2.1 Analysis of Power Flow

Figure 2a shows the existing split type mechanical hydraulic hybrid transmission (S-MHHT) structure. The hydraulic pump is connected to the input planetary carrier, and the hydraulic motor is connected to the outer ring gear. After the power captured by the wind rotor passes through the increase gearbox, it is split at the front end of the planetary carrier. Most of the power is transmitted to the planetary carrier through the main axis system. A small portion of power is transmitted to the outer ring gear through the increase gear—closed hydraulic system—reduction gear. Finally, the power is confluent in the planetary structure.

Figure 2b shows the reflux type mechanical hydraulic hybrid transmission (R-MHHT) structure. The hydraulic pump is connected to the output sun gear, and the hydraulic motor is connected to the outer ring gear. This structure does not divert the input power but rather circulates power internally. A small portion of power flows from the sun gear output shaft. Then, this part of the power is transferred to the outer ring gear through the increase gear—closed hydraulic system—reduction gear. The backflow power is combined with the power input from the planetary carrier and eventually transferred to the Sun output shaft. Then, the power returned through the outer ring gear enters the hydraulic system from the sun gear output shaft to complete the power cycle.

In the two MHHT structures, the power flow in the planetary transmission is the same. That is, the power is input by the planetary carrier and the ring gear and output by the sun gear. Therefore, we can obtain the velocity relationship based on the torque relationship of the planetary structure, as shown in Table 1.

2.2 Power Distribution Characteristic

To ensure that the MHHT maintains a high transmission efficiency, we want the power transmitted by the hydraulic system as little as possible. This paper introduces the hydraulic system power ratio $$$ {R}_h $$$, which represents the ratio of the power transmitted by the hydraulic system $$$ {P}_h $$$ to the total power $$$ {P}_{in} $$$.

The variation law of the hydraulic system power ratio $$$ {R}_h $$$ is shown in Figure 3a. When the transmission ratio $$$ {I}_{mhht} $$$ is constant, the larger the $$$ k $$$, the smaller the $$$ {R}_h $$$. When the planetary transmission parameter k is larger, to meet the same transmission ratio, the required hydraulic system power is smaller. When the planetary transmission parameter k is constant, the larger the $$$ {I}_{mhht} $$$, the larger the $$$ {R}_h $$$. The greater the transmission ratio of the speed regulating device, the greater the power ratio of the hydraulic system.

The change rules of the two speed regulating structures are the same. The difference is the hydraulic system power proportion $$$ {R}_h $$$ of the R-MHHT is greater than that of the S-MHHT. The larger the transmission ratio, the greater the hydraulic system power ratio difference between the two structures.

2.3 Speed Regulation Characteristic

The variation law of the $$$ {I}_{mhht} $$$ is shown in Figure 3b. When $$$ {I}_h $$$ is zero, the speed of the ring gear is 0. And the minimum value of $$$ {I}_{mhht} $$$ increases with the increase of $$$ k $$$. In addition, as k increases, $$$ {I}_{mhht} $$$ increases overall. When the planetary transmission parameter k is larger, the same hydraulic system displacement can make the speed regulating device achieve a larger transmission ratio. When $$$ {I}_h $$$ increases, $$$ {I}_{mhht} $$$ increases. The greater the transmission ratio of hydraulic system, the greater the speed regulating range of speed regulating device.

The change rules of the two speed regulating structures are the same. The difference is that the growth rate of the transmission ratio $$$ {I}_{mhht} $$$ of the R-MHHT is obviously higher than that of the S-MHHT. This shows that using the same hydraulic system, the R-MHHT has a wider range of transmission ratio changes and a stronger speed regulation ability.

2.4 Brief Summary

3.1 Control Principle of MPPT

In the following design, ignore the response process of the speed control system. Only consider the mapping relationship between wind speed and speed control system (MHHT) under the steady-state condition.

