# Conceptual Synthesis of Speed Increasers for Wind Turbine Conversion Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Algorithm for Conceptual Synthesis of Wind Turbine Systems

- Statement of the system global function according to the requirements, by defining the correlations between its input and output entities of material, energy and information types. Only the simplified case in which the WT global function is reduced to the energy flow (the main flow) is considered in this paper; the other two secondary flows (of material and information type) are ignored as they have insignificant relevance for the paper subject. The requirements list is defined by a specialized team according to the customer needs and represents the input in the conceptual design process. The requirements are stated in terms of qualitative and quantitative specifications (of structural, geometric, kinematic, static, dynamic and other types), as well as a set of technical and economic criteria used to evaluate the resulting variants.
- Description of the sub-functions structure of the global function based on the structural specifications in the requirements list, i.e., defining the component sub-functions (either elementary or composite) and the relationships established between them. In the structure of WT global function, three main sub-functions are considered relevant in this study: the conversion of wind energy into mechanical energy, followed by the change of the mechanical energy state parameters (speed and torque) and, finally, the conversion of mechanical energy into electricity.
- Identification of known structural variants that solve each sub-function and generation of qualitative solving variants of the global function as compatible combinations of the structural variants for all sub-functions by means of morphological matrix. A structural variant is the qualitative (conceptual) solution of a function, defined as a solving principle by a physical effect implemented by an effect carrier, and its configuration [45,47] (e.g., the one DOF helical planetary gear with one input and one output can be used as a structural variant of the speed increasing sub-function).
- Classification of the obtained qualitative solving variants into categories according to specific features of the main sub-functions, such as the number of wind rotors, speed increaser complexity and electric generator type. A qualitative solving variant of the WT system is composed by the set of structural variants, one for each of the three main sub-functions stated before, compatible between them; since dimensions or other quantitative aspects are not involved at this stage, the solving variants are of “qualitative” type. The obtained WT solving variants are classified into different categories according to the number of inputs and outputs of the speed increaser.
- Selection of the representative qualitative solving variant for each category by rough evaluation, using specific criteria defined in the requirements list. Steps 5–7 apply in this algorithm only for the sub-function of mechanical energy state parameters modification, considered further as “global function” of the speed increaser. The 22 proposed structural solutions of speed increaser are classified into four categories and the best (representative) solution is identified based on a specific set of evaluation criteria.
- Establishment of quantitative solving variants of the global function by kinematic, static and dynamic analysis of the representative qualitative solving variants. The variants that do not quantitatively meet all the requirements are eliminated. The representative qualitative solutions of speed increaser that were previously identified are quantitatively designed in terms of number of teeth and transmission efficiency.
- Selection of the optimal solution (the conceptual solution) by fine evaluation based on a set of criteria also stated in the requirements list. The fine evaluation uses different weight coefficients for the considered evaluation criteria, established in this paper through the Frisco method.

## 3. Conceptual Synthesis at Wind Turbine System Level

#### 3.1. Sub-Functions Structure of the Wind Turbine Global Function

_{2}sub-function represents this paper’s focus and is highlighted in Figure 2 to better emphasize its direct links with the other two sub-functions (FE

_{1}and FE

_{3}) of the WT global function.

#### 3.2. Morphological Matrix

#### 3.3. Categories of Qualitative Solving Variants of the Global Function

_{2}sub-function—Figure 2, are in direct correlation with the structural variants of the wind rotor (sub-function FE

_{1}) and electric generator (sub-function FE

_{3}). The WT speed increaser configurations, with DOF of the speed increaser (M) = 1 or 2 and the total number of the speed increaser inputs and outputs (L) = 2; 3; 4, are generated by a compatible combination of inputs (i.e., the outputs of the FE

_{1}—Figure 2) and their outputs (i.e., the inputs of the FE

_{3}—Figure 2). Thus, the compatible combinations for generating the speed increaser configurations with L ≤ 4, according to the morphological matrix in Figure 3, are systematized in Figure 4.

- The WT uses an electric generator with fixed stator (generator stator (GS) = 0) that is driven by one of the following solutions, Figure 5a,b:
- (a)
- (b)
- two counter-rotating wind rotors, R1 and R2, are connected to the generator rotor, GR, through a one DOF [50] or a two DOF [16,23,26,29,30,31,32,51] speed increaser, A (Figure 5b). R1 is considered the main rotor, always interconnected with the GR. In the first case (M = 1), R2 provides an additional input torque, while in the second case (M = 2), R2 contributes to increasing the output speed.

- The WT uses a counter-rotating electric generator (both GR and GS are mobile and rotate in opposite directions), Figure 5c–e:
- (a)
- (b)
- (c)

- (1)
- one or two wind rotors.
- (2)
- one DOF speed increasers with one input and one output (L = 2) or two outputs (L = 3, where L is the number of inputs and outputs) or two inputs (L = 3—one output, L = 4—two outputs), and/or two DOF speed increasers with one output (L = 3) or two outputs (L = 4).
- (3)
- an electric generator with fixed or mobile stator.

