energy management control of a hybrid electric vehicle...

EVS26Los Angeles, California, May 6 - 9, 2012

Energy Management Control of a Hybrid Electric Vehicle withTwo-Mode Electrically Variable Transmission

Brian Bole1, Samuel Coogan2, Carlos Cubero-Ponce1, Derek Edwards1, Ryan Melsert1,David Taylor1

1Georgia Institute of Technology, Atlanta, GA, 30332, USA.2University of California, Berkeley, Berkeley, CA, 94720, USA.

AbstractA hybrid electric vehicle with a two-mode continuously variable transmission and four fixed gear operating modes

is modeled to enable a comparative investigation of the charge sustaining energy management control problem.

Control strategies are designed using three distinct approaches: (1) explicit operating logic is defined in terms

of comparison threshold values that can be tuned, using heuristic reasoning and/or model-based calculations,

to operate the engine efficiently; (2) a static optimization, applied at each time instant, is used to minimize

some measure of equivalent fuel consumption for the corresponding sequence of time instants; (3) a dynamic

optimization, applied just once over the appropriate time interval, is used to minimize the actual fuel consumption

over a drive cycle. The first approach requires only limited modeling information, and thus has the benefit of

quick and simple programming. The second approach is based on optimization so as to directly exploit available

modeling information. The third approach will find the vehicle control for optimal fuel efficiency, but it does not

lead to an implementable strategy since prior knowledge of the drive cycle is required.

Keywords: power-split hybrid, energy management control, equivalent fuel optimization, dynamic programming

1 IntroductionThe powertrain architecture selected for this investi-gation is a type of power-split architecture. Power-split architectures provide both a direct mechanicalpower path and an electromechanical power path be-tween the internal combustion engine and the wheels.This design combines the advantages of single-patharchitectures such as parallel (mechanical only) andseries (electromechanical only) architectures.

A simply-geared mechanical path is highly ef-ficient, i.e. approximately 95% of the power pro-vided by the engine can be transmitted to the wheelsthrough such a path; however, the engine speed isconstrained by just a few fixed gear ratios, leadingto potentially inefficient engine operation. An elec-tromechanical path allows only about 75% of engine

power to reach the wheels due to the double con-version of power through a generator and a motor;however, the resulting decoupling of the engine andthe wheels can be exploited to ensure efficient oper-ation of the engine. A well-known example of thecombined-path power-split architecture is employedin the Toyota Prius hybrid electric vehicle (HEV),in which one planetary gearset and two electric ma-chines are used to implement a one-mode electricallyvariable transmission (EVT). The particular type ofpower-split architecture considered in this publica-tion is a two-mode EVT with the optional capabilityto operate as a purely mechanical transmission hav-ing four fixed gear ratios. This technology was de-veloped by GM, beginning with a patent on the firstEVT architecture with two modes [1], evolving toa patent on the first EVT architecture with both an

EVS26 International Battery, Hybrid and Fuel Cell Electric Vehicle Symposium 1

input-split mode and a compound-split mode featur-ing planetary gearsets of both the single-planet typeand double-planet type [2], and culminating with apatent on the first EVT architecture as just describedto be extended with four fixed gear ratios [3].

Two-mode EVTs can be designed to operate moreefficiently than one-mode EVTs, due to the exis-tence of additional “mechanical points,” i.e. operat-ing points at which the electromechanical path car-ries no power, as explained in several recent publi-cations that compare recently introduced two-modearchitectures with the one-mode EVT employed inthe market-leading Toyota Prius THS-II [4, 5, 6]. Apresentation of improvements in powertrain perfor-mance and efficiency that are enabled by adding thecapacity for switching between fixed gear and power-split powertrain operation is given in [7]. A descrip-tion of the front-wheel-drive (FWD) GM two-modeinput-split compound-split EVT with four fixed ra-tios can be found in [8]. Mathematical modelingof the powertrain architecture disclosed in [8], i.e.the world’s first front-wheel-drive two-mode hybridtransaxle architecture appears in [9, 10]. The au-thors of [9] simulate an equivalent fuel consump-tion minimization energy management control strat-egy on a FWD THS-II one-mode HEV and a FWDGM two-mode HEV in order to analyze the behaviorof the energy management controller as it operateson each these powertrain architectures over a drivecycle. This paper uses similar vehicle modeling tothat presented in [9], to explore the comparative ef-fect that four distinct types of energy managementcontrol strategies have on overall vehicle fuel econ-omy and the instantaneous operating modes chosenfor powertrain components within a simulated FWDGM two-mode HEV over a particular mandated drivecycle.

2 Speed and Torque ConstraintsA stick diagram of the hybrid transmission consid-ered in this paper is shown in Fig. 1. There aretwo planetary gearsets denoted PG1 and PG2, twomotor-generator units denoted MGa and MGb, twobraking clutches denoted C1 and C4, and two rotat-ing clutches denoted C2 and C3. Power is receivedthrough the external input which is continuously con-nected to the ring gear of PG1, and power is transmit-ted through the external output which is continuouslyconnected to the carrier of PG2. Internally, PG1 andPG2 are continuously connected to each other, MGais continuously connected to PG1, and MGb is con-

MGAMGB

PG1PG2

C1

C2

C3C4

Figure 1: Stick diagram of two-mode EVT with four fixed-gear ratios.

