Evaluation of PMSG-based drivetrain technologies for 10-MW floating offshore wind turbines: Pros and cons in a life cycle perspective

Correspondence Farid K. Moghadam, Department of Marine Technology, Norwegian University of Science and Technology, Trondheim, Norway. Email: farid.k.moghadam@ntnu.no Abstract This paper presents an in depth evaluation and comparison of three different drivetrain choices based on permanent-magnet synchronous generator (PMSG) technology for 10-MW offshore wind turbines. The life cycle approach is suggested to evaluate the performance of the different under consideration drivetrain topologies. Furthermore, the design of the drivetrain is studied through optimized designs for the generator and gearbox. The proposed drivetrain analytical optimization approach supported by numerical simulations shows that application of gearbox in 10-MW offshore wind turbines can help to reduce weight, raw material cost, and size and simultaneously improve the efficiency. The possibility of resonance with the first torsional natural frequency of drivetrain for the different designed drivetrain systems, the influence of gear ratio, and the feasibility of the application for a spar floating platform are also discussed. This study gives evidence on how gearbox can mitigate the torque oscillation consequences on the other components and how the latter can influence the reliability of drivetrain.

1. There are some limitations such as availability of a technology, vendors, and raw materials that influence on companies' drivetrain selection.
The latter will not be discussed in this research.
2. The purpose of analytical design of the drivetrain components is to provide input data for the comparison study between different drivetrain technologies. Complementary component-level detailed design steps such as finite element analysis for more detailed electromagnetic, structural, and thermal designs are not in the scope of this research.
3. Reliability calculations for under consideration drivetrain systems are not in the scope of this paper, but how the failure modes are affected by the drivetrain technology is discussed. 4. For each drivetrain technology, the conventional configuration is considered in this study. Some innovative wind turbine drivetrain systems, such as gearbox integrated main-bearing (used eg in Areva Multibrid M5000 medium-speed technology) and hub-supported drivetrain (used eg in GE Haliade 150-6MW direct-drive technology) based on more compact and lightweight designs, are not the focus of this work.
The study is carried out on the basis of the most conventional drivetrain configurations available in the market, and the results provide a baseline for any further investigations on 10-MW PMSG-based wind turbine drivetrain systems. The life cycle approach for performance assessment of the different drivetrain topologies based on the PMSG technology over the design lifetime is introduced. Over the life cycle, the research focus will be on design, manufacturing, and operation supported by an analytical model of the drivetrain components. On this basis, the contributions of this paper are the following: 1. Drivetrain performance assessment using the life cycle approach is introduced.

2.
A new drivetrain optimization approach is proposed, which ensures an optimized overall cost and weight, improves the reliability and efficiency, and simultaneously assesses the feasibility of the design. 3. A new analytical design approach is proposed for the optimized design of PMSG based on the optimization of the active material cost. The optimization problem is solved numerically, and the global optimizer is determined. The design is validated by ANSYS-RMxprt software.
4. An analytical design model of the gearbox is presented, and a new optimization approach is proposed to optimize the gearbox weight.
KISSsoft software is used to validate the feasibility of the design. 5. A comprehensive comparison between the design, raw material cost, weight, size, efficiency, and reliability of DDPMSG, MSPMSG, and HSPMSG drivetrain technologies is presented. 6. The first torsional natural frequency of three under consideration drivetrain technologies is calculated, and the feasibility of application in floating offshore wind turbines is investigated.
The selection of drivetrain technology is a multidisciplinary task that needs to compromise between the criteria obliged by the life cycle assessment approach. By using DDPMSG, the generator will be larger, heavier, and more expensive, which can also increase the weight and the cost of nacelle, tower, and platform consequently. The latter also affects the installation and maintenance costs. The first research problem would be if the extra weight of a low-speed generator could compensate the weight reduction due to the gearbox removal. The second question is if the elimination of gearbox can improve the reliability because the nullification of one of the subsystems in a serial system seems to help reach a higher reliability if the reliability of components is fixed. The third challenge is if the gearbox removal can improve the drivetrain efficiency using the same justification that was provided for the reliability. The last question is which topology can safely work with floating offshore platform, which is expected to be widely used in high-power offshore wind turbines. This research uses analytical models of the drivetrain components along with numerical optimization techniques and deals with the described research problems. It is analytically proven that for the under consideration range of speed and power, gearbox helps to improve the operation and reduce the raw material cost of drivetrain. The main focus of this work is studying about the design interactions and dynamic couplings between gearbox and generator in offshore floating wind turbines' drivetrain systems. Higher level details of the drivetrain design including the design of main bearings, high-speed and low-speed shafts, and bedplate are not discussed in this work. The weight of bedplate has an important role on the drivetrain overall weight. The bedplate sizing depends on a wide range of parameters including drivetrain technology, rotor weight, rotor overhang, main shaft dimension, main bearings weight, gearbox weight, generator weight, converter weight, and transformer weight. In addition to individual component weights that the bedplate must support, the bedplate model must take into account rotor loads. For the design of bedplate for the under consideration drivetrain systems, readers are referred to Guo et al. 3,4 The paper is organized as follows: The state-of-the-art of wind turbine drivetrain technologies for offshore and onshore applications is presented in Section 2. In Section 3, the drivetrain life cycle performance assessment approach and the proposed drivetrain optimization algorithm are described. The analytical design and optimization procedure of the three under consideration PMSGs, that is, DDPMSG, MSPMSG, and HSPMSG, and the associated gearboxes are elaborated in the same section. Numerical simulations to validate the optimized generators and gearboxes design and a detailed comparison study between the under consideration drivetrain systems concerning cost, weight, operation, and performance are presented in Section 4. The paper is concluded in Section 5.

