Alparslan Emrah Bayrak1

Mechanical Engineering,

University of Michigan,

Ann Arbor, MI 48109

e-mail: bayrak@umich.edu

Yi RenMechanical Engineering,

Arizona State University,

Tempe, AZ 85287

e-mail: yiren@asu.edu

Panos Y. PapalambrosMechanical Engineering,

University of Michigan,

Ann Arbor, MI 48109

e-mail: pyp@umich.edu

Topology Generation forHybrid Electric VehicleArchitecture DesignExisting hybrid powertrain architectures, i.e., the connections from engine and motors tothe vehicle output shaft, are designed for particular vehicle applications, e.g., passengercars or city buses, to achieve good fuel economy. For effective electrification of newapplications (e.g., heavy-duty trucks or racing cars), new architectures may need to beidentified to accommodate the particular vehicle specifications and drive cycles. The ex-ploration of feasible architectures is combinatorial in nature and is conventionally basedon human intuition. We propose a mathematically rigorous algorithm to enumerate allfeasible powertrain architectures, therefore enabling automated optimal powertraindesign. The proposed method is general enough to account for single and multimodearchitectures as well as different number of planetary gears (PGs) and powertrain com-ponents. We demonstrate through case studies that our method can generate the completesets of feasible designs, including the ones available in the market and in patents.[DOI: 10.1115/1.4033656]

1 Introduction

Increasing demand on energy efficiency has motivated expan-sion of powertrain electrification from passenger cars to buses[1,2] and other utility vehicles [3,4]. One critical design decisionto make in the electrification process is the configuration of thepowertrain, which defines connection arrangements amongpowertrain components and thus the dynamics and the powertrainefficiency. While a powertrain design can have a single configura-tion, some powertrain designs switch among multiple configura-tions during vehicle operation, such as the Chevrolet Volt, seeFig. 1. Disengaging clutch 1 (CL1) and engaging CL2 of thepowertrain in this figure give a hybrid configuration (Fig. 1(a)),while engaging or disengaging both CL1 and CL2 achieves pureelectric configurations (Figs. 1(b) and 1(c)). The collection of allthe configurations in the design defines the architecture of apowertrain. Architectures with more than one configuration arecalled multimode architectures, and each particular configurationis called a driving mode. Switching among multiple configurationsallows the powertrain to achieve better performance and fueleconomy under changing driving conditions. While availablepowertrain architectures in the market have achieved significantfuel efficiency, these are designed for the given vehicle specifica-tions and design requirements and may not perform well or evenbe feasible when the vehicle specifications or driving conditionschange [5].

Configuration design has been studied in other domains includ-ing the design of structural systems [6,7], kinematic chains [8,9],electrical circuits [10], and chemical molecules [11]. There arealso general configuration synthesis approaches that apply tovarious design domains relying on a given library of components[12–14]. The combinatorial nature of powertrain architecturedesign requires one to enumerate and identify all the feasible con-figurations to determine the optimal design. In the conventionalpowertrain design domain, methods have been developed to createall the possible design alternatives for four-speed [15] and six-speed [16] automatic transmissions. However, methodologies inmost of these studies remain limited to their specific domain.

Domain-independent approaches cannot capture the kinematicfeasibility constraints essential in PG systems. Thus, hybridpowertrain architectures require a different approach. Due to thelarge number of possible configurations with one PG [17,18], twoPGs [19–23], and three PGs [24,25], existing hybrid powertraindesign research has focused on subsets of the overall designspace, e.g., enumeration of one- and two-PG configurations [5],exploration of the Toyota Prius-like designs [26], and variationsof a baseline configuration [27]. A recent study by Zhang et al.proposes a power-split hybrid architecture design framework toselect configurations by designing the system configuration matrix[28]. The framework relies on a commonly used set of powertraincomponents, i.e., a single engine and two motor/generators(MGs).

This article proposes a general systematic approach to generateall the feasible hybrid and pure electric configurations withan arbitrary number of PGs and any set of powertrain components.The generation process uses a modified bond graph representationof powertrain configurations. An earlier version of this workappeared in Ref. [29]. The generation method can be combinedwith a search in the feasible space of configurations to identifypromising design candidates. A sequel to the present article[30,31] offers an example for such a design methodology.

In Sec. 2, we describe the proposed representation ofpowertrain configurations. In Sec. 3, we formulate the process forgenerating all the feasible configuration alternatives using thisrepresentation. Section 4 describes the creation of multimodearchitectures by combining the generated configurations. InSec. 5, we discuss results and in Sec. 6 we offer conclusions.

2 Representation of hybrid electric vehicle (HEV)

Configurations

Due to the combinatorial nature of configuration optimization,finding a globally optimal design requires a systematic approachfor generating all the feasible design alternatives. A formalrepresentation of powertrain configurations is necessary for such ageneration method. Current studies have utilized a variety ofpowertrain representations for visualization and analysis purposes.These include stick diagrams as commonly seen in patents, leverdiagrams [32], canonical graphs [33], dynamic system matrices[5], and bond graphs [29,34]. Each of these representations hasadvantages and disadvantages determining the computational

1Corresponding author.Contributed by the Design Automation Committee of ASME for publication in

the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 5, 2015; finalmanuscript received May 11, 2016; published online June 13, 2016. Assoc. Editor:Massimiliano Gobbi.

