipate the variations in the TI in a time series. An algorithm that assesses the cross-correlation coefficients between the dissipation rate and TKE in the search for a new turbulence estimation approach was developed. The nights and early mornings exhibited higher cross-correlation coefficients between the TKE and ε for the three sites investigated. It is possible, however, that different sites present high cross-correlation coefficients at different times of the day. This method can also be used to estimate the time delay between those two parameters, which, in our case studies was approximately between 10 and 20 minutes.  5.2 Future Work and Recommendations The results presented in this research thesis were calculated with wind data measured at 1Hz and five Hz in single point meteorological towers. The next step in a study of this kind would be the analysis of multipoint data, with meteorological towers located in the vicinity of each other. In this matter, the standard 61400-12-1 published by the IEC contains specific guidelines for meteorological towers siting [73]. Spatial wind variability is an open field that needs to be addressed to better understand phenomena such as eddy evolution and wake. 101   It is expected that the results of a study in this field will vary according to the topography of the site. This represents an important approach to study how shear influences the turbulence generation in the spatial domain. 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London: Brooks/Cole, 2000.    109  Appendices Appendix A  Wind Energy in Off-Grid Mining Deployments in Northern Canada  A.1 Introduction Power supply for remote communities is typically delivered by diesel generators, as those locations do not have access to conventional transmission lines. Transport of diesel to remote sites increases the price of this fuel and, as a consequence, the final price of energy delivery [74]. There are also environmental costs generated by the exclusive use of diesel generators to satisfy the electricity demand.  A study developed by Arriaga et al [75] revealed that more than 175 communities in northern Canada generate their own electricity off-grid using diesel-fuel generators. Without further detail, the author reported approximately 100,000 persons living in these communities and an average energy cost of $ 1.3 per kWh. With the increasing electric demand, the replacement or upgrade of current power generation systems becomes imminent, and this presents an opportunity for renewable energy sources.  The territory of Nunavut, in Canada, has a population 36,600 as of July 1, 2014 [76]. The latest report by the Government of Nunavut, published in 2007, states that the total petroleum product consumption was 171.7 million liters, from which 38% corresponded to heat and 20% to electric consumption [77]. The largest consumption corresponded to transportation, being 42% of the total petroleum based consumption. Purchase of petroleum products represented 20% of the Government of Nunavut’s annual budget during the fiscal year 2005-2006 [77]. The Government 110  of Nunavut states its commitment to provide economic incentives to both communities and companies that include renewable sources and efficiency strategies in their energy consumption. These incentives are designed to be integrated along with those offered by Natural Resources Canada (NRCan).  There is evidence of previous efforts from the Government of Canada to provide incentives for the deployment of renewable energy technologies in the country. For example, the Wind Power Production Incentive offered one cent per kWh for ten years for wind-powered electrical generation commissioned. Although this program was discontinued on March 31, 2007, it was replaced by a similar program [78].  The proportion of energy costs in mineral extraction and processing represents approximately 15% and 19% of the production costs for metals and non-metals, respectively [79]. Mining extraction itself represents 60% of the energy consumption [80]. The introduction of renewable energy sources is proposed in this study to lower the costs detected by NRCan. The implementation of the proposed measures can bring benefits not only to the economy of mining companies, but also to the communities of First Nations participating in the mining activities. Aboriginal participation has been encouraged by NRCan as part of the Minerals and Metals Policy of the Government of Canada.  The Diavik Diamond Mine is a case example of success in renewable energy technology deployment in the mining industry. The company installed a wind farm consisting of four Enercon E70 wind turbines, with a total capacity of 9.2 MW, and an annual production of 17 GWh. This 111  represents 8.5% of the mine’s power needs [81]. Another success case example is the nickel and copper operation Raglan mine located in northern Quebec. Considering an additional power generation requirement of almost 100%, a waste heat recovery scheme was applied to capture the exhaust heat from their diesel generators. This measure reduced the foreseen installed power supply and its associated greenhouse gas emissions by approximately 50% [79].  