Integrated optimal design of configuration and parameter of multimode hybrid powertrain system with two planetary gears

Mechanism and Machine Theory 143 (2020) 103630

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Research paper

Integrated optimal design of configuration and parameter of multimode hybrid powertrain system with two planetary gears Jiading Gu, Zhiguo Zhao∗, Yi Chen, Lu He, Xiaowen Zhan School of Automotive Studies, Tongji University, Shanghai 201804, China

a r t i c l e

i n f o

Article history: Received 6 July 2019 Revised 19 September 2019 Accepted 23 September 2019

Keywords: Two planetary gears (2-PG) multimode hybrid powertrain system General model and control strategy Automatic configuration screening Stepwise optimization System performance evaluation method

a b s t r a c t The performance of power-split hybrid powertrain systems vary with its configuration due to the differences of the number of planetary gear (PG) set as well as the type, number and installation location of shift actuators (clutch or brake). The 2-PG scheme has some advantages over 1-PG and n-PG (n >=3) systems in terms of structure, performance, and control complexity of the system. The competitive advantages of 2-PG scheme depend on configuration design (analysis, evaluation and screening) and parameter optimization. In this article, systematic methods of configuration design, analysis, evaluation, and screening for 2-PG multimode power-split hybrid powertrain systems are proposed and all potential configurations of 2-PG, connection positions of power components, different number of actuators, and combinations of different operating modes have been taken into consideration. A stepwise optimization design method (DOE sampling with self-adaptive PSO algorithm) is also formulated to optimize the 2-PG system’s characteristics parameters (the two ratios of ring gear to sun gear) and the ratio of main reducer. Finally, an efficient automatic development process and tool based on engineering constraints is designed to optimize the 2-PG system scheme and to achieve the excellent system performance. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Stringent regulations on fuel consumption and emission have promoted rapid development of energy-efficient and emission-reduction automobiles [1]. Power-split hybrid powertrain systems have been widely recognized as one of mainstream solutions of full hybrid and plug-in hybrid electric vehicles [2,3]. These systems use a single or multiple planetary gears sets as a power-split device to produce a variety of system configurations with different combinations of engine, motor and output shaft, and planetary gear elements connected by shift actuators (wet clutch or brake). In recent years, research on system configuration design has increased rapidly and mainly focuses on the following aspects: configuration design and performance of power-split hybrid systems with different number of planetary gears, kinetic and dynamic modeling of power-split hybrid transmissions, optimal control strategies, component sizing, and configuration exhaustive search methods, as shown in Table 1. There are three types of power-split hybrid system structures: one planetary gear (1-PG), two planetary gears (2-PG), and three or more planetary gears (3-PG+). Several recent studies have concentrated on the analysis of 1-PG configuration. Liu et al. [4], Kim et al. [5], and Zheng et al. [9] improved the optimization control algorithm of the Toyota Hybrid ∗

Corresponding author at: Room A317, School of Automotive Studies, Tongji University, 4800 CaoAn Road, Shanghai, PR China. E-mail address: [email protected] (Z. Zhao).

https://doi.org/10.1016/j.mechmachtheory.2019.103630 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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J. Gu, Z. Zhao and Y. Chen et al. / Mechanism and Machine Theory 143 (2020) 103630

Table 1 Literature review summary.

Ref. No.

Year

Obj.

with clutch

Opt. PGx

Opt. FD

Target FC

Target AP

Auto Exhaustive Search

Kinetic & dynamic modeling

Control Strategy

4 2 5 9 7 8 6 10 11 12 3 13 14 15 16 17 21 Proposal

2008 2009 2011 2012 2012 2013 2014 2014 2016 2010 2010 2013 2013 2014 2015 2016 2016

1-PG 1-PG 1-PG 1-PG 1-PG 1-PG 1-PG 1-PG 1-PG 2-PG 2-PG 2-PG 2-PG 2-PG 2-PG 2-PG & 3-PG 2-PG 2-PG

◦ ◦ ◦ ● ◦ ● ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ● ● ● ●

◦ ● ◦ ◦ ● ● ◦ ● ● ◦ ● ◦ ◦ ◦ ◦ ◦

◦ ◦ ◦ ◦ ◦ ● ◦ ● ● ◦ ◦ ◦ ◦ ◦ ◦ ◦

●

●

● ● ● ● ◦ ● ● ● ● ◦ ● ● ● ● ● ● ● ●

◦ ● ◦ ◦ ◦ ◦ ◦ ● ● ◦ ◦ ◦ ◦ ◦ ● ● ● ●

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ● ◦ ◦ ◦ ◦ ◦ ◦ ◦ ● ●

ST ST ST ST LA ST ST ST ST LE ST BO BO BO ST ST ST ST

SDP, ECMS RB DP, PMP DP / PEARS DP, NN ECMS DP / DP BPTT, RB ECMS ECMS DP, PEARS DP, PEARS PEARS ECMS

System (THS-I) configuration. Zaremba et al. [7] and Zhang et al. [8] established model and compared all configurations of 1-PG input power-split transmission (input PS). Zhang et al. [6], Mimkuk et al. [10], and Kim et al. [11] analyzed all possible 12 configurations of 1-PG (6 input PS, 6 output PS) and Zhang et al. [6] also considered the effects of clutch states on operating modes. However, due to the simple structure of 1-PG, the possible operation mode and performance are showing more disadvantages compared with 2-PG configuration, and several existing literature [3,12–17,21] have analyzed the dynamic performance and fuel economy. Besides, only one specific configuration was selected for optimization analysis in these studies. Zhang et al. [16], Zhuang et al. [17] and Zhang et al. [21] considered the effects of clutch states on operating modes. Furthermore, Zhuang et al. [17] analyzed the feasible configurations of 2-PG and 3-PG for Toyota Prius and Chevrolet Volt power-split hybrid systems and argued that configurations with more than 3-PG have less advantages than 2-PG in terms of dynamic performance and cost. However, the analysis only considered configurations with fixed connection of power components and fixed parameters. Comparing the three configurations, 2-PG has more operation modes, better dynamic performance, and fuel economy than 1-PG. Although its dynamic performance is slightly worse than 3-PG, it also meets the requirements of most vehicles. However, the structural design and control of 2-PG is less complex than that of 3-PG. Therefore, the 2-PG structure is the most suitable for power-split hybrid powertrain systems at present. Power-split hybrid systems typically have three operation modes: input power-split, output power-split and compound power-split. Shift actuators are used to combine multiple modes into one configuration, thereby forming a multimode hybrid powertrain system. This article mainly focuses on the configuration design of 2-PG multimode power-split hybrid powertrain system. The methods for dynamic and kinematic modeling of power-split hybrid transmission include Lever analysis (LA) [12], bond graph (BO) [13–15], graph theoretic method [19], state space equation (ST) [4,5,10,12], and Lagrange equation (LE) [6]. And the control strategies applied to specific configurations include RB (rule-based) [2,13], ECMS (equivalent consumption minimization strategy) [4,10,14,15], DP (dynamic programming) [3,5,6,9,11,16,19], SDP (stochastic dynamic programming) [4], PMP (Pontryagin’s minimum principle), PEARS (power-weighted efficiency analysis for rapid sizing) [8,16–18,21] and NN (neural networks) [9]. Through RB is the fastest algorithm, it has the least optimization performance. DP, SDP and PMP are global optimization algorithms, but more computation time and resource requirements are needed and it is far from the algorithm used in the real vehicle. PEARS [16,17,21] is an efficient algorithm which can reduce the computation time greatly. In addition, ECMS is an instantaneous optimization algorithm with high computation speed, which is one of the mainstream control strategies in produced vehicle in the market. So it will be adopted in this article. For the performance objectives of configuration optimization, most researches only focused on FE (fuel economy) [3– 6,8,9,13–15] or TE (transmission efficiency) [7,12], some researchers focused on FE and AP (acceleration performance) [2,10,11,16,17,21]. However, few studies have considered all factors at the same time. Furthermore, only few of previous works considered component parameters as design variables in their study of the configuration screening. The system performance of specific configurations can be further improved by optimizing the parameters of the components. Planetary gear characteristic parameters (PGx) [2,3,7,8,10,11], final drive ratio (FD) [8,10,11], generator power (PEM1) and traction motor power (PEM2) [8] are the main parameters affecting performance optimization results. Zhuang et al. [18] proposed nested optimization and enhanced iterative optimization method, to execute the topology optimization and component sizing alternately, and can converge to the global optimal design generated from the nested optimization. But it still suffers from computational burden. However, there are few studies on the optimization algorithms of these parameters, which use full-range optimization method. This method is suitable only for the analysis