Journal of Mechanical Design AUGUST 2016, Vol. 138 / 081401-1Copyright VC 2016 by ASME

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

effort required to generate the feasible design space of powertrainconfigurations. In this article, we use bond graphs since they pro-vide built-in tools to model the power flow and allow us to checkthe feasibility of a powertrain configuration by the use of causalityrules. Other graphical representations require additional calcula-tions to model the power flow that is used to determine the kine-matic degree-of-freedom (dof) of a configuration. It acts asanother feasibility constraint as discussed in Sec. 3.4.

This section introduces a modified bond graph representationspecifically tailored for generating powertrain configurations. Wealso introduce connectivity tables extracted from the bond graphsto be used for creating multimode architectures.

2.1 Modified Bond Graph Representation. Bond graphs aregraphical models that specify the power flow in a dynamic system[35]. Their application in modeling HEV powertrain systems isexplained in Fig. 2, where the dynamics of the hybrid configura-tion of the Chevrolet Volt architecture is captured by a bondgraph. The key element in this architecture is the PG set repre-sented by a lever in Fig. 1(a), where R denotes the ring, C the car-rier, and S the sun gear. At the steady-state, the gear speedrelationship is

ð1þ qÞxc ¼ qxr þ xs (1)

where q is the ring-to-sun gear ratio, and xc, xr, and xs denotethe speeds of the carrier, ring, and sun gears, respectively. Takingthe particular component arrangement of this powertrain intoaccount, the speed relationship can be rewritten in the followingmatrix form:

1 0

�q ð1þ qÞ

� �xeng

xout

� �¼ xMG1

xMG2

� �(2)

Here, xeng, xMG1; xMG2, and xout are speeds of the engine,motor/generators 1 and 2 (MG1 and MG2), and output shaft to thefinal drive, respectively. The bond graph representation shown inFig. 2(a) models this kinematic relationship as well as the corre-sponding torque relationship ignoring gear losses and gear inertia.A 0-junction imposes all the torques around the junction to be thesame, while the sum of the speeds to be equal to zero. In our

particular application, a 0-junction is used to model the kinematicrelationship among different gears of a PG set given in Eq. (1). Inthe case of 1-junction, torque and speed relationships of a0-junction are swapped. Physically, a 1-junction models the rela-tionship of the components connected to the same gear of PG set.Engine, MGs, and vehicle output are modeled as effort sourcesdenoted as “Se” in the figure. The transformer blocks denoted as“TF” are used to model the gear ratios and the tire radius. Sincethe losses are ignored, the following toque relationship can bederived from the conservation of power:

� 1 0

�q ð1þ qÞ

� ��TTeng

�Tout

� �¼ TMG1

TMG2

� �(3)

Two simplifications to the bond graph representation canbe made in the current context: First, bond graphs for fixedpowertrain components (i.e., engine, MGs, output shaft, andground) are simplified as flow and effort sources (or sinks) and arereferred to as external components. Second, transformerblocks can be represented as bond weights and weight is set to 1 ifno transformer exists between two adjacent junctions. Therepresentation after these simplifications is a modified bond graph.Figure 2(b) shows the modified bond graph representation of thehybrid configuration in the Chevrolet Volt architecture. Further,we refer to the junctions connected to the external components asexternal junctions and the rest as internal ones. The number ofexternal junctions is equal to that of external components. Weshow in Sec. 3 that PGs can be represented by internal junctions,and the number of internal junctions corresponds to the number ofPGs. It should be noted that each bond graph represents a singleconfiguration. Separate bond graphs are needed in a multimodearchitecture.

2.2 Connectivity Table. Recall that multimode architectures(i.e., combination of multiple bond graphs) are often necessary forimproving fuel economy in various driving conditions, withswitching among configurations enabled by a clutch mechanism.For the representation to be physically meaningful during designoptimization, we must calculate the number of clutches needed toenable the switching in a given architecture. To facilitate this

Fig. 1 Lever representation of three modes of a modified Chevrolet Volt architecture(without series hybrid mode): (a) hybrid configuration, (b) first pure electric configura-tion, (c) second pure electric configuration, and (d) combined multimode architecture

081401-2 / Vol. 138, AUGUST 2016 Transactions of the ASME

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

calculation (see details in Sec. 4), we introduce a connectivitytable, i.e., a binary matrix that defines the connections amongpowertrain components and each particular gear in a PGsystem. The size of the connectivity table for a general case isðNext þ 3NPGÞ � 3NPG, where Next is the number of external com-ponents and NPG is the number of PGs in the system. The first Next

rows define the connections from the external components to thePG system, while the remaining 3NPG rows define internal con-nections among the PGs. For example, the connectivity table

corresponding to the hybrid configuration of the Chevrolet Voltarchitecture is given in Fig. 3. Since it is a one-PG system, the lastthree rows are all zeros.

The connectivity table can be seen as the matrix form of the“simple” lever representation (i.e., lever diagrams with threenodes), while the so-called “compound” levers (i.e., diagramswith more than three nodes, see Ref. [32], for example) are com-monly used to represent multiple-PG systems. Nonetheless, eachcompound lever corresponds to a combination of simple levers.

