Unveiling Inertia Constants by Exploring Mass Distribution in Wind Turbine Blades and Review of the Drive Train Parameters

Citation: Gonzalez-Rodriguez, A.G.; Roldan-Fernandez, J.M.; Nieto-Nieto, L.M. Unveiling Inertia Constants by Exploring Mass Distribution in Wind Turbine Blades and Review of the Drive Train Parameters. Machines 2023,11, 908. https://doi.org/ 10.3390/machines11090908 Academic Editors: Guowei Qian and Davide Astolfi Received: 7 August 2023 Revised: 8 September 2023 Accepted: 11 September 2023 Published: 13 September 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). machines Article Unveiling Inertia Constants by Exploring Mass Distribution in Wind Turbine Blades and Review of the Drive Train Parameters Angel Gaspar Gonzalez-Rodriguez 1,* , Juan Manuel Roldan-Fernandez 2,† and Luis Miguel Nieto-Nieto 1,† 1Department of Electronic Engineering and Automation, University of Jaen, 23009 Jaen, Spain; [email protected] 2Department of Electrical Engineering, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, 41092 Sevilla, Spain; [email protected] *Correspondence: [email protected] † These authors contributed equally to this work. Abstract: In studies of dynamic stability and power quality, it is necessary to know the values of the mechanical parameters determining the transient response of wind turbines. Their exact values are not as decisive as the power curve, but an inaccurate estimate can distort or even invalidate the simulation results. From a review of the literature, it has been found that, despite their importance, the values of inertia, stiffness and damping are hardly available for any turbine model. Another detected problem is the lack of confidence in the data origin. This article aims to solve the issue of the scarcity and unreliability of data on inertia, and gathers the information found on the remaining mechanical parameters. Available blade inertia values in kg · m 2 are presented. Special treatment has been given to those providing the mass distribution along the blade span, for which the provided values of inertia have been compared with those obtained numerically, showing good matching. With this, different reliable relations are obtained that allow for the calculation of the turbine rotor inertia, based on the mass and length of the blade. When the center of gravity is also available, a very correlated expression ( r2= 0.975) is provided to obtain the inertia. The references to the stiffness and damping constant of the drive train, which are even more rare, will also be presented. In addition, the study includes a revision of gearboxes, generators and blade weight, according to their IEC-class and material. Keywords: wind turbine; inertia; mass distribution; density distribution; stiffness constant; values in p.u. 1. Introduction 1.1. Context and Purpose of the Work In power system stability studies, it is important to have an appropriate model for the characterization of a physical phenomenon of interest. Depending on the analysis to be carried out, the corresponding models may be different. According to the time scale of interest for stability studies, the models can be classified as electromagnetic models (to scale short time frames) or electromechanical models to investigate slower events. The influence that modeling can have on the results of the study can be seen in [1]. With regard to the electromechanical models, the most important components to take into account are the turbine inertia, the generator inertia and the coupling between both moving masses. In general, the inertia in AC electrical power systems represents the kinetic energy stored in large rotating generators and synchronous motors, providing them with the tendency to maintain continuous rotation. In the case of additional rotating masses (steam turbine in the case of thermal units, Francis/Kaplan/Pelton turbine in the case of hydro units or blades in the case of wind turbines), the corresponding inertia must be added to the generator value. Machines 2023,11, 908. https://doi.org/10.3390/machines11090908 https://www.mdpi.com/journal/machines Machines 2023,11, 908 2 of 22 A system that possesses a sufficient natural rotational inertia is able to maintain the grid frequency in its rated values, since this frequency is really an electromechanical variable that is linked to the mechanical speed of rotation of the generators. When a sudden power imbalance between generation and demand occurs in the grid, a frequency deviation typically appears. If the contingency is not serious, this stored kinetic energy allows for a rapid response, permitting a transfer of energy to balance generation and load, before the frequency deviation exceeds permitted values. On the other hand, in normal operation, small variations are permitted in the electrical frequency in real time, to ensure an adequate balance of active generation and load. In both cases, abrupt or slow power imbalance, inertia plays a determining role during the balancing transient. This kinetic energy is: Ek=1 2JΩ2 G(1) where J is the system inertia and ΩG is the rotational speed of the electrical