STATE-OF-THE-ART TECHNOLOGIES
A review on commercialized drivetrain system technologies of wind turbines with a power higher than 4 MW is performed in this section. The dominant drivetrain technologies for the under consideration power are DDPMSG, MSPMSG, direct-drive wound rotor synchronous generator (DDWRSG), medium-speed wound rotor synchronous generator (MSWRSG), high-speed doubly fed induction generator (HSDFIG), and high-speed squirrel cage induction generator (HSSCIG) ( less efficient due to a high excitation loss. 6 Brushless DFIG is not an efficient machine due to a low-power torque density compared with typical induction and synchronous generators. 7 Superconducting direct-drive synchronous generators are still far from being commercialized regarding standardization and some open paths to explore regarding cost and reliability. 8 An overview on the commercialized turbines (see Table 2) and the reference turbines developed by research institutions and universities 9,10 shows a unanimous interest in PMSG for high-power offshore wind turbine drivetrain systems. Higher efficiency due to external excitation circuit removal, higher reliability and availability, and less maintenance costs due to the compactness and lightweight design and brushes elimination are the motivations. 10 Different designs for permanent-magnet generators are proposed in the literature. The latter includes vernier/conventional generators, axial/radial flux, surface-mounted/interior rotor magnets, slotted/slotless stator, inner/outer rotor, distributed/concentrated winding, and integer-/fractional-slot winding machines. 11 However, still, the commercialized designs for wind turbine drivetrain systems, that is, radial flux, inner rotor, and surface-mounted PMSGs, are the focus of this research.
In the continued section, the priorities which may be considered by the different turbine manufacturers in selection of the PMSG drivetrain technology is analytically studied. It is aimed to find which PMSG-based drivetrain topology will probably be the future trend based on the analysis of the available choices. The power range and turbine platform are emphasized as the other important criteria which affect the drivetrain technology chosen by different manufacturers.

METHODOLOGY
For this study, the DTU 10 MW reference wind turbine is used. 9 The parameters of the DTU turbine used for the drivetrain studies are listed in Table 3. The other parameters related to the drivetrain components are updated based on the designed generators and gearboxes in this paper.

Life cycle approach
In the life cycle performance assessment approach, to have a meaningful comparison between different drivetrain technologies, the overall costs and benefits are evaluated over the design lifetime, and the outcome is expressed by using levelized values as equivalent costs and benefits at the regular intervals during all the system life. By using this method, the costs and revenues of various drivetrain technologies are needed to be studied over design, manufacturing, installation, O&M, life extension, and decommissioning 12 (see Figure 1). The optimal configuration is the one that gives the highest profit over the cycle.
In design, concerns related to design, for example, complexities in design, maximal design utilization, and higher degree of adaptation to site-specific conditions, reliability, and controllability, are important. A preliminary failure modes study about the generator and gearbox in PMSG drivetrain is summarized in Table 4. It is interesting to observe that the most of failure modes can be rooted back to design stage. In reliability-based design, one should look at the reliability of a drivetrain design considering failure modes to ensure a safe and reliable operation over the operating speed and torque range. However, the failure modes depend on the drivetrain technology. Furthermore, regarding specific operating conditions, some modes are of a higher importance. The failure functions will then be defined for the components and subsequently the drivetrain system based on a criticality analysis of the failure modes of the components. The impacts of uncertainty sources such as wind, wave, material strength model, and power grid can be modelled by using safety factors in a deterministic design or applying a multiplicative model of uncertainties in a stochastic design approach. 18 The outcome of reliability-based design will specify the expected lifetime of drivetrain system. For new systems and applications, for example, PMSG-based drivetrain systems for 10 MW spar floating offshore wind turbine, due to the lack of standards for the design load cases, design is more challenging, which requires to leverage the similar experiences accompanied by a wide range of numerical simulations. As a design study in this paper, the first torsional natural frequency of under consideration PMSG-based drivetrain systems is studied, and design remarks are given about the feasibility of using these technologies in floating platforms.
In manufacturing and installation, weight, compactness, and special manufacturing and installation requirements should be taken into consideration. An example of more manufacturing efforts is in manufacturing DDPMSG, where the too high pole count causes small rotor poles with fine stator slotting, which makes manufacturing expensive. About installation efforts, for instance, less weight drivetrain technology and the consequent reduced weight nacelle and tower help to reach less cost transportation and craning requirements. For more clarification, a promising solution for a floating wind turbine transport and installation is to preassemble the whole turbine in one piece in the shipyard, towing to the   offshore deployment site, and using a large floating crane to hook up with mooring systems, 19 where the most important limitation is the turbine weight. A significant part of this paper is dedicated to the optimization of weight, raw material cost, and size for under consideration drivetrain technologies as a necessary input to study manufacturing and installation costs.
In O&M, performance, efficiency, reliability, availability, and maintenance costs are required to be addressed. In offshore wind turbine applications, especially in high-power floating installations in deep waters, due to farther distance from shores, broader ranges of excitation sources and motions, and utilization of large, massive, and expensive components, any improvement in efficiency, reliability, and availability can help to reduce cost of electricity. It is reported that O&M contributes in order of 30% of LCOE in offshore wind turbines. 20 In performance, the quality of generator output power is concentrated, where the generator technology and the power frequency converter play a significant role. In efficiency, improving the efficiency of the drivetrain is discussed. It is needed to evaluate which technology and configuration will give a greater efficiency, which helps to reduce the electricity cost. Therefore, there will be a need to look into all the individual components of the drivetrain system. Then, the overall efficiency in the rated power is calculated/compared for the different technologies. In reliability, the failure rates of different drivetrain components and fault tolerance properties of each technology is required to be analyzed. Therefore, the failure modes of the subcomponents must be listed, so that the components and subsequently the drivetrain systems could be studied/sorted according to the highest to the lowest probability of failure regarding the most critical failure modes. The latter can be based on the failure reports if enough event logs and trouble shooting reports of the operation are available. It can also be based on the post processing of real operational data measured from the system 21 to calculate the fatigue damage and remaining useful life of the system. The latter could be supported by sufficient simulated models under the different load cases dependent on the application. In general, a combination of these two approaches is used. Obtaining the vulnerability map for each component considering the critical failure modes, which indicates the subcomponents from the highest to lowest probability of damage is the basis for reliability-based maintenance. The latter supported by lifetime-predicting models will give an insight about the reliability and remaining lifetime of the system, which is needed to be done for the different drivetrain technologies. In availability, the downtimes of the wind turbine due to the drivetrain shutdowns, including downtimes for periodic maintenance or unscheduled repairs for different technologies, are needed to be discussed. Furthermore, the possibility of using modular design for different gearboxes and generators for reduction of downtime in different drivetrain technologies relates to availability studies. In order to compare reliability and availability of different technologies, a detailed analysis of each component's failure modes, probability of failure, and the downtimes is required. Maintenance costs will relate to the failure rate of components, labor, parts, operations, equipment, and facilities. 22 The technology of high-power floating offshore wind turbines is new, and there are only few operational turbines of this type. An accurate O&M comparison between different technologies relies on an access to operational data of the same system under the same load cases; however, approximated analyses can be attained based on available experiences in lower power ranges, similar experiences from other industries, some limited data in access from the same operational turbines and applications, scaled-down laboratory-based systems, supported by high-fidelity simulation models and theoretical analyses. Each drivetrain technology is finally needed to be scored based on O&M criteria, namely, efficiency, reliability, availability, and maintenance costs. The other fact that makes O&M analysis more difficult is that each drivetrain technology is a different dynamic system with different responses.
To maintain the tip speed ratio of larger rotors in higher powers to reach the highest power coefficient, the nominal rotational speed of rotor is reduced, which causes different dynamic responses even for the turbines of the same technology but a different rated power. Moreover, floating offshore platforms encounter turbines into different motions induced by the synergistic impacts of wind and wave, the gravity of the turbine, and the floating platform. The lower rotor speed and more diverse and higher amplitude excitations necessitate a special O&M study for the drivetrain in high-power floating applications. In this research, for the under consideration PMSG-based drivetrain topologies, the overall efficiency is calculated/compared considering the efficiency of the individual components. As a reliability analysis, the drivetrain failure modes affected by the rotor torque and generator electromagnetic torque oscillations of different technologies are discussed and compared. Unequal Abbreviations: DDPMSG, direct-drive permanent-magnet synchronous generator; HSPMSG, high-speed permanent-magnet synchronous generator; MSPMSG, medium-speed permanent-magnet synchronous generator; PMSG, permanent-magnet synchronous generator.  Figure 1.