- the conversion systems without a speed increaser usually have a reduced capacity as the wind rotor speed must be compatible with the generator speed (which has a special construction that allows lower operating speeds than usual, and low electric power, implicitly).
- the gearbox (speed increaser) size and complexity increase with the multiplication ratio and power increase.
- the use of two counter-rotating wind rotors allows higher output power at the generator, either by summing up the input motions in the case of two DOF speed increasers or by summing up the torques in the case of one DOF speed increasers.
- the systems using counter-rotating generators (where both rotor and stator are mobile) allow either the reduction of the multiplication ratio or a decrease in the rotor(s) input speed(s).
- the use of a multi-stage gearbox with a high-speed generator increases both complexity and cost of the conversion system, but the system is compact for higher multiplication ratios, whereas the use of a single-stage transmission reduces complexity, but the size and weight of the conversion system increases with the multiplication ratio.

## 4. Conceptual Synthesis of Speed Increasers for Wind Turbines

- Structural specifications:
- -
- number of wind rotors: one or two.
- -
- simple or complex gear transmission as a speed increaser; a simple transmission contains one satellite carrier, while the complex transmission has at least two distinct carriers.
- -
- electric generator with fixed or mobile stator.

- Geometric, constructive and kinematic specifications used in the rough evaluation:
- -
- the same radial size of the speed increaser in any solving variant.
- -
- lower size of the intermediary gears for a minimal inertial effect.
- -
- imposed ratio of the largest gear and smallest gear radii.
- -
- reduced structural complexity and simpler construction of the conversion system.
- -
- increased multiplication ratio of the speed increaser.

- Kinematic, static, dynamic and constructive specifications for the selection of the concept:
- -
- imposed multiplication ratio of the speed increaser (10 ± 0.5% for the analyzed case study).
- -
- highest efficiency of the speed increaser.
- -
- highest mechanical power on the generator shaft.
- -
- smallest axial size of the speed increaser.
- -
- complexity degree of the conversion system as low as possible.

#### 4.1. Qualitative Solving Variants for Speed Increasers

#### 4.2. Selection of the Representative Qualitative Solving Variants

- all speed increasers have the same radial size.
- the speed diagrams are built considering that the ratio between the largest gear radius and the smallest gear radius is equal to six.
- the input speed is considered equal to one.
- in the case of the solutions depicted in Figure 6f,g, the speeds of the two counter-rotating wind rotors are considered equal but in opposite directions; therefore, the ratio (${\mathrm{k}}_{\mathsf{\omega}}$) of the input speeds is: ${\mathrm{k}}_{\mathsf{\omega}}={\mathsf{\omega}}_{\mathrm{R}2}/{\mathsf{\omega}}_{\mathrm{R}1}=-1$.
- in the case of the solutions with counter-rotating outputs, the equivalent speed of an electric generator with fixed stator will be further used in the selection process: ${\mathsf{\omega}}_{\mathrm{G}}={\mathsf{\omega}}_{\mathrm{G}\mathrm{R}}-{\mathsf{\omega}}_{\mathrm{G}\mathrm{S}}$ (i.e., the relative speed between the generator rotor and stator).
- as the input speed is equal to one, the multiplication ratio (${\mathrm{i}}_{\mathrm{a}}$) of the speed increaser is given by the output speed (${\mathsf{\omega}}_{\mathrm{G}}$), the sign “−/+” indicating that the GR is rotating in the opposite/same direction to the main rotors, R1/R.

- C 1 = multiplication ratio, which must be as high as possible;
- C 2 = the size of the intermediate gears has to be as low as possible to have a minimal inertial effect;
- C 3 = simpler construction.

- Category I—the conversion system containing a single wind rotor, a simple transmission and a generator with a fixed or mobile stator: the scheme from Figure 6b has the highest absolute value of the transmission multiplication ratio of this category for both functioning cases (fixed or mobile generator stator)—3 (L = 2)/6 (L = 3); the speed increaser can be with one or two outputs (one DOF, L = 2/one DOF, L = 3).
- Category II—the conversion system containing two counter-rotating rotors, a one DOF simple transmission and a generator with a fixed or mobile stator: the variant illustrated in Figure 6e is the representative solution of this category; it contains a gearbox with a multiplication ratio of 3.76/6.72 and a simple construction for the two functioning cases—one DOF, L = 3 (two inputs and one output); one DOF, L = 4 (two inputs and two outputs).
- Category III—the conversion system containing a single wind rotor, a two DOF simple transmission and a generator with a fixed or mobile stator: the variant from Figure 6g is the representative solution of the third category; the transmission allows a higher multiplication ratio of 29.87/38.89 for the two functioning situations—two DOF, L = 3 (two inputs and one output); two DOF, L = 4 (two inputs and two outputs).
- Category IV—the conversion system containing one or two rotors, a one DOF complex transmission and a generator with a fixed or mobile stator: the scheme from Figure 6j is the representative solution in terms of multiplication ratio; the transmission is characterized by a multiplication ratio of 6.42/13.71 for the two functioning situations—one DOF, L = 2 (one input, one output) and L = 3 (one input, two outputs)/one DOF, L = 3 (two inputs, one output) and L = 4 (two inputs, two outputs).

#### 4.3. Quantitative Solving Variants of Speed Increasers: A Case Study

#### 4.3.1. Quantitative Evaluation of the Representative Qualitative Solving Variants

- the speed increaser multiplication ratio: ${\mathrm{i}}_{\mathrm{a}}=10\pm 0.5\%$;
- the radial dimension of the gearbox is imposed by limiting the number of teeth of the biggest gear: ${\mathrm{z}}_{\mathrm{max}}=400$;
- the efficiency of a gear pair: ${\mathsf{\eta}}_{\mathrm{g}}=95\%$.