generator [ 2 ]. It allows us to define the inertia time constant H of a generator as the ratio between the stored kinetic energy Ek and its rated power. It determines the time interval during which an electrical generator can supply its rated power, by using solely the kinetic energy stored in its rotating masses [2]. Additionally, the current paradigm shift, with the incorporation of wind power and photovoltaics into the power system, is achieved using power-electronic converters that do not possess natural rotational inertia, unlike a synchronous generator (SG). Consequently, renewable generators displace synchronous generation and reduce the amount of rotating mass in the system. Therefore, system operators need to model these electrical systems with low rotational inertia, which presents significant challenges to controlling the stability of the system [3,4]. The availability of precise inertia data permits the system operator to make appropriate decisions and facilitates the improved planning and operation of the system. This ensures the stability and resilience of the network and enables the efficient integration of larger amounts of renewable generators, without compromising system security or power quality. 1.2. Literature Review The importance of inertia in stability studies is addressed by many authors. Ekanayake in [ 5 ] shows how, with the correct control of the power electronics connected to the rotor of a doubly fed induction generator (DFIG), inertia allows the recovery of a significant amount of kinetic energy. This alleviates the effect of a frequency drop in the network system. In the case of fixed speed induction generators (squirrel-cage type), the recovered kinetic energy is much lower since the range of operating rotation speeds is reduced to +1 ÷ 3% above the synchronous speed. At the other extreme, SGs allow a much wider speed range, and there is a decoupling between the mechanical part and the transient phenomena that occur on the network side [ 6 ], due to the IGBT-converters. Initially, this decoupling could lead to a reduction in inertia seen by the network, and therefore, less damping against abrupt changes in generation and charging patterns [ 7 , 8 ]. This has given rise to a multitude of articles on frequency control strategies to effectively integrate wind energy systems into the grid. In any case, for the evaluation of the kinetic storage capacity, it is necessary to know, as precisely as possible, the inertia value for the turbine and the generator. It is also mandatory to be familiar with the mechanism for the transmission of kinetic energy between both components through the coupling, for which it is necessary to have an understanding of its characteristics of stiffness and damping. Guillamon in [ 9 ] performs a review of inertia data, and although providing its dimensionless value, it is difficult for some references to trace those which have been reliably obtained. Other authors such as Morren in [ 6 ] assume, in the absence of reliable data, that the geometry of the blade is very complicated, and approximate this to a bar of constant Machines 2023,11, 908 3 of 22 chord and constant, with which they obtain a simple expression to calculate inertia as a function of power. 1.3. Research Gap and Motivation Despite their importance in system stability and power quality studies, these parameters, especially that of inertia, but also the stiffness and damping constant of the drive train, are rarely available for the turbine model under investigation. Therefore, the researcher must often carry out a study on which parameters to use in order to faithfully reproduce the actual situation. The alternative is to rely on the value used by other researchers, without any assurances that these can be adapted to the characteristics of power, length or weight of their own system. As a result, it is highly desirable that future studies on stability and, in general, on the dynamic behavior of wind turbines contain reliable data, so that the modeling of the system is as realistic as possible. These parameters logically depend on the size of the turbine. Even using a dimensionless formulation, which converts the magnitudes into per unit (p.u.), these parameters have a certain dependence on power and, as power grows over the years, it is necessary to update the available databases with the values that can be extracted from modern machines. 