Proposed drivetrain design optimization
The selection of drivetrain technology is a multidisciplinary task that needs to make compromises between the criteria obliged by the life cycle approach. As explained earlier, PMSG is the promising technology for high-power offshore wind turbines. Because 10 MW PMSG-based drivetrain can be realized by different gear ratios, it is needed to assess different gear ratios over the life cycle. Gear ratio as per definition in IEC 61400-4 is n n r , where n and n r are the speeds of input and output shafts, respectively. 23 The three under consideration drivetrain topologies are DDPMSG, MSPMSG, and HSPMSG technologies, which are specified in Table 5). The 2D view of PMSG overall design that is used for the three under consideration drivetrain technologies is shown in Figure 2A. DDPMSG is a direct-drive technology. MSPMSG is the medium-speed drivetrain technology, which can be realized by either two or three stages. The operational high-power medium-speed technologies are often based on hybrid gearboxes containing three stages, including planetary gears in the upwind stages and parallel gear pairs, which can be used to realize the gear ratio 1 ∶ 36 to 1 ∶ 108. The mentioned gearbox topology is selected for the MSPMSG drivetrain technology in the paper.
HSPMSG is a high-speed drivetrain technology. The gearbox for the high-speed technology can be realized by three planetary stages with a gear ratio variable from 1 ∶ 108 to 1 ∶ 216. The gearbox topology for the MSPMSG and HSPMSG drivetrain technologies is shown in Figures 2B and   2C. In industry, based on the rule of thumb, the conventional parallel stage is used for realization of the inverse of gear ratio 1 < < 3, and a conventional planetary stage is used for 3 < < 6. For 1 < < 3, the industry interest is in using a parallel stage. The reason that planetary gear is not dominant for 1 < < 3 is due to more complex design and manufacturing, more difficult access for maintenance, and difficulty in fault detection and condition monitoring. Therefore, for the realization of the gear ratio 1 ∶ 50 suggested in DTU 10-MW turbine design, the gearbox topology based on two planetary gear stages and one parallel stage is studied in the paper as the most conventional topology to realize the similar gear ratios in the medium-speed wind turbine drivetrain systems but for lower power ranges. In the power range 10 MW in a medium-speed application, there is no agreed topology and gear ratio in the gearbox design. Some manufacturers have recently used other design topologies to meet better the requirements of offshore application by a less weight and more robust design under load variations, for example, Vestas V164-9.5MW and Adwen AD 8MW-180 have used three-stage compound planetary stages and two-stage compound planetary stages, respectively, to realized the gear-ratios 1 ∶ 38 and 1 ∶ 41. Because no reference was found, which prohibits the utilization of three planetary stages to realize an MS gearbox, another MS topology based on three planetaries is also studied. For the HS design with the assumed gear ratio 156, the most practical way to realize the gearbox is using three planetary stages.
Larger generator in DDPMSG technology causes more costly manufacturing, installation, and maintenance. Even though the general idea is that a direct-drive generator is more reliable, available, and efficient due to a gearbox removal (eg, in Polinder et al 20 Because the three generators are designed for the same power factor and consequently apparent power, the rated power of the converters is the same. However, the increased design voltage in MSPMSG and HSPMSG drivetrains compared with DDPMSG helps to reach a higher efficiency for the power converter with less effort due to reduction of switching losses, but the increased voltage raises the power converter cost due to the increased cost of the switches, DC link, and some auxiliary circuits. The changes in the overall drivetrain efficiency and cost due to power converter are neglected. The algorithm of the proposed PMSG drivetrains evaluation is demonstrated by the flowchart in Figure 3. From the conceptual design perspective, the gearbox affects both the input torque and speed of the generator. The latter are among the critical parameters in design of generator so that the size of generator is directly proportional to these parameters. In each iteration of the proposed drivetrain optimization approach, the gear ratio is fixed by the outer loop. Therefore, the optimization problem of each drivetrain technology is broken into two decoupled problems of gearbox and generator optimizations. As a result, for each drivetrain topology, two internal component-level optimization problems are solved. The first problem looks for the optimized design of the generator. The generator optimization problem is designed to ensure the minimization of cost of active material while maximizing the generator torque density. The second problem is a cost (weight) optimization of the  technologies. Finally, the designed drivetrain systems are compared with respect to total weight, cost, size, efficiency, and dynamic responses.