Fig. 2 Representation of the hybrid configuration of the Chevrolet Volt architecturein Fig. 1(a): (a) original bond graph representation of the hybrid configuration of theChevrolet Volt architecture and (b) modified bond graph representation of the hybridconfiguration of the Chevrolet Volt architecture

Fig. 3 Example of a modified bond graph representation and its connectivity table

Journal of Mechanical Design AUGUST 2016, Vol. 138 / 081401-3

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Thus, we can standardize the size of the connectivity table basedon the number of external components and PGs involved in thesystem.

3 Configuration Generation

In this section, we describe the automated process to generateall the feasible powertrain configurations. Figure 4 summarizesthe process: We, first, generate all the simple, connected, andundirected graphs based on the external components and numberof PGs. Then, we assign junction types and causalities to convertthe undirected graphs to bond graphs. Finally, we assign the graphweights to every bond in order to represent a PG system. Once abond graph corresponding to a configuration is created, the quasi-static system of equations is extracted and exported to a vehiclemodel for design evaluation.

The following bond graph properties will help us in formulatingthe generation process:

Property 1. A valid bond graph must be simple and connected.A simple graph does not have more than one bond between any

pair of junctions and contains no loops around a single junction,i.e., a junction is not directly connected to itself. Also, in a con-nected graph no junction remains disconnected.

Property 2. A junction with n> 3 bonds is equivalent to n� 2junctions of the same type with three bonds each.

In the original bond graph representation, a junction can haveany number of bonds connected to it. However, in this study, werestrict the internal junctions to have three bonds and externaljunctions to have one bond connected to an internal junction forsimpler implementation. This property shows that such a restric-tion does not limit modeling capabilities. Figure 5 explains thisproperty for a 1-junction with five bonds: In a chain of bonds,both the first and last junctions will have two bonds from the orig-inal model, leaving n� 4 bonds for the remaining junctions. Eachintermediate junction can only have one bond from the originalmodel, since two bonds are needed to connect to other junctions.Therefore, in total 2þ ðn� 4Þ ¼ n� 2 junctions are needed torepresent the original model.

Property 3. A bond graph must have an even number of junc-tions if each internal junction has three bonds and each externalone has one bond connected to an internal junction.

From Properties 1 and 2, a bond graph with Jext number ofexternal junctions and J internal junctions has ð3J � JextÞ=2þ Jext

bonds. Since the number of bonds has to be an integer, 3J � Jext,or equivalently J þ Jext, must be even.

Property 4. The number of 0-junctions is equal to the number ofPGs in the system.

Since we restricted the number of bonds around a junction tothree, a separate 0-junction is necessary to represent the kinematicrelationship among sun, ring, and carrier gears given in Eq. (1) foreach PG system. Note that if our restriction on the number ofbonds is relaxed, the systems where two or more 0-junctions areconnected to each other can be represented a single 0-junction.This restriction makes formulating the graph generation problemeasier.

A step-by-step bond graph generation process is described asfollows.

3.1 Graph Generation. We start with generating simple,connected, undirected graphs. Recall that each external junction isconnected to an external component, and the number of externalcomponents is usually small. In a typical hybrid powertrain, theexternal components are an engine, two MGs, one vehicle output,and a ground. Here, ground refers to an immobile node that isgenerally achieved by connecting a gear to the stationaryhousing frame. Four types of configurations can be defined in thiscontext: (1) Hybrid with ground, (2) hybrid without ground, (3)pure electric with engine engaged, and (4) pure electric withoutengine. The first two are the configurations in which the engineactively provides motive power to the system. The only differencebetween them is that the former has one of the PG nodes con-nected to a fixed ground, providing zero speed. In configurationtype (3) and (4), the engine is deactivated by either being con-nected to a fixed ground or being disconnected from the system bya clutch, respectively. Connecting a PG node to ground may allowthe architecture to achieve higher torque outputs. The output torqueobtained at the vehicle shaft is constrained by the torque limitationsof the powertrain components actively engaged in the architecturebut a ground connection does not have any torque limitation. Thestudy by Zhang et al. [26] can be given as an example exploitingthis benefit of grounding in HEV architecture design.

Following the bond graph properties introduced earlier, Table 1summaries the number of junctions needed for each type of con-figuration. Note that these numbers can be generalized to any setof external components and any number of PGs. We picked thesecases as examples to demonstrate the application of the proposedmethodology to designs in common practice.

Given the number of junctions, let GðJextþJÞ�ðJextþJÞ ¼½0 A; ATB� be the adjacency matrix representing a graph, whereAJext�J and BJ�J are the binary matrices. The matrix A representsthe connections between external and internal junctions, and Brepresents the connections among internal junctions. The corre-sponding value in the matrix is 1 when there is a connectionbetween two junctions and 0 otherwise. We can see that if thejunctions i and j are connected, Bij ¼ Bji ¼ 1. Therefore, thematrix B is symmetric. Also, the 0Jext�Jext block in the adjacencymatrix prevents any direct connection among external junctions.

The designs of matrices A and B determine the graph under thefollowing constraints:

(i) An external junction must have only one bond

XJ

j¼1

Aij ¼ 1; 8i ¼ 1; 2; 3; :::; Jext (4)

(ii) Each internal junction must have exactly three bonds

XJext

i¼1

Aij þXJ

i¼1

Bij ¼ 3; 8j ¼ 1; 2; 3; :::; J (5)Fig. 4 Configuration generation process flow

081401-4 / Vol. 138, AUGUST 2016 Transactions of the ASME

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

In order to enumerate all the possible graphs satisfying the con-straints given by Eqs. (4) and (5), all the possible As are first enum-erated from Eq. (4). Then, for each enumerated A, all the possibleBs are found from Eq. (5). Since Jext and J do not take large values,i.e., the number of powertrain components is usually limited, thecomputational cost of this enumeration process is tractable.