1.4. Contribution In this article, in addition to reviewing realistic data on inertia, mainly from multimegawatt turbines, it is intended to trace the references that provide data on the density distribution (DD) of real blades, preferentially with designs based on realistic dynamic and structural analyses. From these, the calculation of the inertia of the blade will be made, and a relationship with some specific magnitude of the geometry will be deduced, such as the position of the center of gravity (CoG). This method of obtaining inertia is novel, and is intended to be the initial stage of a more extensive database to provide reliable data on inertia, unlike inertia data obtained by transient analysis or other non-direct methods. Although minor, other components also influence the dynamics of the mechanical system, the precision of which will depend on the number of masses with which the mechanical part of the turbine is modeled [ 10 , 11 ]: generator inertia, hub inertia and stiffness/damping of the power transmission shafts between the turbine and generator. The relationship between the different mechanical variables will be reviewed through these parameters, and the corresponding expressions converted into dimensionless values. In [ 11 ], it is stated that it is possible to reduce the number of masses of the drive train model, without significant deviations. The theoretical and technical contributions of this paper are to: • Provide general expressions that allow the weight of the blade to be estimated based not only on its length, but also on the IEC wind class of the turbine and the material of the blade. • Provide an expression that accurately estimates the blade inertia starting from the position of the CoG, its weight and its length. • Formulate in depth a complete dimensionless framework to relate magnitudes and parameters (such as inertia, damping, stiffness and friction) with respect to their base references, valid for systems of three masses and two masses. 1.5. Paper Organization The rest of the article is explained below. In Section 2, the mechanical equations that govern the rotation movement of each component are reviewed, the expressions that transform each variable and parameter into its value in p.u. are shown and certain general, distinguishing characteristics of each parameter are described. In Section 3, all the data retrieved on the DD of 21 blade models are organized, and a relationship is identified that links the inertia with the position of the CoG of the blade. The values obtained for the other Machines 2023,11, 908 4 of 22 parameters reviewed are also organized. Finally, in Section 4, the results are evaluated, highlighting the achievements obtained as well as their limitations. At the end of the paper, a list of abbreviations and variables is included. 2. Methods This section formulates the expressions governing the dynamics of the interaction between rotor and generator and how they can be converted into p.u. It also anticipates certain issues to be taken into account prior to listing the revised data. Hereinafter, the set of blades plus hub will be named as the turbine rotor or simply turbine, and this will appear referenced in the expressions as T . A blade will be designated as B , the generator as G and the gearbox as GB . Magnitudes and parameters with dimensions will appear in uppercase, and their conversion to p.u. will be displayed in lowercase. 2.1. Dynamics Rotor-Generator Once the blades extract the aerodynamic energy from the wind and transform it into a rotation torque TW exerted at the turbine speed ΩT , these magnitudes will drive the dynamics of the system until they produce a torque on generator TG , which will rotate at speed ΩG . This speed will be measured by the grid frequency, f , divided by the number of pole pairs, npp , and will be exactly equal to this value in the case of SGs or slightly higher (due to slip) in the case of IGs in normal operation. With regard to IGs, a high number of poles would increase the magnetization losses, so this value is reduced to no more than npp = 2 or npp = 3. Since the rotational speed in rpm is equal (or very close) to 60 f/npp with f being the grid frequency, this yields an impermissible value for blade rotation. It is, therefore, necessary to include a gearbox that accommodates both speeds. This new element introduces new equations into the dynamics of the system. Permanent Magnet Synchronous Generators (PMSG), which are the most frequent type of SGs in wind turbines (WT), allow for a high number of poles that considerably reduce the rotational speed. In the latter case, it would not be necessary to use a gearbox. This setup is referred to as direct drive. In other schemes with PMSGs, the number of pairs of poles is not high enough to obtain the adequate speed, although in any case, the required transmission ratio decreases and with it, the dimensions and stages of the gearbox. Other elements that form part of this transmission are the brake and the couplings that account for the misalignment between the rotation axes, although they are generally not included in the dynamic analysis of the system. Figure 1represents the interior of the nacelle, with the transmission chain from the turbine rotor to the generator. The scheme corresponds to an IG or a PMSG with a reduced number of poles. In this case, to accommodate the high speed in the generator shaft with the low speed of the blades, it is necessary to include a gearbox. Its ratio increases with the turbine rotor diameter and decreases with the number of poles. Figure 2corresponds to this dynamic scheme, called three masses, where: •JT is the inertia of the turbine rotor due to the distribution of masses in the blades and, to a lesser extent, in the hub. •DT is the coefficient of friction due to the aerodynamic resistance