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of specific configurations or a small number of configurations. It has low efficiency and high computation time when the number of configurations is large. Hence, this traditional method is not suitable for global optimization scheme. There are two ways to perform exhaustive search of configurations: manual and automatic method. If shift actuators are not considered, there are 12 different 1-PG configurations [2,6,7, and 10] with two motors not in the same planetary gear node, and 24 different 1-PG configurations [11] with two motors in the same planetary gear node. Cipek et al. [13] considered three shift clutches and analyzed only 9 configurations for Input PS. However, the configurations were realized by manual exhaustive search due to small configuration numbers. For the large configuration number of 2-PG, automatic exhaustive search method is required. Kim et al. [11] and Zhuang et al. [17] presented an automated modeling method for planetary gear transmission based on fixed connection of power components, while neglecting all possible connections of power components with shift actuators. Zhang et al. [21] considered part of possible connections of power components, which were similar to the configuration of GM Volt2 and found some better solutions in their research, but also not for allpossible connections. The latter has higher requirements on the number of optimization samples, the difficulty of automatic modeling and the generality of optimization control strategy. In other words, although the above mentioned literatures differ in the design and analysis of power-split system configuration, optimization control algorithm, and optimization method of planetary gear set parameters, they are attempting to find a better configuration to meet design objectives. In practical engineering design, original equipment manufacturers (OEMs) or transmission suppliers generally specify the design requirements as vehicle parameters, performance objectives, power components parameters, design boundary, and certain design constraints during the design of power-split hybrid transmissions with optimal performance. In this case, power component connection position, shift actuator configuration, operating mode are all design variables to be considered. Thus, it is necessary to propose a systematic design method for power-split hybrid systems to meet these practical engineering requirements and design objectives. Therefore, a method of configuration exhaustive search, evaluation, screening, and parameter optimization for 2-PG multimode power-split hybrid system is proposed in this article. This method covers all potential configurations of 2-PG transmission and all connection positions of power components. Besides, the number of shift actuators and the corresponding combination of operating modes are considered to obtain the optimal configuration of the system. Moreover, an efficient and automatic optimization design process and tool based on engineering design constraints are developed. In addition, a stepwise optimization method (design of experiment (DOE) sampling with self-adaptive particle swarm optimization (PSO) algorithm) is formulated, which will optimize the design parameters of transmissions (such as PGx and FD) to solve the problems in configuration screening and improve the overall optimization efficiency. This article is organized as follows: Section 2 gives the automatic design method and flowchart of 2-PG power-split hybrid system configuration, and the graphical description of the configuration, mathematical transformation method, automatic generation and screening, dynamic modeling and general energy management strategy. Section 3 describes a stepwise optimization algorithm for optimizing the parameters of PG characteristic of the optimal configuration. In Section 4, the automatic configuration design process and optimization algorithm are applied to a case study, and the design results are obtained. Section 5 gives the conclusion and the prospect of future research.

2. Overview of systematic design process Problems in configuration optimization arise from the requirements of OEMs and transmission suppliers. To meet these requirements, the configuration and parameter optimization design flow of 2-PG hybrid powertrain system is shown in Fig. 1. The automatic optimization process begins by generating an initial set of planetary gear set parameters (PG1, PG2, and FD) to increase the efficiency of the configuration screening process and to escape local optimum in the early stages of optimization, as discussed in Section 3.1. However, the computation of traditional loop optimization is very intensive and time-consuming. Hence, it is necessary to calculate the dynamic performance and fuel economy of all feasible 2-PG configurations for each initial parameter group in order to obtain the results of all configurations. General modeling of all 2-PG configurations and performance optimization (optimal control algorithm) are introduced in Sections 2.1 and 2.2. The modes that do not meet the requirements are eliminated from the results by stepwise performance simulation during the computation of various configuration modes. Then, cross-comparison (multi-objective Pareto-optimal) is carried out to obtain the set of optimal configurations corresponding to the initial parameters set of the feasible parameters. Afterwards, global optimization of planetary gear set parameters (PGx, FD) is required for each optimal configuration. DOE sampling with self-adaptive PSO stepwise optimization algorithm is introduced to screen and identify the optimal configuration and its corresponding optimal parameters that satisfy all performance requirements and design constraints.

2.1. Auto exhaustive search method In the automatic design of all 2-PG configurations, two preconditions are set: 1) all 2-PG configurations must be automatically searched exhaustively based on design constraints; 2) the standardized performance of all configurations must be computed using general models such as system dynamic model and control strategy model.

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Fig. 1. Process of proposed systematic design.

Fig. 2. Graph theory of 2-PG transmission.

