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A method for the exploration of hybrid electric powertrain architectures with two planetary gearsets.

ABSTRACT

The goal of this paper is to explore the complete set of single mode hybrid electric powertrain designs that can be generated with one and two planetary gearsets (PGs). Contrary to an automated design exploration approach, an analytically-based manual method is developed to identify all unique design modes for each hybrid electric powertrain architecture (parallel, series, power-split) that can be created with two planetary gearsets, one engine, one vehicle output shaft, two electric machines, and at most two brake clutches. Feasible design modes are generated according to a procedure that provably covers the entire design space. The procedure systematically creates all feasible combinations according to the number of connections between PGs; the number of brake clutches; whether brake clutches, vehicle output shaft, and engine are connected to the nodes that combine two PGs; whether engine and vehicle output shaft are connected to the same PG; whether they are connected to the PG with brake clutches and whether electric machines are collocated with the engine and/or vehicle output shaft. The results of this approach not only show the number of unique design modes for each hybrid electric vehicle architecture and their torque and speed relationships, but also provide a means to validate the results obtained through automated design mode exploration methods.

CITATION: Dagci, O. and Peng, H., "A Method for the Exploration of Hybrid Electric Powertrain Architectures with Two Planetary Gearsets," SAE Int. J.Alt. Power. 5(1):2016.

INTRODUCTION

Fuel economy regulations in the U.S. will likely increase on average 4.9% per year from 2017 to 2021 and 2.6% per year from 2021 to 2025 [1]. The National Highway Traffic Safety Administration (NHTSA) projects that fuel economy regulations can be met with the application of new technologies to conventional engines and the introduction of a higher number of hybrid electric vehicles (HEVs), plug-in hybrid electric vehicles (PHEVs) and electric vehicles (EVs) into the market [2].

As of the end of 2014, the share of HEVs in the U.S. light-duty vehicle market is 2.8%, with 80% of these sales belonging to HEV architectures realized with planetary gearsets (PGs), similar to those of the Toyota Prius [3]. The remaining 20% of the HEVs belong primarily to simple parallel hybrid architectures, where a conventional transmission is supplemented with an electric machine. The key reason behind the high customer acceptance of PG-based powertrains is the design superiority and flexibility achieved with PGs. The price that comes with these advantages is the complications in the design and analysis of PG-based hybrid electric powertrains, which necessitate further research in this topic.

The feasible design modes with one PG are limited in terms of architecture variety and number of unique modes. Thus, the potential of PG-based architectures can be fully exploited with at least two PGs. However, the design space with multiple PGs consists of thousands of modes that have to be generated and evaluated. Two approaches that exist in the literature for this analysis can be categorized as bottom-up and top-down methods.

In the bottom-up approach, all combinations that can be achieved with a given set of powertrain components are generated and evaluated [4, 5, 6, 7]. Each study in this type differs from each other in terms of the number of available powertrain components, the evaluation criteria, the methods to evaluate the design candidates, the ease of automation, and the assumptions applied during the design process. The commonality in this approach is automating the task of generating and evaluating all design combinations through software programs due to the very high number of combinations. The software assigns the powertrain components to the PG nodes, eliminates the infeasible and incapable design modes in the background, and gives the final results. The drawbacks in this approach are the inability to assess the characteristics of the feasible design space in terms of design topology and governing speed and torque equations, and the difficulty in validating the results of the computer-based search analytically, which may miss some design modes or may assign a design mode to a wrong category due to an undetected bug in the software or insufficiency in the search algorithm.

In the top-down approach, first a target HEV powertrain architecture is chosen and then its generalized representation is derived through a generic topology or generic set of equations [8, 9]. At the final stage, the parameters of this generic model are optimized to meet the design criteria. The drawbacks of this approach include the inability to cover all architectures in a single generic model or to investigate all possible design topologies in the design space that give the same set of equations.

The content of this paper is not an alternative to the top-down and bottom-up methods. Rather it complements these approaches by presenting the structure of all feasible unique modes for each HEV architecture (parallel, series, and power-split) that can be achieved with two PGs, one engine, one vehicle output shaft, two electric machines, and at most two brake clutches. The results of this study can be used not only to validate the results of top-down and bottom-up methods but also to understand the size and performance limits of the design space for a given set of components.

The paper is organized as follows. In the first section, the basics and modeling procedure of PG-based HEV designs are explained. In the second section, the process that identifies all unique design modes is introduced. The fourth section explicitly shows the number, topology and governing steady-state speed and torque equations of all feasible unique design modes. The fifth section interprets the results of the proposed design process. The sixth section shows how the results of this paper can be used in HEV powertrain design through an example. The paper ends with the concluding remarks.

MODELING PROCEDURE OF PG-BASED HEV MODES

A HEV design mode studied in this paper consists of two planetary gearsets (PGs), one engine, one vehicle output shaft, two electric machines, and brake clutches. The planetary gearset is in the center of a design mode because all the remaining components are connected to its nodes and the characteristics of a design mode is determined by these connections.