Generator optimization
As discussed in Section 2, PMSG is the dominant technology for high-power offshore wind turbines. Some motivations are as follows: 1. Armature reaction is smaller in a permanent-magnet generator with surface-mounted magnets than in electrically excited generators, due to a larger air gap; 2. Losses of the field winding in electrically excited generators cause lower efficiency in these generators compared with permanent-magnet generators; 3. Besides reducing losses, the permanent-magnet generators lead to a lighter design. For the same power and frequency, compared with an electrically excited generator, a permanent-magnet generator can be realized with a smaller diameter, which is really helpful for design of high-power direct-drive generators for wind turbines 26 ; and 4. It is believed that permanent-magnet generators are more reliable than electrically excited generators. One reason is that excitation circuits, the associated power electronics semiconductor devices, and the commutation rings and brushes cause a high portion of failures and downtimes in electrically excited generators, while they are not needed for permanent-magnet generators. The possible faults regarding the application of magnets, including magnets demagnetization and detachment can, to a great extent, be avoided by a proper design approach.
A three-phase radial flux, inner rotor, surface-mounted, slotted machine with laminated stator/rotor cores and distributed winding is the reference generator technology studied in this section. Reluctance torque, which is an important source of torque pulsations, is negligible in surface-mounted PMSGs. Low weight, simple design, and low-armature reaction are some other benefits of surface-mounted technologies. The cost function C of the generator optimization problem is defined as where x is a vector that represents the optimization variables. cost active is the cost of active materials in generator construction in Euro. The cost of active materials consists of the cost of iron used in rotor and stator yokes and stator teeth, copper of the stator windings, and the surface-mounted permanent magnets. cost fe is the total cost of iron, cost cu is the cost of copper, and cost magnet is the cost of magnet material. T N is the nominal torque in kN.m, and V is the generator active volume in m 3 . T d is the generator torque density in kN.m∕m 3

. The variables in Equation
(1) are defined by where the five optimization variables D s , L s , b s , h s , and h m are the air gap diameter, active length of generator, slot width, slot height, and magnet height, respectively. c fe , m fe , c cu , m cu , c pm , and m pm are the specific costs and total weights of core, copper, and magnet, respectively. The design of surface-mounted PMSG deals with the determination of a high number of variables related to the generator geometry and operation.
In the proposed optimized design approach, it was observed that all the generator design variables can be either defined as a function of the five geometrical optimization variables, or considered constant, or can change in optimization problem outer loops. The latter helped to turn a nonlinear and strongly nonconvex problem to a convex problem to be able to find a global optimizer. The other generator design variables, which are not assumed as a optimization variable, are classified into three different groups: first, those that can be written as a function of the five optimization variables, such as flux densities in air gap and teeth; second, the design variables that are assumed constant, for example, air gap thickness, which is fixed at the minimum value regarding the mechanical construction considerations of large permanent-magnet generators, and maximum flux densities in rotor and stator yokes; third, those variables that are updated/changed in outer loops to find their optimal values, for example, magnetic pole embrace is written in terms of the optimization variables.
The definition of the variables used in the the above equations is given in Table 6. For more details on the design variables and their dependencies with the defined optimization variables, readers are referred to Grauers et al. 26 The results of the described optimization problem provide the detailed geometry, material consumption, cost, and the operational criteria such as efficiency, decoupled generator losses, power factor, flux densities, electrical and thermal loadings, induced and terminal voltages, and torque oscillations. The generator optimization problem including the objective function and the constraints is highly nonlinear. Because the resultant is a constrained nonlinear multivariable nonconvex problem in terms of the optimization variables, convex programming tools cannot be useful. Because the constraints are of a high number and mostly nonlinear, duality theories and associated dual problems add to the complexity of optimization. Problem reduction, using linear optimization, 11 and application of heuristic approaches, for example, genetic algorithm 27 are some alternatives suggested in the literature. Our observations show that the aforedescribed disciplined nonconvex problem shows a convex shape in a multidimensional graphical illustration of the objective function and constraints in the predefined range of optimization variables so that the problem is convex on a portion of its domain that is the feasible region regarding the constraints. Therefore, a global optimizer for this optimization problem could be found. MATLAB fmincon numerical optimization solver using sequential quadratic programming (SQP) algorithm is used to find the globally optimized design.
The objective function C is optimized subject to a wide range of specific electrical loading, magnetic loading, insulation and mechanical forces mitigation requirements, current density, armature thermal loading, power factor, and efficiency-based constraints to ensure a feasible design for the three generators. The carter factor is modelled in terms of the design variables in the optimization problems. The generators are assumed  to be equipped with integrated water cooling, which helps to increase the limits of the electrical loading constraint. The winding insulation class N with an average temperature rise of 130 • C for the rated operation and the maximum hot-spot temperature 200 • C is considered for the design limits and power loss calculations. No restriction regarding generator frame sizes is imposed to the problem. It is assumed that the generators outputs are connected to infinite bus. In a real case, the outputs are connected to a converter as an external circuit. The converter's dedicated controller controls the phase angle and amplitude of armature output current to attain higher power factors, maximizing the generator output active power. The generators are all designed for an induced voltage less than the rated terminal voltage, which helps to minimize the permanent-magnet material consumption and weight of generator and to reduce the generator core losses. The optimized design based on the proposed analytical model is performed by MATLAB and validated using ANSYS-RMxprt software to ensure that the generators can stably deliver the desired output power and voltage.
The generator structure weight consists of cooling system, beams, cylinder, shaft, and bearings. Different models for the weight is suggested in the literature. 26 In Hartviksen, 29 the generator structure weight is estimated with a sum of cylinder and beams weights. Because the beams weight is proportional to the generator diameter and the cylinder weight has a relation with the multiplication of the diameter and length, the following experimental equation is used to estimate the weight of structure compared with a known structure weight of a similar generator with the same range of power: where m 0 str , D 0 so , and L s 0 are the weight, outer diameter, and length of the reference generator. The data of reference generator, which is a 10-MW DDPMSG, are obtained from Polinder et al. 30

