Validation of wind turbine wake models

Imagem António Nunes Vicente Validation of wind turbine wake models Dissertação de Mestrado em Engenharia Mecânica na Especialidade de Energia e Ambiente July/2018 DEPARTAMENTO DE ENGENHARIA MECÂNICA Validation of wind turbine wake models Submitted in Partial Fulfilment of the Requirements for the Degree of Master in Mechanical Engineering in the speciality of Energy and Environment Validação de modelos de esteira para turbinas eólicas Author António Henrique Seabra Nunes Vicente Advisors PhD. António Manuel Gameiro Lopes PhD. Omar Herrera Sanchez (menzio GmbH) Jury President PhD. Pedro de Figueiredo Vieira Carvalheira Assistant Professor, University of Coimbra Vowel PhD. Almerindo Domingues Ferreira Assistant Professor, University of Coimbra Advisor PhD. António Manuel Gameiro Lopes Assistant Professor, University of Coimbra Universidade de Coimbra menzio GmbH Coimbra, July, 2018 Acknowledgements António Nunes Vicente i ACKNOWLEDGEMENTS First of all, I would like to thank my advisor, Professor António Gameiro, for having given me the opportunity to work on an interesting and relevant topic. His dedication and guidance were crucial to my thesis work. I much appreciate the fact that I have always been able to made progress in a simple and steady way. I would also like to thank my co-advisor, Professor Omar Herrera Sanchez, for his useful advice and continued support. My time at University of Coimbra has just come to an end, and thus I would like to express my gratitude to my dearest classmates who have made my journey a pleasant one. Last, but not the least, I would like to thank my Parents for always believing in me and pushing me to do my best, and my siblings Laura and Vasco for never letting there be a dull moment. EVALUATION OF WIND TURBINE WAKE MODELS ii 2018 ABSTRACT António Nunes Vicente iii Abstract Wind turbine wakes have a strong impact on wind farms given that they affect the power output and the level of turbulence that determines the turbines lifetime. Thus, wake modelling is of critical importance to the wind energy industry, having a central role in the optimization of wind farm layouts. The main objective of this work is the validation of the analytical wake models implemented in the software package WindStation. Such validation was based on measurement data recorded in an onshore wind farm with eight wind turbines, and supported by results obtained by the software package WindSim. Conclusions were drawn by analyzing the computed velocity deficit of the air flow downstream of the wind turbines and the effective power of a single wind turbine. Keywords: Wind, Turbine, Wake, Turbulence, WindStation, WindSim. EVALUATION OF WIND TURBINE WAKE MODELS x 2018 LIST OF TABLES Table 4.1 – Wind turbine technical specifications .............................................................. 28 Table 5.1 - TI correction ...................................................................................................... 42 Table 5.2 – Error obtained for all five wake models. .......................................................... 45 Table 5.3 – Offset correction: updated error obtained for all five wake models ................ 50 Table 0.1 – Wind turbines details: row number, turbine name, turbine type, altitude (z) in meters and coordinates .......................................................................................... 57 Wind farm and measurement data António Nunes Vicente 11 SIMBOLOGY AND ACRONYMS Simbology  – Wind turbine rotor swept area  – Thrust coefficient  – Wind turbine rotor diameter  – Turbulence kinetic energy  – Wake radius  – Turbulence intensity  – Wind speed  – Wind speed corrected with the wake effect  – Wind speed deficit  – Free stream wind speed  – Wake decay constant  – Wind direction Acronyms GWEC – Global Wind Energy Council SCADA – Supervisory Control And Data Acquisition WFDT – Wind Farm Design Tool WMM – Wind Meteorological Mast wspd – Wind speed EVALUATION OF WIND TURBINE WAKE MODELS 12 2018 Wind farm and measurement data António Nunes Vicente 13 1. INTRODUCTION Altough fossil fuels are still the dominant source of energy, there has been a gradual shift towards renewable energies. One of the most reliable sustainable energy is wind energy, which nowadays is used in large scale for electrical power production. The global cumulative installed wind power capacity in 2017 has overcome the value of 500,000 MW (see Figure 1.1). This power is produced in onshore and offshore wind farms containing large numbers of wind