2.1.1. Configuration description The 2-PG multimode power-split hybrid powertrain system is composed of 3 power components (engine, generator, and traction motor), output shaft, 2 planetary gears sets, shift actuators which include clutches and brakes and some connecting rods. In the previous studies, Peng et al. [20] used graph models to synthesize the schemes of multiple operating DOFs PGs. However, in this article, graph models are used to describe configurations. Fig. 2 describes the connection between components of 2-PG multimode power-split system designed based on graph theory. The numbers represent the moving parts (1–6 represent the planetary components, 8–11 represent the output shaft, engine, generator, and traction motor respectively, while 7 represents the transmission housing), while the shapes of nodes represent different types of planetary components (circle, triangle, square, hollow circle and fork represent ring gear, planet carrier, sun gear, power components, and transmission housing respectively). Meanwhile, the line types represent different connection attributes (solid and dashed lines represent solid connection and connection with clutch or brake actuator respectively). Fig. 3 describes a multimode power-split hybrid powertrain system with 2 clutches and 1 brake. Exhaustive search of 2-PG configurations includes two aspects: the traversal of the connection position of power components (including output shaft) and the traversal of the connection position of shift actuators (clutch or brake). The possible connection position of the power components and shift actuators are P64 = 360 andC72 = 21. However, the total number of shift actuators mainly affects the traversal number of connections of shift actuators. Considering the design complexity and 3 = 1, 330 for systems manufacturing cost, the number of shift actuators should be less than or equal to three. Thus, it is C21

J. Gu, Z. Zhao and Y. Chen et al. / Mechanism and Machine Theory 143 (2020) 103630

5

Fig. 3. Example of one 2-PG configuration.

Fig. 4. Example of mathematical description of one 2-PG configuration.

2 = 210for two shift actuators and 21 for one-shift actuators. Therefore, the total number of the with three shift actuators, C21 2-PG system with the restriction of three actuators is 360 × 1, 330 = 478, 800 based on the possible connection positions of power components and the optional configuration of actuators connection (regardless of rationality). The mathematical description of 2-PG configuration is the precondition for realizing automatic traversal of configurations. Mathematical description methods such as adjacency matrix and sparse matrix have been analyzed. However, due to the large number of 2-PG components and planetary gear set nodes, the dimension of the total adjacency matrix was 11 × 11. This results in complex traversal generation process such that the resultant matrix cannot be accommodated by computer storage, thereby resulting in a large amount of data repetition that wastes resources. Therefore, the description of an arbitrary configuration should take into account various states (joint or separation) of the shift actuator. Hence, a sub-configuration set is adopted to describe the configuration of 2-PG system with different number of shift actuators, which would meet the requirements of configuration integrity description and subsequent dynamic modeling. As such, the mathematical description is divided into two parts: the power component part and the shift actuator part. Fig. 4 gives a mathematical description of the configuration shown in Fig. 3. The left side represents the connection between the power components and the planetary gear set node, while the right side represents the connection between the shift actuators and the planetary gear set node. Label 1 can be used as an expression for this particular configuration or as an expression for the operating mode when all three actuators of this configuration are connected. Labels 2–8 are expressions of the operating modes of the shift actuators in other states in this configuration. Based on the former analysis, all the 2-PG configurations (includes the feasible and infeasible system configurations) can be produced. Later, the infeasible configuration would be filtrated through other design limitation that is described in Section 2.1.3.

2.1.2. Sub-configuration with shift actuators limitation Fig. 4 shows that the partial sub-configurations of two different configurations may be the same. However, if they are stored as mathematical expressions, a lot of duplication may occur. Therefore, a method is selected to traverse and store the non-repetitive sub-configurations as a unit to form a 2-PG sub-configuration library, to be used for later configuration traversal and analysis. Table 2 lists the number of 2-PG sub-configuration libraries with different number of shift actuators constraints. The number of shift actuators constraints is determined by the designer.

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J. Gu, Z. Zhao and Y. Chen et al. / Mechanism and Machine Theory 143 (2020) 103630 Table 2 Number of sub-configuration libraries. Number of Shift Actuators

Number of Configurations

4 3 2 1

P64 P64 P64 P64

4 × C21 3 × C21 2 × C21 1 × C21

Number of Sub-Configurations

= 2, 154, 600 = 478, 800 = 75, 600 = 7, 560

2,716,560 561,960 83,160 7,560

The kinematics model of the sub-configuration can reflect the motion relationship between the power components and the output shaft. Eq. (1) gives the dynamic model of the sub-configuration corresponding to label 5 in Fig. 4.

⎡ ⎤

⎡

TO JO + JC2 ⎢ TE ⎥ ⎢ 0 ⎢ TG ⎥ ⎢ 0 ⎢ ⎥=⎢ ⎢TM ⎥ ⎢ 0 ⎣ ⎦ ⎣ 0 0 0 1 + P G2

0 JE + JC1 0 0 1 + P G1 0

0 0 JG + JR1 0 −P G1 0

0 0 0 JM + JR2 + JS2 0 −1 − P G2

0 1 + P G1 −P G1 0 0 0

⎤⎡

⎤

1 + P G2 ω˙ O 0 ⎥⎢ ω˙ E ⎥ ⎥⎢ ω˙ G ⎥ 0 ⎥⎢ ⎥ −1 − P G2 ⎥⎢ω˙ M ⎥ ⎦⎣ ⎦ 0 F1 0 F2

(1)

where Tx is the output torque of power components, ω˙ x is the angular acceleration of the components,Jx is the inertia of the power components and planetary gear, Fx is the force between the planetary gear and other gear. PGx is a characteristic parameter of the planetary gear set whose value is the ratio of the number of ring gear teeth to the number of the sun gear teeth .(Rx /Sx )., TO is the load torque of the vehicle which can be calculated using the final drive ratio (FD). It also includes roll resistance, wind resistance, ramp resistance, acceleration resistance and braking force. For a particular configuration and its corresponding mathematical description, it is easy to write the equation of state space, as shown in Eq. (1). A series of matrix transformations are introduced to ensure traversal of all configurations and automatically generate a basic matrix B based on the parameter matrix in Eq. (2) with dimension 12 × 12, a matrix E which represents power components connection, and matrix C that represents shift actuators connection. Finally, the basic matrix is transformed into the corresponding parametric matrix MatrixP, which represents the connection of the configuration as shown in Eq. (3).

⎡T ⎤ O

⎡J

O

⎢ TE ⎥ ⎢ 0 ⎢ TG ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎢TM ⎥ ⎢ 0 ⎢ 0 ⎥ ⎢0 ⎢ ⎥ ⎢ ⎢ 0 ⎥ ⎢0 ⎢ ⎥=⎢ ⎢ 0 ⎥ ⎢0 ⎢ ⎥ ⎢ ⎢ 0 ⎥ ⎢0 ⎢ 0 ⎥ ⎢0 ⎢ ⎥ ⎢ ⎢ 0 ⎥ ⎢0 ⎣ ⎦ ⎣ 0 0