3.2 Design of Mechanical Systems

Under the rated condition, the rotor speed reaches the maximum, and the transmission ratio of the transmission chain is the smallest. At this point, the hybrid transmission ratio $$$ {I}_{mhht} $$$ is the smallest.

As shown in Figure 5a, $$$ {I}_{mhht} $$$ decreases with the increase of $$$ v $$$. The larger the parameter $$$ k $$$ of MHHT, the smaller $$$ {I}_G $$$ and the larger $$$ {I}_{mhht} $$$. As shown in Figure 5b, the speed of the ring gear $$$ {\omega}_r $$$ decreases with the increase of v. The input planetary carrier speed of MHHT $$$ {\omega}_c $$$ increases with the increase of $$$ v $$$. The larger $$$ k $$$, the smaller $$$ {\omega}_c $$$ and $$$ {\omega}_r $$$.

The speed of the hydraulic pump cannot be lower than 600 rpm. When the pump speed is too low, the efficiency of the hydraulic system will be greatly reduced. The pump of R-MHHT is connected to the output shaft $$$ {\omega}_s $$$, and its speed is always above 600 rpm. The pump of S-MHHT is connected to the input shaft $$$ {\omega}_c $$$, and its speed is always below 600 rpm.

Therefore, the pump of S-MHHT requires a larger speed ratio $$$ {i}_P $$$ to meet the minimum speed requirement. Moreover, the larger the parameter k, the larger the required $$$ {i}_P $$$. So the mechanical structure of S-MHHT is more complex compared to R-MHHT.

3.3 Design of Hydraulic Systems

3.3.1 Hydraulic System Power

Combining Equations (5) and (13), we can obtain the relationship between the hydraulic system power ratio $$$ {R}_h $$$ and wind speed $$$ v $$$:

$$$$ {R}_h=\left\{\begin{array}{c}1-\frac{k+1}{I_{mhht}}=1-\frac{\lambda_iv}{R_b{\omega}_{b\_\max }}\kern2em \mathrm{S}-\mathrm{MHHT}\\ {}\frac{I_{mhht}}{k+1}-1=\frac{R_b{\omega}_{b\_\max }}{\lambda_iv}-1\kern2em \mathrm{R}-\mathrm{MHHT}\end{array}\right.. $$$$(14)

It can be seen from Equation (14) that in the overall design of the MHHT transmission chain, $$$ {R}_h $$$ is only related to $$$ v $$$ and has nothing to do with $$$ k $$$. The power transmitted by the hydraulic system is

$$$$ {P}_h={P}_{in}{R}_h. $$$$(15)

Figure 6 shows the changes of $$$ {R}_h $$$ and $$$ {P}_h $$$ during MPPT. The hydraulic power ratio $$$ {R}_h $$$ of the two structures decreases with the increase of wind speed $$$ v $$$ and eventually tends to zero. Accordingly, the hydraulic power $$$ {P}_h $$$ of both structures eventually tends to zero.

3.3.2 Constant Displacement Motor

In the closed hydraulic system, the torque of the hydraulic motor can be expressed as [23]

$$$$ {T}_M=\frac{P_h}{\omega_M}=\frac{\varDelta p{V}_M}{2\pi }, $$$$(16)

where $$$ \Delta p $$$ represents the working pressure difference of the hydraulic system. Because the pressure at the low-pressure end is close to zero, this paper calls $$$ \Delta p $$$ the working pressure of the hydraulic system.