**▷**input—connection to the wind rotor; ▶ output to the generator rotor or stator. According to the speed diagrams from Figure 7, Figure 8, Figure 9 and Figure 10 that are valid for both functioning cases (fixed or mobile stator), the angular speeds ($\mathsf{\omega}$) of the sun and ring gears, and of the planetary carrier (H) in each case are obtained from their peripheral linear speed (v) in the contact point $\left(\mathrm{v}=\mathrm{x}\right)$: $\mathsf{\omega}=\mathrm{v}/\mathrm{b}=\mathrm{t}\mathrm{g}\mathsf{\delta}=\mathrm{x}/\mathrm{y}$, where $\mathrm{b}=\mathrm{y}$ is the distance from the contact/center point to the central (fixed) axis of rotation, $\mathsf{\delta}$—angles associated to the angular speed $\mathsf{\omega}$, defined according to Figure 7, Figure 8, Figure 9 and Figure 10 (1–6—fixed axis or planetary gears).

_{0I}/

_{II}—interior kinematic ratio of the planetary gearbox I/II, η

_{0I}/

_{II}—interior efficiency of the planetary gearbox I/II, ${\mathrm{k}}_{\mathrm{t}}={\mathrm{T}}_{\mathrm{R}2}/{\mathrm{T}}_{\mathrm{R}1}$—ratio of the input torques, z

_{i}—teeth number of the gear i.

- (a)
- Representative solving variant (RSV) 1 (Figure 6b)
- (b)
- RSV 2 (Figure 6e), for which is considered the case $\left|{\mathrm{T}}_{\mathrm{R}2}\right|>\left|{\mathrm{T}}_{\mathrm{G}\mathrm{S}}\right|$
- (c)
- RSV 3 (Figure 6g), in the premise ${\mathrm{T}}_{\mathrm{R}2}>{\mathrm{T}}_{\mathrm{G}\mathrm{S}}$
- (d)
- RSV 4 (Figure 6j)

_{R}or P

_{R1}) is 1 kW, and the speed and the torque on the transmission main input shaft is equal to one. The diagrams obtained for the two distinct operating cases of the generator (with a fixed and mobile stator) are presented superposed in order to facilitate the comparison. The main conclusions can be drawn based on the results obtained for each quantitative solving variant (Figure 11, Figure 12, Figure 13 and Figure 14), supporting their evaluation under additional restrictions.

#### 4.3.2. Results and Discussions

- the efficiency of the gearbox with L = 2 and a fixed stator generator is steadily rising towards the interior efficiency (${\mathsf{\eta}}_{0}$) with the increase of the absolute value of the multiplication ratio (${\mathrm{i}}_{\mathrm{a}}$) (Figure 11a).
- the system with L = 3 (containing a generator with mobile and counter-rotating stator and rotor) is characterized by the following advantages compared to the conversion system with one input and one output (L = 2).
- the gearbox efficiency is constant and equal to ${\mathsf{\eta}}_{0}$, not being influenced by the transmission interior kinematic ratio; therefore, the case with L = 3 is preferred to the system with L = 2 for small–medium values of the multiplication ratio ($\left|{\mathrm{i}}_{\mathrm{a}}\right|$ < 30); for instance, for $\left|{\mathrm{i}}_{\mathrm{a}}\right|=10$ and ${\mathsf{\eta}}_{0}=0.857$ the efficiencies are ${\mathsf{\eta}}_{\mathrm{L}=2}=0.844$ and ${\mathsf{\eta}}_{\mathrm{L}=3}=0.857$, which lead to a relative increase of the efficiency of the L = 3 case with approximately 1.4%, versus the L = 2 system, and, therefore, a higher mechanical power of the counter-rotating generator (Figure 11a).
- the increase with one unit of the relative speed between the rotor and stator of the counter-rotating generator (L = 3) brings a significant contribution to the multiplication ratio in the medium range; for instance, the multiplication ratio for ${\mathrm{i}}_{0}=4$ is ${\mathrm{i}}_{\mathrm{a}}=-3$ in the case L = 2, and ${\mathrm{i}}_{\mathrm{a}}=-4$ in the case L = 3, leads to a relative increase with 33% of the multiplication ratio for the same gearbox type. Thus, for the same multiplication ratio, the case L = 3 allows the decrease of the interior kinematic ratio ${\mathrm{i}}_{0}$ and of the radial dimension, implicitly, or a smaller input speed is required when using the same gearbox and the same relative speed in the counter-rotating generator (Figure 7); additionally, in the case L = 3 (Figure 11b), the generator input power (${\mathrm{P}}_{\mathrm{g}}$) and its output electric power, implicitly, are higher at the same input power of the conversion system.
- for the same input torque $\left|{\mathrm{T}}_{\mathrm{R}\text{\hspace{0.17em}}(\mathrm{L}=2)}\right|\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left|{\mathrm{T}}_{\mathrm{R}\text{\hspace{0.17em}}(\mathrm{L}=3)}\right|$, the torque on the generator shaft is lower in the case L = 3, ${\mathrm{T}}_{\mathrm{g}\text{\hspace{0.17em}}\left(\mathrm{L}=3\right)}<{\mathrm{T}}_{\mathrm{g}\text{\hspace{0.17em}}\left(\mathrm{L}=2\right)}$, the differences between the two values being higher at smaller interior kinematic ratios. For instance, for ${\mathrm{i}}_{0}=10$ and ${\mathsf{\eta}}_{0}=0.857$, the torque on the generator shaft in the case L = 3 is smaller by approximately 9% and the speed is higher by approximately 11% compared to the case L = 2; therefore, for the same wind power, the electric power generated by the conversion system in the case L = 3 (Figure 11b) is higher by approximately 1.4% than in the case L = 2, meaning that the counter-rotating generator starts sooner to produce energy than in the other case, as it has to overcome a smaller resistance.