turbines. Figure 1.1 – Global cumulative installed wind capacity. Adapted from GWEC (2018) The concept of energy conservation dictates that if a wind turbine extracts kinetic energy from the wind, then the downstream flow will diminish in momentum. The turbine wake is the region affected by this momentum deficit. Due to wake effects, wind turbines positioned downstream of others will have its performance considerably affected: lower wind speeds reduce the turbine power generation, and the raise of turbulence intensity causes fatigue loads, shortening the turbine life span. Owing to the cost of land, wind turbines are being grouped together in tighter spacing, which leads to increased wake effects. Hence, wake modelling plays a central role in developing optimized wind farm layouts. The goal of this work is to validate and evaluate the wake models included in the software package WindStation. Measurement data from a small onshore wind farm will be used to assess the prediction of the velocity deficit for each wake model. The validation of these models is then corroborated by the software package WindSim. The outline of this thesis is as follows. Chapter 2 will give an introduction to turbine wakes and its modelling background. Chapter 3 will provide a description of WindStation and its available wake models, as well as a short overview of WindSim. The EVALUATION OF WIND TURBINE WAKE MODELS 14 2018 measurement data of the wind farm will be presented in Chapter 4, together with the filtering process made with such data. Chapter 5 then discusses the validation of the wake models: the focus is on comparing the velocity deficit results obtained by a panorama calculation in WindStation and WindSim with measurement data. Both single wake and multiple wake situations are analysed. The effective power of single turbines will also be analysed. TURBINE WAKES António Nunes Vicente 15 2. WIND TURBINE WAKES This chapter will provide a brief description of the wake behavior, from its beginning to a further downstream position, as well as a summary of the wake modelling background. 2.1. Wake behavior As the air flow approaches a wind turbine, it starts to slow down and the pressure increases. Then, when it crosses the turbine rotor, there is a sudden pressure drop (see Figure 2.1, cut A-A). Figure 2.1 – Wind speed and pressure variation. Adapted from Janssen (2012). A turbine wake region is commonly divided into a near wake and a far wake. The region immediately downstream of the rotor is called the near wake and it extends for 2 to 5 rotor diameters. This region is dominated by the turbulence created by the turbine itself: there are non-uniform deficits of pressure and wind speed associated with the axial thrust and torque of the machine. The air circulation along the turbine blades leads to the formation of vortices with helical trajectories that quickly expand, forming a cylindrical shear layer. This shear layer is what separates the inside of the wake from the outside ambient flow. EVALUATION OF WIND TURBINE WAKE MODELS 16 2018 Figure 2.2 depicts a sketch of this situation. The wake growth and shear layer expansion are represented based on an axisymmetric flow. Figure 2.2 – Wake growth based on an axisymmetric flow. Adapted from Crespo et al. (1999). Further downstream, the wake starts to recover: the pressure increases and the velocity inside the wake decreases until ambient pressure is reached (Figure 2.1, cut B-B). As turbulent diffusion of momentum becomes the dominant mechanism, the near wake region ends when the shear layer thickness increases until it reaches the wake axis. The far wake region starts approximately 5 diameters behind the rotor, where the wake flow is completely developed. The wind velocity starts then to recover and the flow will decay to its free stream conditions. The topographic effects and ambient turbulence become dominant over the turbulence caused by the rotor. 