Combining Equations (15) and (16), $$$ \Delta p $$$ can be expressed as

$$$$ \varDelta p=\frac{2\pi {P}_h}{\omega_M{V}_M}=\left\{\begin{array}{c}\frac{\rho \pi A{C}_p{i}_Pk}{\omega_s{i}_M{V}_M}{v}^3\kern5.5em \mathrm{S}-\mathrm{MHHT}\\ {}\frac{\rho \pi A{C}_p{i}_Pk}{\omega_s{i}_M{V}_M}\frac{R_b{\omega}_{b\_\max }}{\lambda_i}{v}^2\kern2em \mathrm{R}-\mathrm{MHHT}\end{array}\right.. $$$$(17)

Taking $$$ {i}_P={i}_M=1 $$$ and $$$ {V}_M=2\times {10}^4\mathrm{mL}/\mathrm{r} $$$, the hydraulic pressure $$$ \Delta p $$$ changes of the two structural during MPPT stage are shown in Figure 7.

It can be seen from Figure 7 that the hydraulic system pressure $$$ \Delta p $$$ of both structures increases with the increase of wind speed. $$$ {\omega}_r $$$ and $$$ {\omega}_M $$$ of the two structures are the same. Because the $$$ {P}_h $$$ of R-MHHT is greater than that of S-MHHT, the $$$ \Delta p $$$ of R-MHHT is greater than that of S-MHHT. The larger the k, the larger the $$$ \Delta p $$$.

At the rated wind speed $$$ {v}_R $$$, the $$$ \Delta p $$$ of the two structures reaches the maximum $$$ \Delta {p}_{\mathrm{max}} $$$ and is the same. According to $$$ \Delta {p}_{\mathrm{max}} $$$, $$$ {V}_M $$$ of the two structures can be expressed as

$$$$ {V}_M=\frac{\rho \pi A{C}_p{i}_Pk{v}_R^3}{\omega_s{i}_M\varDelta {p}_{\mathrm{max}}}. $$$$(18)

In Section 2.2, it is concluded that S-MHHT requires a larger $$$ {i}_P $$$ to meet the minimum speed of the hydraulic pump. Therefore, the $$$ {V}_M $$$ required by S-MHHT is greater than that of R-MHHT.

3.3.3 Variable Displacement Pump

Combining Equations (9) and (13), the relationship between the hydraulic system transmission ratio $$$ {I}_h $$$ and the wind speed $$$ v $$$ is

$$$$ {I}_h=\left\{\begin{array}{c}\frac{I_{mhht}-\left(k+1\right)}{k}=\frac{\left(k+1\right)\left({R}_b{\omega}_{b\_\max }-{\lambda}_iv\right)}{k{\lambda}_iv}\kern2em \mathrm{S}-\mathrm{MHHT}\\ {}\frac{I_{mhht}-\left(k+1\right)}{k{I}_{mhht}}=\frac{\left({R}_b{\omega}_{b\_\max }-{\lambda}_iv\right)}{k{R}_b{\omega}_{b\_\max }}\kern4.75em \mathrm{R}-\mathrm{MHHT}\end{array}.\right. $$$$(19)

Combined with Equation (18), we can get the expression of the hydraulic pump displacement $$$ {V}_P $$$:

$$$$ {V}_P=\frac{i_M{V}_M}{i_P}{I}_h=\left\{\begin{array}{c}\frac{\rho \pi A{C}_pi{v}_R^3\left(k+1\right)\left({R}_b{\omega}_{b\_\max }-{\lambda}_iv\right)}{\omega_s\varDelta {p}_{\mathrm{max}}{\lambda}_iv}\kern2em \mathrm{S}-\mathrm{MHHT}\\ {}\frac{\rho \pi A{C}_p{v}_R^3\left({R}_b{\omega}_{b\_\max }-{\lambda}_iv\right)}{\omega_s\varDelta {p}_{\mathrm{max}}{R}_b{\omega}_{b\_\max }}\kern4.75em \mathrm{R}-\mathrm{MHHT}\end{array}.\right. $$$$(20)

Taking $$$ \Delta {p}_{\mathrm{max}}=40\ \mathrm{MPa} $$$, the change of $$$ {V}_P $$$ during MPPT is shown in Figure 8.