The enumeration process described above might create repli-cated graphs with different ordering of junctions. Figure 6 showssuch a case. By changing the labels of the junctions, the same sys-tems can be obtained from each other. This phenomenon is knownas graph isomorphism [36]. Isomorphism exists in any graphicalrepresentation and must be identified to eliminate the replicates.Algorithms exist to identify graph isomorphism, see, for example,Refs. [37] and [38]. We use the available implementation of iso-morphism detection named graph isomorphism in MATLAB [39] tofilter the isomorphic graphs.

3.2 Junction and Causality Assignment. After the enumera-tion of all the undirected graphs with no replicates, it is necessaryto assign 0- and 1-junctions to the nodes of the graphs. Thisassignment process is linked with the causal stroke assignmentsince the causality restricts the junction assignment. A causal

stroke identifies the side of a bond that imposes flow (speed) onthe system, whereas effort (torque) is imposed by the side withouta causal stroke. The assignment process starts with the externaljunction assignment. All the external junctions are assigned to be1-junctions. We also need to assign causalities to the internal junc-tions. It can be seen from Fig. 2(b) that in a hybrid configuration,the causality stroke is on the internal junction side for MG1 andMG2 and on the external junction side for engine and vehicle out-put. This assignment implies that engine and vehicle outputimpose speed on the system, while MGs impose torque.

For pure electric modes, the causality assignment of the engine(if it exists) is opposite since the engine is grounded. The groundimposes zero speed to the engine. The causality of one of theMGs is also flipped as necessary to make the assignment consist-ent, i.e., the speed of one of the MGs is freely determined by thecontroller if the MG is not grounded.

The causality assignment employed here restricts some of theconfigurations. For instance, since engine and vehicle output havethe same causality, they can never be connected to the same gearof a PG set directly. However, such a configuration links theengine speed to the vehicle speed directly, leaving no possibilityto control engine speed. Thus, these less efficient designs are notconsidered in the configuration generation stage. When the

Fig. 5 A junction with five bonds is equivalently replaced by three junctions with threebonds each

Table 1 Number of junctions needed for one engine, two MGs, and one ground

Number of PGs HEV mode without ground HEV mode with ground Pure EV mode with engine Pure EV mode without engine

One PG Jext ¼ 4 — Jext ¼ 5 Jext ¼ 3J¼ 2 — J¼ 3 J¼ 1

Two PGs Jext ¼ 4 Jext ¼ 5 Jext ¼ 5 Jext ¼ 3J¼ 4 J¼ 3 J¼ 3 or 5 J¼ 3

Three PGs Jext ¼ 4 Jext ¼ 5 Jext ¼ 5 Jext ¼ 3J¼ 6 J¼ 5 J¼ 5 or 7 J¼ 5

Fig. 6 Two replicates generated from the enumeration process. Both graphs result inthe same equation sets after assigning the junction type and bond weights.

Journal of Mechanical Design AUGUST 2016, Vol. 138 / 081401-5

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

designs of interest include parallel configurations, the engine andoutput shaft must be allowed to be connected to the same node ofPG set by flipping the causality of the engine. Recall that, in orderto allow some variety of speed ratios, this design requires anadditional transmission at the vehicle output.

Next, we assign junction types and causal strokes for the in-ternal junctions. Let b ¼ ð3J � JextÞ=2 denote the number ofbonds connecting the internal junctions to each other. Also,denote the junction type for the junction j as tj. Let tj¼ 1 whenthe junction type is 1 and tj ¼ �1 when the junction type is 0.This assignment is useful when formulating the junction typeassignment problem. We denote the causality on the bond con-necting the junctions j1 and j2 as cj1j2 and assign cj1j2

¼ 1 if thestroke is on the junction j1 side, implying cj2j1 ¼ �1. With thesedefinitions and according to the bond graph rules, the followingequation is stated:

�tj1 þX

j2jGj1 j2¼1

cj1j2¼ 0; 8j1 ¼ 1; 2; 3; :::; J (6)

Since the total number of equations is J and the number ofunknowns to be determined is bþ J, multiple solutions exist.Summing all the equations from Eq. (6) gives

XJ

j1¼1

�tj1 þX

j2jGj1 j2¼1

cj1j2

!¼ 0 (7)

The sum of all the causalities between internal junctions is zerosince for every cj1j2 there exists a cj2j1 such that cj1j2 þ cj2j1 ¼ 0.Therefore, sum of the junction types is equal to the sum of thecausalities of the external bonds. This result leads to the numberof junctions in Table 1, assuming that the powertrain consistsof one engine, two MGs, and one ground node. Combined withProperty 4, we can determine how many 0- and 1-junctions toassign to the system. This enables the enumeration of all the possi-ble junction and causality assignments.

Note that in Table 1, under our causality assumptions for exter-nal components, it is not possible to obtain a feasible hybrid con-figuration with ground engaged among one PG designs. Also, forpure electric modes with the engine connected to the groundedgear, the number of internal junctions can take two different val-ues, due to the causality assignment of the MGs. If one of the

MGs is assigned to impose flow on the system, fewer internaljunctions are required to make the bond graph feasible.