2.2. Wake modelling A significant amount of research has been done over the past 50 years in wake modelling. A comprehensive literature survey on wake models can be found in Crespo et al. (1999). They distinguished two classic approaches to the problem. A common approach was to assume that the turbines acted as distributed roughness elements. These models used a logarithmic wind profile, modified by an increase in roughness due to the presence of the turbine itself; see, e.g., Bossanyi et al. (1980), Emeis and Frandsen (1993). TURBINE WAKES António Nunes Vicente 17 However, the traditional approach to wake modelling is based on the description of a single wake, succeeded by a calculation of its interaction with the neighbouring ones. These type of models are known as individual models. The classical work by Lissaman (1979) was one of the pioneers of this method. The author described a computer model for an arbitrary array of turbines, using basic fluid mechanics expressions and self-similar wake profiles derived from the experimental work done by Abramovich (1963) on co-flowing jets. Individual wake models are divided into two categories: analytical models and computational models. Other authors call them kinematic models and field models, respectively. Computational models are very time consuming and computationally expensive, as they make the least simplifications of the Navier-Stokes equations to fully characterize the turbine wake and turbulence. These models calculate the flow magnitudes at every point of the flow field, with resource to Computational Fluid Dynamics (CFD). Relevant field models were developed by Taylor (1980), Ainslie (1985), and Crespo and Hernández (1989). According to Réthoré (2009) there are three main CFD wind turbine wake models: full-rotor computations, the actuator line method and the actuator disk method. However, these type of models will not be studied in this work. Analytical wake models are based on semi-empirical functions and simplifications of the Navier-Stokes equations. They apply analytical expressions to calculate the wind speed deficits after the calculation of wind fields. Different models have been presented in the past years; see, e.g., Lissaman (1979), Jensen (1983), Frandsen (2007) and Ishihara et al. (2004). These models can be very effective in modelling the wake expansion and the velocity deficit, and are usually preferred due to its computational efficiency and fastest resolution. However, as the change in ambient turbulence is not considered, a turbulence model has to be coupled with analytical models. WindStation provides three (analytical) wake models: Jensen, Jensen 2D and Larsen. These models will be introduced in the following chapter. EVALUATION OF WIND TURBINE WAKE MODELS 18 2018 SOFTWARE PACKAGES António Nunes Vicente 19 3. SOFTWARE PACKAGES In this chapter the software packages that were used on this work are described: WindStation and WindSim. The main focus is on describing WindStation, with reference to the main theoretical foundation concepts and available wake models. 3.1. WindStation WindStation is a software package for the numerical simulation of turbulent flow over complex topography, complemented with a recent update of turbine wake modelling. The numerical wind fields are calculated with provided solutions for the non-linear fluid dynamics equations, coupled with turbulence models. Detailed information about WindStation is available in the WindStation manual by Lopes (2018). 3.1.1. Theoretical background 3.1.1.1. Transport equations The numerical calculation is supported by the non-linear fluid dynamics equations, more specifically the Navier-Stokes equations, the continuity equation and the energy equation. A summary of these equations will be made next. The Navier-Stokes equation describes the conservation of momentum for a fluid flow, with the assumption that it is a function of a pressure term and a diffusion viscous term. The generic WindStation steady state formulation of these equations is:            = −     +     󰇩 Γ 󰇧 2       − 2 3   󰇍  󰇨 󰇪 +     󰇩 Γ 󰇧 2       +       󰇨 󰇪 + 󰇡    −    󰇢    +   +   (3.1) where  [kg/m3] is the fluid density,  [m] is a generic Cartesian coordinate,  [N/m2] is the pressure, and Γ==+ [N s/m2] is the time diffusion coefficient for momentum, i.e., the effective viscosity. EVALUATION OF WIND TURBINE WAKE MODELS 26 2018  Wind Resources – the wind field numerical results are coupled with climatology data to provide a wind resource map;  Energy – the annual energy production, AEP, is calculated for all turbines, including wake losses. In this work, WindSim was used as an alternative approach to WindStation for assessing the wake models. The procedure was to replicate, as far as possible, the simulation parameters used on WindStation. WindSim provides three wake models: Jensen, Larsen, and a third one with a turbulent dependent rate of wake expansion (which was not used in this work). More details about WindSim can be found in the WindSim Getting Started manual by Meissner (2015). WIND FARM AND MEASUREMENT DATA António Nunes Vicente 27 4. WIND FARM AND MEASUREMENT DATA 4.1. Wind farm The wind farm under study in this