Table 1: Clutch Table for Mode SelectionMode M C1 C2 C3 C4

EVT-1 1 onEVT-2 2 onFG-1 3 on onFG-2 4 on onFG-3 5 on onFG-4 6 on on

tinuously connected to both PG1 and PG2. The fourclutches allow additional connections to be made se-lectively, in order to impose the operating modes in-dicated in Table 1. The external output is a centrallylocated transfer gear rather than a longitudinally ori-ented drive shaft, since this transmission is part ofa transaxle assembly for a front-engine, front-wheel-drive vehicle.

The interconnections indicated in Fig. 1 and Ta-ble 1 lead to transmission constraint equations thatmust be derived separately for each mode of oper-ation. The speed and torque at the output (ωo,To)are dictated by driver demand, whereas the speedsand torques at the input (ωi,Ti) and at the motor-generator units (ωa,Ta) and (ωb,Tb) are not all prede-termined; design freedoms vary with mode selection.

As shown in Fig. 1, the transmission consideredhere is composed of two distinct types of planetarygearsets. For either type, modeling equations are de-rived using the following concepts. Kinematic speedconstraints are derived by enforcing speed agreementat points where gear meshing occurs; carrier speedωc, ring gear speed ωr and sun gear speed ωs areassumed positive for a common direction (e.g. allspeeds directed clockwise). Dynamic torque con-


straints are derived by imposing a power balance, andthese reduce to steady-state torque constraints if ac-celerations are neglected; carrier torque Tc, ring geartorque Tr and sun gear torque Ts are assumed posi-tive for a common direction (e.g.all torques directedinward).

The first planetary gearset is of the double-planettype. The kinematic speed constraint is

ωc1 =1

1−ρ1ωr1−

ρ1

1−ρ1ωs1 (1)

and the steady-state torque constraints are

−Tc1 = (1−ρ1)Tr1 =1−ρ1

−ρ1Ts1 (2)

if losses are neglected. Both relations are character-ized by the ratio of sun gear teeth to ring gear teeth,ρ1 = ns1/nr1.

The second planetary gearset is of the single-planettype. The kinematic speed constraint is

ωc2 =1

1+ρ2ωr2 +

ρ2

1+ρ2ωs2 (3)

and the steady-state torque constraints are

−Tc2 = (1+ρ2)Tr2 =1+ρ2

ρ2Ts2 (4)

if losses are neglected. Both relations are character-ized by the ratio of sun gear teeth to ring gear teeth,ρ2 = ns2/nr2.

Two relevant facts regarding planetary gearsetsare: if any two terminals are connected together, thenall three terminals will rotate at the same speed in thesame direction, i.e. the gearset becomes locked-up;if any one terminal is held stationary, then no powerflows through that terminal, i.e. the gearset reducesto a conventional two-terminal gearset. The speedsof planet gears internal to the planetary gearsets arenot expressed directly in the kinematic equations pre-sented in this publication; however, the maximum al-lowable rotation speeds of these gears will imposelimitations on the maximum allowable speed ratiobetween any two terminals of the planetary gearset,as described in [10].

2.1 Variable-Ratio Operating ModesFor the transmission modeling equations presentedhere, the positive directions for mechanical powerflow are taken to be into the input terminal, out of theoutput terminal, and out of both motor-generators.

In the EVT modes, a fraction of the input powercan be diverted to an electromechanical path, soit is possible to choose (ωi,Ti) independently from(ωo,To). Setpoints for MGa and MGb are imple-mented by imbedded speed controllers to enable thedesired input-output decoupling.

2.1.1 EVT-1

This is an input-split mode, achieved by holding thering gear of PG2 stationary. The input power is splitby PG1 whereas PG2 serves as a conventional two-terminal gearset. Using (1) and (3) with ωs2 = ωc1and ωr2 = 0, the speed constraints are

ωa =1ρ1

ωi−(1−ρ1)(1+ρ2)

ρ1ρ2ωo (5)

ωb =1+ρ2

ρ2ωo (6)

and using (2) and (4) the steady-state torque con-straints are

Ta =−ρ1Ti (7)

Tb =−(1−ρ1)Ti +ρ2

1+ρ2To (8)

From (5)–(6), there are two speed ratios at which oneof the electric machines does not rotate, and they aredetermined from ρ1 and ρ2 according to

ωo

ωi=

0 , ωb = 0

ρ2

(1−ρ1)(1+ρ2), ωa = 0

(9)

These speed ratios are called mechanical points be-cause they correspond to power taking a direct me-chanical path to the output instead of being partiallydiverted to the electromechanical path by one of theelectric machines.

