Strojniški vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535 © 2014 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2013.1492 Original Scientific Paper Received for review: 2013-10-08 Received revised form: 2014-03-27 Accepted for publication: 2014-04-14 Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model Feng Li1 - Yumo Qin1 - Zhao Pang1 - Lei Tian2 - Xiaohua Zeng3* 1 Jilin University, School of Mechanical Science and Engineering, China 2 BAIC MOTOR, China 3 Jilin University, State Key Laboratory of Automobile Dynamical Simulation, China A new power-split device (PSD) is designed as the core component of the multi-power coupling system in a hybrid electric vehicle. It is important to consider the influence of the factors of weight and stiffness on performance of PSD when designing PSD housing. In this paper, the overall arrangement of the power distribution system and PSD housing are redesigned. The finite element analysis is conducted to test PSD housing stiffness. According to the results of the analysis, this study adopts a hybrid optimization strategy based on surrogate models to obtain the least weight under the constraint of PSD housing stiffness. The surrogate models are established using a responsive surface method based on the data obtained by the optimal Latin hypercube design (OLHD). The hybrid optimization strategy combines a multi-island genetic algorithm (MIGA) with a nonlinear programming quadratic line search (NLPQL), which ensures obtaining optimal design parameters for PSD housing. In comparison with optimization using a single MIGA, this hybrid optimization strategy is more efficient and feasible for optimizing housing. Keywords: hybrid electric vehicle, power-split device, hybrid optimization, surrogate model, multi-island genetic algorithm, nonlinear programming quadratic line search 0 INTRODUCTION The hybrid electric vehicle (HEV), an eco-friendly and energy-saving vehicle, has been a very popular in vehicle design and manufacturing [1] to [4]. The power-split device (PSD) is a core component of a multi-power coupling system in a series-parallel HEV [5], because it can achieve energy coupling and conversion among the engine, motor, and generator under different working modes. Based on the principle of differential, Yu et al. [6] proposed a PSD by which the multiple power coupling and decoupling can be achieved. In order to avoid the difficulty in the arrangement and installation of transmission gears in the PSD housing, the design of a PSD is explored in this paper. In a traditional drive vehicle, the transmission is very compact, and the space is insufficient. However, new drive source and drive components are added in the hybrid electric drive vehicle, so the minimum optimization of the volume and quality is more important. The design and optimization of this kind of PSD are challenging. In particular, lightweight design has attracted intensive effort due to its great contribution to cost, material volume and time savings in engineering design [7] and [8]. With the field of complex engineering, numerous single optimization algorithms, such as the Karush-Kuhn-Tucker method [9], non-gradient optimization [10], genetic algorithm [11], and particle swarm optimization [12] are proposed to optimize structures. However, a single optimization algorithm does not always achieve a global solution [13] and can be time-consuming. Hybrid optimization algorithms, such as the combination of the global search (genetic algorithm or evolution strategies) and the local search (descent method) [14], the SASP method (hybridization of simulated annealing and the descent method) [15], the simulated annealing and the local proximal bundle method [16], the combination of the particle swarm optimization and the genetic algorithm [17], are of significant interest for the rapid speed of convergence and robustness in seeking globally optimal solutions [18]. In order to obtain a globally optimal solution and reduce computing time, this study adopts a hybrid optimization strategy based on the combination of a multi-island genetic algorithm (MIGA; a global optimizer), and a nonlinear programming quadratic line search (NLPQL; a local optimizer). In the MIGA, each group is divided into several subgroups called "islands". The selection, crossover, and mutation operations are performed in the subgroups, and the immigration operation is periodically performed among different targeted islands. The MIGA is a pseudo-parallel genetic algorithm, which can both avoid a local optimal solution as far as possible and accelerate convergence [19]. When solving constrained nonlinear mathematical