A PG shown in Figure 1 is a mechanical device, which has four rotating elements called the sun gear, the planet gears along with their carrier, and the ring gear. It is represented as a lever with three nodes, each of which corresponds to ring, carrier and sun gears [10]. The lengths between ring-carrier nodes and carrier-sun nodes on the lever are taken as 1 and [N.sub.R]/[N.sub.S], where [N.sub.R] and [N.sub.s] are the number of teeth on the ring and sun gears. In this paper, a and [beta] represent [N.sub.R]/[N.sub.S] ratios of two PGs. The rotational motion of the gears is governed by equation 1, where [[omega].sub.R], [[omega].sub.C], and [[omega].sub.S] are the speed of ring, carrier, and sun gears, respectively. As observed from the equation 1, the degrees of freedom in a planetary gearset are two i.e. the speed of any two gears determine the speed of the third. The torque relationship between the gears of a PG is also determined by [N.sub.R]/[N.sub.S] as shown in equation 2, where TR, TC, and TS are the torque at the ring, carrier, and sun gears, respectively, and F1 is the tangential force between these gears.

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In this study, the steady-state speed and torque equations of a design mode will be used to assess its characteristics and to draw conclusions. These equations will be derived leveraging the speed and torque equations of PGs in equations 1-2 and the constraints imposed by the connection of the powertrain components.

The derivation process of the equations will be shown through an example in Figure 2, where the engine (ICE), the vehicle output shaft (Vehicle) and the first electric machine (EM1) are connected to the ring (R1), carrier (C1), and sun (S1) gears of the first PG (PG1), respectively. Similarly, the brake clutch (B1) and the second electric machine (EM2) are connected to the ring (R2) and sun (S2) gears of the second PG (PG2). The connection between PG1 and PG2 is provided through their carrier gears (C1, C2). The torque exerted by the components is equal to the torque at the nodes of the PGs in the static equations.

First, the speed and torque equations of the PGs are written in equation 3, where TL1 is the transferred torque from PG1 to PG2 through their connection. Then, speed constraints due to the brake, and the connection between PGs are expressed in equation 4. Through variable substitutions, the speed and torque relationships between Vehicle, ICE, and electric machines are unveiled in equation 5. This final equation is used to determine the feasibility, uniqueness, and HEV architecture type of each design mode throughout this paper.

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DESIGN IDENTIFICATION PROCESS

One way of identifying the design space is to generate and evaluate all combinations with two PGs, one ICE, one vehicle output shaft, two electric machines, and brake clutches. The combinations can be categorized according to the number of connections between two PGs that varies between one and three. In the case of three connections between two PGs, the speed and torque are the same at each PG node. Thus, the investigation of the combinations in this category can be omitted. In the case of a single connection, any node of the first PG can be connected to any node of the second PG. Hence, 9 connection options are possible. There are five nodes, to which the components (ICE, two EMs, Vehicle, Brake Clutches) can be assigned. The ICE and Vehicle cannot be assigned to the same node but EMs can be assigned to any node. Brake clutches can also be assigned to any node, except the ones for the ICE and Vehicle. As a result, the total number of design modes that needs to be evaluated for the case of a single connection becomes 9x5x4x5x5x[2.sup.3] = 36,000. Conducting a similar analysis for the case of two connections, 18x4x3x4x4x[2.sup.2] = 13,824 design modes must be evaluated. As seen from this analysis, approximately 50,000 design modes should be generated and investigated in order to cover the entire design space. This approach is impractical if the analysis is performed manually. Thus, the feasible design space should be analyzed using a different approach, which will be introduced in this paper.

The first step in the proposed approach is to represent a group of design combinations with a single set of speed and torque equations, and graphical representation. The steady-state speed and torque equations are written in the form of equation 6, where [[omega].sub.i], i= 1,2,3 can be the speed of any PG node and [[psi].sub.i], i= 1,2,3 can take any of the values (1+[alpha]), -[alpha], and -1. For example, all 1-PG design mode combinations, where ICE, Vehicle, and an electric machine are connected to the different nodes of the PG can be represented with a single set of equations in equation 7. This set of combinations can also be represented with a single lever diagram in Figure 3. The interpretation of the lever diagram in Figure 3 is different from the one in Figure 1. In Figure 1. the top, middle, and bottom nodes are always the ring, carrier, and sun gears of the PG, respectively. However, the three nodes in Figure 3 can be any gear of the PG depending on to which node ICE, Vehicle, and EM are connected. If ICE, Vehicle, and EM are connected to the carrier, ring, and sun gears of the PG, [[psi].sub.1], [[psi].sub.2], and [[psi].sub.3] would take 1+[alpha], -[alpha], and -1, respectively.