2.1.2 EVT-2

This is a compound-split mode, achieved by lockingthe ring gear of PG2 to the sun gear of PG1. The inputpower is split twice, by PG1 at the input and by PG2at the output. Using (1) and (3) with ωs2 = ωc1 andωr2 = ωs1 , the speed constraints are

ωa =−ρ2ωi

1−ρ1−ρ1ρ2+

(1−ρ1)(1+ρ2)

1−ρ1−ρ1ρ2ωo (10)

ωb =1

1−ρ1−ρ1ρ2ωi−

ρ1(1+ρ2)

1−ρ1−ρ1ρ2ωo (11)


and using (2) and (4) the steady-state torque con-straints are

Ta =−ρ1Ti +1

1+ρ2To (12)

Tb =−(1−ρ1)Ti +ρ2

1+ρ2To (13)

From (10)–(11), there are two speed ratios at whichone of the electric machines does not rotate, and theyare determined from ρ1 and ρ2 according to

ωo

ωi=

ρ2

(1−ρ1)(1+ρ2), ωa = 0

1ρ1(1+ρ2)

, ωb = 0(14)

These speed ratios are called mechanical points forthe same reason stated above; note that both variable-ratio modes share a common mechanical point.

2.2 Fixed-Ratio Operating ModesAs indicated in Fig. 1 and Table 1, fixed-ratio modesare also available. Their inclusion enables enginepower to be transmitted mechanically to the output,without the necessity of power flow through the elec-tromechanical path; this permits use of electric ma-chines with lower power ratings. By disabling theless-efficient electromechanical path, fixed-ratio op-eration also provides lower fuel consumption if theengine operating point is near optimum. Moreover,if desired, the electric machines can also be used toprovide additional output power in fixed-ratio modes.

2.2.1 FG-1

Using (1) and (3) with ωs2 = ωc1 = ωr1 and ωr2 = 0,the speed constraints are

ωi = ωa = ωb =1+ρ2

ρ2ωo (15)

From (2) and (4) the steady-state torque constraint is

To =1+ρ2

ρ2(Ti +Ta +Tb) (16)

2.2.2 FG-2

Using (1) and (3) with ωs2 = ωc1 and ωr2 = ωs1 = 0,the speed constraints are

ωi =(1−ρ1)(1+ρ2)

ρ2ωo (17)

ωa = 0 , ωb =1+ρ2

ρ2ωo (18)


To =(1−ρ1)(1+ρ2)

ρ2Ti +

1+ρ2

ρ2Tb (19)

The speed ratio of this mode matches the boundarybetween the normal ranges of EVT-1 and EVT-2 op-eration. Since MGa does not rotate, it should not beenergized.

2.2.3 FG-3

Using (1) and (3) with ωs2 = ωc1 = ωr1 and ωr2 =ωs1 , the speed constraint is

ωi = ωa = ωb = ωo (20)


To = Ti +Ta +Tb (21)

2.2.4 FG-4

Using (1) and (3) with ωs2 = ωc1 = 0 and ωr2 = ωs1 ,the speed constraints are

ωi = ρ1(1+ρ2)ωo (22)ωa = (1+ρ2)ωo , ωb = 0 (23)


To = ρ1(1+ρ2)Ti +(1+ρ2)Ta (24)

Since MGb does not rotate in this mode, it should notbe energized.

3 Power Flow Analysis

3.1 Variable-Ratio Operating ModesThe idealized case of lossless power transmission forpropulsion with zero battery power is briefly consid-ered in order to illuminate the role of the electricmachines in the EVT modes, and to identify oper-ating conditions that should be avoided. Fig. 1 is re-drawn as shown in Fig. 2 to more clearly illustrate thepower flow paths as well as the preferred directionsfor power flow.

Operating speeds at PG1, PG2, MGa and MGb areall computed using (5)–(6) in EVT-1 or (10)–(11) inEVT-2. Torque splits at PG1 and PG2 are computedusing (2) and (4), respectively, whereas the torquesplits at MGa and MGb are computed using (7)–(8)in EVT-1 or (12)–(13) in EVT-2. Following this pro-cedure, the fraction of input power flowing througheach branch of Fig. 2 may be expressed as an explicitfunction of the ratio of output speed to input speed.


PG1

PG2

MGB MGA

PG1

PG2

MGB MGA

r r

c

rr

cc

c

s s

s s

Figure 2: Preferred propulsive power directions (left = EVT-1,right = EVT-2).

EVT-1:

Pa

Pi=−1+

(1−ρ1)(1+ρ2)

ρ2

ωo

ωi(25)

Pb

Pi=

Po

Pi− (1−ρ1)(1+ρ2)

ρ2

ωo

ωi(26)

EVT-2:

Pa

Pi= c1−

12+ c2

ωo

ωi+

(12+ c3 + c4

ωi

ωo

)Po

Pi(27)

Pb

Pi=

ωo

ωi−1

2−c1−c2+

(12− c3− c4

ωi

ωo

)Po

Pi(28)

Normalized power flow is shown as a function ofinput-output speed ratio in Fig. 3, where the sign con-vention for power flow direction is defined in Fig. 2.The values of ρ1 and ρ2 used in Fig. 3 are listed in Ta-ble 3; the black circles correspond to the mechanicalpoints defined by (9) and (14). Inspection of Figs. 2and 3 reveals the possibility of undesired circulatingpower loops, which require planetary gearsets to sup-port power levels exceeding the applied input power.