3.3 Weight Assignment. In order to complete the modifiedbond graph enumeration process, the proper bond weights(weights for the TF blocks) that correspond to feasible configura-tions need to be assigned. Let q be the ring-to-sun PG ratio. AsFig. 2(b) also shows, a PG set is modeled by 0- and 1-junctionstogether, with weights around a 0-junction being 1, q and 1þ q,and the rest of the bond weights being 1. There are six possibleways to assign these bond weights around a 0-junction as shownin Fig. 7.

Assignment of bond weights must be consistent. When connect-ing two 0-junctions together, bond weight assignments might con-flict and scaling of the weights might be necessary to resolve theconflict. Figure 8 shows an example for a 0-to-0 junction connec-tion when scaling is needed. In the figure, the original bondweights connecting two 0-junctions, i.e., weights of 2 and 3,conflict with each other. These weights are scaled to the leastcommon multiple of the conflicting bond weights. In this example,bond weights of the left junction are multiplied by 3 and those ofthe right junction are multiplied by 2.

3.4 Equation Extraction. Once enumeration and weightassignment are complete, the link between the graph representa-tion and the system equations must be formed to evaluate thedesigns.

Let xext be the vector of speeds from external components.Define x as the b� 1 vector of speeds on the bonds connectinginternal junctions. Following the bond graph rules for 0- and 1-junctions, i.e., speeds around a 0-junction sum up to 0 and speedsaround a 1-junction are the same, the following system of equa-tions can be written:

W0xext þWx ¼ 0 (8)

where W0 and W are the matrices containing the bond weightsfrom the graph. Let J0 and J1 denote the number of 0-junctionsand 1-junctions, respectively. Then, W is a matrix of sizeðJ0 þ 2J1Þ � b since a 0-junction defines a single relationship forthe speeds and 1-junction defines two. When W has rank b,Eq. (8) can be rewritten as

Fig. 7 All possible six combinations for the bond weight assignment around a0-junction

081401-6 / Vol. 138, AUGUST 2016 Transactions of the ASME

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

x ¼ �ðWTWÞ�1WTW0x

ext (9)

Substituting Eq. (9) into Eq. (8) gives

ðW0 �WðWTWÞ�1WTW0Þxext ¼ 0 (10)

Let fW ¼ ðW0 �WðWTWÞ�1WTW0Þ be a matrix of size

ðJ0 þ 2J1Þ � Jext that contains all the linear kinematic equationsrelating all the external components. The rank of fW that deter-mines the kinematic dof of the PG system ranges from 1 to 3.From here, separate analyses need to be performed for hybrid andpure electric configurations.

(i) For hybrid configurations, a rearrangement is necessaryto obtain the kinematic relationship matrix of Eq. (10)in Sec. 2. Consider a specific case with xext ¼½x1;x2;x3;x4� being the vector of speeds from engine, ve-hicle output, MG1, and MG2, respectively. Let

½fW1;fW2� ¼ fW, where both fW1 and fW2 are of size

ðJ0 þ 2J1Þ � 2. When both fWT

2fW1 and fWT

2fW2 are inverti-

ble, rearrangement on Eq. (10) gives

x3

x4

� �¼ �ðfWT

2fW2Þ�1ðfWT

2fW1Þ

x1

x2

� �(11)

where the kinematic relationship matrix is Cmode ¼�ðfWT

2fW2Þ�1 ðfWT

2fW1Þ.

In the case where xext contains speeds from ground nodes,the same idea holds after the following additional step:Since the speed of a ground node is always zero, columns

of the matrix fW to be multiplied by the ground speed can

be removed. Then, the definition of fW1 and fW2 must be

modified as ½fW1;fW2;fWgnd� ¼ fW, where fWgnd are the col-umns to be removed. With that definition, Eq. (11) holdsfor the cases that include ground nodes.

(ii) For pure electric configurations, if an engine exists in theconfiguration, the following definitions can be made

½fWeng;fW1;fWgnd� ¼ fW, where fWeng is the column multi-plied by the engine speed. Since the engine is alwaysgrounded in a pure electric configuration, i.e., it is always

at zero speed, both fWeng and fWgnd can be removed. Then,

linearly independent rows of fW1 are used to model thekinematic relationship of pure electric configurations. Notethat while 1–3 dofs exist among the configurations gener-ated by the process, we are only interested in kinematically1-dof and 2-dof configurations. Three dof configurationsrequire a sophisticated control algorithm, and they are not

used in common practice. Therefore, only fW1 matrices withranks 1 and 2 are used for practicality. For a 1-dof configura-tion, the kinematic relationship matrix is defined as

1

0

� �x2 ¼ Cmode

x3

x4

� �(12)

Similarly, for a 2-dof configuration, the same kinematic matrixbecomes

x2 ¼ Cmodex3

x4

� �(13)

4 Multimode Architectures

We create multimode architectures by combining two or moreconfigurations under a single arrangement of its components.Recall that we call each particular configuration in a multimodearchitecture a driving mode. The main motivation to design multi-mode architectures is to benefit from switching among a variety of“specialized” modes, each of which operating efficiently underdifferent driving conditions. For instance, a mode that is designedto operate in highway driving conditions is not necessarily effi-cient in city driving conditions. Incorporating the two in onepowertrain can achieve better efficiency than a single modedesigned for both driving conditions.

