A Comparative Study of Supervisory Control Strategies for Hybrid Electric Vehicles

506 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 3, MAY 2007

A Comparative Study Of Supervisory ControlStrategies for Hybrid Electric Vehicles

Pierluigi Pisu and Giorgio Rizzoni, Fellow, IEEE

Abstract—Hybrid electric vehicles (HEVs) improvements infuel economy and emissions strongly depend on the energy man-agement strategy. The parallel HEV control problem involvesthe determination of the time profiles of the power flows fromthe engine and the electric motor. This is also referred to as thepower split between the conventional and the electric sources.The objective of HEV control is in fact to find out the sequenceof optimal power splits at each instant of time that minimizes thefuel consumption over a given driving schedule. Big obstacles tothe control design are the model complexity and the necessity of“a priori” knowledge of torque and velocity profiles. This paperpresents three different energy management approaches for thecontrol of a parallel hybrid electric sport-utility-vehicle that donot require a priori knowledge of the driving cycle. The consideredapproaches are: a rule-based control, an adaptive equivalentfuel consumption minimization strategy (A-ECMS), and thecontrol. Results, compared with the optimal solution given by thedynamic programming, show that the A-ECMS strategy is thebest performing strategy.

Index Terms—Automotive, hybrid electric vehicles, robust con-trol, supervisory control.

NOMENCLATURE

Uncertainty matrix.

State matrix.

Input matrix.

Output matrix.

Feedforward matrix.

Available chemical energy.

Maximum electrical energy that can bestored in the batteries.

Stored electrical energy.

Lower linear fractional transformation.

Upper linear fractional transformation.

Low heating value of fuel.

Identity matrix.

Identity matrix of size .

Manuscript received May 24, 2006; revised October 30, 2006. Manuscriptreceived in final form February 2, 2007. Recommended by Associate EditorK. Fischbach.

P. Pisu is with the CU-ICAR, Department of Mechanical Engineering,Clemson University, Clemson, SC 29634 USA (e-mail: [email protected]).

G. Rizzoni is with the Center for Automotive Research, The Ohio State Uni-versity, Columbus, OH 43212 USA (e-mail: [email protected]).

Color versions of Figs. 2, 4-8, and 14-18 are available online at http://ieeex-plore.ieee.org.

Digital Object Identifier 10.1109/TCST.2007.894649

Total optimal cost.

Cost associated with the policy .

Optimal cost at time step .

Complex matrix.

Available chemical power.

Battery output electric power.

Available electrical power.

Electric motor power.

Power provided by the electric motor atthe engine shaft.Engine power.

Power demand.

Nominal battery capacity.

Engine stroke.

State of charge.

, Minimum and maximum state of charge.

State of energy.

, Minimum and maximum state of energy.

Desired state of energy.

Sample time.

Electric motor available torque.

Electric motor effective torque.

Electric motor torque loss.

Engine available torque.

Engine effective torque.

Engine torque loss.

Driver torque request.

Final time.

Battery voltage.

Active rotor volume.

Engine displacement.

, Uncertainty weight.

Control policy from to .

Optimal control policy from to .

, Normalized uncertainty.

, Small positive number.

Engine efficiency.

Electric motor efficiency.

Battery efficiency.

1063-6536/$25.00 © 2007 IEEE

PISU AND RIZZONI: A COMPARATIVE STUDY OF SUPERVISORY CONTROL STRATEGIES FOR HYBRID ELECTRIC VEHICLES 507

Average battery charging efficiency.

Average battery discharging efficiency.

Belt efficiency.

Efficiency of the electrical path.

gain.

Unit step function.

, Constant weight.

Belt coupling ratio.

Final drive ratio.

Gear ratio.

Weight for NO .

Time-varying parameter vector.

Electric motor speed.

Engine speed.

Reduced state vector.

Speed at the rotor surface.

Mean piston speed.

Internal motor efficiency.

Internal engine efficiency.

Penalty function.

Battery current.

Engine fuel rate.