Case 1: All speed ratios in the range

0 <ωo

ωi<

ρ2

(1−ρ1)(1+ρ2)(29)

result in preferred power flow directions for EVT-1,but circulating power loops on the lower mesh andthe outer mesh for EVT-2. For this reason, EVT-1is generally the preferred variable-ratio mode withinthis speed ratio range.


ρ2

(1−ρ1)(1+ρ2)<

ωo

ωi<

1ρ1(1+ρ2)

(30)

result in preferred power flow directions for EVT-2, but a circulating power loop on the upper mesh

0 0.5 1 1.5 2

-1

0

1

Pc

1 / P

i

0 0.5 1 1.5 2

-1

0

1

Ps

1 / P

i

0 0.5 1 1.5 2

-1

0

1

Pb / P

i

0 0.5 1 1.5 2

-1

0

1

Pa / P

i

0 0.5 1 1.5 2

-1

0

1

Ps

2 / P

i

o /

i

0 0.5 1 1.5 2

-1

0

1

Pr2 / P

i

o /

i

Figure 3: Normalized propulsive power values (blue = EVT-1,green = EVT-2).

for EVT-1. For this reason, EVT-2 is generally thepreferred variable-ratio mode within this speed ratiorange.


1ρ1(1+ρ2)

<ωo

ωi< ∞ (31)

result in a circulating power loop on the upper meshfor EVT-1 and circulating power loops on the uppermesh and the outer mesh for EVT-2. For this reason,this speed ratio range should generally be avoidedwhen using variable-ratio modes.

Following this logic for propulsion with constantengine speed, vehicle launch would occur in EVT-1with all the engine power flowing through the elec-tromechanical path, with MGa generating and MGbmotoring. As the vehicle speed increases, the enginepower flowing through the electromechanical pathwould decrease until it reaches zero at the mechan-ical point shared by EVT-1 and EVT-2. At this point,whether in EVT-1 or EVT-2, no power flows throughthe electromechanical path or through the ring gearof PG2 and, consequently, a smooth shift from EVT-1 to EVT-2 is made possible. As the vehicle speedincreases up to the second mechanical point of EVT-2, now with MGb generating and MGa motoring, thepercent engine power flowing through the electrome-chanical path increases from zero up to a peak valueof 27.2% and then decreases back to zero.


4 Energy Management Control

4.1 Problem FormulationThe purpose of the transmission is to provide a de-manded output torque To at a given output speed ωo,within limits. The required output power Toωo maybe derived from engine power Tiωi and/or from bat-tery power Pb, both of which are magnitude limited.The battery is an energy buffer that is not rechargedby external means, so the battery state of charge(SOC) x must be maintained within acceptable lim-its at all times. Any recharging of the battery takesplace on-board by operating one or both electric ma-chines as generators. The only sources of mechan-ical energy for on-board recharging are: the kineticand potential energies recovered from the vehicle’smass during braking, and the thermal energy releasedwithin the engine by the fuel combustion process.

The energy management controller is responsiblefor selecting a transmission operating mode M, theengine-fuel on/off status E, battery power Pb, andany speeds and torques that are free to assign to theengine and electric machines for the selected mode.Ideally, these selections would be made so as to sat-isfy driver power demands, maintain battery state ofcharge, and minimize fuel consumption and emis-sions. The sensor signals that could be used in theselection process include: vehicle speed, acceleratorand brake pedal positions, and battery SOC.

Fixed-gear modes afford the energy managementcontroller degrees of freedom in the selection of en-gine and electric machine torques that may be ex-ploited to efficiently meet a given driver demandedoutput torque, which is interpreted from the gas andbrake pedal positions. The EVT modes will affordthe energy management controller even more degreesof freedom; in this case, both engine torque and speedcan be selected without regard to the output torque-speed operating point, since the electric machinescan then be operated to impose the desired operatingpoint.

Energy usage within the vehicle’s powertrain willalways be greater than propulsive energy delivered tothe vehicle’s wheels, due to inefficiencies associatedwith the instantaneous power conversions occurringwithin the powertrain’s components. The thermal ef-ficiency of the internal combustion engine, the powerefficiency of the electric machines, and the electricalenergy loss that is associated with sinking and sourc-ing power from the battery, are each dependent on themanner in which the energy management controllerselects from available operational degrees of freedom

at each instant on a given drive cycle.The thermal efficiency of the engine is denoted

ηi(Ti,ωi) =Tiωi

Hlm f(32)

where Hl denotes the lower heating value of the fueland m f denotes the mass flow rate of the fuel.

The electric machines and their power electronicconverters are characterized by their power efficien-cies

η?(T?,ω?) =

T?ω?

Pe,?, T?ω? > 0

Pe,?