offered by the blades. •KHGB is the stiffness constant in the slow axis that joins the hub and the gearbox. •CHGB is the damping constant of the torsional movement of the slow axis. •JGB is the inertia of the gearbox discs, measured from the slow shaft. •DGB is the coefficient of friction due to friction in the gearbox, measured from the slow shaft. •JGis the inertia of the rotor of the electric generator and the brake. •DGis the coefficient of friction due to friction in the generator and ventilation losses. •KGBG is the stiffness constant in the fast axis that joins the gearbox and the generator. •CGBG is the damping constant of the torsion motion of the fast axis. Machines 2023,11, 908 5 of 22 Figure 1. Typical components in a wind turbine drive train. JT Keq Ceq n2gb JG DT n2gb DG JT KHGB CHGB ngb:1 KGBG CGBG JG DT DG JGB DGB ngb = 1 for direct drive (a) (b) Figure 2. ( a ) Schematic diagram of the traditional three-mass and ( b ) two-mass equivalent model of a wind turbine. In the two-mass equivalent model, HSS-side values are translated to the LSS-side. 2.2. Mechanical Equations The equations that govern the dynamics in a system formed by the turbine rotor, the slow shaft, the gearbox, the fast shaft and the generator are those that correspond to the left side of the following expressions: TW=TLSS +DTΩT+JT˙ ΩT→tW=tLSS +dTωT+2HT˙ ωT(2) TLSS =KHGB ΘLSS T+CHGB ˙ ΘLSS T→tLSS =kHGB θT+cHGB ˙ θT(3) ˙ ΘLSS T=ΩT−ΩLSS GB →˙ θT=ωT−ωGB (4) TLSS GB =TLSS −DLSS GB ΩLSS GB −JGB ˙ ΩLSS GB →tGB =tLSS −dGB ωGB −2HGB ˙ ωGB (5) ΩHSS GB =nGBΩLSS GB (6) TLSS GB =nGBTHSS (Ideal) →tGB =tHSS (7) THSS =KGBGΘHSS G+CGBG ˙ ΘHSS G→tGB =kGBG θG+cGBG ˙ θG(8) ˙ ΘHSS G=ΩHSS GB −ΩG→˙ θG=ωGB −ωG(9) THSS =TG+DGΩG+JG˙ ΩG→tHSS =tG+dGωG+2HG˙ ωG(10) where superscripts LSS and HSS stand for low-speed shaft and high-speed shaft, respectively (as an example, TLSS is the torque available in the LSS), Θ is the twist angle in the LSS or HSS and nGB is the gearbox ratio. The variables and magnitudes on the right hand side are the same expressions, although referred to their base magnitudes; they are explained in the following section. 2.3. Referring to Base Magnitudes On many occasions, the value of a magnitude expressed in a measurement system (e.g., MKS) does not provide information on the operating range in which it is working and must be compared with its rated value. For this reason, it is customary to compare magnitudes, electrical or mechanical in our case, to their rated values, so that when working below rated conditions, the value will be between 0 and 1. This usually entails the adimensionaliza- Machines 2023,11, 908 6 of 22 tion of the parameters that intervene in the relationships between magnitudes. In most cases, the new magnitudes and parameters lose their dimensions and we then work in p.u. This facilitates computational manipulation, makes the variables independent of the capacity of the turbine to a certain extent or offers a better view of how close the operation is to overloading or idling. In the case of an ideal gearbox, this disappears when operating in p.u. The expressions, once transformed into p.u, are those found on the right side of (2)–(10), where uppercase has been used for values with dimension and lowercase for values in p.u. In the following subsections, a set of dimensionless values for the different parameters of these equations (inertia, damping coefficients and friction coefficients) will be listed, and it will be easy to identify outlier values and mark them as unreliable. If the adimensionalization is not carried out, the normality range for each parameter will depend on other variables (power, speed, nGB), so its identification is more difficult. The relationships between variables with dimensions and in p.u. are the following: TW=tWTLSS Bwith TLSS B=P ΩLSS B and ΩLSS B=2πf nGB npp (11) TLSS =tLSSTLSS B(12) ΩR=ωTΩLSS B(13) ΩLSS GB =ωGB ΩLSS B(14) ΘLSS =θ1 npp nGB (15) TLSS GB =tGB TLSS B(16) ΩHSS GB =ωGB ΩHSS Bwith ΩHSS B=2πf npp (17) THSS =tGB THSS Bwith THSS B=P ΩLSS B (18) ΘHSS =θ1 npp (19) ΩG=ωGΩHSS B(20) TG=tGTHSS B(21) where P is the turbine rated capacity. This determines the transformation to p.u. of several parameters (inertias, stiffness, friction or damping), the values of which will be different according to whether they are given in the LSS or the HSS. It should be mentioned that there are cases such as that of the inertia constant J in which a complete adimensionalization does not make practical sense, and although it is transformed to a value referred to rated conditions, it has a magnitude of s . In any case, and although it would be more accurate to speak of magnitudes referred to base values, the term p.u. will continue to be used, despite not being strictly correct for some parameters. Its reference to the rated values (H) also appears as an exception, in capital letters. For the parameters in the LSS, the transformations yield: H[s]=JLSS 2 (ΩLSS B)2 P=J 2 (2πf)2 P n2 pp n2 GB (22) kp.u. radel =KLSS ΩLSS B P npp nGB =KLSS 2πf P n2 pp n2 GB (23) c[p.u.]=CLSS (ΩLSS B)2 P=C(2πf)2 P n2 pp n2 GB (24) d[p.u.]