From the figure, we can analyze the following: The $$$ {V}_P $$$ of both structures decreases with the increase of v. The larger the k, the larger the $$$ {V}_P $$$ of the S-MHHT. The $$$ {V}_P $$$ of R-MHHT is independent of $$$ k $$$ and much smaller than that of S-MHHT. This is consistent with the conclusion drawn in Section 2.3 that R-MHHT has better speed regulation capability.

3.4 Brief Summary

Therefore, R-MHHT is superior to S-MHHT in terms of structural complexity and displacement requirements, and R-MHHT is slightly weaker than S-MHHT in transmission efficiency. Considering the advantages and disadvantages of the two structures, the R-MHHT structure is finally rotated.

4.1 Introduction of Cosimulation Model

As shown in Figure 9a, the wind model, the rotor aerodynamic model, the grid-connected generator model, and the control module were all built in Simulink. The main parameters of the wind turbine are shown in Table 2. As shown in Figure 9b, the transmission system was built in AMEsim [24, 25], including the rotor inertia, the two-stage planetary transmission of fixed gearbox, and the third stage reflux type mechanical hydraulic hybrid transmission (R-MHHT).

The closed hydraulic system of R-MHHT is composed of a main circuit, a control circuit, a protection circuit (two high-pressure relief valves), a make-up circuit (slippage pump and make-up relief valve in parallel), and a flushing circuit (flushing valve and flushing relief valve in series) [26]. The primary parameters of the hydraulic system are shown in Table 4. The main parameters of the transmission chain are shown in the appendix.

As shown in Figure 9, through the cosimulation interface, data are exchanged between the rotor aerodynamic model in Simulink and the input shaft in AMEsim. Additionally, data are exchanged between the high-speed shaft (HSS) output end in AMEsim and the generator input end in Simulink. Finally, according to the rotor speed in AMEsim, the control module in Simulink controls the displacement of the hydraulic pump.

4.2 Verification of Theoretical Design

To verify the speed regulation function of R-MHHT, slope wind is used as the input of the co-simulation, as shown in Figure 10a. Before 30 s, the wind speed is lower than 7.1 m/s, and the wind turbine does not exert control; After 30 s, when the wind speed is higher than 7.1 m/s, the wind turbine begins to perform MPPT control. The operating data of each parameter of the wind turbine are shown in Figure 10.

Before 30 s, the pump displacement $$$ {V}_P $$$ is unchanged, and the transmission ratio of R-MHHT and transmission chain is unchanged (Figure 10b). Therefore, the rotor speed $$$ {\omega}_b $$$ is basically unchanged (Figure 10c). With the increase of wind speed, the tip speed ratio $$$ \lambda $$$ is more and more close to the optimal tip speed ratio $$$ {\lambda}_i $$$; thereby, $$$ {C}_P $$$ gradually increases (Figure 10c).

As the wind speed increases, more power $$$ {P}_{in} $$$ is captured by the wind turbine (Figure 10d). Because the pump displacement $$$ {V}_p $$$ is unchanged, the power distribution ratio between the mechanical system $$$ {P}_m $$$ and the hydraulic system $$$ {P}_h $$$ at this stage is unchanged (Figure 10d,e). In addition, as the power transferred by the hydraulic system $$$ {P}_h $$$ increases, the pressure of the hydraulic system $$$ \Delta p $$$ correspondingly increases (Figure 10f).

After 30 s, the wind turbine began to control the rotor speed by controlling the pump displacement $$$ {V}_P $$$ to achieve MPPT control. The rotor speed $$$ {\omega}_b $$$ increases with the increase of the wind speed; thus, $$$ {C}_P $$$ is always maintained near the optimum value (Figure 10c). As the wind speed increases, the pump displacement $$$ {V}_P $$$ decreases, resulting in a reduction in the transmission ratio of the transmission chain (Figure 10b), which is consistent with the theoretical derivation (Figures 5a and 8).