T?ω?, T?ω? < 0

?= a or b (33)

where Pe,? denotes the electrical power flowing intothe power electronic converter of electric machine ?.

Clutch-selectable interconnections within thetransmission will determine specific constraints onfeasible speeds and torques of powertrain com-ponents for each mode of operation, as has beendescribed. On the basis of this type of transmissionmodel, three approaches to energy management con-troller development are explored: heuristic control,real-time optimization-based control, and optimalcontrol via dynamic programming.

4.2 Heuristic ControlA straightforward method for designing an energymanagement controller is to rely on explicit operat-ing logic defined in terms of comparison thresholdvalues. In this case, the threshold values are definedin an attempt to ensure efficient operation of the en-gine over a drive cycle, while keeping the batterySOC within specified upper and lower bounds; thesethreshold values may be tuned heuristically, or withthe aid of modeling information.

The simplistic heuristic strategy explored in thispublication makes heavy use of the EVT modes,which allow the engine operating point to be cho-sen independently of vehicle state. Because highefficiency operation of the engine corresponds to ahigher output power than is typically needed by thedriver, the heuristic strategy is designed to alternatebetween a charging mode, in which the engine is runat a high power (high efficiency) operating point, anda discharging mode, in which the engine is turned offor run at a lower power (lower efficiency) operatingpoint. Switching from one mode to the other occurswhen upper or lower SOC boundaries are reached.


Also, if the engine would be operated efficiently atany time in a fixed gear mode, with no electric as-sist, then that mode is chosen. This strategy is imple-mented by employing the rules listed in Fig. 4. Thedesign parameters in this strategy are the low-powerengine setpoint (TiL ,ωiL) and the high-power enginesetpoint (TiH ,ωiH ), as well as the threshold values η∗i ,v1/2, voff, xmin, and xmax, that are defined in Table3. The variables v and x are used to represent ve-hicle speed and battery state of charge respectively;T k

i and ωki are used to represent the engine torque

and speed that would be required to produce a givencommanded powertrain output torque and speed ifthe fixed gear ratio k were active.

Heuristic or fuzzy logic based controls are com-monly used in industry and academic applicationsof hybrid vehicle controls [11, 12]. While heuristiccontrollers can provide an effective means of exploit-ing heuristic knowledge available to controls design-ers regarding efficient operation of the hybrid power-train, it should be expected that optimization-basedcontrols will result in significant improvements inoverall efficiency, assuming the availability of suf-ficient vehicle modeling information and on-boardcomputational power. In Section 5, the results ob-tained using the simplistic heuristic control strategyare contrasted with the comparatively more complexcontrols resulting from direct numerical optimizationof a cost function at each control time-period.

4.3 Optimization-Based ControlThe energy management control problem for maxi-mizing the fuel efficiency of a charge sustaining ve-hicle can be expressed as the following constrainedoptimization problem:

minimize m f (T ) =ˆ T

0m f (u,w) dt (34)

subject to x = f (x,u,w), x(0) given (35)xmin ≤ x≤ xmax, for 0≤ t ≤ T (36)x(T )≈ x(0) (37)

where the objective of this optimization problem is tominimize the mass of fuel m f (T ) consumed during adrive cycle of duration T . The drive cycle is charac-terized by the exogenous input w, which dictates a ve-hicle speed versus time requirement; or equivalently,if vehicle and physics are known, then w dictates thetransmission output speed and toque required of theenergy management controller at each time instantover a given drive cycle. The rate of fuel consump-tion at any time instant will depend on the nature of

Criterion for fixed-ratio modes (engine only):

set δk = ηi(T ki ,ω

ki )−η∗i , k = 1, . . . ,4

set ∆ = max{δk : k = 1, . . . ,4}if ∆ > 0

select FG-k such that ∆ = δkelse

select an EVT modeend

Criterion for variable-ratio modes (engine on/off):if v < v1/2

select EVT-1else

select EVT-2endif charge discharge mode = discharge

if v < voffturn engine off

elseoperate engine at low power, (TiL ,ωiL)

endelse

operate engine at high power, (TiL ,ωiL)end

Battery operation mode:if x < xmin

set: charge discharge mode = chargeelse if x > xmax

set: charge discharge mode = dischargeend

Figure 4: Heuristic control operating logic.

the drive cycle, through w, as well as on the controlinput u, which includes all choices to be made by theenergy management controller. The battery’s SOCsatisfies a dynamic equation involving battery power,and hence is also influenced by w and u.

An inherent trade-off is evident in the optimiza-tion problem: over-reliance on engine power will un-duly degrade the objective (34) but will allow theconstraints, (36) and (37) to be more easily satisfied;on the other hand, generous use of battery power en-hances the objective but cannot be sustained withoutviolating the constraints. Solution of this problem istherefore non-trivial.