The industrial partner, whose identity will not be disclosed, is a mining company which has a residential facility in North West Nunavut and a processing plant to be built in the next three years. The residential facility operates typically from February to October and it is normally closed in the months of extreme weather conditions. They are interested in reducing their diesel consumption by including renewable sources in their energy budget. The industrial partner was also interested in reducing generation costs by looking at the opportunities to increase the efficiency of their residential facility.  The objective of this section is to assess the economic feasibility of wind energy installation in mining deployments.  A.2 Case Study The economic feasibility of including wind energy based power production in the site study was assessed. A bank of batteries for energy storage in the system was considered. For the analysis the software HOMER Energy, from the National Renewable Energy Laboratory (NREL, USA) [82] was used. The mining project is foreseen as a 10 year project. The interest rate considered in this study was 6%, but a sensitivity analysis that considered a wider range was performed. There was 112  no public wind data available for the exact location of the site study, so the data from the nearest location in the database of the software RETScreen, from NRCAN [83] was considered. The monthly wind speed average was set equal to that at Robertson Lake Airport, located at approximately 200 kilometers of the site study. There is an evident need for real data, and the installation of a meteorological tower is being considered by the author’s industrial partner. These resources were used to calculate the economic feasibility of including solar and wind technology in the company’s energy budget. The averaged data are illustrated in Table A.1.  Month Wind Speed Average [m s-1] January 6.3 February 6.4 March 6.3 April 6.2 March 6.0 May 5.2 June 5.7 July 6.2 August 6.8 September 7.0 October 6.0 November 5.8 December 6.3 Table A.1 Solar and Wind resource assumed for the site study  A.3 Methodology A constant power generation of 23 MW will be required to supply the power needs in the mine’s processing plant and residence during the extraction phase. The demand will reach the total power 113  consumption when the processing plant enters into operations, during the transition between 2018 and 2019. This means that the mine’s power consumption will be 50 times higher than in the residential facility, where all the exploration tasks will be carried out. The load profile will change to a constant load throughout the day all year long, since the mining plant operations will be continuous. The annual variability in the residential facility’s power consumption will be insignificant compared to the continuous power demand of the mining machinery. This phase would require large power generation and energy storage systems. Under these conditions, wind energy becomes an ideal candidate to supply the future power demand of the site under study when the processing plant becomes operational. The option of installing a wind farm to reduce diesel consumption and the consequent costs of fuel transportation, O&M of the engines, was analyzed.  For study, wind speed average data illustrated in Table A.1 was considered to make the simulations in HOMER software. The wind speed averages were measured at a 10 m height, and the annual average is considered as wind class 5, according to the classification made by Jain [2]. The scale is defined by seven classes, 1 being the lowest potential and seven the highest one. Class 5 is considered ‘Excellent’ by the previously mentioned author. Three possible scenarios were considered: diesel generators, storage system, and wind turbines. The wind speed correction at the turbine’s hub height was considered by the software.  A.4 Results Table A.2 illustrates some of the results obtained in the simulations performed by HOMER software. 114  Scheme Energy Generated [GWh/year] Annualized Capital Cost [$ USD per year] Operating Cost [$ USD per year] COE (Cost Of Electricity) [$/kWh] Diesel Consumption [L Per Year]  20 MW diesel generator  10 MW diesel generator  10 3.3 MW wind turbines  43 MWh battery bank  15 MW converter  70.0   35.7   121.2 $19,048,642 $113,263,690 0.657 36,093,492  20 MW diesel generator  11 MW diesel generator  5 3.3 MW wind turbines  43 MWh battery bank  15 MW converter  95.3   50.3   60.5 $8,793,558 $154,622,128 0.821 48,963,296  25 MW diesel generator  11 MW diesel generator  43 MWh battery bank  7 MW Converter  183.2   21.6 $13,193,596 $207,400,785 1.095 68,898,856  30 MW diesel generator  11 MW diesel generator  193.4   8.1 $12,950,083 $220,186,069 1.157 72,971,648 Table A.2 Diesel consumption and costs of different power generation deployments for the extraction phase. The amounts are based on a 10 year project lifetime at 6% interest rate per year  115  It is observed that the introduction of large scale wind turbines in the system brings significant fuel and economical savings. The capacity factor used for the wind farm deployment was set at approximately 41.9%, since HOMER calculates the best scenario given the wind conditions and the technical information provided by the turbine manufacturer.  