work is located in northern France and is composed by eight wind turbines and one meteorological mast (WMM), displayed as in Figure 4.1. A 3D layout from WindSim of the wind farm is also available in Figure 0.1 of APPENDIX A. The turbines are arranged in two rows: row 1 (composed by 21,20,9,6) and row 2 (composed by 22,0,8,23). The meteorological mast is placed southwest of the array. Detailed information regarding turbine coordinates and mean sea level height is available in Table 0.1 of APPENDIX A. Figure 4.1 – Wind farm layout. The turbine types are  90 − 2.0 MW and  112 − 3.075 MW. Row 1 is composed by the 90 type and row 2 by 112. Its main technical specifications are displayed in the following table: EVALUATION OF WIND TURBINE WAKE MODELS 28 2018 Table 4.1 – Wind turbine technical specifications Turbine Rated power (kW) Cut - in wind speed (m/s) Cut - out wind speed (m/s) Rotor diameter (m) Hub height (m) V90 2000 4 25 90 105 V112 3075 3 25 112 94 The wind farm array is irregularly spaced. The spacing between turbines in a row range from a minimum distance of 548 m for 21−20, which corresponds to 4.9 rotor diameters (4.9 ), to a maximum distance of 601 m (6.2 D) for 0−8 (see Figure 4.1). On the other hand, adjacent turbines are separated by a minimum distance of 1642 m (14,7 ) for 6−23 and a maximum distance of 1872 m (16,7 ) for 21−22. As for the meteorological mast, its closest wind turbine is 21 at 1599 m. Note that this wind farm is neighbored by 3 other ones, which will not be considered throughout this study due to inexistent measurement data. 4.2. Measurement data For this investigation, the available data was SCADA data, recorded during the month of August 2016. The dataset is composed by 10-minute mean values measured in the 8 wind turbines and in the meteorological mast. The wind turbine measurements were made at hub height. The variables measured for each turbine were the following:    [m/s]    [°󰇠     [°󰇠    [kW]        ℎ  [°󰇠     [rpm]    [°C] The meteorological mast measurements were made at 5 different heights: 40 m, 60 m, 80 m, 99 m and 101 m. The variables measured for each height were the following:     [m/s]    [°󰇠 WIND FARM AND MEASUREMENT DATA António Nunes Vicente 29           The turbulence intensity in the meteorological mast at a height of  can be defined by equation 4.1:   ,  =  (  )   (  ) (4.1) where () is the wind speed standard deviation and () is the free stream wind speed, both at height . This way of computing the turbulence intensity will be discussed later. Furthermore, two  files with information about the 90 and the 112 wind turbines were provided. The information included the hub height, rated power, rotor diameter, and measured values for both power and thrust coefficient curves as function of wind speed. 4.2.1. Filtering measurement data Filtering measurement data is an important part of the validation process. It is known that several external factors can influence the measurements accuracy, such as turbulence, air density, wind speed gradients, wind turbine technical problems, etc. In a report of flow and wakes in large wind farms by Barthelmie et al. (2011), a description is provided for the authors’ experience in organizing and filtering data from large wind farms. A previous paper (Réthoré et al., 2009) proposed a general guideline for data validation. This section describes all the filtering process made in this work when using SCADA data. The starting point was to eliminate all wind speed values lower than the cutin wind speed, i.e., <3 m/s for 112 turbines and <4 m/s for 90 turbines. The negative power production values were also all eliminated. Nacelle misalignment is an important factor to take in consideration. It can be defined as the difference between the ambient wind direction at hub height and the nacelle direction. Time records with registered values above 5° were eliminated. Furthermore, a comparison between the real power curve of each wind turbine (obtained with measurement data) and the one provided by the trb file was made. Taking turbine 22 as an example, Figure 4.2 shows both power curves (real and trb file) in one chart. The points away from the power curve were eliminated. EVALUATION OF WIND TURBINE WAKE MODELS 30 2018 Figure 4.2 – Power curve of turbine T22. Other parameters taken into consideration were the rotor rotation speed and the blades pitch angle. By plotting these variables with ambient wind speed, one can evaluate whether the turbine is working normally. See for instance the rotor rotation speed in Figure 4.3 - the