=DLSS (ΩLSS B)2 P=D(2πf)2 P n2 pp n2 GB . (25) Machines 2023,11, 908 7 of 22 For the parameters in the HSS, the transformations yield: H=JHSS 2 ΩHSS B P=JHSS 2 (2πf)2 P n2 pp (26) k=KHSS ΩHSS B P npp =KHSS 2πf P n2 pp (27) c=CHSS (ΩHSS B)2 P=CHSS (2πf)2 P n2 pp (28) d=DHSS (ΩHSS B)2 P=DHSS (2πf)2 P n2 pp . (29) It is worth mentioning that some authors use the turbine rotation speed provided by the manufacturer as ΩLSS B to transform the dimensioned parameters to p.u. parameters. This is true for turbines driving PMSGs. However, for turbines with IGs, the catalogs generally do not show the exact speed value at the rated conditions, but ΩLSS B , so once again, it is correct to use the value obtained from the catalogs as ΩLSS B . However, certain other sources have also been identified in which these two values differ, although the deviation is minimal, less than 5%. This deviation is due to the slip s , of reduced value in high-power generators since this slip determines the rotor losses due to the Joule effect. Consequently, there is a certain error in turbines with IGs when considering the rotational speed of the turbine as the base of the slow shaft speed. This error, although small, can be avoided by adopting ΩLSS B=2πf nGB npp as the base speed. 2.4. System of Two Masses In the case of multipole PMSGs with direct drive, there would be no gearbox and all the measurements would refer to a single axis. In this case, it is usual to group the different constants in such a way that one works with a model of two masses, as in Figure 2b. The error committed is not usually significant in stability studies [11]. Even when there is a gear ratio, all constants tend to refer to a single axis, thus simplifying the analysis of the dynamics and making it possible to compare parameters between turbines with different gearbox ratios. In addition, the system is simplified if, as indicated in [11], the coupling on the fast axis is considered infinitely rigid, with which ΩHSS GB =ΩG . As shown in Figure 2b, the parameters measured in the HSS can be converted to the LSS (or vice versa), simply by multiplying (or dividing) by n2 GB. This is ParamLSS =ParamHSS n2 GB (30) where Param is a coefficient of inertia, stiffness, friction or damping. This is coherent with the expressions (22)–(29). The different constants could be grouped into equivalent values according to [ 10 ], which, in the case of using variables in p.u., is more simplified: Jm2≃JLSS GB +n2 GB JG→Hm2≃HGB +HG(31) Dm2≃DGB +n2 GBDG→dm2≃dGB +dG(32) 1 Keq =1 KHGB +1 KGBG n2 GB →1 keq =1 kHGB +1 kGBG (33) 1 Ceq =1 CHGB +1 CGBG n2 GB →1 ceq =1 cHGB +1 cGBG (34) where m 2 refers to the equivalent value that includes all the components to the right of the rotor. This would constitute mass 2 against mass 1 which is that of the turbine. Machines 2023,11, 908 8 of 22 From the grouped values of Jm2 , JT and Keq , the critical damping of the torsional dynamics can be obtained: Cc=2qKeq Jeq. (35) where for the study of damping in torsional dynamics, the following expression must be used [10,12]. Jeq =JTJm2 JT+Jm2 →Heq =HTHm2 HT+Jm2(36) From (22) and (23), Cc can be obtained as a dimensionless value. Assuming, without loss of generality, that the values refer to the LSS, this would be: Cc=2skeq P n2 pp n2 GB 2πf2Heq P n2 pp n2 GB (2πf)2=4P n2 pp n2 GB (2πf)2qkeq Heq πf(37) From (24), this is as follows: cc=4qkeq Heq πf(38) where f is the network frequency. In the event that the values referred to the fast axis, it would be necessary to multiply Jeq and Keq by n2 GB , but at the same time, the damping would also need to be divided by n2 GB, so the expression would still be valid. The resonant frequency is: ωnhel.rad si=v u u tkeq P n2 pp n2 GB 2πf (2πf)2 2Heq P n2 pp n2 GB =skeqπf Heq (39) similar to that obtained in [13]. 2.5. Evaluation of the Blade Inertia Figure 3represents the typical blade aspect composed of sections of different airfoils. Different regions can be established: root, a transition zone and the aerodynamic zone from which the tip part can also be distinguished. The root region is the zone that is connected to the hub through a bolted joint. It is cylindrical for structural reasons and to facilitate pitch control; hence, it has lower aerodynamical efficiency. It consists of thick aerofoil profiles to provide greater structural integrity, since this region supports the largest edgewise moments (due to the weight) and flapwise bending moments (due to aerodynamical forces). A transition region adapts the geometry from the cylindrical shape to the aerodynamical profile. This geometry is not as structurally efficient as the circular variant, but it has to withstand practically the same moments. Consequently, it is the most structurally requested region and carries the highest loads, especially at the high-pressure side [ 14 , 15 ]. Next is the aerodynamic zone, where the geometry is designed to be able to resist the design loads according to the IEC class in which it is framed. Once this restriction is overcome, the design focus is on maximizing the lift to drag ratio. In general, a larger chord length close to the root would increase energy capture, but this