problems, the NLPQL algorithm shows stability, rapid convergence, and the capacity to seek globally optimal solutions [20]. As the NLPQL can quickly determine the local optimal solution near the starting point, it can reduce the computing time during the optimization process. *Corr. Author's Address: Jilin University, 5988 Renmin Street, Changchun, China. zengxh@jlu.edu.cn 525 Strajniski vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535 This study proposes a new PSD and focuses on the sufficient level of stiffness of PSD housing, which ensures the minimum meshing misalignment between planet gear and half-axle gear. In such a case, a hybrid optimization strategy combining the MIGA and NLPQL based on a surrogate model is adopted. In this optimization, finite element analysis (FEA) is used to investigate the PSD housing stiffness. The surrogate models of weight and stiffness of the PSD housing are established. The hybrid strategy demonstrates higher efficiency and a stronger ability to find better solutions compared with a single MIGA. 1 DESIGN OF A NEW PSD 1.1 Transmission System with a PSD An integrated power distribution system including a PSD is shown in Fig. 1. The main parts of this system are the engine, the engine clutch, the gear reduction unit, the input clutch, the PSD, the motor, the gear acceleration unit for the generator, the generator clutch and the generator. Fig. 1. Integrated power distribution system of parallel-series HEV with PSD The torque generated by the engine is introduced by the input axle and is passed to the PSD through the gear reduction unit. Then, according to the principle of the differential mechanism, the torque drives three planet gears and two half-axle gears. The input clutch connects with the left-housing, so that the parts of PSD housing cannot rotate. In this way, the bevel planet gear in PSD only rotates on own axis. The left side of the motor connects with the half-axis gear, and the right side connects with the rear axle. The electricity, which is produced by the generator, is stored in the battery. When the motor is working, the battery supplies electricity to it. The generator is connected to the generator clutch, and its operation can be stopped when the generator clutch disengages. 1.2 Structure of PSD The PSD and the gear transmission train are shown in Fig. 2. The thrust bearing is applied at the shafts of each bevel planet gear to decrease the sliding friction between the bevel planet gears and the PSD housing; furthermore, the uneven distribution of the loading on the gear tooth is also improved. c) mid-housing Fig. 2. PSD; 1) left-housing; 2) mid-housing; 3) right-housing; 4) bevel planet gears; 5) left half-axle gear; 6) planet gear axle; 7) right half-axle gear; 8) outer ring 9) inner ring; 10) threaded hole; 11) axle holes; 12) radial plate; 13) ladder hole The left and right-housing is connected to the mid-housing of the PSD, which is composed of an outer ring and an inner ring. The ladder hole, which is opened on the inner ring, accommodates the roller bearing as well as makes the left spline axle through it. Hence, the driving torque transmitted to left and right half-axle gears becomes an output from one side of PSD. 2 FEA OF PSD HOUSING 2.1 Forces and Boundary Conditions of PSD Housing Bevel planet gears and half-axle gears are the core transmission components of the PSD. The PSD 526 Li, F. - Qin, Y. - Pang, Z. - Tian, L. - Zeng, X. Strojniski vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535 housing deflection, which is introduced by the insufficient stiffness, usually leads to the meshing misalignment of the gears. The meshing misalignment increases the transmission error and transmission noise, and decreases transmission efficiency and gear life. In this paper, FEA is adopted to evaluate the PSD housing stiffness. Before the finite element models are created, the loads applied to the PSD housing are computed. The engine torque M (100 Nm) is transmitted to the PSD housing by the left spline axle, and then M is allocated to the bevel planet gears through planet gear axles. The forces applied to PSD housing are shown in Fig 3. Fig. 3. Forces applied to PSD housing Fal and Far are the axial forces of the half-axle gears, which are applied to the left- and right-housing by half-axle gears. Ftx (Ft1, Ft2, Ft3) is the resultant force of the circumferential forces of the bevel planet gears, which is applied to the PSD housing. Fax (Fa1, Fa2, Fa3) is the resultant force of the axial force created by the bevel planet gear meshing transmission and the centrifugal force created by the bevel planet gear revolution, which is also