0 0

0 JE 0 0 0 0 0 0 0 0 0 0

0 0 JG 0 0 0 0 0 0 0 0 0

0 0 0 JM 0 0 0 0 0 0 0 0

0 0 0 0 JR1 0 0 0 0 0 −P G1 0

0 0 0 0 0 JC1 0 0 0 0 1 + P G1 0

MatrixP = C × E × B × E T × C T

0 0 0 0 0 0 JS1 0 0 0 −1 0

0 0 0 0 0 0 0 JR2 0 0 0 −P G2

0 0 0 0 0 0 0 0 JC2 0 0 1 + P G2

0 0 0 0 0 0 0 0 0 JS2 0 −1

0 0 0 0 −P G1 1 + P G1 −1 0 0 0 0 0

⎤⎡

⎤

0 ω˙ O 0 ⎥⎢ ω˙ E ⎥ ⎢ ⎥ 0 ⎥ ⎥⎢ ω˙ G ⎥ 0 ⎥⎢ ω˙ M ⎥ ⎥⎢ ⎥ 0 ⎥⎢ω˙ R1 ⎥ ⎢ ⎥ 0 ⎥ ⎥⎢ω˙ C1 ⎥ 0 ⎥⎢ ω˙ S1 ⎥ ⎥⎢ ⎥ −P G2 ⎥⎢ω˙ R2 ⎥ ⎢ ⎥ 1 + P G2 ⎥ ⎥⎢ω˙ C2 ⎥ ⎥ ⎢ ⎥ −1 ⎦⎣ω˙ S2 ⎦ 0 F1 0 F2

(2)

(3)

Matrices E and C are generated from the mathematical description given in Eq. (2). The initial matrix E is a 12 × 12 unit matrix, which is converted into an 8 × 12 matrix after processing the information of the connection of power components (regardless of the same node connection between power components). The initial matrix C is 8 × 8 unit matrix, which is processed into 5 × 8, 6 × 8 or 7 × 8 matrices according to the connection of the shift actuators. Zhang, et al. deduces the transformation matrices calculation method [16] and Eq. (4) shows the processes when the nodeX is connected with the nodeY. Using the mathematical description of the sub-configurations of label 5 in Fig. 4 as an example, the transformation matrices E and C are shown in Eqs. (5) and (6) respectively.

rownodeX = rownodeX + rownodeY , rownodeY = [ ]

(4)

where rownodeX is the row corresponds to nodeX in unit matrix, and nodeX can be the node of power source component, shift actuator or PG set node. Row number of nodeX should be small than that of nodeY while calculating and eliminating rows should begin with large-numbered rows. If one of the node is transmission housing, directly eliminate the row corresponds

J. Gu, Z. Zhao and Y. Chen et al. / Mechanism and Machine Theory 143 (2020) 103630

to the other node.

8 9 node 10 11

⎡1

5 row1 2 ( 3, 7 ) row2 → 1 ( 4, 6 ) row3 4 row4

⎢0 ⎢0 row9 ⎢ row6 ⎢0 →E=⎢ row5 ⎢0 ⎢0 row8 ⎣ 0 0 ⎡

8 9 node 10 11

5 2 ( 3, 7 ) row5 → 1 ( 4, 6 ) row4 4

1 ⎢0 ⎢0 hou sin g →C =⎢ ⎢0 row6 ⎣ 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 1 0 0 0 0

0 0 0 1 0 0 0 0 0 0 1 0 0 0

0 0 1 0 0 0 0 0 0 0 0 1 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 1 0 0

0 0 0 1 0 0 0 0 0 0 0 0 1 0

1 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0

7

⎤

0 0 0 0 0 0 1 0

0 0⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎦ 0 1

(5)

⎤

0 0⎥ 0⎥ ⎥ 0⎥ ⎦ 0 1

(6)

Thus, all the transformation matrices E can be obtained by traversing of the connection of the power components and summing P64 types. In the same vein, all the transformation matrices C can be obtained by the permutation and combination of the upper limit of the number of shift actuators Nex and the lower limit. However, special attention should be given to the joint between the transmission housing and the actuator during the traversal of the combination of actuators to obtain the conversion matrix C. 2.1.3. Multimode power-split contained mode screening The coordination of operating modes in hybrid power systems during driving determines the final system efficiency. Parallel mode has higher output power and torque but low system efficiency. However, series mode decouples the engine from the output shaft but has a long power transmission path. Meanwhile, the efficiency of input power-split mode is high in low-speed region and low in high-speed region while output power-split mode has circulatory power flow leading to lower efficiency in the low-speed region and good performance in high-speed region. Similarly, the compound power-split mode has higher efficiency in a wider region. The multiple operating modes of 2-PG can exist in the same configuration to form multimode hybrid powertrain system due to the addition of shift actuators. As such, [17] gives 14 types of operating modes and pattern recognition methods that may be included in 2-PG sub-configurations. The corresponding labels of the operating mode can be added to each configuration in the sub-configuration library by traversal calculation. OEMs impose constraint requirements on the operating modes of 2-PG configuration design. Through benchmarking of well-known models such as Prius III, GM Voltec II and I, we found that the designed configuration includes at least traditional engine mode and more than two types of EV and eCVT modes is mainstream requirement. These constraints ensure the basic functions of the designed configuration and rapidly reduce the total number of analysis samples to improve optimization efficiency. The specific screening process is shown in Fig. 5. The corresponding sub-configurations are obtained from the sub-configuration library based on the selected configurations, and the properties of their operating modes are compared with design requirements. Finally, all configurations satisfying the constraints of work mode are screened as samples for subsequent optimization analysis, and infeasible configuration will not be selected. 2.2. Dynamic modeling To realize the automatic simulation of all the selected configurations, a general dynamic model and general energy management strategies are essential. This model should describe all the configurations for different connection positions of the power components and evaluate the dynamic performance, fuel economy and driving comfort of the configurations. 2.2.1. Power-split transmission The demand of the driveline (torque and rotational speed) of the 2-PG system is determined by the driving cycle and the vehicle parameters. The kinetic equation is shown in Eq. (7), where FD is final drive ratio, ηFD is the transmission efficiency of final drive, Vreq is the target speed,

dVreq dt

is the target acceleration, Tb is the mechanical braking torque, mgfcos α is

the rolling resistance, mgfsin α is the ramp resistance, 0.5ρ ACD Vreq 2 is the wind resistance and δ m resistance, δ is the rotating inertia equivalent coefficient.



To_req =

1 T + F D ηF D b



mg f cos α + mg sin α + 0.5ρ ACDVreq 2 + δ m



dVreq dt

is the acceleration



dVreq Vreq Rtire , ωo_req = FD dt Rtire

(7)

The general model of power-split transmission reflects the transmission ratio for each sub-mode in each configuration. For a given configuration with few modes or explicit degrees of freedom (DOF), the transmission ratios for each mode can be obtained manually. However, the possibility of various modes of any configuration should be considered in advance and the

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Table 3 Gear ratio for all mode types. No.

Work Mode Type

Ratio Equation[io , ie , ig , im ]T

No.