results in higher moments when the turbine is parked in extreme wind conditions. The tip is a compromise between aerodynamics, aeroacoustics and deflection control. A less tapered tip can increase the lift but also the noise and thrust forces, which will give larger tip deflections and bending moments. In addition to these design requisites, manufacturers also introduce modifications into the design and skin material to reduce the effect of leading-edge soiling on airfoil performance. Machines 2023,11, 908 9 of 22 Figure 3. The position of the CoG with respect to the blade root is the position with respect to the AoR summed to the hub radius. In order to obtain reliable blade inertia data, and also to be able to deduce realistic expressions for their estimation, the repositories and the literature have been reviewed in search of blade models that provide mass distribution along the blade span. This has been interpolated to produce uniform distributions across 101 positions of each blade, and the result has been uploaded to [ 16 ] as a csv file. It is also accompanied by the Matlab/Octave code that allows them to be recovered. Section 3shows the result of the analysis performed. In the event that the moment of inertia is provided with respect to the root of the blade Jroot , instead of the axis of rotation (AoR) JAoR , this inertia would have to be transferred by the amount equivalent to the radius of the hub RH (see Figure 3). Applying Steiner’s theorem, and passing first through the CoG, we have: Jroot =JCoG +MBL2 CoG JAoR =JCoG +MB(LCoG +RH)2⇒JAoR =Jroot +MB(R2 H+2RHLCoG)(40) where JCoG is the inertia around the CoG, MB is the blade mass and LCoG is the distance from the CoG up to the blade root. 2.6. Modelling the Drive Train A precise study of the behavior of the drive train dynamics requires a model of five masses [ 10 ], relating to the turbine rotor Figure 2b, the minimum requirements of which are outlined in IEC 61400-4. However, as mentioned in Section 2.4, a two-mass model is preferentially used in stability studies. Even for this simplified model, it is very difficult to find data for stiffness or even data regarding the damping of the coupling between the rotor and the generator. In fact, the drive train torsional damping is often estimated, as in [ 17 ] for the NREL-5MW reference model, by assuming a relative damping ζ=0.05. Hence: ceq =ζcc⇒ceq =0.05 cc(41) with ccdeduced from (38). 2.7. Generator Inertia The different types of generator used in megawatt WTs are listed in [ 18 ] along with their generic characteristics and are resumed in Table 1. Typical speeds, in concordance with [ 19 ], appear in the third column. Relationships between the mass and the power of the generator have also been extracted from this reference (fifth column). The data of the remaining columns have been extracted from [20], from [21] and from other scattered catalogs. Machines 2023,11, 908 16 of 22 In the following, the blade geometry will be included in the inertia estimation through the CoG position. Accordingly, a more precise estimation of the blade inertia is obtained by comparing columns 7 ( CoG/L ) and 8 ( kJ ) of Table 4. This is represented in Figure 7with a good correlation between them (r2=0.975), in the form: kJ=JB MB(LB+Rhub)2=JB MBR2 T (47) and CoG2 pu =CoG2 (LB+Rhub)2=CoG2 R2 T . (48) From Figure 7, it can be seen that they are related through: kJ=0.042 +1.114CoG2 R2 T . (49) or JB=MB0.042 R2 T+1.114 CoG2(50) The results of applying (50) are observed in Figure 6as the items with the red marker, and are compared to the results of applying (45) and (46). As can be seen, the values obtained from the CoG (red marker) are very similar to those calculated from the DD (in blue). Consequently, expression (50) allows for a precise estimate of the inertia once the mass, length and CoG positions are known. It should be mentioned that when the value of (CoG −RH)/LB is used as abscissa instead of CoG ·/Rrotor , a similar relationship is obtained, although with a somewhat lower correlation ( r2= 0.972). However, there is no observed dependence between the turbine capacity, identified through the marker colour in Figure 7and kJ. 0.06 0.08 0.1 0.12 0.14 (CoG / RT)2 0.08 0.1 0.12 0.14 0.16 0.18 0.2 kJ = J / (M RT 2) Relationship between CoG and Inertia P <= 1.5 MW 2.4 <= P <= 3.4 MW P = 5 MW 6 MW <= P <= 10 MW 10 MW <= P <= 13.2 MW P = 15 MW KJ = 0.042 + 1.114 (CoG/RT)2 r2 = 0.975 Figure 7. Correlation between the position of the CoG and the inertia, obtained from the mass distributions found in the literature. The value, L, is half the rotor diameter. 