points away from the curve were eliminated. Figure 4.3 – Rotor rpm average for turbine T22. -200 300 800 1300 1800 2300 0 5 10 15 Power production (kW) Wind Speed (m/s)  22 - Power curve T22 SCADA data .trb file 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 Rotor rpm average (rpm) Wind speed (m/s)  22 - Rotor rpm average SIMULATION OVERVIEW António Nunes Vicente 31 5. SIMULATION OVERVIEW 5.1. Input data The input data for both WindStation and WindSim consisted on the terrain data of the site, the meteorological mast data, and the wind turbine data stored in the trb files already mentioned in Section 4.2. The terrain data was composed by the elevation and roughness files which were converted from WindStation file format, ArcInfo ASCII, to WindSim format gws, using the Global Mapper software package (see Global Mapper 19.1). In WindStation, the initialization of the wind fields is done by assigning velocity, turbulence and temperature values for the whole domain. In this case, those values are based on the meteorological mast data. Then, a reconstruction of the vertical profiles is done for both wind speed and turbulence quantities. This reconstruction may be done with two different approaches, depending whether the Coriolis forces are considered or not. In this work, Coriolis forces were always considered. The remaining calculation process depends on the boundary conditions and other parametrization. For more details please see Lopes (2018). 5.2. Parametrization Domain extension. The calculation domain is nearly parallelepipedic. It is delimited at the bottom by the ground and at the top by a horizontal plane. The area covered by the whole terrain data is huge (roughly 6659 ), which naturally led to a reduction of the calculation domain extension (into an area of 37 ). Figure 5.1 shows the top view of the actual calculation domain in WindStation, and a lateral view through cut A-B. EVALUATION OF WIND TURBINE WAKE MODELS 32 2018 Figure 5.1 – Top view and lateral view of the calculation domain. Mesh. The mesh in WindStation is defined by a constant horizontal spacing and a variable vertical spacing. In order to chose a value for the horizontal spacing, a mesh refinement analysis was performed by reducing the horizontal spacing from 1000 m to 20 m. The expectation was that a mesh refinement would not have much influence on the results, because the terrain is rather smooth and the measurements are done at a relevant distance from the ground. The chosen turbine was 22, with an ambient wind direction corresponding to an undisturbed incoming flow (=234.2°). Figure 5.2 plots both the measured wind speed and the one obtained in WindStation, together with the number of nodes. As expected, the mesh influence on result accuracy is hardly perceptible (note that the vertical axis values only range between 4,9 m/s and 5,1 m/s). However, the optimal solution fell on a horizontal spacing of 40 m, with a number of nodes approximately equal to 500,000. Although a 20 m spacing could provide a vaguely better agreement with measured data, one would increase significantly the number of nodes, leading to an excessive computational time. The 40 m spacing showed a good balance between accuracy and simulation time. SIMULATION OVERVIEW António Nunes Vicente 33 Figure 5.2 – Mesh grid analysis. The vertical spacing in WindStation is defined by the altitude (above sea level) of the calculation domain top (Ztop), by the vertical distance between the first calculation point and the ground (First node), and by the number of calculation levels (Levels). The Max vertical spacing is the height of last control volume. As shown in Figure 5.3, these parameters slightly differ from WindStation to WindSim. The difference between Ztop and Height above terrain is given by:  ℎ    =   −  (5.1) The Height distribution factor gives the fraction between the cell at the ground and the cell at the upper boundary:  ℎ    =   Max   (5.2) 1000 m 500 m 400 m 300 m 200 m 100 m80 m 60 m 40 m 20 m 4,9 4,95 5 5,05 5,1 1000 10000 100000 1000000 Wind Speed (m/s) Number of Nodes Mesh refinement - T22 Wind Speed Measured WindStation EVALUATION OF WIND TURBINE WAKE MODELS 34 2018 5.3. Important issue concerning measurement data The major setback when using the meteorological WMM mast data as an input parameter was dealing with discrepancies between its values and the ones from wind turbines. In fact, if a wind direction measured in the mast is considerably different from the one measured in the wind turbines, the modelled wind flow can account for a multiple