Work Mode Type

1-1

Series Mode 2DOF (E + G,M + O) Series Mode 2DOF (E + M,G + O) Compound split 3DOF

io = [1, 0],ie = [0, 1], ig = R3 /R2 ∗ ie , im = R4 /R1 ∗ io

7

Parallel with EVT 2DOF (E,G + M,EVT)

io = [1, 0], ie = [0, 1], ig = R3 /R1 ∗ io , im = R4 /R2 ∗ ie

8

Pure Engine

io = [1, 0, 0], ie = [0, 1, 0], R1 ig = [0, 0, 1], im = R4 [R2 ]−1 R3 io = [1, 0], ie = [0, 1], R R ig = R3 [ 1 ]−1 , im = R4 [ 1 ]−1 R2 R2 io = [1, 0], ie = [0, 1], R ig = R3 [ 1 ]−1 , im = R4 /R1 ∗ io R2 io = [1, 0], ie = [0, 1], R ig = R3 /R1 ∗ io , im = R4 [ 1 ]−1 R2 io = [1, 0], ie = [0, 1], R ig = R3 /R2 ∗ ie , im = R4 [ 1 ]−1 R2 io = [1, 0], ie = [0, 1], R ig = R3 [ 1 ]−1 , im = R4 /R2 ∗ ie R2 io = [1, 0], ie = [0, 1], R ig = R3 [ 1 ]−1 , im = [0, 0] R2 io = [1, 0], ie = [0, 1], R ig = [0, 0], im = R4 [ 1 ]−1 R2

9

Parallel 2DOF (E,G,M)

io = [1, 0], ie = R2 /R1 ∗ io , R ig = [0, 1], im = R4 [ 1 ]−1 R3

10

Parallel 1DOF (E + O,G + O,M + O)

io = 1 , ie = R2 /R1 ∗ io , ig = R3 /R1 ∗ io , im = R4 /R1 ∗ io

11-1

Parallel 1DOF (E + O,G + O)

io = 1 , ie = R2 /R1 ∗ io , ig = R3 /R1 ∗ io , im = 0

11-2

Parallel 1DOF (E + O,M + O)

io = 1 , ie = R2 /R1 ∗ io , ig = 0 , im = R4 /R1 ∗ io

12

EV 2MGs 2DOF

13

EV 2MGs 1DOF

io = [1, 0], ie = [0, 0], R ig = [0, 1], im = R4 [ 1 ]−1 R3 io = 1 , ie = 0 , ig = R3 /R1 ∗ io , im = R4 /R1 ∗ io

14-1

EV 1DOF(G)

io = 1 , ie = 0 , ig = R3 /R1 ∗ io , im = 0

14-2

EV 1DOF(M)

io = 1 , ie = 0 , ig = 0 , im = R4 /R1 ∗ io

1-2

2

3

Compound split 2DOF

4-1

Input split 2DOF (E + G,M + O)

4-2

Input split 2DOF (E + M,G + O)

5-1

Output split 2DOF (E + G & M diff)

5-2

Output split 2DOF (E + M & G EVT)

6-1

Parallel with EVT 2DOF (E + G,EVT) Parallel with EVT 2DOF (E + M,EVT)

6-2

Ratio Equation[io , ie , ig , im ]T io = [1, 0], ie = [0, 1], R R ig = R3 [ 1 ]−1 , im = R4 [ 1 ]−1 R2 R2 io = 1 , ie = R2 /R1 ∗ io , ig = 0 , im = 0

transmission ratio must be calculated. A division of 14 types of operating modes is given in [17], which is modified in this study, and the sub-types are refined for some cases of 1DOF and 2DOF systems. The mode segmentation is due to the fact that the transmission ratio expressions for the two cases are different and need to be distinguished during system analysis and calculation. Table 3 gives the transmission ratio results for different operating modes. The transmission ratio expression is obtained by matrix operation between row vectorsRx , where Rx , x = 1, 2, 3, 4 represents the row vector of MatrixQ as

Fig. 5. Contained mode screening from sub-configuration library.

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shown in Eq. (8). MatrixQ is obtained from part of the inverse matrix deformation of MatrixP

⎡

⎤ ⎡ ⎤ ⎡ ω˙ O TO R1,1 ⎢ ω˙ E ⎥ ⎢ TE ⎥ ⎢R2,1 ⎣ ω˙ ⎦ = MatrixQ ⎣ T ⎦ = ⎣R G G 3,1 ω˙ M TM R4,1

R1,2 R2,2 R3,2 R4,2

R1,3 R2,3 R3,3 R4,3

⎤⎡ ⎤

⎡ ⎤⎡ ⎤

R1,4 TO R1 TO R2,4 ⎥⎢ TE ⎥ ⎢R2 ⎥⎢ TE ⎥ = R3,4 ⎦⎣ TG ⎦ ⎣R3 ⎦⎣ TG ⎦ R4,4 TM R4 TM

(8)

2.2.2. Battery, engine, and motors Rint model is adopted for battery power modeling, and mathematical expressions of equivalent circuit are shown in Eqs. (9) and (10):

SO˙ C = −

IBAT =

IBAT , disch arg e_ mod e : IBAT > 0, ch arg e_ mod e : IBAT < 0 QMAX

VOC −

(9)



VOC 2 − 4RBAT PBAT , VOC = f (SOC, TBAT ), RBAT = f (SOC, TBAT ) 2RBAT

(10)

where IBAT is the current charging or discharging current of the battery [A], QMAX is the capacity of the battery [Ah], VOC is the open-circuit voltage of the battery, RBAT is the internal resistance of the battery, PBAT is the real-time charging or discharging power of the battery, which is obtained by Eq. (11),

PBAT = PBAT _GEN + PBAT _MOT + PBAT _AUX = TG ωG ηG+INV −k + TM ωM ηM+INV −k + f (V eh_Class )

(11)

where PBAT _GEN is the battery power demand of the generator, PBAT _MOT is the battery power consumption of the traction motor, PBAT _AUX is the power consumption of the high-voltage battery for high-voltage electrical appliances and other vehicle loads, which are mainly affected by the vehicle electrical configuration, ηG+INV and ηM+INV are the e-drive system efficiencies of the generator and traction motor (which is the product of motor and inverter efficiency), respectively; k = 1 and k = −1for driving and regeneration modes, respectively. The instantaneous fuel consumption [g/s] and electric drive system efficiency [%] of engine and motor can be obtained using the rotation speed and torque of power components.

F uelrate = F 1(TE , ωE ), ηG+INV = F 2(TG , ωG ), ηM+INV = F 3(TM , ωM )

(12)

2.2.3. Optimal control strategy There are many optimization objectives of power-split hybrid powertrain systems which include vehicle fuel economy, dynamic performance, driving comfort and other indicators. In the research of energy management strategies, DP algorithms are limited due to large sample configuration analysis and more optimization variables. Hence, ECMS algorithm is still widely used in engineering. The discussions below are based on ECMS energy management strategy. 2.2.3.1. Optimal objective function. Dynamic performance is an important indicator of power-split hybrid powertrain systems, which guarantees the basic driving demand. It mainly includes three parts: 0–100 km/h acceleration time, maximum velocity, and maximum grade ability. The optimal objective function of 0–100 km/h acceleration performance is shown in Eq. (13), where Acc|Mode is the instantaneous acceleration value for each possible sub-mode. By comparing the maximum velocity and maximum grade ability with the target value, the feasible configuration modes can be screened.