An example for an architecture with multiple modes is given inFig. 1. Switching among the modes is achieved by engaging ordisengaging clutches. We represent every mode with a differentmodified bond graph. Since the number of clutches involved mustbe limited in practice, a systematic way to evaluate the requiredclutches for a multimode architecture is necessary during architec-ture design. In this section, we will describe the process to calcu-late the clutching solutions for given three modes and thengeneralize to any number of modes.

4.1 Clutching Solutions. We introduced the connectivitytable earlier in Sec. 2.2. A link must be formed between modifiedbond graph representation and connectivity table in order to auto-mate the process of finding clutching solutions.

We explain the process of extracting this information from themodified bond graph using a color analogy. The weights around a0-junction represents the kinematic relationships among sun, ring,and carrier gears and every bond around 0-junction can beassigned to a different color representing each of these respectivegears. In Fig. 3, the edge weights representing sun, ring, and car-rier gear are noted by the color designations red, green, and blue,respectively. As seen in this figure, since the speeds around the 1-junction are the same, the same coloring is kept around a 1-junction. When this process is repeated for all the bonds we cansee the connection information of each external component bychecking the color of the bond connecting an external componentto its corresponding external junction. In the case of a two-PGmode, we need to start with six colors; for three-PG modes, weneed nine colors, and so on. When multiple 0-junctions imposedifferent colors on a bond, i.e., when colors are mixed on a bond,it means this bond is connected to multiple PGs. Such color mix-tures give information on the internal connections among PGs.

Once the connectivity table is extracted from the modified bondgraph, one can calculate the number of clutches needed by com-paring the connectivity tables of the driving modes to be com-bined. Basically adding or removing a connection requires oneclutch, and changing a connection of an external component fromone gear to another requires two.

Figure 9 shows an example that compares connectivity tablesof three one-PG modes and identifies the clutches required. Weperform a pairwise comparison of the modes in the architecturesequentially. Starting with modes A and B, in the figure, mode Bhas an additional ground connected to the carrier that does notexist in mode A. Adding this ground connection requires oneclutch at the carrier gear. Also, MG2 is connected to the ring gearin mode A and connected to carrier gear in mode B. Disconnect-ing MG2 from the ring requires a clutch at the ring and connectingit to the carrier requires another clutch at the carrier gear. Compar-ing modes B and C, we can see that engine and ground, which areconnected in mode B, are disconnected in mode C. Disconnecting

Fig. 8 Bond weight scaling for a 0-to-0 junction connection

Journal of Mechanical Design AUGUST 2016, Vol. 138 / 081401-7

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

these components requires one clutch for each. We take the unionof these clutches to calculate the final number. In this example,we need four physical clutches to allow shifting among the threemodes. These four clutches are located at the connections from(1) MG2 to R1, (2) MG2 to C1, (3) ground to C1, and (4) engineto C1.

Note that this solution is independent from the order of themodes. If we follow the same steps switching from mode A tomode C and from mode C to mode B, we obtain the same set ofclutches, for instance. These steps can be automated in anarchitecture design process.

5 Results

The formulation presented in this paper can be used to generateall the possible configurations for systems with any number ofPGs and any number of external components. In addition to theHEV modes, by removing the engine from the external compo-nents or connecting the engine and ground to the same gear, pureelectric modes can also be generated.

The results we present in this section follow the causalityassignment for external junctions discussed in Sec. 3.2. Recall

that this assignment limits configurations to only power-splitdesigns.

Also, as discussed in Sec. 3.4, the rank of the matrix fW deter-mines the kinematic dof of the configurations generated. In thissection, we present results for 2-dof hybrid configurations andboth 1-dof and 2-dof pure electric configurations.

The process described above results in 52 unique configurationsfor one PG assuming one engine, two MGs, one ground, and onevehicle output. Among these 52 configurations, 16 of them arehybrid and 36 are pure electric ones. The results include someavailable designs in the market such as hybrid and pure electricconfigurations of the Chevrolet Volt architecture shown in Fig. 1.

The number of feasible configurations for two PGs with thesame powertrain components is 3420, where 2124 of them arehybrid and the rest 1296 are pure electric. Among these two-PGconfigurations, there are some designs that look different but givethe same quasi-static matrices Cmode. Their graph representationsare not isomorphic but are kinematically equivalent. When thegear ratios of the two PGs are both equal to 2, these 2124 hybridconfigurations have 1178 unique kinematic matrices, for instance.These kinematically equivalent designs should not be eliminatedin a design process since they might be useful for multimode

Fig. 9 Three sample connectivity tables and the corresponding clutching solution indicatedby dashed boxes

Fig. 10 All the modes of the dual-mode architecture by Ai and Anderson [23]: (a) firsthybrid configuration and (b) second hybrid configuration

Fig. 11 All the modes of the dual-mode architecture by Holmes et al. [22]: (a) first hybridconfiguration and (b) second hybrid configuration

081401-8 / Vol. 138, AUGUST 2016 Transactions of the ASME

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

architecture designs. For instance, two kinematically equivalentdesigns might result in different number of clutches needed in amultimode architecture. Also some of the configurations are onlyfeasible when PG ratios are different. Thirty-six of all the two-PGhybrid configurations give noninvertible Cmode matrices whentwo-PG ratios are the same. This situation occurs when engineand vehicle output shaft are connected to carrier gears of two dif-ferent PG sets, and ring-to-ring and sun-to-sun connection exitsbetween these two-PG sets. When PG ratios are the same, thevehicle and engine speeds are imposed to be the same due to thekinematics of the both PG systems, and thus, dof of this hybridconfiguration is reduced to 1. It appears as a noninvertible Cmode

in this study.The configuration generation process can create some available

designs in the literature such as those by Ai and Anderson [23]given in Fig. 10 and by Holmes et al. [22] given in Fig. 11.