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Once the design modes are assigned to groups according to the characteristics of their steady-state speed and torque equations, the design analysis can be conducted over these groups instead of the individual design modes. However, these groups should be created methodically in order not to miss any group in the design space. To construct the entire set of design groups, the following rules that take into account all feasible combinations of connections are evaluated in the given order:

1. How many connections are there between two PGs?

2. How many brake clutches exist in the design group?

3. Is any brake clutch assigned to any connection between two PGs?

4. Is any component other than the brake clutches assigned to any connection between two PGs?

5. Is ICE or Vehicle assigned to the connection between two PGs?

6. Are both ICE and Vehicle connected to the nodes of the same PG?

7. Are both ICE and Vehicle connected to the nodes of the PG to which a brake clutch is also assigned?

8. How many ways can EMI and/or EM2 be collocated with the ICE and Vehicle?

Once the design mode groups are generated, their HEV mode types are determined using the steady-state speed and torque equations that are evaluated according to the following criteria where [f.sub.ij]'s i=1,2 and j=1,2,3; [g.sub.mn]'s m=l and n=l,2,3; and [h.sub.pq]'s p=l,2 andq=l,2,3 are the coefficients of the speed and torque variables in the equations, and are the functions of [[psi].sub.1].,.., [[psi].sub.6].

Parallel HEV Mode:

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Series HEV Mode:

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Power-Split HEV Mode:

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Power-split modes can be subcategorized according to the following equations in addition to equation 10:

Output-Split:

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Input-Split:

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Compound-Split:

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Special Type Power-Split:

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At the last stage of the process, all combinations of (1+[alpha]), -[alpha], -1 and (1+[beta]), -[beta], -1 that are assigned to [[psi].sub.1] [[psi].sub.2], [[psi].sub.3] and [[psi].sub.4], [[psi].sub.5], [[psi].sub.6] are analyzed. The combinations, which make [h.sub.11] coefficient negative in the static torque equations are counted as feasible design modes because [T.sub.Vehicle] should be negative for forward motion and positive [T.sub.ICE] is desired to contribute to forward motion.

IDENTIFIED HEV POWERTRAIN DESIGN MODES

Using the eight rules described in the previous section, the charts in Appendices A and B are created. Appendix A covers the design groups that have one connection between two PGs, whereas Appendix B consists of design groups with two connections between two PGs. Each item in these charts is going to be analyzed in this section.

Although the focus of this paper is two-PG HEV designs, some design groups in Appendices A and B have the same characteristics as a one-PG HEV design. Thus, the identification of HEV powertrain design modes should start first with the one-PG case.

One-PG HEV Design Modes

In a one-PG HEV design, there are three PG nodes. Since ICE and Vehicle are assigned to different nodes, there is only one node that can be assigned to either a brake clutch or an electric machine.

Brake Clutch is Assigned to the Third Node

EMI and EM2 can be collocated with ICE and Vehicle as seen in Figure 4. After the speed and torque equations in equation 15 are derived, the HEV powertrain type and the number of feasible design modes are determined according to the process described in the previous section.

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Since the structure of equation 15 is the same as the one in equation 8, this design mode group is a parallel HEV. [[psi].sub.3]/[[psi].sub.2] should be negative for the forward motion supported by [T.sub.ICE]. Thus, the number of feasible parallel HEV design modes in this group is four. Table 1 shows these four combinations.

EM is Assigned to the Third Node

In this case, EM2 can be collocated with either ICE or Vehicle as seen in Figure 5. The components in the parentheses in the figure mean that their assignment is optional.

EM2 on ICE

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The structure of speed equations in equation 16 is the same as the one in equations 10 and 11. Thus, this design mode group is an output-split HEV. Since [[PSI].sub.3]/[[PSI].sub.2] should be negative for forward motion, the number of feasible output-split design modes in this group is four.

EM2 on Vehicle

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The structure of speed equations in equation 17 is the same as the one in equations 10 and 12. Thus, this design mode group is an input-split HEV. The number of feasible input-split design modes in this group is four.

As a result of this analysis, four parallel and eight power-split feasible HEV design modes can be generated with one PG.

Two-PG HEV Design Modes

Although the focus of this paper is HEV design modes with two PGs, all HEV design modes with one PG were explored first in the previous subsection since some 2-PG combinations end up to have the functionality of a one-PG mode. In this subsection, the design mode groups that are categorized according to the set of rules described in the previous section are analyzed. The first condition of these rules is the number of connections between two PGs. The next condition is the number of brake clutches in the design mode group. In the following subsections, each design mode group is identified by two numbers and one letter, each of which is separated by underscores. For example, 1_0_A represents the first (A) design mode group under the category of one connection between PGs (1_) and no brake clutch (0_). These codes are also written under each design mode group in Appendices A and B.

One Connection between Two PGs

1_0_A

In this design mode group, there is not any component assigned to the connection between PGs. Furthermore, ICE and Vehicle are connected to the nodes of the same PG as shown in Figure 6. EM1 and EM2 should be connected to the remaining two nodes of the second PG. Otherwise, this design mode group becomes identical to a 1-PG design.