3.3. Drive Train Based on data obtained from [ 20 ], it can be seen that medium-speed wind turbines use planetary gears, with one or two stages. Clipper wind turbines, with four synchronous generators per turbine, and those that mount the WinDrive system are practically the only high-speed models with two stages. All the others with a speed above 1000 rpm have three stages, mainly combining spur and planetary, and specifically comprise 181 models of 241. Machines 2023,11, 908 17 of 22 There are some other models (37/241) with planetary gears along with helical or spur gears only (7/241) or just planetary (6/241). Each type has its own transmission structure that gives different stiffness constants [43]. The values found are very scarce and, in many cases, the reliability of their origin is questionable. They are represented in Table 6, where the dimensional values are in uppercase and the p.u. values are in lowercase. The parameters are those indicated in Figure 2. Most of these values are in the range specified by Gonzalez-Longatt in [ 11 ]: LSS stiffness, KLSS (p.u./ radel ) 0.35–0.70; HSS stiffness, KHSS (p.u./ radel ) (p.u.) 2.00–4.00. In general, the coupling on the slow shaft is less rigid than that on the fast shaft. In addition, within their wide ranges, the turbines with active-stall control typically occupy the upper values and those with pitch control occupy the lower values. Table 6. Values for the drive train components of Figure 2, obtained from the literature. Values in cursive are in pu or pu/el.rad. MW ΩT(rpm) nGB KHGB (Nm/rad) kHGB (pu/el.rad) KGBG (Nm/rad) kHGBG (pu/el.rad) CHGB/CGBG or CGB (Nms/rad) cHGB/cGBG or cGB (pu) DT/DGB/DG (Nms/rad) dT/dGB/dG(pu) Ref. 0.180 42.00 24 - 2700.0 0.52 -/- -/-/- [44] 0.200 57.69 26 - - 3.500/10.00 0.022/0.020/0.010 [45] 0.225 42.74 23 5.10 ×106 1.4 - -/- -/-/- [35] 0.225 41.00 23 - 2242.0 0.33 -/- 334/-/0.61 0.027/-/0.027 [36] 0.330 34.00 - 3.18 12.30 132.19/- 0.004/-/0.004 [46] 0.500 - - 54.8 1834.1 3.500/10.00 0.022/0.022/0.035 [11] 0.600 - - 50.0 1834.1 1.000/10.00 0.005/0.022/0.005 [47] 0.750 28.65 63 1.30 ×108 4.1 -2.78 ×105 3.3 -/-/- [24] 0.900 22.22 68 6.00 ×107 1.1 -1.00 ×106 6-/-/- [38] 1.000 41.78 22 - 1.00 ×106 24 -/- -/-/- [48] 1.270 20.00 90 2.74 ×108 2.5 -5.02 ×105 1.7 -/-/- [39] 1.500 20.46 88 4.83 ×108 3.9 -1.36 ×106 4.2 -/-/- [24] 1.500 20.70 1.0 2.00 - -/- -/-/- [49] 1.670 16.00 75 0.60 -1.200 -/-/- [13] 2.000 18.00 83 1.60 ×108 0.9 -2.50 ×105 0.44 -/-/- [40] 3.000 14.47 124.4 1.04 ×109 2.1 -4.99 ×106 3.8 -/-/- [24] 5.000 12.10 97 8.68 ×108 0.74 -6.22 ×106 2/- -/-/- [17] 5.000 11.19 160.8 2.30 ×109 1.7 -1.49 ×107 4.1 -/-/- [24] 5.000 12.37 145.5 0.30 -0.0037 2-/-/- [50] 6.000 11.84 93 3.29 ×108 0.22 2.78 ×106 16 -/- -/-/- [51] 1 The values, KLSS and KHSS in [ 46 ] have been divided into 2 πf= 314 rad/s , because the values were given in p.u. instead of p.u./el.rad. 2 The value of CGB in [ 50 ] has been multiplied by 2 πf= 377 rad/s , because the values were given in p.u. s/el.rad, instead of p.u., although these are still very low. Machines 2023,11, 908 18 of 22 3.4. Hub Inertia The hub is the weightier rotating component and, therefore, its inertia must be taken into account or at least analyzed. Table 7lists certain values found in the literature: Table 7. Values for hub inertia obtained from the literature. MW Ωrot (rpm) nGB J (kg ·m2)H (s) Ref. 0.750 28.65 62.832:1 5160 0.031 [24] 1.500 20.46 87.965:1 2.998 ×1040.046 [24] 3.000 14.47 124.407:1 1.980 ×1050.076 [24] 5.000 11.19 160.85:1 6.685 ×1050.092 [24] 5.000 12.10 97.1:1 1.160 ×1050.019 1[31] 5.000 12.10 97.1:1 1.159 ×1050.019 1[17] 6.000 11.84 92.873:1 5.070 ×1040.006 [51] 10.000 8.560 137.256:1 4.640 ×1050.019 1[31] 13.200 7.440 157.918:1 8.120 ×1050.019 1[31] 15.000 6.990 168.084:1 1.040 ×1060.019 1[31] 1 The references of [ 31 ] are equal to one another and are equal to that of [ 17 ] since, for this study, the NREL 5 MW model has been taken as a reference and the mass moments of inertia were scaled with the fourth power of the diameter scale factor, which in turn inversely determines the speed of rotation. Although the weight of the hub is around twice that of a blade, its concentration around the axis of rotation means that the inertia, expressed in s, does not have a representative value with respect to the general dynamics. 3.5. Generator Inertia This component, although not as heavy as the hub, will add significant inertia to the system dynamics, since, except in the case of direct coupled turbines, there will be a gearbox to accommodate the rotor and generator speeds. Thus, although the inertia of the generator in kg m2 is not that high, when viewed from the slow axis of the gearbox, its apparent inertia is multiplied by the gear ratio squared. In this case, the time constant is obtained from the expression (26). Table 8lists certain representative values: Table 8. Values of generator inertia obtained from the literature. MW Ωrot (rpm) nGB J (kg ·m2)H (s) npp f (Hz) Ref. 0.180 42.00 23.75:1 4.500 0.136 3 - [44] 0.225 41.00 23.40:1 10.00 0.224 3 50 [36] 0.750 28.65 62.80:1 16.65 0.394 2 60 [24] 0.900 22.22 67.50:1 35,184 10.106 2 50 [38] 1.270 20.00 90.00:1 84.08 1.176 2 - [39] 1.500 20.46 88.00:1 56.44 0.669 2 60 [24] 2.000 18.00 83.33:1 416.6 2.570 2 - [40] 3.000 14.47 124.40:1 177.9 1.053 2 60 [24] 3.000 16.67 3.00:1 21.400 ×1066.397 60 50 [41] 5.000 12.10 97.10:1 534.1 0.809 3 - [17] 5.000 11.19 160.80:1 438.9 1.558 2 60 [24] 5.000 12.10 97.10:1 534.1 0.809 3 - [31] 5.000 12.10 1.00:1 3.790 ×1050.061 248.0 - [52] 10.000 8.560 137.26:1 2140 1.620 3 - [31] 13.200 7.440 157.92:1 3740 2.145 3 - [31] 15.000 6.990 168.08:1 4800 2.422 3 - [31] 1 