The power captured by the wind turbine $$$ {P}_{in} $$$ increases with the increase of wind speed. Because part of the power is used for the speed increase of the rotor, the output power $$$ {P}_{out} $$$ is slightly lower than the captured power $$$ {P}_{in} $$$ (Figure 10d). As the wind speed increases, the power transferred by the hydraulic system $$$ {P}_h $$$ decreases, whereas the power transferred by the mechanical system $$$ {P}_h $$$ increases (Figure 10d). Both the distributed power $$$ {P}_h $$$ and power proportion $$$ {R}_h $$$ of the hydraulic system decrease with the increase of wind speed (Figure 10e), which is consistent with the conclusion of theoretical derivation (Figure 6). The pressure of the hydraulic system $$$ \Delta p $$$ increases with the increase of the wind speed (Figure 10f), and its variation trend and range are consistent with the conclusion of theoretical derivation (Figure 7).

4.3 Brief Summary

In this section, R-MHHT transmission chain parameters at $$$ k=5 $$$ are used to build the cosimulation model of 10-MW wind turbine by AMEsim and Simulink. With the slope wind as the input of the system, the parameters' variation of R-MHHT transmission chain was observed and analyzed. The parameters during MPPT stage, such as pump displacement, R-MHHT transmission ratio, power distribution, and hydraulic system pressure, are consistent with the conclusions of the theoretical derivation, which verifies the correctness of the theoretical derivation in Section 2.

5.1 Modeling of Closed Hydraulic Transmission

The addition of the hydraulic system increases the complexity of the transmission system. To facilitate the establishment of the overall torsion model, the closed hydraulic system is regarded as a parallel gear transmission, as shown in Figure 11b.

5.2 Modeling of R-MHHT

As shown in Figure 10a, $$$ c,r,s,{p}_{n\left(n=1,2,3,4\right)} $$$ represent the planetary carrier, the ring gear, the sun gear, and the planetary gears, respectively; the shafts inside the MHHT are short and thick compared to the whole transmission chain, so the internal shafts are not considered. Therefore, $$$ s $$$ and $$$ {P}_2 $$$ are considered as a whole, and $$$ r $$$ and $$$ {M}_2 $$$ are considered as a whole. The detailed model of planetary gears in MHHT is shown in Figure 10b.

5.3 Y Modeling of Transmission Chain

As shown in Figure 12a, this model mainly considers four segments of shafts, which are the low-speed shaft (LSS), the high-speed shaft (HSS), and two segments of shafts inside the gearbox. The stiffness of the four shaft segments is defined as $$$ {k}_1,{k}_2,{k}_3,{k}_4 $$$.

As shown in Figure 12b, three stiffness matrices ($$$ {\mathbf{K}}_{\mathbf{\mathrm{I}}},{\mathbf{K}}_{\mathbf{\mathrm{I}\mathrm{I}}},{\mathbf{K}}_{\mathbf{\mathrm{I}\mathrm{I}\mathrm{I}}} $$$) of the three stages are combined by means of the shaft stiffness matrix to obtain the overall 23 × 23 stiffness matrix (K). Finally, we obtained the 23 degrees of freedom pure torsion model of the entire transmission chain.

5.4 Brief Summary

The 23 DOF pure torsion model of R-MHHT transmission chain was established by using the lumped mass method. First, the model of closed hydraulic transmission was established by considering it as a parallel gear transmission. Then, the R-MHHT 9 DOF torsion model was established by combining planetary transmission with parallel gear transmission. Finally, the rotor, the three-stage drive, and the generator were connected in series through the four shafts. Thereby, the R-MHHT transmission chain 23 DOF pure torsion model was formed.

6.1 Frequency Characteristic

By solving the characteristic equation, the natural frequencies of each mode can be obtained.