After the manipulation of the steady-state speed and torque equations, equation 18 is obtained, where [[psi].sub.i], i=l,2,3 and [[psi].sub.j], j=4,5,6 can take any value from the sets {1+[alpha], -[alpha], -1} and {1+[beta], -[beta] -1}, respectively. The structure of equation 18 is the same as the one in equations 10 and 14. Thus, this design mode group is a special type power-split HEV. Since [[psi].sub.3]/[[psi].sub.2] should be negative for forward motion, and there is no restriction on the selection of [[psi].sub.4], [[psi].sub.5], and [[psi].sub.6], the number of feasible design modes in this group is 4 x 6=24.

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1_0_B

In this design mode group, there is not any component assigned to the connection between PGs. Furthermore, ICE and Vehicle are connected to the nodes of the different PGs as shown in Figure 7. EMI and EM2 should be connected to the remaining two nodes of the two PGs.

After the manipulation of the steady-state speed and torque equations, equation 19 is obtained. This design mode group is a special power- split HEV. [[psi].sub.1] [[psi].sub.1]/[[psi].sub.2] [[psi].sub.4] should be positive for forward motion and there are 20 combinations that can meet this requirement.

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1_0_C

The characteristics of this design mode group shown in Figure 8 is the same as the ones of 1-PG designs. Thus, there is no need for its further analysis.

1_0_D

The characteristics of the design mode group in Figure 9 are the same as the ones of 1-PG designs. Thus, there is no need for its analysis.

1_0_E

PGs are ineffective due to the free rotating gears of both PGs in the design mode group shown in Figure 10. Thus, there is no need for its further analysis.

1_1_A

The characteristics of this design mode group shown in Figure 11 is the same as the ones of 1-PG designs. Thus, there is no need for its further analysis.

1_1_B

In this design mode group, brake clutch is assigned to the connection between PGs. Furthermore, ICE and Vehicle are connected to the nodes of different PGs. In this group, EM1 and EM2 can be connected in four different ways (I, II, III, IV) as shown in Figure 12.

The first variant (I) can have 36 design modes (six combinations for each PG), the second and third variants (II, III) have six design modes each and the last one (IV) has just one design mode. As a result, this design mode group has 49 different design modes. Since the speed and torque equations are in the form of equation 9, all design modes are series HEVs.

1_1_C

In this design mode group, ICE or Vehicle is assigned to the connection between PGs. Moreover, ICE, Vehicle, and brake clutch are connected to the nodes of the same PG. In this group, EMI and EM2 can be connected in two different ways (I, II) as shown in Figure 13.

After the manipulation of the steady-state speed and torque equations of the first variant (I), equation 20 is obtained. This design mode group is a parallel HEV Since [[psi].sub.3]/[[psi].sub.1] should be negative for forward motion, this variant of the design group has 24 design modes (4 x 6). Furthermore, 24 more unique design modes can be generated by interchanging ICE and Vehicle assignments. As a result, the total number of design modes of this variant is 48.

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The second variant (II) of this design mode group has the same properties of a 1-PG design. Thus, there is no need for its further analysis.

1_1_D

In this design mode group, ICE or Vehicle is assigned to the connection between PGs. Moreover, ICE or Vehicle is on the PG, to which no brake clutch is connected as shown in Figure 14.

After the manipulation of steady-state speed and torque equations, equation 21 is obtained.

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The structure of equation 21 is the same as the one in equations 10 and 11. Thus, this design mode group is an output-split HEV Since [[psi].sub.5]/[[psi].sub.4] should be negative for forward motion and there is no restriction on the selection of [[psi].sub.4], [[psi].sub.5], and [[psi].sub.6], the number of feasible design modes in this group is 4 x 6=24. Furthermore, 24 more unique design modes can be generated by interchanging ICE and Vehicle assignments. However, these additional 24 design modes are input-split type due to the direct dependence of [[omega].sub.Vehicle] on [[omega].sub.EM1] To sum up, in this design mode group, there are 24 input-split and 24 output-split design modes.

1_1_E

In this design mode group, EM1 is assigned to the connection between PGs, and ICE and Vehicle are on the same PG as shown in Figure 15.

As seen from the equation 22, which is derived from the steady-state speed and torque equations, this design mode group is a special power-split HEV. The number of design modes in this group is 24 due to the sign restriction on [[psi].sub.3]/[[psi].sub.2].

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1_1_F

In this design mode group, EM1 is assigned to the connection between PGs, and ICE and Vehicle are on different PGs as shown in Figure 16.