- for $\left|{\mathrm{k}}_{\mathrm{t}}\right|>1$, the secondary rotor (R2) becomes the main rotor, which is less viable from a functional point of view; therefore, the case $\left|{\mathrm{k}}_{\mathrm{t}}\right|<1$ is further considered.
- the gearbox efficiency is strongly influenced by the ratio ${\mathrm{k}}_{\mathrm{t}}$, obtaining higher values of the efficiency for lower absolute values of this ratio; the efficiency decreases with the increase of the torque generated by R2 (Figure 12a); if the input torques on the two rotors are equal in absolute values ($\left|{\mathrm{k}}_{\mathrm{t}}\right|=1$), the gearbox efficiency (Figure 12a) and the input power on the generator shaft, which is due to R1 (Figure 12b), decreases by approximately 10% relative to the case in which the torque given by the secondary rotor is null (${\mathrm{k}}_{\mathrm{t}}=0$).
- the advantage of higher efficiency of the solutions with a mobile stator relative to the fixed stator case is practically insignificant for higher values of the multiplication ratio (${\mathrm{i}}_{\mathrm{a}}$) (Figure 12).
- a linear increase of the power generated by the secondary rotor (Figure 12c) and of the total output power (Figure 12d) with the increase of the absolute value of ${\mathrm{k}}_{\mathrm{t}}$ is obtained by bringing into operation the secondary rotor ($\left|{\mathrm{T}}_{\mathrm{R}2}\right|>0,\text{\hspace{0.17em}}\left|{\mathrm{k}}_{\mathrm{t}}\right|>0$).
- the mechanical power provided by the secondary rotor generates an increase in electric power, which can reach up to approximately 80% (Figure 12b) when the torque of the secondary rotor becomes equal to the torque given by the main rotor ($\left|{\mathrm{k}}_{\mathrm{t}}\right|=1$).
- the system with L = 4 (two inputs and two outputs) has the following advantages compared to the system with L = 3 (2 inputs and 1 output): the efficiency of the L = 4 speed increaser is higher to that of the L = 3 transmission, regardless of the value of input torques ratio (${\mathrm{k}}_{\mathrm{t}}$). For the same ratio (${\mathrm{k}}_{\mathrm{t}}$), the differences between the efficiency values are higher with up to 6% for low multiplication ratios and decrease with the increase of the multiplication ratio; for instance, the difference between efficiency values reaches approximately 3% for ${\mathrm{i}}_{\mathrm{a}}=10,\text{\hspace{0.17em}}{\mathsf{\eta}}_{\left({\mathrm{k}}_{\mathrm{t}}=-1,\text{\hspace{0.17em}}\mathrm{L}=4\right)}=0.845$ $\text{\hspace{0.17em}}{\mathsf{\eta}}_{\left({\mathrm{k}}_{\mathrm{t}}=-1,\text{\hspace{0.17em}}\mathrm{L}=3\right)}=0.821$ (Figure 12a), which implies a slight increase in the output mechanical power that justifies the use of a counter-rotating generator (L = 4).

- R2 becomes the main rotor for ${\mathrm{k}}_{\mathsf{\omega}}<-1$, which is not viable from a functional point of view; therefore, the case ${\mathrm{k}}_{\mathsf{\omega}}\in \left[-1,0\right)$ is further analyzed.
- the efficiency of the gearbox increases with the increase of the ratio $\left|{\mathrm{k}}_{\mathsf{\omega}}\right|$ for the same multiplication ratio (${\mathrm{i}}_{\mathrm{a}}$). For instance, considering ${\mathrm{i}}_{\mathrm{a}}=10$ and L = 3, the efficiency almost triples in the case ${\mathrm{k}}_{\mathsf{\omega}}=-1$ ($\mathsf{\eta}=70\%$) compared to the case ${\mathrm{k}}_{\mathsf{\omega}}=0$ ($\mathsf{\eta}=25\%$) (Figure 13a).
- the efficiency values decrease with an increase in the multiplication ratio by a higher gradient as the module of the k
_{ω}ratio is lower. Thus, these gearboxes can be implemented in wind turbines of medium and low power (with low multiplication ratios). - the gearbox efficiency decreases with the increase of the multiplication ratio (Figure 13a).
- the conversion system with L = 4 has the following advantages compared to the system with L = 3:
- -
- the solution containing a generator with a mobile stator (L = 4) ensures the efficiency increase with the increase of the ratio $\left|{\mathrm{k}}_{\mathsf{\omega}}\right|$; for instance, for ${\mathrm{k}}_{\mathsf{\omega}}=-1$, the efficiency in the case L = 4 is higher by approximately 10% to the case L = 3, regardless of the value of the multiplication ratio (${\mathrm{i}}_{\mathrm{a}}$) (Figure 13a).
- -
- the power generated by the secondary rotor (${\mathrm{P}}_{\mathrm{R}2}/{\mathrm{P}}_{\mathrm{R}1}$) (Figure 13c) and the surplus of mechanical power on the generator shaft due to R2 (${\mathrm{P}}_{\mathrm{g}}/{\mathrm{P}}_{\mathrm{R}1}$) (Figure 13b,d) increases with the increase of the ratio $\left|{\mathrm{k}}_{\mathsf{\omega}}\right|$ module, these values being higher in the case L = 4.