Table 5 shows the frequency changes of each mode of the 9 DOF R-MHHT system under different pump displacement. Overall, the change of hydraulic pump displacement has little influence on the frequency of each mode. The higher the order, the smaller the effect of displacement change on frequency. Among them, the first-order mode frequency of R-MHHT increases with the increase of displacement, which is the same as the change law of closed hydraulic system.

6.2 Damping Characteristic

6.2.2 System of R-MHHT

According to Equation (34), when the hydraulic motor displacement is constant, the hydraulic system damping is related to the speed and displacement of the hydraulic pump. Take different pump displacement $$$ {V}_P $$$ and pump speed $$$ {\omega}_P $$$ to observe the first-order damping changes of R-MHHT, as shown in Figure 13.

The first mode damping of the R-MHHT is approximately proportional to the speed and displacement of the hydraulic pump, which is consistent with that of closed hydraulic system.

The damping of each mode of R-MHHT at $$$ {V}_P $$$ maximum and $$$ {\omega}_P=1500\ \mathrm{rpm} $$$ is shown in Table 8. Compared with first-order mode damping, the damping of other modes is almost negligible.

TABLE 8. Mode damping of R-MHHT at $$$ {V}_P $$$ maximum and $$$ {\omega}_P=1500\ \mathrm{rpm} $$$.

Mode	1	2	3	4	5	6
Damping (%)	36.63	< 0.01	0.16	0.02	< 0.01	< 0.01

Therefore, the damping of hydraulic system has great influence on the first-order mode damping of R-MHHT but little influence on the higher order mode damping.

6.2.3 Transmission Chain System of R-MHHT

During the MPPT stage of the wind turbine, the speed of the generator at the end of the transmission chain remains constant. Because the hydraulic pump of R-MHHT is connected to the output shaft, the speed of the hydraulic pump remains constant.

Under the condition that $$$ {\omega}_P=1500\ \mathrm{rpm} $$$, the damping changes of each mode are observed by taking different pump displacement. Among them, the variation of the damping of the first two modes is shown in Figure 14.

It can be seen from the figure that with the increase of pump displacement, the damping of the first two modes increases significantly. The damping of the first-order mode is approximately half that of the second-order mode. The second-order mode corresponds to the first-order mode of R-MHHT, and its damping change is proportional to the pump displacement.

The damping of each mode of R-MHHT transmission chain at $$$ {V}_P $$$ maximum and $$$ {\omega}_P=1500\ \mathrm{rpm} $$$ is shown in Table 9. Compared with the first two orders of modal damping, the damping of other modes is almost negligible. Therefore, the damping of hydraulic system has great influence on the first two-order mode damping of R-MHHT but little influence on the higher order mode damping.

TABLE 9. Mode damping of R-MHHT transmission chain at $$$ {V}_P $$$ maximum and $$$ {\omega}_P=1500\ \mathrm{rpm} $$$.

Mode	1	2	3	4	5	6	7	8
Damping (%)	18.62	34.20	< 0.01	0.02	< 0.01	0.16	0.02	< 0.01

6.3 Brief Summary

From two levels of R-MHHT system and R-MHHT transmission chain system, the influence of hydraulic system on frequency and damping characteristics has been analyzed. Finally, it is found that the addition of the hydraulic system in the R-MHHT has less influence on the frequency and more influence on the damping of the transmission chain.

Frequency: The lowest order frequency of the transmission chain is slightly reduced and decreases with the increase of the hydraulic pump displacement.

Damping: The damping of the first two modes of the transmission chain increases significantly with the increase of the hydraulic pump displacement, and the second-order damping is about twice the first-order damping.

In the MPPT stage, with the increase of wind speed, the displacement of the hydraulic pump decreases, as shown in Figure 10b. Therefore, when the wind speed is greater than the rated wind speed, the hydraulic pump displacement is the smallest, and the damping of the system is also the smallest. In the design of the R-MHHT structure, the minimum displacement of the hydraulic pump should be appropriately increased to ensure that the low-order mode damping of the transmission chain is large enough.