The value seems to refer to the LSS and, consequently, expression (22) has been used to obtain H. 2 Appears as a direct drive, but with nGB = 1, the tip speed is 227 m/s. Assuming nGB = 3, the tip speed is 75, which is more feasible. The value of JGhas been correspondingly divided by n2 GB. Machines 2023,11, 908 19 of 22 Other values, supplied directly in s, are: H= 0.685 s for a 2 MW DFIG [ 53 ]; H= 0.142 s for a 0.6 MW DFIG [ 47 ]; H= 2 s for a 0.75 MW DGIG [ 10 ]; H= 0.5 s for a 1.5 MW PMSG [ 49 ]; H= 0.75 s for a 1.67 MW DFIG [ 13 ]; H= 1.150 s for a 2 MW PMSG [ 54 ]; H=0.4 s for a 5 MW DFIG [50]. These values are more significant than those from the hub, and should be considered in dynamic studies. It was also observed that there is a great variability in the value of H , which will prove difficult in extracting some generic law, as the inertia will depend on the constructive characteristics of the rotor, probably whether the generator is an IG or a PMSG. 4. Discussion 4.1. Article Contribution The main focus of this article has been to provide reliable inertia data for the wind turbine. For this reason, HT values recovered from the bibliographic search have not been included. An expression has been proposed that quite accurately links inertia to the mass of the blade, its length and the position of the center of gravity, although the latter is only slightly easier to find than the inertia itself. Two expressions have also been proposed, which are very similar in terms of their results and which do not require CoG data, although there is a deviation for some blade models with less conventional geometry. In order to model the mechanical power transmission dynamics, this work has two additional purposes. The first is to establish an end-to-end dimensionless framework of the mechanical magnitudes that come into play in the turbine dynamics. The second is to collect existing data on the inertia of the remainder of the components and other mechanical parameters, such as stiffness, friction and damping. Since the data found are scarcer, the values in p.u. found in the literature have been incorporated into the review. However, many are not entirely reliable or have not been precisely defined in the base magnitudes of the adimensionalization. 4.2. Limitations and Benefits of the Proposed Work As already mentioned, the present work provides an expression that presents the designer with a very accurate value of the blade inertia. Its limitation is that one of the arguments required is the position of the CoG. This value is easier to find than the inertia value, but even so, it is not often found for each turbine model. 4.3. Future Work The aim of future work is to study in depth the characteristics of the coupling between the hub and the generator, especially in the slow shaft for non direct-drive couplings. This will allow to link the values of stiffness and damping more precisely. The constructive characteristics of squirrel cage, wound rotor and PMSGs will also be studied in order to deduce in each case the expressions that link its rated capacity with the rotor weight and, if possible, with its inertia. Author Contributions: Conceptualization, A.G.G.-R. and J.M.R.-F.; Data curation, A.G.G.-R. and L.M.N.-N.; Formal analysis, A.G.G.-R. and J.M.R.-F.; Investigation, A.G.G.-R. and L.M.N.-N.; Methodology, A.G.G.-R. and L.M.N.-N.; Project administration, A.G.G.-R.; Resources, L.M.N.-N.; Software, A.G.G.-R. and L.M.N.-N.; Supervision, J.M.R.-F.; Validation, A.G.G.-R. and L.M.N.-N.; Visualization, A.G.G.-R. and J.M.R.-F.; Writing—original draft, A.G.G.-R.; Writing—review and editing, J.M.R.-F. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Data Availability Statement: Data with the DD and other data related to blade models can be found in [16]. Conflicts of Interest: The authors declare no conflict of interest. Machines 2023,11, 908 20 of 22 Abbreviations The following abbreviations are used in this manuscript. They can appear as subscript or superscript for some variables or magnitudes AoR Axis of rotation B Blade CoG Center of gravity DD Density distribution DFIG Doubly-fed induction generator eq Equivalent of m2 and turbine G Generator GB Gearbox GBG Shaft joining gearbox and generator HGB Shaft joining hub and gearbox HSS High-speed shaft IG Induction generator LSS Low-speed shaft m2 Components at the high-speed side P Turbine rated capacity PMSG Permanent magnet synchronous generator T Turbine or turbine rotor W Wind WT Wind turbine The following variables and parameters are used in this manuscript: Magnitude Symbol [units] Referred to base [units] Rotational speed Ω[rad/s]ω[radel/s] Torque T [N m]t[p.u.] Twist angle Θ[rad]Θ[radel] Inertia J kg m2H[s] Mutual damping C hN·m·s radmec ic[p.u.] Shelf damping D hN·m·s radmec id[p.u.] Torsion stiffness K hN·m radmec ikhp.u. radel i EkKinetic energy [J] fGrid frequency [Hz] PTurbine rated capacity [W, also MW when explicitly specified] kJShape coefficient, defined in (47) npp Number of pole pairs ngb Gearbox ratio r Correlation index ζRelative damping References 1. 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