- due to the relatively good efficiencies, the solution can be applied to systems characterized by high absolute values of the multiplication ratio and limited radial dimensions; in these cases, a higher speed of the generator can be obtained by using two or more planetary gear trains as a speed increaser (Figure 10).
- the gearbox efficiency is decreasing up to 1.5% for the case L = 2 and up to 4% for the L = 3 variant with the increase of the absolute value of the multiplication ratio ($\left|{\mathrm{i}}_{\mathrm{a}}\right|$) (Figure 14a); the mechanical power at the generator input has the same characteristic (Figure 14b).
- the system with L = 3, containing a counter-rotating generator has the advantage of a higher efficiency compared to the system with L = 2; this advantage becomes insignificant for higher values of the multiplication ratio:
- -
- the solution with L = 3 has a higher efficiency (Figure 14a) and a higher mechanical power (Figure 14b) than the L = 2 case for small values of the multiplication ratio. For instance, the efficiencies are ${\mathsf{\eta}}_{\mathrm{L}=2}=0.855$, and ${\mathsf{\eta}}_{\mathrm{L}=3}=0.87$ (Figure 14a) for $\left|{\mathrm{i}}_{\mathrm{a}}\right|=10$.
- -
- the variant with L = 2, containing a fixed stator generator achieves lower efficiencies and mechanical power but close to those of the mobile stator solutions (L = 3) for high absolute values of the multiplication ratio, the difference being approximately 1% (Figure 14); therefore, in these cases, the fixed stator variant is recommended to be used.

#### 4.4. Selection of the Conceptual Solution

- C
_{A}: Highest mechanical power of the generator; - C
_{B}: Highest efficiency of the speed increaser; - C
_{C}: Smallest axial size of the gearbox; - C
_{D}: Degree of complexity as low as possible.

_{k}—the global grade of the k criterion = the sum of grades from k row, S

_{k}= number of criteria whose global grades are inferior to the global grade of the current k criterion, P

_{min/max}—minimum/maximum value of P

_{k}.

_{k}) and the imposed evaluation criteria (Table 6):

## 5. Conclusions and Recommendations

- (a)
- the degree of freedom (one or two DOF).
- (b)
- the number of inputs: one or two wind rotors.
- (c)
- the number of outputs: one output (electric generator with a fixed stator) or two outputs (mobile and counter-rotating rotors and a stator).
- (d)
- the number of external links (L), i.e., the sum of inputs and outputs.
- (e)
- the (minimum) multiplication ratio, defined as the ratio between the equivalent generator speed (rotational speed of the rotor relative to the stator) and the wind rotor speed (in the case of two rotors: the higher rotor speed).
- (f)
- the mechanical efficiency, defined as the ratio between the output and input mechanical powers.
- (g)
- the (structural and technological) degree of complexity.

- (a)
- one DOF speed increaser with one input and one output (generator rotor) (L = 2).
- (b)
- one DOF speed increaser with one input (single wind rotor) and two outputs (counter-rotating generator) (L = 3); the mechanical driving of the rotor and stator of the electric generator is in opposite directions with speeds that are inversely proportional to their mechanical moments of inertia.
- (c)
- one DOF speed increaser with two inputs (counter-rotating wind rotors) and one output (generator rotor) (L = 3) and two independent torques; the mechanical summation of the two torsional torques (from the wind rotors) allows the increase of the output mechanical power.
- (d)
- one DOF speed increaser with two inputs (counter-rotating wind rotors) and two outputs (counter-rotating generator) (L = 4) and two independent torques.
- (e)
- two DOF speed increaser with two inputs (counter-rotating wind rotors) and one output (generator rotor) (L = 3) and two independent motions.
- (f)
- two DOF speed increaser with two inputs (counter-rotating wind rotors) and two outputs (counter-rotating generator) (L = 4) and two independent motions.

- (a)
- maximization of the speed increaser efficiency.
- (b)
- increase of the speed at the generator by:
- -
- using a counter-rotating generator
- -
- summing up the motions of two counter-rotating wind rotors with a two DOF planetary transmission as a speed increaser
- -
- branching out the wind rotor motion with the help of a one DOF transmission and summing up the obtained motions with a two DOF planetary transmission

- (c)
- increase of the torque on the generator shaft by summing up the torques of two counter-rotating rotors through a one DOF transmission.
- (d)
- combination of the above solutions.

- (a)
- the gearbox should be designed to be installed in a specific system that allows the conversion of the wind energy into electricity, based on the location and the implementation conditions.
- (b)
- if a speed increaser with a high multiplication ratio is requested, it is recommended to use a gearbox consisting of a one DOF planetary gear that allows branching out of the rotor speed, and a two DOF planetary gear that sums up the two motions, increasing the speed in this way (Figure 10).
- (c)
- although the use of a counter-rotating generator increases, to a certain extent, the complexity of the conversion system, the mobile stator of the generator adds additional speed and power, which becomes significant mainly for small and medium values of the multiplication ratio.