If driver desired performance were known fromto to T , then the optimization problem expressedin (34)-(37) could be solved directly. The require-


ment of a priori knowledge means that a direct so-lution of (34)-(37) is generally non-causal and non-implementable in practice; however, computing theoptimal control solution for certain reference drivecycles is useful for comparative purposes. Dy-namic programming [13], linear programming [14],and optimal control theory [15] have been shown togenerate solutions for the non-implementable opti-mal HEV control strategy. Implementable optimiza-tion approaches will be causal and real-time exe-cutable. At each control time-step the supervisorycontroller will minimize an engineered cost function,using only information regarding past operating con-ditions to estimate the best present powertrain set-points for a presently desired transmission torque-speed output. Section 4.4 discusses an optimal yetnon-implementable approach to supervisory controland this section will introduce strategies for real-timeimplementable optimization-based controls.

The most common form of the real-timeoptimization-based control for charge sustain-ing parallel-hybrid vehicles is an equivalent-consumption minimization (ECM) strategy [16].This strategy will select a single powertrain operatingpoint at each time-instant from a range of feasibleoperating points by minimizing an instantaneousperformance function,

P = Pf + sPb (38)

where Pf = Hlm f represents the instantaneous fuelpower consumed by potential powertrain operatingmodes, Pb represents instantaneous battery powerdraw, and s is an equivalence factor used to equatethe cost of instantaneous electrical power and fuelpower consumption. Assuming that sufficient com-putational power is available to solve for the power-train control that minimizes (38) at each control time-instant, then the performance of this type of energymanagement controller is determined solely by themanner in which the equivalence factor is specified.

Because all electrical energy consumed in a chargesustaining vehicle ultimately comes from burningfuel, the equivalence factor could be considered tohave a “true” evaluation, which would exactly con-vert present electrical power consumption into anequivalent future liquid fuel consumption. Although,evaluation of this “true” equivalence factor is gener-ally impossible in practice, due to the dependence ofthermal, electrical, and mechanical energy exchangeson future driver inputs and future powertrain controls.In fact, if evaluation of the “true” equivalence factorwas possible in practice, then minimization of (38) ateach control time-step would be equivalent to a direct

minimization of the global optimization problem ex-pressed in (34). Several recent publications on ECMhave explored strategies for equivalence factor eval-uation that attempt to approximate the “true” equiva-lence factor at a given control time-step using proba-bilistic estimation of future vehicle demands and cor-responding future powertrain controls [17, 18] .

The more typical approach to ECM, which is alsoutilized in this publication, does not explicitly in-corporate modeling of future powertrain operation inthe evaluation of an equivalence factor, but ratheruses an equivalence factor that is based only onthe present operating environment and the battery’spresent SOC. In general, such ECM strategies willrely on empirical testing procedures to tune the pa-rameters of an equivalence factor evaluation model.Ideally, the equivalence factor will result in an energymanagement controller that makes heavy use of thebattery as an energy sink/source to enable efficientoperation of all powertrain components at each con-trol time-step, while also ensuring that battery SOCwill always remain within specified upper and lowerbounds. Publications on this topic vary widely in themethodology used to specify the equivalence factor;common strategies include: a constant value foundexperimentally [19], a function that varies the equiva-lence factor with battery SOC [20], and mixed strate-gies that use the current output speed and torque de-manded of the drivetrain as well as SOC deviationsto adjust the equivalence factor [21]. This publica-tion explores two approaches for equivalence factorselection; the first approach represents a simplisticstrategy in which the equivalence factor is varied asa linear function of battery SOC, and the second ap-proach introduces a more complex, but more effec-tive, mixed strategy, in which the equivalence factoris varied based on the efficiencies of presently avail-able engine and electric machine operating points, aswell as the present battery SOC.

4.3.1 SOC-Based Equivalence Factor

This strategy updates the equivalence factor at eachcontrol time-instant using a piecewise linear functionof SOC,

s =

smax , x < xmin and Pb > 0smin , x > xmax and Pb < 0mx+b , otherwise

(39)

Observe that for m < 0, an increasing SOC will causethe equivalence factor to decrease, which will putmore weight on fuel consumption minimization, re-sulting in higher electrical power consumption and


less energy storage in the battery; conversely, a de-creasing SOC will result in higher fuel consumptionand greater energy storage in the battery. Repeatedsimulations of the real-time energy management con-trols on a range of sample drive cycles were used tofind values for m and b that resulted in high long-run fuel efficiencies, without violation of the batterySOC constraints. Boundary conditions were added to(39) to heavily penalize SOC excursions beyond thespecified bounds; however, if m and b are chosen ap-propriately the battery SOC should rarely approachthese bounds. The values of m and b used in the sim-ulations described in Section 5 are given in Table 3.