NO equivalent fuel consumption.

Electric motor equivalent fuelconsumption.Gear number.

System matrix.

Auxiliary input vector.

Electric motor mean effective pressureloss.Mean effective pressure.

Engine mean effective pressure loss.

Auxiliary output vector.

Rotor radius.

Tunable parameter.

Input vector.

Disturbance vector.

State vector.

Normalized state of charge.

Measured output vector.

Controlled output vector.

I. INTRODUCTION

HYBRID electric vehicles (HEVs) can achieve better fueleconomy and lower exhaust emissions by using two

sources of energy, namely, a fuel and stored electrical energy.At any time and for any vehicle speed, the control strategy hasto determine the power distribution between primary energy

Fig. 1. Mechanical powertrain configuration of the Ohio State Future Truck2004.

converter and renewable electrical storage system (RESS),as well as the optimal gear ratio of the transmission, if any.Regardless of the vehicle topology, the primary objective ofany control strategy is to satisfy the driver’s power demandby managing the power flows from the various energy storagedevices to minimize fuel consumption and simultaneouslysatisfying other constraints such as regulation of the RESSstate of charge (SOC) or state of energy (SOE), emissions anddrivability. In order to meet these requirements, many optimalcontrol strategies for HEVs have been proposed in the past.In particular, they can be classified in three groups: dynamicprogramming approaches [1]–[3], intelligent control techniquessuch as rule-based, fuzzy logic [4], and neural networks [5],and methods based on the conversion of the electric power intoequivalent fuel consumption [6], [7].

In this paper, a pretransmission belt-coupled parallel architec-ture has been chosen for the hybrid configuration (see Fig. 1).This vehicle was designed at the Ohio State University for theFutureTruck 2004 competition. The engine chosen for the OhioState FutureTruck was a 103-kW, 2.5-L, direct injection Dieselengine made by VM Motori. A Siemens 18/42-kW inductionmotor was chosen as the electric machine. The engine is the pri-mary source of power while the electric machine assists it pro-viding extra torque when requested and serving as a generatorfor both braking energy recuperation and for converting fuel en-ergy to electric to recharge the batteries. A control strategy op-timizing the instantaneous power split between the engine andelectric machine was developed in order to meet the followingenergy management objectives:

• minimization of fuel consumption;• lower NO emissions;• good drivability;• performance comparable to production model;• battery state of charge management.A rule-based control technique is first described [8]. A

finite-state machine (FSM) with eight states switches amongthe possible driving situations according to event-triggeredrules that depend on the brake and accelerator pedal angle, thestate-of-charge (SOC) of the battery and the request of torque.An adaptive equivalent fuel consumption minimization strategy(A-ECMS) is introduced next. The problem of a global opti-mization over a given cycle is brought back to an instantaneousminimization of the fuel consumption [6], [7]. The cost of theuse of the electric motor is expressed in terms of fuel throughtwo parameters representing the efficiency of the energy pathin charge and discharge mode. The choice of these parameters


is critical and an adaptive algorithm estimates their optimalvalues in order to satisfy the charge-sustaining constraint andto achieve best performance. Next, we propose an controltechnique in order to determine an output feedback controllerthat minimizes fuel consumption with respect to a family ofpossible torque/power input profiles. From a quasi-static modelof the vehicle [9] a linear fractional representation (LFR) isobtained [10]. The model is described by a “nominal” lineartime-invariant (LTI) system connected to an uncertainty blockand the controller is finally derived [11]. Finally, the last ap-proach proposed in this paper assumes the knowledge of thedriving cycle. In this case, the solution to the global optimiza-tion problem can be found by means of dynamic programming[1]. The optimal torque split policy can be used as a referencefor suboptimal strategies [12]. For computational reasons, asimplified model of the vehicle is adopted in this case.

II. VEHICLE MODEL

In general, at the design and optimization stage, it is highlydesirable to perform extensive simulations of HEVs with dif-ferent powertrain configuration and components. To realisticallyassess the best benefits which can be realized with each configu-ration, the control strategy must be adapted to the characteristicsof the components considered.