According to the equation 23, which is derived from the steady-state speed and torque equations, this design mode group is an input-split HEV. The number of design modes in this group is 20 due to the constraint on [[psi].sub.1] [[psi].sub.6] [[psi].sub.2] [[psi].sub.4] being positive. Furthermore, 20 more unique design modes can be generated by interchanging ICE and Vehicle assignments. However, these additional 20 design modes are output-split type due to the direct dependence of [[omega].sub.ICE] on [[omega].sub.EM1]

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1_1_G

In this design mode group, no component is assigned on the connection between PGs. Furthermore, ICE and Vehicle are on the same PG as shown in Figure 17.

Depending on to which node EM2 is assigned, the HEV architecture type of the design mode group changes. If EM2 is collocated with ICE, equation 24 is obtained and the design modes become output-split type. If EM2 is collocated with the Vehicle, the design modes become input-split type. Considering the sign constraint on [[psi].sub.3]/[[psi].sub.2], this design mode group has 24 input-split and 24 output-split design modes.

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1_1_H

In this design mode group, no component is assigned on the connection between PGs. Furthermore, ICE and Vehicle are on different PGs as shown in Figure 18.

Depending on which node EM2 is assigned to, the HEV type of the design mode group changes. If EM2 is collocated with ICE, equation 25 is obtained and the design modes become output-split type. If EM2 is collocated with the Vehicle, the design modes become input-split type. Considering the sign constraint on [[psi].sub.1] [[psi].sub.6]/[[psi].sub.2] [[psi].sub.4], this design mode group has 20 input-split and 20 output-split design modes.

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1_2_A

In this design mode group, there are two brake clutches, one of which is connected to the connection between PGs. The equations of this design mode group are the same as the ones of 1_1_A.

1_2_B

In this design mode group, two brake clutches are assigned to different PGs. ICE and Vehicle are on different PGs and there is not any component assigned to the connection between PGs as shown in Figure 19.

The governing speed and torque equations in equation 26 reveal that all design modes in this group are parallel HEVs. When the sign of [[psi].sub.1] [[psi].sub.6]/[[psi].sub.3] [[psi].sub.4] is considered, the number of design modes in this group is 20.

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1_2_C

In this design mode group, two brake clutches are assigned to different PGs. ICE or Vehicle is assigned to the connection between two PGs as shown in Figure 20.

Due to the speed relationship in equation 27, all design modes in this group are parallel HEVs. The torque equation depends on the connection of EM2. If EM2 is collocated with ICE, the torque equation in equation 28 is obtained. Considering the sign of [[psi].sub.3]/[[psi].sub.1], there are 24 design modes in this configuration. When the cases, where EM2 is collocated with Vehicle, and ICE and Vehicle node assignments interchange are considered, the total number of design modes in this group becomes 96.

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1_2_D

In this design mode group, two brake clutches are assigned to different PGs. EMI is assigned to the connection between two PGs as shown in Figure 21.

Due to the speed relationship in equation 29, all design modes in this group are parallel HEVs. The torque equation depends on the connection of EM2. If EM2 is collocated with ICE, the torque equation in equation 29 is obtained. Considering the sign of [[psi].sub.1] [[psi].sub.6]/[[psi].sub.3] [[psi].sub.4], there are 20 design modes. If EM2 is collocated with the Vehicle, 20 more design modes can be generated.

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1_2_E

In this design mode group, there are two brake clutches, both of which are assigned to the same PG. However, no brake is connected to the connection between PGs. The equations of this design mode group are the same as the ones of 1_1_A.

Two Connections between Two PGs

2_0_A

In this design mode group, there is not any component assigned to the connection between PGs. Static torque equations of this design mode group do not have a solution.

2_0_B

In this design mode group, ICE or Vehicle is assigned to one of the connections between PGs. No component is assigned to the second connection as shown in Figure 22.

As seen in equation 30, when EM2 is collocated with ICE, the design mode group becomes output-split type. When EM2 is collocated with the Vehicle, the group is input-split type. Considering the sign of [[psi].sub.3]/[[psi].sub.1] and two possibilities of EM2 colocation, this group has 48 output-split, 48 input-split design modes. When the interchange of ICE and Vehicle node assignments is taken into account, 96 additional design modes are also available. Furthermore, the denominator term [[psi].sub.1] [[psi].sub.5] - [[psi].sub.2] [[psi].sub.4] is introduced to the speed and torque equations for the design mode groups with two connections between two PGs. Since this term can be equal to zero just for a specific [alpha], [beta] value pair, the cases where [[psi].sub.1] [[psi].sub.5] - [[psi].sub.2] [[psi].sub.4]=0 are not considered in this study.

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2_0_C

In this design mode group, EMI is assigned to one of the connections between PGs. No component is assigned to the second connection as shown in Figure 23.

As seen in equation 31, when EM2 is collocated with ICE, the design mode group becomes an output-split type. When EM2 is collocated with the Vehicle, the group is an input-split type. Considering the sign of [[psi].sub.2] [[psi].sub.5]/[[psi].sub.2][[psi].sub.6] and two possibilities of EM2 colocation, this group has 20 output-split, 20 input-split design modes.