If wind energy is not considered for the final project, it is expected that a battery bank alone can bring reductions in the fuel consumption of more than 4 million liters of diesel per year. This reduction represents more than $100,000,000 USD in the NPC (Net Present Cost). Wind energy related costs illustrated in Table A.2 were extrapolated from the amounts provided in [84], where ten 3.3 MW wind turbines were considered for a wind farm. Since the costs from the electric infrastructure and the AC/DC converter were considered separately, the difference between them was calculated. The final cost of the electric infrastructure was set at $1,008,185 USD per turbine. Based on the information provided in [84], the capital cost of a 3.3 MW wind turbine was set at $3,085,500 USD. Equipment transportation costs were considered to double the capital costs of the deployment, similar to that of diesel transportation. In this way, the total cost for each wind turbine deployment, including the electric infrastructure (excluding the converter) was set at approximately $ 8,200,000 USD. This value was interpolated for an initial five wind turbine deployment and then for 15, 20, 25 and 30 turbine, as illustrated in Figure A.1. The 30 wind turbine deployment option lacks of technical feasibility in this case study, due to existing constraints in land availability. However, the plot is merely illustrative and only provides an economic scenario for such option. The annualized capital cost illustrated in Table A.2 positions the five wind turbine deployment as one of the best options for the extraction phase, as it is the lowest value in the table.   116  0 5 10 15 20 25 30306090120150180210240270300Number of TurbinesO&M and Initial Capital ($ MM USD) O&M Costs Initial Capital Net Present Cost75090010501200135015001650 Net Present Cost ($ MM USD) 0 5 10 15 20 25 302025303540455055606570Number of TurbinesDiesel Consumption (L per year) Diesel Renewable Fraction0.00.10.20.30.40.50.60.7 Renewable Fraction Figure A.1 Left: Costs involved in a power generation plant that includes both wind energy and conventional sources. Right: Diesel consumption for the same power plant relative to the number of wind turbines included in it, and the wind farm participation in the power generation  The left side of Figure A.1 exhibits a tendency to have a minimum NPC cost of approximately $850,000,000 USD, as the number of wind turbines increases. However, the larger the wind deployment is, the more stability issues in the power delivery (voltage and frequency) will arise. The stability constraints of the wind farm installation were not addressed, as this is out of the author’s scope. O&M cost involved those related to the entire system, including both conventional and renewable equipment. Reduction in the use of the diesel generators results in a decrease of O&M costs, thus it explains the inverse proportionality between the number of turbines and the mentioned costs. This is a result of lower O&M costs of the wind turbines than the analogous costs of the generators plus the actual diesel price. Right side of Figure A.1 shows the energy participation of the wind power generation deployment. The results indicate that 50% of the energy would be generated from wind energy if ten wind turbines are considered, and 30% for the five wind turbine option. The largest increment is observed when ten wind turbines are installed, 117  compared to the five wind turbine installation. It represents approximately 40% increase relative to the energy produced by five wind turbines, but only and approximate 10% increase is observed when fifteen wind turbines are deployed. An analogous decrease occurs with the diesel consumption, having its largest reduction when ten wind turbines are installed. From the same plot it can also be inferred that the maximum energy fraction supplied by a wind farm should not be larger than 70%, for economic optimality.  Since the interest rate was varied with the number of installed wind turbines, the NPC decreased when the interest rate increased, as illustrated in Figure A.2 left. However, the overall COE increased, making the project more expensive. The wind farm consisting of five wind turbines is exhibited as the best option at different interest rates because the difference in the NPC (savings) decreases when more turbines are added.  