4.3.2 Adaptive Piecewise-Linear EquivalenceFactor

After some tuning, the SOC-based linear equiva-lence factor can result in high fuel efficiencies anda charge sustaining behavior over a wide range ofdrive-cycles; however, more optimal results can beobtained by supplementing the linear dependence onSOC with additional linear dependencies on the pow-ertrain and engine efficiencies corresponding to po-tential powertrain operating points. The improvedstrategy adjusts the parameters of (39) based on theengine thermal efficiency and electrical power trans-mission efficiencies of feasible powertrain operatingpoints, using the following expressions:

ηe =

Taωa +Tbωb

Pe,a +Pe,b−Pb,loss, discharge

Pe,a +Pe,b−Pb,loss

Taωa +Tbωb, charge

(40)

α =c1(1−ηi)+ c2(1−ηe)

Pb(41)

m = m′+α (42)

b = b′+α (43)

where Pb,loss denotes the electrical power dissipationwithin the battery, ηe represents the power efficiencyof the combined the electrical path, which includesboth electric machines and the battery, and c1, c2, m′,and b′ are weighting coefficients found via repeatedsimulations on a variety of drive cycles. A discussionof parameter tuning for a similar ECM approach isgiven in [21]. The values of c1, c2, m′, and b′ usedin the simulations described in Section 5 are given inTable 3.

4.4 Optimal ControlTo determine an upper bound on achievable perfor-mance, the solution of the optimal control problemgiven in (34)–(37) was approximated using dynamicprogramming. The constraints on x, given in (36),are enforced over a given drive-cycle, and an equal-ity constraint is imposed on the initial and final valuesof battery SOC, x(T ) = x(0).

The dynamic programming implementation in-volves two steps. In the first step, an exhaustivesearch is performed in order to construct a table thatidentifies feasible values of

{M,(Ti,ωi),(Ta,ωa),(Tb,ωb)}

that minimize the instantaneous value of m f for aquantized set of (To,ωo,Pb), where M represents theopen or closed state of the transmission’s clutches.By resolving this issue first, the number of choices re-maining to completely specify the transmissions op-erating state at each time instant on a drive cycle isreduced to just the battery power. The only compo-nent of u that directly links to (35)–(37) is batterypower, so it makes intuitive sense to first solve forall components of u that minimize m f , in terms ofPb, and then to manipulate Pb to minimize (34) usingoptimal control methods.

In the second step, the well-known dynamic pro-gramming method is employed to approximate thesolution of optimal control problem. The table pro-duced in the first step is leveraged to permit efficientcomputation of the optimizing time trajectory of bat-tery power Pb, at which point all components of thetransmission operating state are determined.

5 Simulation ResultsDrive cycle simulation results are provided based onthe standard model of road load, the transmissionmodel of Section 2, the energy management con-trollers of Section 4, and the parameter values listedin Table 3. The engine and motor-generator unitsare modeled by their steady-state efficiency mapsηi(Ti,ωi), ηa(Ta,ωa) and ηb(Tb,ωb).

The drive cycle is translated into (To,ωo) demandsat one-second intervals. To facilitate comparisons,initial and final values of battery state of charge wereforced to be nearly equal, meaning that the net en-ergy used over the drive cycle is accurately repre-sented by fuel consumption alone. This constraintis explicit in dynamic programming, but for the othercontrol methods it was imposed through appropriate


Table 2: Fuel Consumption on UDDS Drive CycleControl Method mpg

Heuristic 48SOC-Based Equivalence Factor 48

Adaptive Piecewise-LinearEquivalence Factor 54

Non-Causal Dynamic Programming 56

Table 3: Numerical Values for Simulationsparameter value unit

mass 1746 kgfrontal area 2.642 m2

drag coefficient 0.386tire radius 0.352 mfinal drive 3.29(ρ1,ρ2) (44/104,37/83)

Eb 9.6 kWh(Pb,min,Pb,max) (−50,100) kW(xmin,xmax) (0.3,0.7)

η∗i 0.3(v1/2,voff) (30,30) mph(TiL ,TiH ) (20,120) Nm(ωiL ,ωiH ) (900,3000) rpm(smin,smax) (0,10)

(m,b) (−1.69,3.46)×103

(m′,b′) (−1.33,3.51)×103

(c1,c2) (26.6,4.44)×103 kWHl 44.4 MJ/kg

choice of the initial battery SOC. Since all of the con-trollers are forced to follow the drive cycle exactly,the energy required to move the vehicle is identicalfor each; the differences in the energy expended byeach are due solely to differences in waste heat losscaused by inefficient operation of components.

Table 2 summarizes the miles-per-gallon (mpg) es-timates obtained using the EPA urban dynamome-ter driving schedule (UDDS). The dynamic program-ming method results in the largest mpg estimate, asexpected. The heuristic approach, which was de-signed to maximize engine operating efficiency, re-turns an impressive mpg estimate. The ECM strat-egy with an SOC-based equivalence factor returns thesame mpg estimate as the heuristic approach, using anumerical method rather than expert rules. Finally,the adaptive equivalence factor used in the secondECM method gives an mpg estimate that is far su-perior to the other implementable methods.

Figure 6 shows engine operating points superim-posed on the engine efficiency map and a histogramof the amount of time spent in each mode for eachof the control strategies executed on the UDDS.

Figure 5: Electric machine efficiency map (top) and operatingpoints dictated by dynamic programming for the UDDS.