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2_0_D

In this design mode group, ICE and Vehicle are assigned to the connections between two PGs as shown in Figure 24. The governing speed and torque equations in equation 32 show that all design modes in this group are compound split type. Considering the sign of [[psi].sub.5]/[[psi].sub.4], the number of design modes in this group is 24.

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2_0_E

In this design mode group, EMI and EM2 are assigned to the connections between two PGs as shown in Figure 25. The governing speed and torque equations in equation 33 show that all design modes in this group are compound split type. Considering the sign of [[psi].sub.3][[psi].sub.4]/[[psi].sub.1] [[psi].sub.6] the number of design modes in this group is 20.

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2_0_F

In this design mode group, EMI, and ICE or Vehicle are assigned to the connections between two PGs as shown in Figure 26. The governing speed and torque equations in equation 34 show that all design modes in this group are compound split type. Considering the sign of [[psi].sub.3][[psi].sub.4]/([[psi].sub.1][[psi].sub.5]-[[psi].sub.2][[psi].sub.4]), the number of design modes in this group is 29. When the node assignments of ICE and Vehicle are interchanged, there are 29 more compound-split design modes.

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2_1_A

In this design mode group, the brake clutch is assigned to one of the two connections between PGs. ICE or Vehicle is assigned to the other connection as shown in Figure 27. The topology of this design mode group is the same as the one of 1_2_C. Thus, the analysis of this group is skipped.

2_1_B

In this design mode group, the brake clutch is assigned to one of the two connections between PGs. EMI is assigned to the other connection as shown in Figure 28. The topology of this design mode group is the same as the one of 1_2_D. Thus, the analysis of this group is skipped.

2_1_C

In this design mode group, the brake clutch is assigned to one of the two connections between PGs. No component is assigned to the other connection as shown in Figure 29. The topology of this design mode group is the same as the one of 1_2_B. Thus, the analysis of this group is skipped.

2_1_D

In this design mode group, the brake clutch is not assigned to any connection between PGs. ICE and Vehicle are connected to the connections as shown in Figure 30.

Due to the speed relationship in equation 35, all design modes in this group are parallel HEVs. The torque equation depends on the connection of EM2. If EM2 is collocated with ICE, the torque equation in equation 35 is obtained. Considering the sign of [[psi].sub.5]/[[psi].sub.4], there are 24 design modes in this configuration. If EM2 is collocated with the Vehicle, 24 more design modes can be generated.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

2_1_E

In this design mode group, the brake clutch is not assigned to any connection between PGs. ICE or Vehicle is connected to one of the connections and there is not any component on the other connection as shown in Figure 31.

Due to the speed relationship in equation 36, all design modes in this group are parallel HEVs. Considering the sign of [[psi].sub.3][[psi].sub.5]/([[psi].sub.2][[psi].sub.4]-[[psi].sub.1][[psi].sub.5]), there are 29 design modes in this configuration. Moreover, when ICE and Vehicle node assignments are interchanged, 29 more design modes can be generated.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

2_1_F

In this design mode group, the brake clutch is not assigned to any connection between PGs. ICE or Vehicle and EMI are connected to the connections as shown in Figure 32.

Due to the speed relationship in equation 37, all design modes in this group are parallel HEVs. The torque equation depends on the connection of EM2. If EM2 is collocated with ICE, the torque equation in equation 37 is obtained. Considering the sign of [[psi].sub.3][[psi].sub.5]/([[psi].sub.2][[psi].sub.4]-[[psi].sub.1][[psi].sub.5]), there are 29 design modes in this configuration. If EM2 is collocated with the Vehicle, 29 more design modes can be generated. Moreover, when ICE and Vehicle node assignments are interchanged, 58 additional design modes are created. As a result, 116 parallel design modes exist in this group.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

ANALYSIS OF THE RESULTS

In the previous section, all design mode groups in Appendices A and B are analyzed and the number of unique design modes in each group is determined. Table 2 groups the number of these unique design modes according to their HEV type. As seen in the table, series HEV modes have the least number of design modes, whereas parallel modes have the most.

The HEV type specific observations about the results will be described here. In this analysis, the notations [[psi].sub.i], [[psi].sub.j], [[psi].sub.k], [[psi].sub.m], [[psi].sub.n], [[psi].sub.p] where i, j. k[member of]{l,2,3}, i[not equal to]j[not equal to] and m, n,p[member of]{4,5,6}, m[not equal to]n[not equal to]p will be used.

In the observations below, the coefficients in eqs. 8-14 are referred.

1-PG Design Modes

[h.sub.1], [h.sub.12], and [h.sub.13] terms in equations 8, 9, 10, 11, 12, 13, 14 can take [[psi].sub.i]/[[psi].sub.j] or-1, whereas [f.sub.11] can only be - [[psi].sub.i]/[[psi].sub.j]'

1-PG design modes can only be parallel, input-split, and output-split types.