6 Conclusions

In this paper, we presented a formulation to generate all the fea-sible hybrid and pure electric configurations relying on a modifiedbond graph representation. The formulation is general enough tobe extended to any set of powertrain components and any numberof PGs. We also presented the methodology to create multimodearchitectures using the modified bond graph representation ofthe generated designs. We identified sample designs among thegenerated configurations available in the literature.

The generation process can be integrated with a design evalua-tion and optimization framework to discover design architectureswith improved fuel economy for a target vehicle application.Examples for such a design process can be found in Refs. [30,31],and [40]. These studies design the architecture and gear ratios in ahybrid powertrain by relying on a given set of feasible configura-tions such as the configurations generated in this study.

Acknowledgment

This research has been partially supported by the GeneralMotors Corp. and the Automotive Research Center, a U.S. ArmyCenter of Excellence in Modeling and Simulation of GroundVehicle Systems headquartered at the University of Michigan.This support is gracefully acknowledged.

References[1] Folkesson, A., Andersson, C., Alvfors, P., Alak€ula, M., and Overgaard, L.,

2003, “Real Life Testing of a Hybrid PEM Fuel Cell Bus,” J. Power Sources,118(1), pp. 349–357.

[2] Gao, D., Jin, Z., and Lu, Q., 2008, “Energy Management Strategy Based onFuzzy Logic for a Fuel Cell Hybrid Bus,” J. Power Sources, 185(1),pp. 311–317.

[3] Evans, D. G., Polom, M. E., Poulos, S. G., Van Maanen, K. D., and Zarger,T. H., 2003, “Powertrain Architecture and Controls Integration for GM’sHybrid Full-Size Pickup Truck,” SAE Technical Paper No. 2003-01-0085.

[4] Lin, C.-C., Peng, H., Grizzle, J. W., and Kang, J.-M., 2003, “Power Manage-ment Strategy for a Parallel Hybrid Electric Truck,” IEEE Trans. Control Syst.Technol., 11(6), pp. 839–849.

[5] Liu, J., 2007, “Modeling, Configuration and Control Optimization of Power-Split Hybrid Vehicles,” Ph.D. thesis, Department of Mechanical Engineering,The University of Michigan, Ann Arbor, MI.

[6] Kicinger, R., Arciszewski, T., and Jong, K. D., 2005, “Evolutionary Computa-tion and Structural Design: A Survey of the State-of-the-Art,” Comput. Struct.,83(23), pp. 1943–1978.

[7] Bendsøe, M. P., and Kikuchi, N., 1988, “Generating Optimal Topologies inStructural Design Using a Homogenization Method,” Comput. Methods Appl.Mech. Eng., 71(2), pp. 197–224.

[8] Roth, B., 1967, “Finite-Position Theory Applied to Mechanism Synthesis,”ASME J. Appl. Mech., 34(3), pp. 599–605.

[9] Raghavan, M., 1989, “Analytical Methods for Designing Linkages to MatchForce Specifications,” Ph.D. thesis, Department of Mechanical Engineering,Stanford University, Stanford, CA.

[10] Koza, J. R., Bennett, F. H., III, Andre, D., Keane, M. A., and Dunlap, F., 1997,“Automated Synthesis of Analog Electrical Circuits by Means of GeneticProgramming,” IEEE Trans. Evol. Comput., 1(2), pp. 109–128.

[11] Schneider, G., and Fechner, U., 2005, “Computer-Based De Novo Design ofDrug-Like Molecules,” Nat. Rev. Drug Discovery, 4(8), pp. 649–663.

[12] Wu, Z., Campbell, M. I., and Fern�andez, B. R., 2008, “Bond Graph BasedAutomated Modeling for Computer-Aided Design of Dynamic Systems,”ASME J. Mech. Des., 130(4), p. 041102.

[13] Starling, A. C., and Shea, K., 2005, “A Parallel Grammar for Simulation-Driven Mechanical Design Synthesis,” ASME Paper No. DETC2005-85414.

[14] Campbell, M. I., Cagan, J., and Kotovsky, K., 2000, “Agent-Based Synthesis ofElectromechanical Design Configurations,” ASME J. Mech. Des., 122(1),pp. 61–69.

[15] Hsieh, H.-I., and Tsai, L.-W., 1996, “A Methodology for Enumeration ofClutching Sequences Associated With Epicyclic-Type Automatic TransmissionMechanisms,” SAE Technical Paper No. 960719.

[16] Kahraman, A., Ligata, H., Kienzle, K., and Zini, D., 2004, “A Kinematics andPower Flow Analysis Methodology for Automatic Transmission Planetary GearTrains,” ASME J. Mech. Des., 126(6), pp. 1071–1081.