In other words, the property of scalability is very desirablein all aspects of the HEV design and optimization problem, forcomponent sizing, powertrain architecture, and control policy aswell. This property enables the automation of the HEV designproblem which is required because of the size of the HEV designspace.

Scalable models for the efficiency characteristics of compo-nent sizing have been developed in [9] and [13]. To avoid de-pendence on the availability of specific efficiency maps for theinternal combustion (IC) engine and for the electric motor (EM)to optimize and formulate optimal control HEV strategies, a uni-versal representation of these devices has been used [3], [9],[13]. A brief summary of this parametrization is given in thenext few paragraphs.

A. IC Engine Parametrization

Let be the chemical power available in the fuel. Then,for each time instant, the following equation holds

(1)

where is the IC engine angular speed, is the IC engineeffective torque, is the fuel mass flow rate, is thefuel low heating value, and is the engine’s efficiency thatdepends in some not yet specified way on some parameters. Asa first approach, the engine’s efficiency can be approximated byassuming that a linear relationship between torque and fuel massavailable in one cycle exists

(2)

where is the “available torque” thatwould be generated by the engine if all the chemical energy were

converted into mechanical form, while is the torque lossdue to friction and other loss mechanisms [6].

Introducing the concept of mean effective pressureand the concept of mean piston speed

(3)

where is the engine’s displaced volume and its stroke.The variable can be interpreted as an available meaneffective pressure, i.e., the maximum mean effective pressurethat could be produced by an unit of fuel using an engine withefficiency equal to one.

Using definitions (3) and inserting them into (1) and (2), adimensionless definition of engine efficiency can be found

(4)

where is the mean effective pressure loss defined by

Equation (4) is the key element of a dimensionless formulation.The two new parameter and are function of en-gine speed and load; the following parametrization have beenexperimentally validated on different engines:

(5)

The unknown coefficients and , are obtainedfrom curve fits of actual experimental engine data

(6)

where

Despite its apparent simplicity, this representation of the in-ternal combustion engine (ICE) efficiency or fuel consumptionmap is a fairly accurate representation of actual engine data


Fig. 2. Willans lines for VM Motori engine.

and has been verified on many different types of engines, fromconventional spark ignition (SI) to compression ignition directinjection (CIDI) [9], [13]. An example of this representation,known also as Willians lines, is shown by the dashed lines inFig. 2 for a 103-kW, 2.5-L VM Motori Engine.

B. Electric Motor Parametrization

In this section, the same concepts introduced earlier for ICEsare developed for EMs. Let be the battery output electricalpower. Then, for each time instant, the following equation de-fines the efficiency of the electric machine:

(7)

where and are battery voltage and current, respectively,is the speed of the electric motor, is the electric motor effi-ciency, and is the EM effective torque.

Again, as a first approach, the motor’s efficiency can be ap-proximated by assuming that an affine relationship between ef-fective torque and input energy exists

and, thus, (7), for the discharge/charge case, can be rewritten as(see [9] for details)

ifif

(8)where

Fig. 3. Abstract representation of a hybrid powertrain.

For an electric motor, the mean effective pressure isdefined as the mean force acting on the motor’s rotor divided bythe rotor surface, the mean speed is the speed at the rotorsurface, and is the rotor radius.

III. ENERGY MANAGEMENT PROBLEM

For the purposes of this paper, an abstract form of Rizzoni,Guzzella, and Baumann’s [9] model is proposed (see Fig. 3). Inthis framework, the supervisory control problem can be formal-ized as follows. We have the following state space model:

(9)

(10)

with

if (11)

if (12)

where is the belt efficiency, while the control inputs are thepower provided by the IC engine, , and the power pro-vided by the electric motor at the engine shaft .