wake superposition situation, when in reality it is not. Figure 5.4 plots the    at a height of 101 m with the    [°󰇠 at turbine 0. This was done for the wind direction interval of ∈[200°,280°󰇠. The offset in wind direction was found to be large and for same cases reaches a value of 40°. In this way, it was concluded that it is rather difficult to use the meteorological mast data as a reference value for the wind turbine when analysing the simulated results. Instead, the upstream wind turbine was used. Figure 5.3 – Calculation domain parameters in WindStation (left) and WindSim (right). Note that the altitude of the calculation domain top is equal to Ztop=1200 m. SIMULATION OVERVIEW António Nunes Vicente 35 Figure 5.4 – Offset in wind direction. 5.4. Panorama simulation Figure 5.5 is a wind rose taken from the climatology report of WindSim. It gives the wind speed distribution in the WMM at a height of 101 m, divided in bins of 2 m/s and wind direction sectors of 30°. It is clear that the most common wind direction sectors correspond to an air flow from southwest. Figure 5.5 – WMM wind rose at 101 m (WindSim). Based on the sector availability of the WMM measured data (displayed in the wind rose) a panorama simulation was performed for a wind direction interval of ∈ [225°;270°󰇠 and the results were separated in two wind speed bins: 4−6 m/s and 6−8 m/s. Wind speeds above 8 m/s were not considered due to lack of measurement data. The goal of this simulation was to investigate the velocity deficit at hub height for each turbine. 200 210 220 230 240 250 260 270 280 200 210 220 230 240 250 260 270 280 Wind direction (°) Wind direction (°) WMM vs T0 WMM wind direction T0 ambient wind direction EVALUATION OF WIND TURBINE WAKE MODELS 42 2018 represent its transition. The increase in the normalized wspd value led to the conclusion that the turbulence intensity in the mast was higher than the one predicted by Equation 3.8. Whether this correction is applied in WindSim or not, it is yet unknown. Table 5.1 -  correction WindStation  [ ° 󰇠 243.9 246.9 251.9 Modelled   11.6% 12.5% 11.2% Corrected   (z=99 m) 22.3% 16.3% 11.8% Modelled normalized wspd 0.78 0.72 0.82 Normalized wspd with   correction 0.93 0.88 0.86 Figure 5.13 - Larsen (WindStation) and Larsen (WindSim) panorama results for turbine T20 and wspd bin of 4-6 m/s. 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1,10 1,20 1,30 215 220 225 230 235 240 245 250 255 260 265 270 275 280 Normalized Wind Speed Wind direction (°) Turbine T20 Measurement Data Larsen (WindStation) Larsen (WindSim) SIMULATION OVERVIEW António Nunes Vicente 43 Figure 5.14 - Larsen (WindStation) and Larsen (WindSim) panorama results for turbine T20 and wspd bin of 6-8 m/s. Finally, in Figure 5.15 and Figure 5.16 it was possible to display all the results for turbine 20. When comparing the Jensen 2D (WindStation) with the Larsen (WindStation), it is clear that the Larsen curve has a lower steepness and a wider wake width. Note that there were no significant differences in the modelled wake width or velocity deficit when changing from a wind speed bin of 4−6 m/s to 6−8 m/s. However, when looking only at the measurement data points, the velocity deficit seems to be higher for 4 − 6 / than for 6 − 8 /. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 215 220 225 230 235 240 245 250 255 260 265 270 275 280 Normalized Wind Speed Wind direction (°) Turbine T20 Measurement Data Larsen (WindStation) Larsen (WindSim) Corrected TI EVALUATION OF WIND TURBINE WAKE MODELS 44 2018 Figure 5.15 – Panorama results obtained in all wake models for turbine T20 and wspd bin of 4-6 m/s. Figure 5.16 - Panorama results obtained in all wake models for turbine T20 and wspd bin of 6-8 m/s. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Normalized Wind Speed Wind direction (°) Turbine T20 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 215 220 225 230 235 240 245 250 255 260 265 270 275 280 Normalized Wind Speed Wind direction (°) Turbine T20 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) SIMULATION OVERVIEW António Nunes Vicente 45 In order to evaluate how each model has adjusted to the measurement data, a polynomial function that best fits the corresponding model points was first computed. For that purpose, the Matlab function polyfit was used. Several degrees for the polynomials were tried and the value chosen was 6 for every model except for the Jensen (WindStation). In this last one, a linear function was enough, since its values are approximately constant. Then, the polynomial