[17] Conlon, B. M., Savagian, P. J., Holmes, A. G., and Harpster, M. O., 2011,“Output Split Electrically-Variable Transmission With Electric PropulsionUsing One or Two Motors,” U.S. Patent No. 7,867,124.

[18] Sasaki, S., 1998, “Toyota’s Newly Developed Hybrid Powertrain,” 10th Inter-national Symposium on Power Semiconductor Devices and ICs, IEEE Press,Kyoto, Japan, pp. 17–22.

[19] Schmidt, M., 1996, “Two-Mode, Split Power, Electro-Mechanical Trans-mission,” U.S. Patent No. 5,577,973.

[20] Schmidt, M., 1996, “Two-Mode, Input-Split, Parallel, Hybrid Transmission,”U.S. Patent No. 5,558,588.

[21] Holmes, A., and Schmidt, M., 2002, “Hybrid Electric Powertrain Including aTwo-Mode Electrically Variable Transmission,” U.S. Patent No. 6,478,705.

[22] Holmes, A., Klemen, D., and Schmidt, M., 2003, “Electrically VariableTransmission With Selective Input Split, Compound Split, Neutral and ReverseModes,” U.S. Patent No. 6,527,658.

[23] Ai, X., and Anderson, S., 2005, “An Electro-Mechanical Infinitely VariableTransmission for Hybrid Electric Vehicles,” SAE Technical Paper No. 2005-01-0281.

[24] Schmidt, M., 1999, “Two-Mode, Compound-Split Electro-Mechanical Vehicu-lar Transmission,” U.S. Patent No. 5,931,757.

[25] Raghavan, M., Bucknor, N. K., and Hendrickson, J. D., 2007, “ElectricallyVariable Transmission Having Three Interconnected Planetary Gear Sets, TwoClutches and Two Brakes,” U.S. Patent No. 7,179,187.

[26] Zhang, X., Li, C.-T., Kum, D., and Peng, H., 2012, “Priusþ and Volt�: Config-uration Analysis of Power-Split Hybrid Vehicles With a Single PlanetaryGear,” IEEE Trans. Veh. Technol., 61(8), pp. 3544–3552.

[27] Cheong, K. L., Li, P. Y., and Chase, T. R., 2011, “Optimal Design of Power-Split Transmissions for Hydraulic Hybrid Passenger Vehicles,” 2011 AmericanControl Conference, IEEE Press, San Francisco, CA, pp. 3295–3300.

[28] Zhang, X., Li, S. E., Peng, H., and Sun, J., 2015, “Efficient Exhaustive Searchof Power-Split Hybrid Powertrains With Multiple Planetary Gears andClutches,” ASME J. Dyn. Syst., Meas., Control, 137(12), p. 121006.

[29] Bayrak, A. E., Ren, Y., and Papalambros, P. Y., 2013, “Design of Hybrid-Electric Vehicle Architecture Using Auto-Generation of Feasible DrivingModes,” ASME Paper No. DETC2013-13043.

[30] Bayrak, A. E., Kang, N., and Papalambros, P. Y., 2015, “Decomposition BasedDesign Optimization of Hybrid Electric Powertrain Architectures: Simultane-ous Configuration and Sizing Design,” ASME J. Mech. Des. (accepted).

[31] Bayrak, A. E., Kang, N., and Papalambros, P. Y., 2015, “Decomposition-BasedDesign Optimization of Hybrid Electric Powertrain Architectures: Simultane-ous Configuration and Sizing Design,” ASME Paper No. DETC2015-46861.

[32] Benford, H. L., and Leising, M. B., 1981, “The Lever Analogy: A New Tool inTransmission Analysis,” SAE Technical Paper No. 810102.

[33] Chatterjee, G., and Tsai, L.-W., 1996, “Computer-Aided Sketching ofEpicyclic-Type Automatic Transmission Gear Trains,” ASME J. Mech. Des.,118(3), pp. 405–411.

[34] Kim, N., Kim, J., and Kim, H., 2008, “Control Strategy for a Dual-Mode Elec-tromechanical, Infinitely Variable Transmission for Hybrid Electric Vehicles,”Proc. Inst. Mech. Eng., Part D, 222(9), pp. 1587–1601.

[35] Karnopp, D. C., Margolis, D. L., and Rosenberg, R. C., 2012, System Dynam-ics: Modeling, Simulation, and Control of Mechatronic Systems, 5th ed., Wiley,Hoboken, NJ.

[36] Read, R. C., and Corneil, D. G., 1977, “The Graph Isomorphism Disease,”J. Graph Theory, 1(4), pp. 339–363.

[37] Fortin, S., 1996, “The Graph Isomorphism Problem,” University of Alberta,Edmonton, AB, Technical Report No. 96-20.

[38] McKay, B. D., 1981, “Practical Graph Isomorphism,” Congressus Numeran-tium, 30, pp. 45–87.

[39] MATLAB, 2014, “Graph Isomorphism,” The MathWorks, Natick, MA.[40] Bayrak, A. E., 2015, “Topology Considerations in Hybrid Electric Vehicle

Powertrain Architecture Design,” Ph.D. thesis, Department of MechanicalEngineering, The University of Michigan, Ann Arbor, MI.

Journal of Mechanical Design AUGUST 2016, Vol. 138 / 081401-9

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use