J = min

100 0

1 dv = Acc

1 dv max(Acc|Mode )

(13)

St. Slopemax > T arg etuphill St. Vmax > T arg etmax speed Fuel economy is another key indicator for evaluating power-split hybrid powertrain systems. It depends on many factors for a given driving cycle, including the operating mode of the vehicle, the working point of the engine, generator and motor and the efficiency of power battery and transmission at any time. Based on these factors, the optimization objective function of fuel economic performance is shown in Eq. (14):







Jk = min(J/Mode ) = min min F uelRAT E + α × CF E −1 PBAT + β × ClutchACT + γ × ModeSHIF T |Mode



(14)

where Jk represents the instantaneous optimal economic indicators, J|Mode represents the instantaneous optimal economic value in the overall possible sub-mode and is composed of four parts, FuelRATE represents the instantaneous fuel consumption of the engine, PBAT represents the instantaneous energy consumption of the battery, CFE represents the gas-electric conversion coefficient, α represents the penalty coefficient which is the difference between the current State of Charge (SOC) and target SOC, i.e., α = f (SOC, SOCt arg et ). The more the target value deviates, the higher the electricity costs will be.

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J. Gu, Z. Zhao and Y. Chen et al. / Mechanism and Machine Theory 143 (2020) 103630 Table 4 Recommended logic of clutch/brake shift. Shift Actuator Status 1 2 3 4 5 6 7 8

K1 ● ● ● ● × × × ×

●Engaged × Disengaged Shift Actuator Mode 1 1 – 2 ● 3 ● 4 × 5 ● 6 × 7 × 8 × ●Recommended × Penalty

K2 ● ● × × ● ● × ×

K3 ● × ● × ● × ● ×

2 ● – × ● × ● × ×

3 ● × – ● × × ● ×

4 × ● ● – × × × ●

5 ● × × × – ● × ×

6 × ● × × ● – × ●

7 × × ● × × × – ●

8 × × × ● × ● ● –

Fig. 6. Process of optimal control strategy.

ModeSHIF T =

γ1 [ωE |Mode(now ) − ωE |Mode( prev )]2 2 2 + γ2 [ωG |Mode(now ) − ωG |Mode( prev )] + γ3 [ωM |Mode(now ) − ωM |Mode( prev )]

(15)

where ClutchACT and ModeSHIFT are the indicators of the transmission control which is highly associated with the driving comfort, ModeSHIFT is the speed difference of the components before and after mode switch. For a large speed difference, the mode switch control is more complex and it is difficult to guarantee the smoothness as shown in Eq. (15). γ 1 , γ 2 , γ 3 are variable weighting factors, which can be determined by the response character of the power components. Normally, the faster the speed response time of power components, the smaller the value of γ . ClutchACT is the switch process of shift actuators that fulfills the requirements of smooth process, simple operation and fast response time. It requires less mode switch of the clutch or the brake while changing the operating mode of configuration. The penalty factor β is used to punish the mode switch solutions that do not meet the constraints. Table 4 is an example of the recommended switching logic of

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Table 5 Computation cost under loop iteration optimization. Parameters

Optimization Scope and Accuracy

Number

PG1 PG1 FD 2-PG configurations (shift actuators limitation) 2-PG configurations (work mode limitation) Total number of optimization cycles (considering shift actuators limitation)

[2.0: 0.1: 4.0] [2.0: 0.1: 4.0] [1.0: 0.1: 5.0] Solutions under max. 3 shift actuators Solutions under different work mode requirements

21 21 41 561,960 ca. <1000 10,160,798,760 18,081,000

Total number of optimization cycles (considering shift actuators and work mode limitation)

the three shift actuators where K1 , K2 , K3 represent the shift actuators numbers. Table 4 gives the numbers of the two modes before and after the recommended mode switch. β is imposed when the shift mode is not the recommended mode. The larger the value of β , the more likely it is to prohibit non-recommended shifting. The setting of weighting factors α , β and γ can be adjusted according to the designer’s preferences for fuel consumption, control and mode switch smoothness. 2.2.3.2. Energy control strategy programming. The main challenge of general energy management strategy program is the construction of the configuration library. However, the screening process of the configuration has been introduced in Section 2.1 and configuration attributes have been added to each selected configuration, including sub-configuration number, parent configuration number, operating mode label, transmission ratio expression, power source component connection combination number, and shift actuator connection combination number information. Fig. 6 shows the input and output structure of the general energy management strategy module. The relevant information of the configuration library will be used in the computation as a workspace parameter through the automatic input of the configuration number. 3. Planetary gear component sizing stepwise optimization This article finds the optimal configuration of the connections of power components and shift actuators, and the corresponding optimal planetary gear set parameters (PG1 , PG2 , FD) to meet the optimization objectives of design constraints. Fig. 6 describes the overall multi-step optimization process. The set of initial planetary gear set parameters were constructed and all the configurations were traversed through the optimal control to find the best solution of the primary configuration set. Then, global parameter optimization was applied to the primary configurations to find best configurations and corresponding best parameters set. However, global parameter optimization suffers from a large number of calculation samples and long computation time. The computation cost is shown in Table 5 by using simple loop iteration optimization method. The stepwise parameter optimization method (DOE sampling with improved self-adaptive PSO algorithm) is proposed below to improve the optimization efficiency. 3.1. DOE thinking The first step in parameter optimization is the construction an initial set of optimization parameters. DOE sampling is used as a reference to select the initial optimization parameter set in order to obtain the initial parameters to be uniformly distributed and fully reflect the requirements of the design space. The orthogonal method used is fractional factorial design (FFD), in which some representative points from the comprehensive experiment are selected for the experiment. It has uniform dispersion, neatness, and comparability. Although the system precision is slightly reduced, the number of experimental samples is greatly reduced. k = 3 factors (PG1 , PG2 , FD) were studied in which the optimization precision for each factor is 0.1. Since stepwise optimization was adopted, the level number of each factor was defined as L = 5 for the initial sample. Table 6 gives the initial parameter sets obtained by orthogonal sampling method (N = 25) from the first three columns of orthogonal table L25(56 ). 3.2. Self-adaptive multi-objective particle swarm optimization The screening configuration number of the first round of the optimal non-dominant solution (dynamic and economic performances) can be obtained by the traversal simulation of the feasible configuration. Particle swarm optimization (PSO) has the advantages of simple implementation, small population, and fast convergence speed. In PSO, a group of particles are initialized in the feasible solution space where each particle represents a potential non-dominant solution, and three indicators (position, velocity, and fitness) represent the particle characteristics. The particle moves in the solution space by tracking individual extreme Pbest and global extreme Gbest to update its position. The update expressions of particle for each iteration are shown in Eq. 16.







k Vidk+1 = ωVidk + c1 r1 Pidk − Xidk + c2 r2 Pgd − Xidk



(16a)

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J. Gu, Z. Zhao and Y. Chen et al. / Mechanism and Machine Theory 143 (2020) 103630 Table 6 Orthogonal Array L25(56 ). Orthogonal Arrays (3 Factors,5 Levels) No.