- -
- one DOF transmission that sums up the speeds of two rotors (Figure 6d1,d2,e1,e2,k1,k2).
- -
- two DOF transmission that sums up the speeds of two rotors (Figure 6f1,f2,g1,g2).
- -
- complex transmission containing a one DOF planetary gear, in which the motion of a rotor branches into two other motions, and a two DOF planetary gear, which sums up the two motions (Figure 6h1,h2,i1,i2,j1,j2).

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The proposed algorithm for the conceptual synthesis of speed increasers for wind turbines (WTs).

**Figure 2.**The sub-function structure of WT global function: FE

_{1}= transformation of wind energy into mechanical energy; the structural variant for this function is represented by the wind rotor(s); FE

_{2}= appropriate modification of mechanical energy parameters (transmission of mechanical energy with speed increase); the structural variant is the speed increaser; FE

_{3}= transformation of mechanical energy into electrical energy, sub-function fulfilled by the electric generator. E = wind energy. E* = electric energy.

**Figure 4.**WT speed increaser configurations. R = wind rotor. GR = generator rotor. GS = generator stator.

**Figure 5.**Conceptual configuration of WT conversion systems with speed increasers generated in Figure 4: (

**a**) single-rotor R—one degree of freedom (DOF), speed increaser A—electric GR; (

**b**) counter-rotating wind rotors R1, R2—one DOF/two DOF speed increaser A—electric GR; (

**c**) single-rotor R—one DOF speed increaser A—counter-rotating generator (GS—generator stator, GR—generator rotor); (

**d**) counter-rotating wind rotors R1, R2—one DOF/two DOF speed increaser A—counter-rotating generator GS, GR; (

**e**) counter-rotating wind rotors R1, R2—dual distinct speed increasers A1, A2—counter-rotating generator GS, GR.

**Figure 7.**Structural scheme, speeds diagram and the main parameters of the representative solving variant of Category I (1—sun gear, 2 and 3—planetary gears, 4—ring (internal) gear, H—planetary carrier, x

_{R}= x

_{GS}—linear speed of the center point of the planetary gear 3, x

_{GR}—linear speed of the contact point of the gears 1 and 2, y

_{R}= y

_{GS}—distance from the center point of the planetary gear 3 to the central axis, y

_{GR}—radius of the gear 1, δ

_{R}= δ

_{GS}—angle associated to the angular speed ω

_{R}= ω

_{H}= ω

_{GS}, δ

_{GR}—angle associated to the angular speed ω

_{GR}).

**Figure 8.**Structural scheme, speeds diagram and the main parameters of the representative solving variant of Category II (1—sun gear, 3 and 4—planetary gears, 2 and 5—ring gears, x

_{R1}—linear speed of the center point of the planetary gear 4, x

_{GR}—linear speed of the contact point of the gears 1 and 4, x

_{R2}= x

_{GS}—linear speed of the contact point of the gears 2 and 3, y

_{R1}—distance from the center point of the planetary gear 4 to the central axis, y

_{R2}= y

_{GS}—distance from the contact point of the gears 2 and 3 to the central axis, δ

_{R1,2}—angle associated to the angular speed ω

_{R1,2}).

**Figure 9.**Structural scheme, speeds diagram and the main parameters of the representative solving variant of Category III (1—sun gear, 3 and 4—planetary gears, 2 and 5—ring gears, x

_{R1}—linear speed of the contact point of the gears 4 and 5, x

_{GR}—linear speed of the contact point of the gears 1 and 3, y

_{R1}—distance from the contact point of the gears 4 and 5 to the central axis).

**Figure 10.**Structural scheme, speeds diagram and the main parameters of the representative solving variant of Category IV (1 and 6—sun gears, 2 and 5—planetary gears, 3 and 4—ring gears, H

_{1}and H

_{2}—carriers, x

_{R}—linear speed of the contact point of the gears 2 and 3, x

_{GS}—linear speed of the contact point of the gears 4 and 5, x

_{Gr}—linear speed of the contact point of the gears 5 and 6, y

_{R}—distance from the contact point of the gears 2 and 3 to the central axis, y

_{GS}—distance from the contact point of the gears 4 and 5 to the central axis, y

_{GR}—radius of the gear 6).

**Figure 11.**The diagrams for the variation of: (

**a**) efficiency $\mathsf{\eta}$ and (

**b**) generator input power ${\mathrm{P}}_{\mathrm{g}}$ as function of the multiplication ratio ${\mathrm{i}}_{\mathrm{a}}$.

**Figure 12.**Diagrams for the variation of: (

**a**) efficiency $\mathsf{\eta}$; (

**b**) generator input power relative to the main rotor power ${\mathrm{P}}_{\mathrm{g}}/{\mathrm{P}}_{\mathrm{R}1}$ as functions of the multiplication ratio ${\mathrm{i}}_{\mathrm{a}}$; (

**c**) the power of the secondary rotor ${\mathrm{P}}_{\mathrm{R}2}/{\mathrm{P}}_{\mathrm{R}1}$; (

**d**) the input power of the generator ${\mathrm{P}}_{\mathrm{g}}/{\mathrm{P}}_{\mathrm{R}1}$ relative to the main rotor power as functions of the ratio ${\mathrm{k}}_{\mathrm{t}}$.