0 5 10 15 20 25 300.40.60.81.01.21.41.61.8NPC ($ BB USD)Number of Turbines6%10%15%20%5 10 15 20 25 3050001000015000200002500020%15%10%6%Savings per kW ($ USD/kW)Number of Turbines Figure A.2 Left: NPC variation with different values of the interest rate. The installation of ten wind turbines exhibits the highest rate of change in the NPC at all interest rates. Right: Savings per kW installed of wind generated power 118   Right side of Figure A.2 illustrates the calculated savings that every installed kilowatt of wind power represents. These savings were calculated by subtracting the NPC at all wind turbine deployment cases from the NPC without wind generation. This difference was divided by the installed power of each wind turbine deployment. It is observed that the largest amount was calculated for the five wind turbine deployment at all interest rates. The five wind turbine deployment is then positioned as the option to generate a fraction of the plant’s power consumption.  In the tornado chart illustrated in Figure A.3, the diesel price is the main factor that impacts the economics of the extractions phase. The other parameters have a very low impact compared to that of the diesel price, considering a 10 wind turbine deployment. The interest rate of the base case was set at 10% in order to include $0.41$0.64$0.65$0.70$0.66$0.94$0.71$0.70$0.67$0.69$0.30 $0.50 $0.70 $0.90 $1.10Diesel Price ($/L)Capital CostInterest Rate (%)Mine Life (Yr)O&M ($)Cost of Electricity  ($USD)-50%50%119  a wider interest rate range to impact on the COE. The variation in the O&M costs were also included since those are significantly higher for a wind farm deployment than for a PV (Photovoltaic) installation. The diesel price is the parameter whose variations affect more the overall COE, as illustrated in Figure A.3. These parameters are directly proportional and they have approximate constant rate of 1 to 1.    $0.41$0.64$0.65$0.70$0.66$0.94$0.71$0.70$0.67$0.69$0.30 $0.50 $0.70 $0.90 $1.10Diesel Price ($/L)Capital CostInterest Rate (%)Mine Life (Yr)O&M ($)Cost of Electricity  ($USD)-50%50%120  Figure A.3 Tornado chart of five parameters’ impact on the COE for the extraction phase. COE of base case was calculated as $0.674 USD  A.5 Summary The installation of wind turbine for the extraction phase can bring even larger savings, due to the high energy demand that the processing plant will have. A wind farm deployment of 5 to 10 wind turbines can supply between 30% and 40% of the total plant’s power demand, saving between 25 and 35 million liters of diesel. The environmental benefits of not burning such amount of fuel are very important in terms of greenhouse gases that will not be emitted to the atmosphere. It is very important to highlight the negative impact that a large renewable source penetration in a remote grid can have on the supply-side stability. From a strictly economic perspective, the total savings in a renewable energy project must be reassessed depending on the diesel price variations $0.41$0.64$0.65$0.70$0.66$0.94$0.71$0.70$0.67$0.69$0.30 $0.50 $0.70 $0.90 $1.10Diesel Price ($/L)Capital CostInterest Rate (%)Mine Life (Yr)O&M ($)Cost of Electricity  ($USD)-50%50%121  throughout the year. This will determine the option that will bring the highest revenue to the consumer.  The issues that the electric line may experience were not addressed, as they are out of this study’s scope. However, during the implementation of the renewable technology, special attention must be focused on the effects that an intermittent power source will have. State of the art regulators with fast response and constant renewable source monitoring are tools that can contribute to mitigate the influence of unexpected variations in the renewable power supply. The economic values calculated in this study may be affected by the measures taken to address the aforementioned issues.  Significant savings were calculated in the energy demand costs during the extraction works if a wind farm installation with 16.5 MW of nominal capacity is installed. The calculated savings for a processing plant and residential facility with a demand of 23 MW on a continuous basis are approximately $403,000,000 USD for ten years of operations. A 33% reduction of the carbon dioxide emissions during the extraction phase can be achieved with the operation of a wind farm. The added expenses of transportation were also considered, which doubled the capital costs of the wind turbines and the energy storage systems compared to average installation sites.   