Gearbox optimization
In this section, the objective function is minimization of the cost of active material in gearbox by optimizing the stage gear ratios. The gearbox cost function for a typical three-stage configuration is defined by where u is vector of the optimization variables. cost gear is total raw material cost of the gears of gearbox stages in Euro. The variables in Equation (5) are described as where Q S and Q P are the input torques to the sun and pinion for planetary and parallel stages, respectively. u is the gear ratio, B is number of planets of the planetary stage, and k r is a ring scaling factor of the planetary stage. k is the intensity of tooth loads factor. The overall active material weight for a sample three-stage gearbox with two planetary and one parallel stages is calculated as Equation (8) is obtained by the sum of the weights of two planetary stages and one parallel stage as described in Equation (7). The input torque applied on the sun/pinion of each gearbox stage is replaced in the weight function of the stage by applying the gear ratio of the previous stages on the main shaft torque Q 0 . To derive Equation (8), it is important to notice the sun/pinion is the output gear of the planetary/parallel gear stage.
The above described optimization problem is solved subject to constraints related to the gear ratio of each stage and the overall gear ratio.
The outputs of the problem are the optimized gear ratios, weight, and cost. The resultant is a constrained nonlinear multivariable nonconvex optimization problem. Because the aforedescribed disciplined nonconvex problem shows a convex behavior in multidimensional graphical illustration of the objective function and constraints in the defined range of optimization variables, it is possible to find the global optimizer.
MATLAB fmincon numerical optimization solver is used to find the global optimizer.
The replacement of the third parallel stage in MSPMSG drivetrain with a planetary stage in HSPMSG helps to reach higher gear ratios and reduce weight and size with compact design features of planetary gears. The disadvantages of planetary gearboxes are their complexity and sensitivity to manufacturing errors and elastic deformations in the shafts, bearings, and gearbox cage so that a planetary gearbox performs the best only if equal load sharing between planets is achieved. 32 The optimized three-planetary stages gearbox for HSPMSG drivetrain and two-planetary and one-parallel stage gearbox for MSPMSG drivetrain based on the explained model and the proposed optimization approach are evaluated as the case studies in Section 4. The gearboxes optimized designs are validated using the KISSsoft gear design software, and the overall geometry, weight, cost, and efficiency are compared for the different topologies.

The first torsional natural frequency of drivetrain
To ensure that the different designed drivetrain technologies execute a safe operation without any danger of resonance due to external/internal excitations, the first torsional natural frequency of three under consideration drivetrain systems is studied. The first torsional frequency is calculated by using a simplified two-mass model. The gearbox is modelled with the gear ratio and the stiffness of the connecting links to the rotor and generator are modelled, but the moment of inertia and stiffness of the gearbox are neglected. The moment of inertia of the gearbox is negligible compared with the moment of inertia of the generator so that it does not have any considerable impact on the first drivetrain torsional mode. The first torsional frequency by using a two-mass model is calculated by 33 where k eq is the equivalent shaft stiffness in the rotor side, J r and J gen are the moment of inertia of rotor and generator, k r and k gen are the shaft stiffness of rotor and generator, and is inverse of the gear ratio.
The first torsional natural frequency of drivetrain of the three under consideration topologies obtained by the above model is validated by using a three-mass model in Simpack software, which models the moment inertia of the gearbox to see how it can influence the first mode.