Note that although the heuristic and piecewise-linearequivalence factor approaches both give the samempg estimate, they control the engine and transmis-sion much differently over the drive cycle. It wouldseem from the results that the heuristic strategy doesthe best job of optimizing engine efficiency on thedrive cycle, even surpassing that of dynamic pro-gramming; however, this approach fails to accountfor inefficient operation of the electric machines andthe battery. As can be seen from the optimization-based results, it is sometimes best to operate the en-gine inefficiently if that will facilitate a more efficientoperation of the other components.

Figure 5 shows the operating points for MGaand MGb selected by dynamic programming on theUDDS. This figure shows that the optimal control op-erates the electric machines well within their powerlimits and, as with engine operation, it seems thatit is sometimes best to operate the electric machinesin lower efficiency regions in order to maximize theoverall efficiency of the powertrain on the drive cy-cle.


(a) Heuristic control.

(b) Optimization-based control using a piecewise-linear equivalence factor.

(c) Optimization-based control using an adaptive piecewise-linear equivalence factor.

(d) Optimal control using non-causal dynamic programming.Figure 6: Comparative results for energy management methods on the UDDS. Left plots show engine operating points at eachsecond of the drive cycle and right plots show the time spent in each mode; the green bars show the time spent with the engine off.


6 ConclusionsHeuristic, optimization-based, and optimal energymanagement control strategies were compared forthe front-wheel-drive GM two-mode input-splitcompound-split EVT with four fixed ratios. Simu-lation results demonstrated that although the degreesof freedom available to the energy management con-troller could be used to largely avoid inefficient oper-ation of the internal combustion engine or the trans-mission’s electromechanical path at any particulartime instant within a drive cycle, it is sometimes nec-essary to operate the components in lower efficiencyregions in order to maximize the overall fuel econ-omy of a charge sustaining HEV over a drive cycle.

The analysis presented in this paper omits somemodeling details, such as mass factor and gearinginefficiencies which could significantly improve theaccuracy and practicality of the developed controlstrategies. The control strategies described in thiswork are based upon static operating points that donot include the vehicle dynamics that come into playwith changing vehicle speed, torque, or mode. Ve-hicle drivability, which includes handling and per-formance as interpreted by the driver were not con-sidered. Also, accessory electrical loads such asthose caused by air-conditioning, engine manage-ment, lighting, and multimedia among others werenot included in the electrical load analysis.

AcknowledgmentThis work was supported by: EcoCAR Challenge,a three-year collegiate advanced vehicle technologyengineering competition established by the Depart-ment of Energy and General Motors and managedby Argonne National Laboratory; and by the Depart-ment of Energy: Award Number DE-EE0002627.

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AuthorsBrian M. Bole graduated from FloridaState University in 2008 with a B.S.in Electrical and Computer Engineer-ing and a B.S. in Applied Mathemat-ics. He received a M.S. degree inElectrical Engineering from GeorgiaTech in 2011 and he is currently pur-suing Ph.D. His research interests in-clude: failure prognostics, stochasticoptimization, and fault adaptive con-trol systems.

Samuel Coogan graduated from theGeorgia Institute of Technology witha B.S. in Electrical Engineering in2010. He is currently pursuing M.S.and Ph.D. degrees in the Electrical En-gineering and Computer Sciences De-partment at the University of Califor-nia, Berkeley.

Carlos Cubero-Ponce was a controlsgroup leader for the Georgia Tech Eco-CAR team. He led the team’s develop-ment of Hardware in Loop (HIL) and asupervisory control implementation ona DSpace controller. He is currentlypursuing a Ph.D in Electrical Engineer-ing at Georgia Tech.

Derek Edwards received his B.S. andM.S. degrees in Electrical and Com-puter Engineering from Georgia Techin 2006 and 2009, respectively. He iscurrently pursuing his Ph.D. with anemphasis on intelligent transportationsystems. His research focuses on opti-mal routing in passenger transportationsystems. Specifically, his work seeksto provide demand-responsive publictransportation options in low-densityurban areas that are unsuited to tradi-tional modes of transit.

Ryan Melsert graduated from PennState University in 2004 with a B.S.in Mechanical Engineering and a B.S.in Engineering Science. He receivedan M.S. in Mechanical Engineering in2007 and an MBA in 2010 from theGeorgia Institute of Technology. Hisresearch at the Strategic Energy Insti-tute focuses on developing innovativeshort-term high-impact solutions to ourglobal energy challenges. Specifi-cally, on the development of cellulosicbiofuels, solar thermal power gener-ation, hybrid-electric vehicle design,and thermal systems design.

David Taylor received his B.S. in Elec-trical Engineering from the Universityof Tennessee, Knoxville, in 1983. Heearned M.S. and Ph.D. degrees in Elec-trical Engineering from University ofIllinois, Urbana-Champaign, in 1985and 1988, respectively. He has beenwith Georgia Tech since 1988, wherehis teaching and research activitieshave encompassed both the theory andapplication of control systems, with aparticular emphasis on control of elec-tromechanical systems such as roboticmotion systems, electric machine sys-tems and power electronic systems.


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