applied to the PSD housing. Fal, Far, Fx and Fax are defined as: 6T F, = F, = — tana sin5 (1) F = 4T 11 d ' 4T Fax = — tanacos5 + F, d„ (2) (3) M where T = — is the torque applied on the bevel planet gears, F = m(^)2 r is the centrifugal force of the bevel planet gear, m is the weight of the bevel planet gear, n is the rotational speed of the revolution, r is the reference radius of the half-axle gear, a is the pressure angle of the bevel planet gear, 8 is the reference cone angle of the bevel planet gear, dm is the reference diameter of the bevel planet gear at the tooth width midpoint. The boundary conditions should be determined before a simulation is conducted. The engine transmits the input torque M (100 Nm) to the left-housing, and the torque is distributed to the two half-axle gears through the differential planetary gear train. In static load analysis, a hypothesis is proposed that the half-axle gears are fixed, so the torque M is applied to the left-housingof the PSD. Fax, Fal and Far can be converted into the pressure P acting on the corresponding plane, the loading results and the boundary conditions of the PSD housing are shown in Fig. 4. Fig. 4. Loads and boundary conditions of PSD housing m Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model 527 Strajniski vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535 2.2 Results of FEA The loads and boundary conditions are applied to the finite element models, and FEA is conducted. The results of Von Mises equivalent stress are shown in Fig. 5. The maximum stress, approximately 92 MPa, occurs on the joint between the radial plate and inner ring of the mid-housing. Displacements are shown in Fig. 6 in the contour and vector forms. The maximum displacement is 0.025 mm. stiffness for decreasing the meshing misalignment when optimizing the weight of the PSD housing. Fig. 5. Von Mises equivalent stress [MPa] Fig. 6. Displacement [mm] Deflections of the inner and outer rings are shown in Fig. 7. The inner ring turns a certain angle relative to the outer ring, which leads the planet gears to deflect p angle. The outer ring expands b along the radial direction, which leads the planet gear to deflect along the axial direction. All of these events lead to the meshing misalignment among the bevel planet gears and the half-axle gears. The insufficient stiffness of the PSD housing affects the accuracy of gear meshing, which is considered to be a key constraint condition in designing the structural parameters of the PSD. Thus, this study aims to secure the PSD housing Fig. 7. Deflections of PSD housing 3 OPTIMIZATION BASED ON SURROGATE MODEL The surrogate model method uses a simple mathematic model to replace a complex structural optimization problem, which can reduce computing costs and simplify problems. This paper adopts it in the design optimization of PSD housing. First, the design variables, the objective functions and the constraint conditions are determined; then, the design of the experiment based on the OLHD is implemented, and the surrogate models are created with response surface method according to the experimental data. 3.1 Mathematical model of optimization 3.1.1 Design Variables Based on the FEA results, the structure sizes of the inner ring, the outer ring and the radial plate have significant effects on the stiffness and weight of the PSD housing. Therefore, their sizes are selected as the design variables, as shown in Fig. 8. Fig. 8. Design variables Design variables are expressed in matrix form, shown in Eq. (4): 528 Li, F. - Qin, Y. - Pang, Z. - Tian, L. - Zeng, X. Strojniski vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535 X -[pX^,X3,X4,X5,X6,X7,X8] . (4) 3.1.2 Objective Function To meet the lightweight design requirement, minimizing the weight of the PSD housing is considered the objective function, as shown in Eq. (5): Min W = f(X), (5) where W is the weight of the PSD housing and X is the design variable matrix related to W. 3.1.3 Constraint Conditions (1) Constraints of Design Variables: The ranges of design variables are determined under the geometric non-interference of the PSD housing structure, as shown in Table 1. Table 1. Ranges of design variables Design variables Xi X2 X3 X4 X5 X6 X7 X8 Upper bound [mm] 20 20 15 12 17 20 8 8 Lower bound [mm] 5 5 5 5 5 5 3 3 (2) Stiffness Constraint of the PSD Housing: According to the results of the FEA, the insufficient stiffness of the PSD housing will make the bevel gears deflect along the axle, which will lead to the stress concentration. The tooth surface is shaped like a drum in order to alleviate the above problems. In this study, the stiffness index is