Optimized generator specification
The results of the three optimized generator designs based on DDPMSG, MSPMSG, and HSPMSG using the proposed analytical approach are listed in Table 6. The three optimized designed generators are validated by ANSYS-RMxprt. The terminal voltage and power-angle characteristics of the designed generators, which show that the three generators can stably deliver the designed power with the target voltage, are shown in  also has a safe distance from the power angle limit = ∕2 rad. Although the designed generators can give higher power especially in DDPMSG, it will not result in a desirable operation regarding a much lower efficiency. The frequency converter can also influence on the power factor and subsequently the power angle. The generator terminal voltage is also an important constraint in the optimized design problem so that the designed generators must be able to deliver the rated power at the rated voltage level, which the generator is designed for. Figure 4B shows one cycle of the terminal line voltage (V Line = √ 3V Phase ) of the three designed generators validated by ANSYS-RMxprt.
As it can be seen in Table 6, a transition from the low-speed to high-speed generator helps to save weight by reduction of diameter and length of the generator although the required stator and rotor yokes height increases. For higher speed generators, the reduction of the number of poles still decreases the diameter and length of machine; however, the rise in the stator and rotor yokes height exceeds the amount of weight saved by the reduced diameter and length. Consequently, the weight of active material in HSPMSG is increased, but the total weight including the weight of structure is still less than MSPMSG technology. Considerable reduction of weight and raw material cost in the generators designed for higher operational speeds and improved efficiency and torque oscillations are observed based on the simulation results.
The efficiency of generator in the optimized analytical design approach is considered as a constraint so that the design solution has to fulfill the minimum requirements regarding the power loss and efficiency. For this purpose, in the design code, the components of generator power loss consisting of winding loss, rotor and stator cores losses (hysteresis and eddy current losses), magnet losses, additional losses (stray losses), and friction and windage losses are modelled at rated load operation. The optimized design is then validated by the ANSYS-RMxprt analytical design software to ensure that the designed generator can supply the designed power at the specified voltage. This software makes it possible to check a wide range of parameters related to the generator operation including but not limited to flux densities in different places, electric loading, thermal loading, induced and terminal voltage, power factor, total harmonic distortion, power losses, and efficiency. The values of efficiency given in the paper are the verified results obtained by ANSYS. It is worth noting that maintaining the same efficiency for the low-speed generator needs more increase in volume compared with higher speed generators. The latter makes high-efficiency DDPMSG technology for high-power applications infeasible or inefficient. The voltage and frequency for the design of DDPMSG are selected less than the two other generators, to help the minimum weight and cost design. The design voltages are selected so that the output voltage of converter place in a standard voltage level based on the standards. 34,35 Regarding the reduced rotational speed in DDPMSG and the direct relationship between the induced voltage and speed, it is not efficient to design the direct-drive generator with the same voltage level used for medium-speed and high-speed generators design. The relative root mean square (RMS) value of the fundamental component of the induced voltage is given by 27 where k w is the winding factor, s is the rotational speed in rad∕s, and B 1 is the RMS value of the fundamental component of the air gap flux density.
Therefore, the increase of induced voltage means the rise in the number of turns (more winding weight), the length of the machine (more core weight), and the air gap flux density (more magnet weight). The latter increases the weight and cost of design. A power transformer in the wind turbine is used to adjust the converter output voltage to the voltage at point of common coupling (PCC) so that the reduced voltage of DDPMSG and consequently the reduced converter output voltage does not cause any problem if the voltage is selected among the standard voltages.
The design for a reduced output frequency also helps to reduce the number of poles and consequently the diameter and weight of generator.
The latter is possible, because the power converter can adjust the output frequency if the frequency is in the converter operating range.
Therefore, the design of DDPMSG for a lower voltage level and frequency is practical and results in the minimum weight design of the direct-drive generator. The other two machines, that is, MSPMSG and HSPMSG are designed for the same standard voltage level. Similar loading conditions are considered for design of the three generators to reach a fair comparison between them. Mechanical losses including friction and windage are neglected.

Optimized gearbox specification
The optimized medium-speed and high-speed gearboxes specifications including the geometrical and technical data of the stages, respectively for 10-MW MSPMSG and HSPMSG wind turbine drivetrain systems using the analytical model described in Section 3 are specified in Table 7.
The results validated using KISSsoft software show that even though the multistage gearbox for the HSPMSG drivetrain is designed for a higher gear ratio, it can still slightly reduce the total weight and improve the efficiency of the gearbox. The change in the weight scaling factor applied to model the structure weight of the two different gearboxes can slightly affect the results about the gearbox weight. As it can be seen, a transition from medium-speed to high-speed gearbox can help to slightly improve the efficiency and weight. To provide a fair comparison, because there was no reference found, which prohibits the utilization of three planetary stages instead of two planetary and one parallel stages to realize a medium-speed gearbox, the optimized results of another medium-speed topology based on three planetary stages is added to Table 7.