[PG1 , PG2 , FD]

No.

[PG1 , PG2 , FD]

No.

[PG1 , PG2 , FD]

1 2 3 4 5 6 7 8 9

[2.0,2.0,1.0] [2.0,2.5,2.0] [2.0,3.0,3.0] [2.0,3.5,4.0] [2.0,4.0,5.0] [2.5,2.0,2.0] [2.5,2.5,3.0] [2.5,3.0,4.0] [2.5,3.5,5.0]

10 11 12 13 14 15 16 17 18

[2.5,4.0,1.0] [3.0,2.0,3.0] [3.0,2.5,4.0] [3.0,3.0,5.0] [3.0,3.5,1.0] [3.0,4.0,2.0] [3.5,2.0,4.0] [3.5,2.5,5.0] [3.5,3.0,1.0]

19 20 21 22 23 24 25

[3.5,3.5,2.0] [3.5,4.0,3.0] [4.0,2.0,5.0] [4.0,2.5,1.0] [4.0,3.0,2.0] [4.0,3.5,3.0] [4.0,4.0,4.0]

Table 7 Vehicle parameter specification. Parameters Specification

Values [unit]

Curb Weight Tire Radius Wheelbase Engine Displacement Engine Power/Max. Torque Generator Power/Max. Torque Traction Motor Power/Max. Torque Battery Voltage/Energy Acceleration Time (0-100 km/h) Fuel Consumption (NEDC)

1610 [kg] 0.345 [m] (235/55 R17) 2.905 [m] 1.8 [L] 94 [kW] / 175 [Nm] 54 [kW] / 140 [Nm] 60 [kW] / 275 [Nm] 288 [V] / 1.5 [kWh] 8.9 [s] 4.7 [L/100 km]

Xidk+1 = Xidk + Vidk+1

(16b)

where ω is the inertia weight, d is the parameter dimension, i is the number of particles, k is the current iteration number Kmax is the maximum iteration number, Vid is the particle speed, X is the particle location, c1 and c2 are acceleration factors, k is the individual extreme, P k is the global extreme, X and r1 and r2 are random numbers distributed in the interval [0,1], Pid gd V are limited generally in [Xmin , Xmax ] and [Vmin , Vmax ] separately to prevent particles from blindly searching. The solution of multi-objective optimization is a Pareto set, which is different from single-objective optimization. A common approach is to record the non-dominant solutions and use them to guide the trajectories of other particles. Several studies have focused on the improvement of the evolutionary operator problems, e.g., by introducing sparse information using weight aggregation method. Thus, the dispersion and uniformity of multi-objective optimization solution set has been improved to some extent. However, premature maturity and slow convergence speed of PSO algorithms in the later stages cannot be avoided. To accelerate the convergence speed of the algorithm and reduce the risk of premature maturity, several improvements are introduced into the algorithm below: 1) Improved generation of initial particle. The optimization result and convergence speed of PSO algorithm are affected by the initial particle position. If initial random generated particles are too concentrated, it may lead to a direct and rapid descent into local optimal results. Hence, orthogonal sampling method has been adopted to ensure uniform, comprehensive, and flexible quantity setting in the generation of initial particle swarm. The detailed steps of generation are discussed in Sections 3.1 and 2) The effect of the adaptive inertia weight control strategy on the results is that a larger inertia weight favors global search, while a smaller inertia weight favors local search.

  ωk = ωbegin + ωbegin − ωend (Kmax − k )/Kmax

(17)

where ωbegin is the initial inertia weight, and ωend is the inertia weight at the maximum iteration number. 4. Case study The case study below fully demonstrates the automatic optimal design process of 2-PG configuration and the optimization results are also discussed. 4.1. Assumptions The parameters of the vehicle and the power components of the case study are based on GM Lacrosse 30H 2017 (with eCVT transmission). The specific datasheet and curves used in this article are shown in Table 7 and Fig. 7, respectively.

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Fig. 7. Characteristic curves of power components. (a) Engine Map, (b) Generator Efficiency Map (c) Traction Motor Efficiency Map.

The market of the hybrid electric vehicle and the specifications of OEMs were taken into consideration. The specifications are as follows: Vehicle performance target: i ii iii iv

0–100 km/h acceleration time should be less than 10 s. Grade ability should be more than 30%. Continuous maximum velocity should be more than 150 km/h. Fuel consumption under NEDC driving cycle should be better than 6 L/100 km.

Design limitations of power-split hybrid system: i Total shift actuators number should be less than 3; ii The following operating modes must be included: 1 traditional engine mode, EV mode (more than 1 type), eCVT mode (more than 1 type and compound power-split). Parameter optimization scope: Transmission ratio optimization scope of PG1, PG2, and FD is found in Table 5.

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Fig. 8. FFD of initial parameters and configuration analysis DOE. a. Initial parameters FFD (35) b. Configuration analysis DOE FFD (35).

Fig. 9. Graph theory description of the four optimal configurations. a. Configuration No. 144 b. Configuration No. 196.

The best configuration and corresponding planetary gear set parameters (PGx, FD) are obtained by generating and analyzing all potential configurations, all connection positions of power components, different number of shift actuators, combination of different operating modes based on the design objectives, and design constraints given by OEMs. 4.2. Results and discussions By limiting the total number of shift actuators, 561,960 configurations were generated. Basic feasibility analysis was then done to screen out 82,694 configurations. Finally, 288 configurations that met the requirements were finally screened out based on the defined requirements with operating modes constraint. The initial parameters of 25 sets of transmission components were selected by DOE orthogonal method (Fig. 8a). Dynamic performance and fuel economy were then calculated for 144 configurations (half of 288 configurations are equivalent configurations and they will not affect the performance calculation results whatever each component is connected to the 1st or the 2nd planetary gear set), and the result is shown in Fig. 8b. The total amount of computation is 25 × 144 = 3, 600 times. The black points represent all simulation results of possible configurations while the red points represent the Pareto front solutions of acceleration performance and fuel consumption from which 4 optimal configurations satisfy the design objectives. Take the optimal configuration #144 and #196 as example, the graph theory descriptions of them are shown in Fig. 9. Meanwhile, the valid working modes of each configuration and the corresponding initial parameters are listed in Tables 8 and 9, respectively. For each selected high-potential optimal configuration, self-adaptive PSO algorithm was used to optimize the planetary gear set parameters. The particle number in each generation was set to 30 and the total iteration time was 40. Then, linearly reduced inertial weight was used 4800 times 40 × 3 × 40 = 4, 800. Fig. 10 shows the optimization results of high-potential

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Table 8 Valid working modes of the four optimal configurations. Optimized Configuration No.