122  Appendix B  Probabilistic Forecast of Wind Speed Peaks Using HFD  B.1 Methodology Stull explains [85] that it is possible to make probabilistic forecasts rather than deterministic ones. The statement is based on the random nature of the atmosphere conditions. In this way, statistical estimations of wind speed can be made within a certain range of confidence. This section illustrates a method that performs wind speed peaks estimation within a specific time period (30s). It is certainly not possible to improve persistence forecast accuracy of the number of peaks within that range of time, but the trend that wind speed has been taking in the past minutes can be reported. It is considered that the atmosphere stays in the same conditions when short time periods are studied. The HFD used for this study was obtained from meteorological towers owned by Vestas, Inc. The designed algorithm was applied for three different locations that will not be disclosed due to a confidential agreement signed with the industrial partner. The frequency of measurements for two of the sites is 1 𝐻𝑧 and 5 𝐻𝑧 for one site. However, only one site will be reported in this appendix for illustrative purposes. For convenience, 1 𝑠 averages in the case of the highest frequency measurements were calculated. The statistic estimation was made in 30 𝑠 time periods; the data was divided accordingly and the average and standard deviation were calculated for each period. Matlab was used to find the peaks and their location in each time period, considering a peak as the local maximum in the wind speed data series. From this point, the occurrence frequency of both the value of the peak and the time in which it happens was tabulated.   Figure B.1 and Figure B.2 illustrate two examples of how frequency distributions evolve for a 4.5 minute sample. The former captures the time in which a local maximum occurs and the latter 123  captures the wind speed value of the maximum. The ordinate axis in both figures capture the cumulative occurrence frequency. This method can be extrapolated to the data series of the any site captured in high frequency measurements. The points in time with higher frequencies are used to estimate where the future peaks are going to be located in the next time period.  For Figure B.1, 10 bins of 3 𝑠 each were considered, while for Figure B.2, bins of 0.5 𝑚𝑠−1 were considered. In this specific example, certain regularity is observed at the approximate half of each period (12 − 15 𝑠). For this reason, it was estimated that a peak will be observed in the half of each period with high confidence. After the program deploys the frequency and cumulative frequency of time and peak value (wind speed), a second procedure is carried out. The estimation consists of evaluating the rate of change in the cumulative frequency of both parameters. The dominant peak and times are those with higher frequencies and they will be preferably considered in the next period estimation. The model also considers that if the modes are not changing in more than two periods, they will not be considered for the next estimation. In case of multimodal peak values, the algorithm considers the smallest wind speed to avoid overestimation. The distributions illustrated in both figures are combined to estimate the 3 𝑠 period in which a local maximum will occur, and the value of such maximum. These estimations were compared to the 60% success rate of a persistence forecast.   124  0 5 10 15 20 25 30012345678  Time [s]Cumulative Frequency   30s 60s 90s 120s 150s 180s 210s 240s 270s Figure B.1 Cumulative frequency histograms of the time when the peaks occur every 30s in a site study. As the number of peaks increases, the histograms show where the majority of peaks occurs  13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.002468101214  Cumulative FrequencyWind Speed [ms-1] 30s 60s 90s 120s 150s 180s 210s 240s 270s Figure B.2 Cumulative frequency histogram of the peaks are highly dependent on the average and the TI.  125  Table B.1 illustrates examples of the method that is carried out to deliver the estimation. It is not expected that all the peaks that will occur in a time period will be predicted in advance, but all the estimated peaks should occur within a range error of 1-2s. The estimation starts after 30s of observations and the first guess is identical to the peak distribution in the first period. In the case of the peak values, the lower wind speed is taken as threshold to avoid over estimation as much as possible.  Time period [s] Predicted occurrence [s] Predicted threshold [ms-1] Real occurrence [s] Real wind speed value [ms-1] 61-90 16-18 15.0-15.5 17 16.3 91-120 16-18 22-24 26-28 15.5-16.0 15 18 20 16.33 16.33 16.94 121-150 14-16 16-20 15.5-16.5 13 20 14.48 14.48 151-180 2-4 12-14 16-18 20-24 26-28 14.0-14.5 2 12 14 21 25 13.86 14.48 14.48 14.48 14.48 181-210 2-4 12-14 20-22 26-28 14.0-14.5 3 13 21 29 14.48 13.86 13.86 13.86 211-240 2-4 14-18 20-24 26-28 13.5-14.5 3 15 21, 24 29 13.86 15.09 14.48 13.86 Table B.1 Estimation examples for a 4 min period in a site study  126  B.2 Conclusions The principle of wind speed estimation consists of guessing that a given wind speed frequency distribution will remain the same in two consecutive time periods. This approach was taken in this study to compare it with the proposed estimation model. It was observed that a 60% of the predicted peaks occurred in the time series. This is not better than the persistence approach, and confirms that reliable forecast is unfeasible when shorter time windows are targeted.  

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