**Figure 13.**Diagrams for the variation of: (

**a**) efficiency$\mathsf{\eta}$; (

**b**) generator input power ${\mathrm{P}}_{\mathrm{g}}/{\mathrm{P}}_{\mathrm{R}1}$ relative to the main rotor power as functions of the multiplication ratio ${\mathrm{i}}_{\mathrm{a}}$; (

**c**) the secondary rotor power ${\mathrm{P}}_{\mathrm{R}2}/{\mathrm{P}}_{\mathrm{R}1}$; (

**d**) generator input power ${\mathrm{P}}_{\mathrm{g}}/{\mathrm{P}}_{\mathrm{R}1}$ relative to the main rotor power as functions of the ratio ${\mathrm{k}}_{\mathsf{\omega}}$.

**Figure 14.**Diagrams for the variation of: (

**a**) efficiency $\mathsf{\eta}$; (

**b**) generator power related to the rotor R power, ${\mathrm{P}}_{\mathrm{g}}/{\mathrm{P}}_{\mathrm{R}}$ as function of the multiplication ratio ${\mathrm{i}}_{\mathrm{a}}$.

**Table 1.**The angular speeds of wind rotors and electric generator for the qualitative solving variants from Figure 6 (${\mathsf{\omega}}_{\mathrm{R}}$ —wind rotor speed, ${\mathsf{\omega}}_{\mathrm{R}1}$ —main wind rotor speed, ${\mathsf{\omega}}_{\mathrm{R}2}$ —secondary wind rotor speed, ${\mathsf{\omega}}_{\mathrm{G}\mathrm{R}}$ —generator rotor speed, ${\mathsf{\omega}}_{\mathrm{G}\mathrm{S}}$ —generator stator speed, Min/Max = the minimum/maximum absolute values of the variation range for angular speed and multiplication ratio).

Speed | ω_{R} = ω_{R1} | ${\mathsf{\omega}}_{\mathbf{R}2\text{}}$ | ${\mathsf{\omega}}_{\mathbf{G}\mathbf{R}\text{}}$ | ${\mathsf{\omega}}_{\mathbf{G}\mathbf{S}\text{}}$ | ${\mathbf{i}}_{\mathbf{a}}={\mathsf{\omega}}_{\mathbf{G}}=$${\mathsf{\omega}}_{\mathbf{G}\mathbf{R}\text{}}\left(\mathbf{G}\mathbf{S}=0\right)$ | ${\mathbf{i}}_{\mathbf{a}}={\mathsf{\omega}}_{\mathbf{G}}=$${\mathsf{\omega}}_{\mathbf{G}\mathbf{R}\text{}}-{\mathsf{\omega}}_{\mathbf{G}\mathbf{S}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Category Figure 6 | Min | Max | Min | Max | Min | Max | Min | Max | Min | Max | ||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | ||

I. | a | +1 | - | - | −1.5 | −6 | +1 | +1 | −1.5 | −6 | −2.5 | −7 |

b | - | - | −2 | −5 | +1 | +1 | −2 | −5 | −3 | −6 | ||

c | - | - | +2.66 | +7 | −0.41 | −2.51 | +2.66 | +7 | +3.08 | +9.51 | ||

II. | d | −0.83 | −1.14 | +1.86 | +5.92 | −0.83 | −1.14 | +1.86 | +5.92 | +2.69 | +7.06 | |

e | −0.63 | −0.72 | +3.12 | +6 | −0.63 | −0.72 | +3.12 | +6 | +3.76 | +6.72 | ||

III. | f | −1 | −1 | +4 | +13 | −1 | −1 | +4 | +13 | +5 | +14 | |

g | −1 | −1.01 | +28.87 | +37.88 | −1 | −1.01 | +28.87 | +37.88 | +29.87 | +38.89 | ||

IV. | h | - | - | +4 | +23 | −1 | −1.49 | +4 | +23 | +5 | +24.49 | |

i | - | - | +3.45 | +13.25 | −0.63 | −1.04 | +3.45 | +13.25 | +4.09 | +14.29 | ||

j | - | - | −5.42 | −12.71 | +1 | +1 | −5.42 | −12.71 | −6.42 | −13.71 | ||

k | −0.63 | −1.04 | +3.45 | +13.25 | −0.63 | −1.04 | +3.45 | +13.25 | +4.09 | +14.29 |

Figure 6 | I. | II. | III. | IV. | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Criterion | a | b | c | d | e | f | g | h | i | j | k | |

C 1 | 9 | 8 | 10 | 10 | 9 | 6 | 10 | 10 | 7 | 7 | 7 | |

C 2 | 7 | 10 | 7 | 9 | 8 | 9 | 8 | 9 | 9 | 9 | 10 | |

C 3 | 10 | 9 | 6 | 7 | 10 | 10 | 8 | 5 | 9 | 10 | 8 | |

$\sum$ | 26 | 27 | 23 | 26 | 27 | 25 | 26 | 24 | 25 | 26 | 25 | |

Place | 2 | 1 | 3 | 2 | 1 | 2 | 1 | 3 | 2 | 1 | 2 |

Representative Solving Variants | RSV 1 | RSV 2 | RSV 3 | RSV 4 | |
---|---|---|---|---|---|