determined by comparing the changes of the contact stress along with the skew angles, which is between standard and modified tooth profiles. The value of the drum-gear profile [21] is presented in Eq. (6): Ag = 0.25b x10 + 0.5 f, (6) where Ag [^m] is the drum size of the tooth profile, b [mm] is the gear width, fg = A(0.1b+10), A is the constant related to the accuracy magnitude of the gear as shown in Table 2. This paper adopts the accuracy magnitude 7, accordingly, A = 2.5. Table 2. Constant A related to the accuracy Accuracy magnitude 0 1 2 3 4 A 0.63 0.71 0.8 1.0 1.25 Accuracy magnitude 5 6 7 8 A 1.6 2.0 2.5 3.15 The gear parameters are used in Eq. (6) to obtain Ag = 14.2 ^m. An axial modification is adopted [22], i.e. the theoretical involute of the two end surfaces of the modification gear rotates from the central axis to the gear tooth. The rotation angle is determined by the size of the drum-shape. Finally, the endpoints of the two theoretical involutes are connected into an arc. The arc radius is equal, which is used as the scanning trajectory of the tooth surface. With the variation of the skew angle p, the maximum contract stress of the modified and unmodified tooth profile is shown in Fig. 9. —O— Maximum stress of unmodified —□— Maximum stress of modified 0 0.01 0.02 0.03 0.04 0,05 0.06 0.07 0.08 0,09 0,1 Skew angle of bevel gear [°] Fig. 9. Maximum contact stress of modified and unmodified tooth profile As shown in Fig. 9, when the axial skew angle is smaller, the contact stress of the modified gear is heavier. However, with the increasing of the skew angle, the maximum contact stress of the modified gear increases dramatically. When the skew angle is more than 0.04°, the maximum contact stress of the unmodified gear exceeds that of the modified gear. When the skew angle is 0.1°, the load distribution along the width of the gear tooth is very uneven, and the gear meshing line is shorter; the maximum contact stress increases by 85% compared with the normal meshing. The skew angle p can be effectively reduced by improving the PSD housing stiffness. The uneven load distribution of the gear tooth can also be improved. Therefore, in the optimization process, the PSD housing stiffness is selected as the constraint condition to ensure that the uneven load distribution of the gear tooth, which is mainly caused by the insufficient housing stiffness, is minimized. According to the above analysis, in the PSD housing optimization, the skew angle of the bevel gear is restricted to less than about 0.04°. The relationship Design and Optimization of PSD Housing Using a MIGA-NLPQL Hybrid Strategy Based on a Surrogate Model 529 Strajniski vestnik - Journal of Mechanical Engineering 60(2014)7-8, 525-535 between the skew angle of the bevel gear axis and the deflection of axis endpoint is presented in Table 3. Table 3. Relationship between the skew angle of the bevel gear axis and the deflection of axis endpoint Skew angle of the bevel gear axis Deflection of axis endpoint [°] [mm] 0.01 0.0049 0.03 0.014 0.05 0.024 0.07 0.034 0.1 0.0489 By fitting calculation based on Table 3, the maximum deflection of the PSD housing is no more than about 0.02 mm. Thus, the constraint can be defined in the following form: U < 0.02 mm. (7) 3.2 Establishment of the Surrogate Model 3.2.1 Design of experiment based on OLHD The surrogate models are created based on the design matrix. In this study, the design matrix is generated using the design of experiment (DOE) and the real responses of the sample points are calculated by the FEA. The typical DOE includes the orthogonal design, uniform design, full factorial design, OLHD, etc. The OLHD improves the uniformity of the LHD by spacefilling and balance [23]. Thus, OLHD is selected as the DOE in this study. 3.2.2 Surrogate Model The surrogate model uses a simple mathematical model to replace a complex relation between the design variables and corresponding response during structural optimization process. At the same time, the calculation cost can be reduced effectively, and the results are very close to the true values [24] and [25]. RSM uses different order polynomials to express the relationship between the design variables and their responses. One of the most used RSM is the second order polynomial model, which has the advantages of low computational cost, better approximation effect, being easy to solve, etc. Therefore, in this paper, the second order polynomial RSM is adopted to establish the surrogate models. The mathematical model of the second-order polynomial RSM can be expressed in Eqs. (8) and (9): n n n j-1 y = ao+Xax++££a»xxj > i=1 i=1 j=2 i=1 (8) A = [a0,a1,^,a„,