Work Mode 1

Work Mode 2

Work Mode 3

Work Mode 4

Work Mode 5

Work Mode 6

NO. 144 NO. 196

infeasible infeasible

EVT Engine

EV1 EV1

infeasible EV2

EVT2 EVT1

Engine EVT2

Work Mode 7 EV2 EV3

Table 9 Configuration design result (preliminary). No.

Parameter [PG1]

Parameter [PG2]

Parameter [FD]

Optimized Configuration No.

Acceleration Time (0-100 km/h) [s]

1 2 3 4

3.5 3.5 4 4

3.5 4 3.5 3

2 3 3 2

NO. NO. NO. NO.

9.28 8.35 7.85 9.37

88 144 155 196

Fuel Consumption NEDC [L/100 km] 4.63 5.02 5.23 4.54

Fig. 10. PSO optimal results of optimized configurations. a. NO. 88 PSO optimal result b. NO. 144 PSO optimal result. c. NO. 155 PSO optimal result d. NO. 196 PSO optimal result.

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Table 10 No. 88 configuration PSO result. No.

Optimized Configuration No.

Parameter [PG1]

Parameter [PG2]

Parameter [FD]

Acceleration Time (0–100 km/h) [s]

1 2 3 4 5

NO. NO. NO. NO. NO.

3.5 2.9 3.7 3.6 3.4

3.8 3.8 3.6 3.8 3.8

2.0 2.1 2.4 2.4 2.6

9.67 9.24 8.79 8.64 8.25

88 88 88 88 88

Fuel Consumption NEDC [L/100 km] 4.35 4.47 4.58 4.74 4.94

Table 11 No. 144 configuration PSO result. No.

Optimized Configuration No.

Parameter [PG1]

Parameter [PG2]

Parameter [FD]

Acceleration Time (0–100 km/h) [s]

1 2 3 4 5

NO. NO. NO. NO. NO.

3.6 3.0 2.7 3.0 2.9

3.4 3.6 3.3 3.3 3.6

2.4 2.3 2.4 2.5 2.6

8.95 8.91 8.87 8.70 8.25

144 144 144 144 144

Fuel Consumption NEDC [L/100 km] 4.68 4.96 4.99 5.09 5.48

Table 12 No. 155 configuration PSO result. No.

Optimized Configuration No.

Parameter [PG1]

Parameter [PG2]

Parameter [FD]

Acceleration Time (0–100 km/h) [s]

1 2 3 4 5

NO. NO. NO. NO. NO.

3.8 4.0 3.8 3.5 3.5

2.8 2.3 3.6 3.1 3.6

2.1 2.2 2.6 2.7 3.2

9.22 8.65 8.28 8.17 7.65

155 155 155 155 155

Fuel Consumption NEDC [L/100 km] 4.80 4.99 5.18 5.74 5.88

Table 13 No. 196 configuration PSO result. No.

Optimized Configuration No.

Parameter [PG1]

Parameter [PG2]

Parameter [FD]

Acceleration Time (0–100 km/h) [s]

1 2 3 4 5

NO. NO. NO. NO. NO.

3.7 3.3 3.8 3.5 3.5

3.6 3.4 3.1 3.1 2.9

2.4 2.6 2.5 2.6 2.6

8.70 8.56 8.37 8.36 8.31

196 196 196 196 196

Fuel Consumption NEDC [L/100 km] 4.22 4.98 5.05 5.12 5.48

configurations. The black points represent the fitness of each generation of particles while the red points represent the best Pareto solutions. The distribution of Pareto solutions is relatively wide and diverse, and the multi-objective optimization results of various high-potential configurations have been improved reasonably. The top 5 Pareto front results of high-potential configurations ranked by lowest fuel consumption are listed in Table 9. The fuel economy of the configurations improved considerably compared with the initial stage with the following values: NO. 88 by 6.0%, NO. 144 by 6.8%, NO. 155 by 8.2%, and NO. 196 by 7.0%. The optimal configuration and corresponding component design parameters are obtained by the transverse comparison of the best results of all high-potential configurations. The optimal configuration for Pareto optimal solution of fuel consumption is No. 196 while the corresponding transmission component parameters are PG1=3.7, PG2=3.6, and FD=2.4. Under this configuration, the vehicle has optimal fuel economy and high power performance level (Tables 10–13). 5. Conclusion A method of configuration design and parameter optimization for 2-PG multimode hybrid powertrain system is proposed in this article based on vehicle performance and design constraints defined by OEMs. We have considered all the potential configurations of 2-PG, all the connections of power components and different number of shift actuators, the corresponding operating mode combination. We have also used an established general simulation model and control strategy to optimize configuration by traversal, screening, and evaluation analysis. Also, DOE orthogonal sampling method was used to generate the initial parameter set while adaptive multi-objective PSO algorithm was used to optimize the transmission parameters of each high-potential configuration. The algorithm accelerated the analysis of the selected configurations to find high-potential configuration modes, thereby obtaining the optimal design parameters of the corresponding system. It was found that configuration No. 196 is the best configuration scheme for Pareto optimal solution of fuel consumption, and the corresponding

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transmission component parameters are PG1=3.7, PG2=3.6, and FD=2.4. Hence, the vehicle under this configuration scheme had high fuel economy and dynamic performance. The essential contribution of this article is to propose an integrated design method of automatic configuration design, screening, and parameter optimization of 2-PG multimode hybrid powertrain systems. The proposed configuration is more comprehensive and can adapt to the engineering requirements of different design objectives. The optimal configuration scheme can be found quickly under any given design boundary, which lays a solid foundation for the commercial product development of power-split hybrid systems. The search for multi-objective global optimal solution set is an important criterion for evaluating the merits of the configuration screening method based on computation cost (huge analysis sample and calculation time). In this article, DOE orthogonal sampling has been adopted to accelerate the screening process of high-potential configurations, thereby reducing the computation cost considerably. In addition, adaptive PSO algorithm has been adopted for global parameter optimization, which has a better optimization effect than traditional PSO algorithm. However, there are some aspects for further improvement. Subsequent researches will focus on the improvement of high-potential configuration screening method and parameter global optimization algorithm. In addition, the frequency of gearshift and mode switching is considered in the application of ECMS algorithm in this article, but due to the complexity of adjusting the parameters of the corresponding control strategy, the robustness of the algorithm has a certain improvement space. Therefore, the author will consider using other algorithms with better robustness of algorithm in future work as well. Declaration of Competing Interest No conflict of interest exits in the submission of this manuscript, and the manuscript is approved by all authors for publication. I would like to declare that this article is original, has not been published previously, and not under consideration for publication elsewhere, in whole or in part. Acknowledgement This research was supported by the National Natural Science Foundation of China [grant number 51675381]. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2019.103630. References [1] Zongwei Liu, et al., CAFC & NEV double-credit and comprehensive research on carbon quota regulation and combination policy, Chin. J. Automot. Eng. 7 (1) (2017) 1–9, doi:10.3969/j.issn.2095-1469.2017.01.01. [2] H. Yang, et al., Analysis of planetary gear hybrid powertrain system part 2: output split system, Int. J. Automot. 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