Comparison between different PMSG-based drivetrain technologies
The overall drivetrain weight, cost, and efficiency for DDPMSG, MSPMSG, and HSPMSG are summarized in Table 8 and graphically compared in Figures 5A to 5C. As it can be seen, moving from the direct-drive technology towards the medium-speed, the drive train weight, raw material cost, efficiency and electromagnetic torque oscillations are significantly improved. A transition from the medium-speed to the high-speed generator still helps to improve all the aforementioned properties.
To reduce fatigue damage and noise and comply with standard limits, a suppression to the permissible level 2% for electromagnetic torque oscillations is required. 37 Torque oscillation in surface-mounted PMSGs mainly comes from two sources 38,39 : 1. Cogging effect due to the variable permeance of the air gap and 2. Distortion of sinusoidal distribution of air gap flux density due to saturation, current ripple resulting from pulse width modulation (PWM), and low power quality of the grid.
Cogging torque resulting from cogging effect is an important component of torque oscillations in PMSGs, which can be improved by the proper design. As it can be seen in Table 8, the designed generators fulfill the torque oscillation requirements. However, the ratio of the cogging to the average torque reduces by the increase of gear ratio. The fundamental cogging frequency for the PMSG with integer-slot windings and where Q is number of slots, f in is generator input frequency, and p is number of poles. It can be seen that the cogging torque frequency depends on the input speed, but in rated operation, the cogging torque frequencies for DDPMSG, MSPMSG, and HSPMSG are 0.96, 57, and 3600 Hz, respectively. The higher cogging torque frequency of higher speed generators reduces the probability of coincidence on the natural frequencies of the drivetrain main bearings and the generator bearings' inner ring, outer ring, and rolling elements (see the associated failure modes in Table 4).
Regarding the higher shaft first torsional natural frequency in DDPMSG, it is doubted that the cogging frequency can coincide this frequency.
The type of main bearings in the wind turbines are mostly spherical roller bearings. These bearings have different defect frequencies due to out-of-round rotating, roller irregularity, and outer and inner race irregularities. For the under consideration 10-MW turbine, the defect frequency of these modes can start from a fraction of hertz and rise up to few hertz. The cogging torque in DDPMSG can cause fatigue damage to bearing components due to the possibility of coincidence with main bearing defect frequencies. For higher speed generators in MSPMSG and HSPMSG technologies, it is also possible that cogging torque coincides the gears mesh frequencies, which can affect the gear teeth fatigue life. However, in the designed medium-speed and high-speed gearboxes, the mesh frequencies of generator side gear stage are 120 and 474 Hz, respectively, which keep a safe distance from the rated fundamental cogging torque frequencies. The presence of gearbox results in a longer transmission path between the external excitation source initiated from the rotor (aerodynamic torque and forces) and the generator side (electromagnetic torque and forces), which are transmitted to the generator and rotor side components, respectively, which can reduce the amplitude of drivetrain external excitations. The latter can affect the probability of failure of generator and rotor side components (eg, main bearings) due to those external excitations. The increased defect frequencies of the generator bearings due to the rise in the rotational speed naturally immunizes these components against the loads applied on the drivetrain by rotor as a result of synergistic impacts of wind, waves, and structural motions, which all appear with a low frequency content. The latter shows some potentials to improve the reliability of main bearings and generator in the drivetrain, although the appearance of gearbox as a new serial component has a negative impact on the reliability of drivetrain. Therefore, it is challenging to compare the overall reliability of the under consideration drivetrain systems. In the continued part, the possibility of resonance by using the three designed drivetrain systems is investigated.
There is a coupling between gearbox and generator in the geared wind turbine drivetrain systems used for reducing torsional vibration and accommodating misalignment. The coupling between generator and gearbox could suppress the high-frequency excitations initiated by the generator due to distortion of air gap flux in order not to be transmitted to the rest of the drivetrain. Those frequencies happen mostly in harmonics of generator output frequency. The cogging frequency in MSPMSG and HSPMSG has a high-frequency nature and is isolated by the coupling from the rest of the drivetrain although there is not any possibility for this frequency to coincide with the first torsional frequency due to the high-frequency nature of cogging torque in geared drivetrain technologies. In DDPMSG, there is normally no mechanical torsional vibration isolator in the drivetrain system. As a result, the generator torsional vibrations including low frequency (due to cogging torque) and higher frequencies (due to saturation, power converter switching, and power grid) transmit to the rotor side of the drivetrain. The latter can influence on the drivetrain lifetime specially in higher powers with smaller values of the first drivetrain torsional frequency, generator cogging torque frequency, and generator output frequency.
Active vibration dampers are another drivetrain torsional vibration damping mechanism, which are used to mitigate the impact of the significant torsional vibrations on the drivetrain due to aerodynamic torque oscillations (usually, the frequencies that are considered to be damped by active dampers are the first drivetrain torsional mode and first blade in plane mode) by compensation of them in the generator electromagnetic torque set point. These dampers are generally band-pass filters, which are tuned around some prespecified frequencies. If these mechanical oscillations do not get compensated in the generator torque, the consequent electric power oscillations not only reduces the power quality but also can interact with power system modes and cause resonances with the electric circuits frequencies, which can cause high voltages and currents and damage the electric components. Instead of these compensations on torque through active dampers, some turbines use passive parallel band-stop filters in the frequency converter system to filter the frequencies of the aerodynamic torque oscillations from the output electric power. To our best knowledge, the active dampers do not influence on the torsional vibrations initiated by generator. In direct-drive technologies, due to the lack of coupling, there is no mechanism to suppress the transmission of generator vibrations to the rotor side. In 10-MW DDPMSG, the cogging frequency can coincide with the defect frequencies of the main bearings and there is no mechanism reported in the literature of direct-drive drivetrain systems to isolate this low frequency range excitations that are initiated from the generator side.