functions were used as a replacement for the model points, to compute the corresponding error of the adjustment to the measurement data. This error was calculated by summing up all the absolute values of the individual errors for each entry of the measurement data, divided by the number of those entries: ∑ 󰇻     ,   −   ,  󰇻      (5.3) where  is the polynomial corresponding to the model at stake, (,, ,) are the measurement data points and  is the number of those points. The range of the abscissa , of the measurement data points was restricted to the already mentioned wind direction interval [235°,260°󰇠. Table 5.2 shows the results obtained: Table 5.2 – Error obtained for all five wake models. T20 Jensen (WindStation) Jensen 2D ( = 0 . 075 ) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) 4 - 6 m/s 0.1012 0.1192 0.1011 0.0947 0.1237 6 - 8 m/s 0.0794 0.0894 0.0651 0.0652 0.0708 The largest errors were found in Larsen (WindSim) for a 4− 6 / wind speed bin and in Jensen 2D (WindStation) for a 6−8 m/s wind speed bin, with values of 0.1237 and 0.0894 respectively. These values strengthened the conclusion that the Larsen (WindSim) underestimated the wake effects and the Jensen 2D (=0.075) overestimated them. On the other hand, lower values were obtained in the Jensen (WindSim) EVALUATION OF WIND TURBINE WAKE MODELS 46 2018 model than in the Jensen (WindStation) model, which can only be explained by the step points in Jensen (WindSim). As already shown by Figure 5.6−Figure 5.8, the wind direction sector ∈ [225°;270°󰇠 accounts for multiple wake superposition situations. The results obtained for turbines 9 and 6, which are respectively placed in a double and triple wake region, are presented in Figure 5.17−Figure 5.20. Naturally, if the distance to the lead turbine increases, the wind direction interval in which the wake effect of turbine 21 is noticed, decreases. See for instance the example of Larsen (WindStation) in the 4−6 m/s wspd bin: for turbine 20 (Figure 5.15) this interval is approximately [236°,260°󰇠, for turbine 9 (Figure 5.17) it is reduced into [238°,257°󰇠 and for turbine 6 (Figure 5.19) into [238°,256°󰇠. However, the effect of the wake superposition in the velocity deficit was hardly noticed, despite the slight increase from turbine 20 to turbine 9. Overall, the normalized wind speed values obtained in turbines 9 and 6 were similar to the ones in turbine 20. Figure 5.17 - Panorama results obtained in all wake models for turbine T9 and wspd bin of 4-6 m/s. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Normalized Wind Speed Wind direction (°) Turbine T9 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) SIMULATION OVERVIEW António Nunes Vicente 47 Figure 5.18 - Panorama results obtained in all wake models for turbine T9 and wspd bin of 6-8 m/s. Figure 5.19 - Panorama results obtained in all wake models for turbine T6 and wspd bin of 4-6 m/s. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Normalized Wind Speed Wind direction (°) Turbine T9 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Normalized Wind Speed Wind direction (°) Turbine T6 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) EVALUATION OF WIND TURBINE WAKE MODELS 48 2018 Figure 5.20 - Panorama results obtained in all wake models for turbine T6 and wspd bin of 6-8 m/s. 5.4.2. Offset in wind direction The same analysis as in the previous section was done to row 2, by normalizing the wind speed with respect to turbine 22. The results obtained are shown in Figure 0.1−Figure 0.6 of APPENDIX B. Although the same conclusions were taken, the offset in wind direction was considerably bigger, due to a larger distance of this row to the meteorological mast. This situation is clearly visible in the results obtained for turbine 0 (wspd bin of 4−6 m/s), displayed in Figure 5.21. In fact, the average value of the difference between the measured and the modelled wind direction was calculated and equals 9.3°. The offset was corrected and Figure 5.21 was updated into Figure 5.22. The error was calculated like in the previous section and updated (see Table 5.3); note that it decreased for all models, except for the Larsen (WindSim). 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Normalized Wind Speed Wind direction (°) Turbine T6 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (windSim) SIMULATION OVERVIEW António Nunes Vicente 49 Figure 5.21 - Panorama results obtained in all wake models for turbine T0 and wspd bin of 4-6 m/s. Figure 5.22 – Offset correction: updated panorama results obtained in all wake models for turbine T0 and wspd bin of 4-6 m/s. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 215 220 225 230 235 240 245 250 255 260 265 270 275 280 Normalized Wind Speed Wind direction (°) Turbine T0 Measured Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 215 220 225 230 235 240 245 250 255 260 265 270 275 280 Normalized Wind Speed Wind direction (°) Turbine T0 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) EVALUATION OF WIND TURBINE WAKE MODELS 50 2018 Table 5.3 – Offset correction: updated error obtained for all five wake models T0 (4-6 m/s) Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) Normal 0.0935 0.1323 0.1125 0.0925 0.0776 W ith offset correction 0.0928 0.0989 0.0906 0.0917 0.0883 5.4.3. Effective power of a wind turbine As wind flows across a wind turbine, the power available () in the wind is given by (Tong et al., 2012): =12 where  is the air density,  is the rotor swept area and  is the incoming wind speed at hub height. Naturally, not all this power is generated by the wind turbine. In WindStation, the effective wind turbine power is computed by interpolating the effective velocity deficit given by Equation 3.25 (see Section 3.1.3) into the measured power curve of the respective wind turbine. It should be interesting to compare results obtained for wind speed deficits with wind turbine effective power. Figure 5.23 plots the power obtained in row 1 wind turbines for the same case showed earlier in Figure 5.6−Figure 5.8, together with measurement data. The values for effective power were normalized with the wind turbine rated power and are displayed in percentage. SIMULATION OVERVIEW António Nunes Vicente 51 Figure 5.23 – Normalized effective wind power computed in all wake models = °;,=. /. Note that for all wake models, there was a significant power drop between turbine 21 and turbine 20. Then, it slightly increased until turbine 6. However, the measurement data showed an unpredictable power behavior. It is interesting to see that the Jensen 2D wake model severely overestimated the power loss, which is coherent with the overestimation of velocity deficit estimated in the previous sections. The opposite conclusion can be also taken from the Larsen (WindSim) wake model. Furthermore, a considerable difference between the Jensen (WindStation) and the Jensen (WindSim) wake model was detected, which goes against the similarity shown in the wspd deficit results. In order to have more detailed conclusions, this analysis should be performed to a large number of cases. 0% 10% 20% 30% 40% 50% 60% Normalized Power Wind Turbine Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) T21 T20 T9 T6 EVALUATION OF WIND TURBINE WAKE MODELS 58 2018 APPENDIX B António Nunes Vicente 59 APPENDIX B Figure 0.1 - Panorama results obtained in all wake models for turbine T0 and wspd bin of 4-6 m/s. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 215 220 225 230 235 240 245 250 255 260 265 270 275 280 Normalized Wind Speed Wind direction (°) Turbine T0 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) EVALUATION OF WIND TURBINE WAKE MODELS 60 2018 Figure 0.2 - Panorama results obtained in all wake models for turbine T0 and wspd bin of 6-8 m/s. Figure 0.3 - Panorama results obtained in all wake models for turbine T8 and wspd bin of 4-6 m/s. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Normalized Wind Speed Wind direction (°) Turbine T0 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Normalized Wind Speed Wind direction (°) Turbine T8 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) APPENDIX B António Nunes Vicente 61 Figure 0.4 - Panorama results obtained in all wake models for turbine T8 and wspd bin of 6-8 m/s. Figure 0.5 - Panorama results obtained in all wake models for turbine T23 and wspd bin of 4-6 m/s. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Wind speed ratio Wind direction (°) Turbine T8 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Corrected Wind Speed Wind direction (°) Turbine T23 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) EVALUATION OF WIND TURBINE WAKE MODELS 62 2018 Figure 0.6 - Panorama results obtained in all wake models for turbine T23 and wspd bin of 6-8 m/s. 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05 1,10 1,15 1,20 1,25 1,30 225 230 235 240 245 250 255 260 265 270 Normalized Wind Speed Wind direction (°) Turbine T23 Measurement Data Jensen (WindStation) Jensen 2D (WindStation) Larsen (WindStation) Jensen (WindSim) Larsen (WindSim) Erro! A origem da referência não foi encontrada. António Nunes Vicente 63