2-PG Design Modes

There should be at least one brake clutch in the design in order to create a parallel architecture type design mode.

All design modes with two brake clutches are parallel or series type.

Compound-split modes exist only in the groups with two connections between PGs.

The possibility of having [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] coefficient in torque equations in 2-PG designs makes them superior in terms of performance to all 1-PG designs, which can have only [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] term in their torque equations.

One Connection between PGs

[f.sub.11], [f.sub.11], [f.sub.12], and [h.sub.13] terms in equations 8. 9, 10, 11, 12, 13, 14 can take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [h.sub.13] can also be -1. Since these terms do not include any subtraction or addition of [psi] variables, the variation of a and [beta] in the allowable range [1.8, 3.8] does not cause any sign change in these terms. As a result, [alpha] and [beta] variation affects only fuel economy and performance of a design.

Series architecture type exists only in the design modes with one connection between PGs.

Special power-split types exist only in the design modes with one connection between PGs.

The degrees of freedom of the speed equation are three in a special power-split type design mode if there is no brake clutch in the design. The degrees of freedom of the speed equation are two in a special power-split type design mode if there is one brake clutch in the design.

Two Connections between PGs

All parallel design modes have one brake clutch.

Compound-split types exist only in the design modes without brake clutch.

Design modes without brake clutch are input-split or output-split or compound-split type.

[f.sub.11], [f.sub.11], [f.sub.12], and [h.sub.13] terms in equations 8, 9, 10, 11, 12, 13, 14 can take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [h.sub.12] and [h.sub.13] can also be -1. Since some of these terms include the subtraction of [psi] variables, the variation of a and [beta] in the allowable range [1.8, 3.8] may cause a sign change in these terms, which converts a feasible design to an infeasible one or vice versa.

APPLICATION EXAMPLE

In this section, an application example will be presented to show how the results of this paper can be used in the design of an HEV powertrain.

It is desired to design a single mode HEV powertrain with superior performance for a vehicle similar in size to Toyota Prius. The 0-60mph time of the vehicle should be less than 10sec. Additional design criteria can also be set during the design process like maximum vehicle speed allowable for the maximum engine and electric machine speed, and/or fuel economy improvement potential. The size of electric machines and/or engine can also be varied during the design process. However, in this study, a single set of components is considered. The vehicle parameters and the characteristics of the engine and two electric machines are shown in Table 3. The maximum torque curves of the engine and electric machines are derived by scaling the curves in [12] to resemble the ones in Toyota Prius 2010 with the third generation Toyota Hybrid System (THS-III).

The HEV powertrain candidates belong to either of parallel, series, input split, output split, and compound split types. Since the requirement is to design a HEV with a single mode, a parallel mode is not capable of covering the entire vehicle speed range. It is well known that series modes cannot be efficient in all driving conditions due to the double energy conversions (mechanical-to-electrical-tomechanical). Output-split and compound split modes require high power electric machines for electronically controlled continuously variable transmission (eCVT) operation at low vehicle speed. Thus, an input-split hybrid mode similar to the one in Toyota Prius is going to be designed in this example.

According to the analysis in this paper, the input-split modes exist under the 1-PG, 1_1_D, 1_1_F, 1_1_G, 1_1_H, 2_0_B, and 2_0_C mode groups. Input-split modes in these groups are generated by taking into account each combination of [[psi].sub.1], [[psi].sub.2], [[psi].sub.3][member of]{1+[alpha], -[alpha], -1} and [[psi].sub.4], [[psi].sub.5], [[psi].sub.6][member of]{1+[beta], -[beta], -1}. The 0-60mph time performance of each feasible mode is evaluated for varying [alpha],[beta][member of][1.8, 3.8] with 0.1 increments according to the algorithm in [7]. At the end of the analysis, it is discovered that none of the modes under the 1-PG, 1_1_G, 1_1_H, 2_0_B, and 2_0_C mode groups can achieve 0-60mph time below 10 seconds. 1_1_D and 1_1_F have five and two input-split modes that meet 0-60mph time requirement, respectively. The topology, PG gear ratios, and 0-60mph time of these modes are shown in Figure 33 and Table 4, where R1-2, C1-2 and S1-2 correspond to ring, carrier, and sun gears of PG1 and PG2, respectively. Note that mode l_l_D_e in Figure 33 has the same configuration as the one in Toyota Prius THS-III. However, it is not the most superior design in terms of 0-60mph time. The designs l_l_D_a and l_l_D_c give the best performance. Another important observation in this application example is the fact that neither of the input-split mode groups with collocated Vehicle and EM2 (1-PG, 1_1_G, 1_1_H, 2_0_B, and 2_0_C) can meet the 0-60mph time performance requirement.