The first torsional natural frequency of drivetrain and resonance analysis
ANSYS-Maxwell moment of inertia calculation package is used to obtain a good estimation of the generator moment of inertia for the designed generators. The first torsional frequency of designed DDPMSG, MSPMSG, and HSPMSG drivetrains is presented in Table 9 and graphically illustrated in Figure 5D. The rotor and tower flexibilities are neglected in the calculation of the first torsional frequencies.
The excitations that influence on the drivetrain operation come from different sources, for example, turbine rotational motion, motions induced by structural resonances, wind-and wave-induced motions, the electromagnetic torque oscillation resulted by cogging torque/saturation/converter system/power grid, and the drivetrain internal excitations due to the gears mesh frequencies and the bearings defect frequencies. For the drivetrain system, the most unfavorable operating condition happens when the rotor rotational frequency (1P) coincides the drivetrain first torsional frequency. The other source of excitation that can affect the rotor torque comes from the tower shadow effect, which occurs with blade passing frequency 3P. The tower bending (TB) frequency excited by the 1P or 3P rotational frequencies is another source of excitation in a spar floating platform. 41 An excited TB natural frequency causes motions that induce a moment component on the rotor torque.
Wave is another excitation source in offshore wind turbines so that waves at sea have significant energy at periods of 5 to 20 seconds. An example of wave impacts is nacelle side-side motion developed by sway and rolling motions, which has a low frequency inherent. The natural periods of wind excitations also change from a fraction of a minute for turbulent fluctuations to some hours due to diurnal effects. The impacts of the generator cogging torque on the drivetrain components was discussed earlier. The power grid and converter system are the other excitation sources that influence on the generator electromagnetic torque and consequently the moment on the shaft. The latter can have different reasons, for example, grid voltage drops, swells and harmonics, and the harmonics in the current resulted by the PWM frequency converter, 42 which happen with higher frequencies potentially starting from the first or the higher order harmonics of np 120 up to a few thousands of hertz depending on the grid power quality, converter topology, the modulation technique, and switching frequency. 43 n is the generator rotational speed. The internal excitations due to any excited bearing frequencies (main bearings, gearbox, and generator bearings) or gear mesh frequencies of the gearbox stages can work as an impact that could excite the system natural frequencies and the defect frequencies of the other components.
The excitation sources with the same frequencies can cause more complicated impacts. Combined impacts of different excitations with the same T p (s) 6.5 7.5 7.5 11.5 frequencies can result in a coupled effect by producing other frequencies described as beating effect in Thomson. 44 An overview of the excitation sources for 10-MW spar floating wind turbine is presented in Table 9.
In order to determine which loads can excite the first torsional natural frequency of drivetrain, an effective approach could be to study the drivetrain response by using a frequency-domain analysis. For this purpose, the rotor torque data of the DTU 10MW reference wind turbine with a spar floating platform obtained from SIMA global simulation software is used, and the impacts on the drivetrain are studied using a decoupled analysis. Decoupled analysis approach is explained in detail in Nejad et al. 45 This model is very useful to study the consequences of the drivetrain torsional excitations induced by aerodynamic loads and structural motions. The specification of the spar floating platform used in the global simulations is given by Hegseth et al. 41 To study the impact of wind and wave, four different environmental conditions are simulated as listed in Table 10. For different wind speeds U w within the operational range of the turbine, the most probable values of significant wave height H s and peak period T p are selected for the global simulations. In order to analyze how the different loads influence on the response, the frequency spectrum of rotor torque for the different operating conditions is shown in Figure 6. The frequency spectrum of the signal in frequencies lower than 1 Hz is selected and shown because the most significant energy of the rotor torque was observed in this range. The possible resonances due to external excitations for the three optimally designed drivetrain systems are demonstrated by the Campbell diagram in Figure 6.
The torque components induced by tower shadow, TB, wind, and wave are depicted in the figure. As it can be seen, the wave-and wind-induced components are very low frequency so that they cannot excite the torsional modes of any of the three under consideration drivetrain systems.
The other excitations with considerable energy (including 1P, TB, 3P, and 6P) happen at the frequencies less than 1 Hz so that none of the designed DDPMSG, MSPMSG, and HSPMSG drivetrain technologies are affected by aerodynamic loads and structural excitations. A larger distance between the drivetrain natural frequency and the aerodynamic loads is an advantage of lower speed generator technologies, where DDPMSG outperforms the other drivetrain systems. On the other side, for the drivetrain with higher first torsional frequency, it would be more susceptible to excitations with generator torsional vibration frequencies, especially in the direct drive technology in which there is no coupling between the generator and rotor to suppress the generator electromagnetic torque vibrations. In lower speed generator technologies, in order to reduce the generator size, the number of poles is reduced. Consequently, the generator output frequency is reduced. For instance, the generator output frequency of the DDPMSG design in the paper is 16 Hz. The generator reduced output frequency and simultaneously increased first torsional frequency (eg, 11.66 Hz for the designed DDPMSG), which is a drawback of drivetrain technologies based on lower speed generators.
During operations at speeds lower than the rated value, it is quite possible that the frequency of the generator-induced voltage coincides with the first natural frequency, which is not a recommended operation. Operations at this speed or any transition over it can cause large vibrations in the system. The nontorsional excitation sources , for example, bearings and gears, have also a potential to induce torque vibrations at bearing defect frequencies and gear mesh frequencies, respectively. The higher the value of the first torsional frequency, the more the possibility of coincidence with those frequencies. For example, in DDPMSG, it is possible that the second harmonic of main bearings roller irregularity frequency coincides with the first torsional frequency. From this perspective, higher speed generators could exhibit a higher performance. Therefore, with respect to dynamic response and resonance possibility, MSPMSG seems to be a better compromise.

DISCUSSIONS AND CONCLUDING REMARKS
The optimized analytical 10-MW drivetrain designs for three different PMSG-based drivetrain topologies, that is, DDPMSG, MSPMSG, and HSPMSG, were explained and compared, and the feasibility of application in spar floating offshore wind turbines was studied. The results showed that utilization of gearbox can improve both the economics and operations of the wind turbine in 10-MW floating offshore wind turbines based on the following proven reasons: 1. It helps to reduce the weight of the drivetrain system, 2. It reduces the raw material cost of the drivetrain, 3. It can result in a better dynamic performance and reduce the possibility of resonance in the drivetrain due to the coincidence of the excitation frequencies (initiated by aerodynamic torque and electromagnetic torque) with the first drivetrain torsional frequency and the individual components defect frequencies, 4. Higher speed generator technologies in MSPMSG and HSPMSG drivetrain systems help to improve the overall drivetrain efficiency.
By considering the interactions between the drivetrain components, it was discussed how a medium-speed gearbox can improve some drivetrain failure modes concerning with the generator and the main bearings. However, new failure modes due to the presence of gearbox will also appear in the geared drivetrain systems. Regarding the lack of any operational data at this power range, the weakness of simulation tools in modelling of such complex dynamical system, and insufficient knowledge on loads in floating applications, it is challenging to compare the reliability of different drivetrain technologies. Future work will be devoted on investigating the optimal gear ratio for the PMSG drivetrain for 10-MWturbine and comparing the expected lifetime of different PMSG drivetrain systems by a thorough reliability analysis for floating offshore applications.