Figure | 7 | 8 | 9 | 10 | |

Characteristics | |||||

Number of teeth and the interior kinematic ratio | z_{1} = 40z _{2} = 90z _{3} = 90z _{4} = 400i _{0} = 10 | z_{1} = 36z _{2} = 400z _{3} = 225z _{4} = 139z _{5} = 314i _{0I} = −8.72i _{0II} =1.27 | z_{1} = 300z _{2} = 400z _{3} = 50z _{4} = 25z _{5} = 375i _{0I} = −0.4i _{0II} = −0.75 | z_{1} = 296z _{2} = 32z _{3} = 360z _{4} = 400z _{5} = 143z _{6} = 114i _{0I} = −1.21;i _{0II} = −3.5 | |

Multiplication ratio | −10 | 9.993 | 10 | −9.992 | |

Efficiency of a gear pair ${\mathsf{\eta}}_{\mathrm{g}}$ | 0.95 | 0.95 | 0.95 | 0.95 |

Representative Solving Variant | RSV 1 | RSV 2 | RSV 3 | RSV 4 | |
---|---|---|---|---|---|

Figure | 7 | 8 | 9 | 10 | |

Characteristics | |||||

Multiplication ratio | −10 | 9.993 | 10 | −9.992 | |

Efficiency of a gear pair η_{g} | 0.95 | 0.95 | 0.95 | 0.95 | |

Efficiency of the speed increaser | 0.857 | 0.845–0.937 ${\mathrm{k}}_{\mathrm{t}}=-\frac{{\mathrm{T}}_{\mathrm{R}2}}{{\mathrm{T}}_{\mathrm{R}1}}=-\mathrm{1...0}$ | 0.237–0.696 ${\mathrm{k}}_{\mathsf{\omega}}=-\frac{{\mathsf{\omega}}_{\mathrm{R}2}}{{\mathsf{\omega}}_{\mathrm{R}1}}=-\mathrm{1...0}$ | 0.87 | |

Mechanical power at the generator input ${\mathrm{P}}_{\mathrm{g}}$ [kW] | 0.857 | 1.69–0.937 ${\mathrm{k}}_{\mathrm{t}}=-\frac{{\mathrm{T}}_{\mathrm{R}2}}{{\mathrm{T}}_{\mathrm{R}1}}=-\mathrm{1...0}$ | 1.39–0.237 ${\mathrm{k}}_{\mathsf{\omega}}=-\frac{{\mathsf{\omega}}_{\mathrm{R}2}}{{\mathsf{\omega}}_{\mathrm{R}1}}=-\mathrm{1...0}$ | 0.87 |

**Table 5.**The relative weight coefficients (L

_{k}= the place of the current k criterion according to the P

_{k}values hierarchy).

k | Criterion | C_{A} | C_{B} | C_{C} | C_{D} | P_{k} | L_{k} | S_{k} | W_{k} | w_{k} |
---|---|---|---|---|---|---|---|---|---|---|

1 | C_{A} | 0.5 | 1 | 1 | 1 | 3.5 | 1 | 3 | 20 | 0.805 |

2 | C_{B} | 0 | 0.5 | 1 | 1 | 2.5 | 2 | 2 | 3.5 | 0.140 |

3 | C_{C} | 0 | 0 | 0.5 | 1 | 1.5 | 3 | 1 | 1.14 | 0.046 |

4 | C_{D} | 0 | 0 | 0 | 0.5 | 0.5 | 4 | 0 | 0.2 | 0.009 |

$\sum $ | 24.84 | 1 |

**Table 6.**Concept selection by ordering the solving variants (N

_{k}—the grade on the scale of 1 to 10 awarded to the k criterion for a given solving variant).

RSV 1 | RSV 2 | RSV 4 | |||||
---|---|---|---|---|---|---|---|

Criterion | w_{k} | N_{k} | w_{k}·N_{k} | N_{k} | w_{k}·N_{k} | N_{k} | w_{k}·N_{k} |

C_{A} | 0.805 | 6 | 4.83 | 10 | 8.050 | 7 | 5.635 |

C_{B} | 0.140 | 8 | 1.12 | 10 | 1.400 | 9 | 1.260 |

C_{C} | 0.046 | 10 | 0.46 | 9 | 0.414 | 8 | 0.368 |

C_{D} | 0.009 | 10 | 0.09 | 9 | 0.081 | 7 | 0.063 |

$\sum $ | 34 | 6.5 | 38 | 9.945 | 31 | 7.326 | |

Place: | 3 | 1 | 2 |

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**MDPI and ACS Style**

Saulescu, R.; Neagoe, M.; Jaliu, C. Conceptual Synthesis of Speed Increasers for Wind Turbine Conversion Systems. *Energies* **2018**, *11*, 2257.
https://doi.org/10.3390/en11092257

**AMA Style**

Saulescu R, Neagoe M, Jaliu C. Conceptual Synthesis of Speed Increasers for Wind Turbine Conversion Systems. *Energies*. 2018; 11(9):2257.
https://doi.org/10.3390/en11092257

**Chicago/Turabian Style**

Saulescu, Radu, Mircea Neagoe, and Codruta Jaliu. 2018. "Conceptual Synthesis of Speed Increasers for Wind Turbine Conversion Systems" *Energies* 11, no. 9: 2257.
https://doi.org/10.3390/en11092257