CONCLUSIONS

In this paper, a manual procedure is developed to explore all unique design modes that can be created with one or two PGs, one engine, one vehicle output shaft, two electric machines, and at most two brake clutches. Due to the large size of the design space, the design modes are categorized under more manageable design mode groups. These design mode groups are created according to a methodic procedure that takes into consideration the number of connections between PGs, the number of brake clutches, and how the engine, vehicle output shaft, and electric machines are assigned to the PG nodes. The introduced method facilitated the deduction of the results about the number of unique design modes for each HEV powertrain type, the characteristics of the steady-state speed and torque equations, the design limitations, and their topologies.

REFERENCES

[1.] Joint Technical Support Document: Final Rulemaking for 2017-2025 Light-Duty Vehicle Greenhouse Gas Emission Standards and Corporate Average Fuel Economy Standards, 2012.

[2.] NHTSA and EPA Set Standards to Improve Fuel Economy and Reduce Greenhouse Gases for Passenger Cars and Light Trucks for Model Years 2017 and Beyond, 2012.

[3.] Cobb, J., December 2014 Hybrid Car Sales Numbers, http://www.hvbridcars.com/december-2014-dashboard/.2015.

[4.] Raghavan, M., Bucknor, N., Maguire, J., Hendrickson, J., et al. The Design of Advanced Transmissions. FISITA World Automotive Congress, Yokohama, Japan, 2006.

[5.] Liu, J. and Peng, H., "A Systematic Design Approach for Two Planetary Gear Split Hybrid Vehicles," Vehicle System Dynamics, 48(11), 1395-1412, 2010.

[6.] Zhang, X., Peng, H, Sun, J., and Li, S., "Automated Modeling and Mode Screening for Exhaustive Search of Double-Planetary-Gear Power Split Hybrid Powertrains," ASME Dynamic Systems and Control Conference, San Antonio, Texas, 2014.

[7.] Dagci, O.H., Peng, H, Grizzle, J., "Power-Split Hybrid Electric Powertrain Design with Two Planetary Gearsets for Light-Duty Truck Applications," IFAC Workshop on Engine and Powertrain Control, Simulation and Modeling, Columbus, OH, 2015.

[8.] Cheong, K.L., Li, P.Y., and Chase, T.R., "Optimal Design of Power-Split Transmissions for Hydraulic Hybrid Passenger Vehicles," American Control Conf., San Francisco, CA, 2011.

[9.] Bayrak, A., "Topology Considerations in Hybrid Electric Vehicle Powertrain Architecture Design," PhD thesis, The University of Michigan, Ann Arbor, 2015.

[10.] Benford, H. and Leising, M., "The Lever Analogy: ANew Tool in Transmission Analysis," SAE Technical Paper 810102. 1981. doi:10.4271/810102.

[11.] Wikipedia, Epicyclic Gearing. http://en.wikipedia.org/wiki/Epicyclic_gearing. [Online; accessed 04-April-2015].

[12.] Hermance, D. and Abe, S., "Hybrid Vehicles Lessons Learned and Future Prospects," SAE Technical Paper 2006-21-0027. 2006.

CONTACT INFORMATION

Oguz H. Dagci

oguzhada@umich.edu

ACKNOWLEDGEMENT

The authors gratefully thank Prof. Jessy W. Grizzle at the University of Michigan for his helpful comments and suggestions.

Oguz H. Dagci and Huei Peng

University of Michigan - Ann Arbor

APPENDIX

Table 1. Feasible [[psi].sub.2] and [[psi].sub.3] Combinations.

[[psi].sub.2]  [[psi].sub.3]

1+[alpha]      -[alpha]
-[alpha]       1+[alpha]
1+[alpha]      -1
-1             1+[alpha]

Table 2. The number of unique design modes for each HEV type.

HEV Type          Unique Number
                 of Design Modes

Parallel               526
Series                  49
Power-Split
 Input-Split           160
 Output-Split          160
 Compound-Split        102
 Special Type           68

Table 3. Characteristics of Vehicle, Engine, and Electric Machines.

Vehicle Mass                  1500kg
Final Drive Ratio                3.5
Aerodynamic Drag Coefficient     0.25
Frontal Area                     2.37[m.sup.2]
Engine
 Maximum Power                  73kW@5000rpm
 Maximum Torque                142Nm@4000rpm
Electric Machines 1 and 2
 Maximum Power                  60kW
 Maximum Torque                207Nm (0-2770rpm)

Table 4. Gear Ratios and Performance of Design Modes.

Design Mode  [alpha]  [beta]  [t.sub.0]-60mph

1_1_D_a      2.4      1.8     7.8sec
1_1_D_b      2.4      1.8     9.8sec
1_1_D_c      2.4      2.8     7.8sec
1_1_D_d      3.8      1.8     9.4sec
1_1_D_e      3.8      2.8     9.4sec
1_1_F_a      1.8      1.8     8.7sec
1_1_F_b      2        2.8     7.9sec
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Author:Dagci, Oguz H.; Peng, Huei
Publication:SAE International Journal of Alternative Powertrains
Article Type:Report
Date:May 1, 2016
Words:7801
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