The determination of the optimal power split requires the def-inition of the system objective to be minimized and it is also sub-ject to constraints. The objective is to minimize the followingcost function:

(13)

As for the constraints we have that, for proper operation, thepower supplied to the vehicle must be at least equal to the powerdemand

(14)

The quantity of energy in the rechargeable energy storage de-vice must be continuously modulated within certain operationalbound

(15)

or if the state of charge is considered

where

(16)

with nominal battery capacity. Additional constraintsexist on the upper and lower bounds of the power and efficiencyratings

(17)

A. Parallel Hybrid Electric Vehicle

The mechanical arrangement of this kind of vehicle is clas-sified as a torque addition-type parallel hybrid. That means thatthe true free variables are the torque produced by both the IC en-gine and the electric motor, while their speeds are imposed bythe instantaneous wheel speed , although with a different me-chanical ratio as illustrated in Fig. 1, ,

, where is the gear ratio of thetransmission, which is a function of the selected gear , isthe gear ratio of the coupling between the electric motor and thedrive shaft, and is the gear ratio of the final differential. Sub-stituting the previous equations in (6) and (8), and choosing

as the single independent control variable, after some manipu-lations, we can write the following expression:

(18)

(19)

where all the coefficients and are functions of the instan-taneous wheel speed .

As it appears in the previous equations, in a double shaft,pretransmission parallel hybrid configuration, the gear ratio ofthe transmission affects both power contributions. It ispossible to separate the control strategy into two substrategies:1) a gear shifting strategy, which selects the gear from adiscrete set of solutions (four in this case) to optimize the op-eration of the engine and 2) a power-split strategy, which de-fines the best power split between the two machines. The gearshifting strategy is based on selecting the highest possible gearcapable of delivering the required power from the IC engine,given the vehicle speed. Practically, this generic principle is suit-ably modified with hysteresis and delays to eliminate transmis-sion “hunting” and provide better drivability.

The power-split strategy can be different, and, in Sections IVand V, we present and compare four different approaches.

IV. CONTROL STRATEGY IMPLEMENTATION

A. Energy Management Strategy 1: FSM

The first strategy implements a rule-based control technique,using heuristic knowledge to develop a set of event-triggeredrules. This strategy, from here on labeled FSM, uses the high-voltage electric motor to assist the engine in torque production,especially during maneuvers requiring high torque (typicallyovertaking or hard acceleration). The engine is always providingthe majority of the torque requested by the driver, leaving the re-mainder to the electric motor. The decision about the split of thetotal torque over the two machines depends on the values of thebrake pedal, accelerator pedal, state of charge (SOC) of the bat-teries, and requested torque. The eight states are as follows.

1) Start.2) Stop.3) Cruise: normal driving, without significant accelerations.4) Hard acceleration: the electric motor provides additional

torque for better performance.


Fig. 4. FSM controller.

Fig. 5. EM operating points for FUDS cycle, 0.7 SOC, FSM.

5) Hard deceleration: hard braking causes the vehicle to stopby activating immediately the conventional braking system(safety is the foremost concern here).

6) Regeneration: the braking action is used to recharge thebatteries unless they are at the maximum charge level.

7) Recharge: when the batteries charge goes below thethreshold minimum value, the engine has to recharge thebatteries. This state has been implemented in order to keepthe battery SOC between 60% and 80%. These values arephysical constraints of the batteries to guarantee properoperation.

8) Reverse: uses only the engine.The FSM is depicted in Fig. 4, where at each state corre-

sponds a different control action dependent on the state. Thevariables “alpha” and “beta” represent the normalized acceler-ator pedal position and brake pedal position, respectively.

In Figs. 5 and 6, the operating points for the EM and ICEduring a federal urban driving cycle (FUDS) are reported. Thestate of charge is reported in Fig. 7.

Fig. 6. ICE operating points for FUDS cycle, 0.7 SOC, FSM.

Fig. 7. SOC versus time for FUDS cycle with FSM controller.

B. Energy Management Strategy 2: Adaptive EquivalentConsumption Minimization

The second strategy is based on the equivalent consump-tion minimization technique [7], [14]. While FSM does notexplicitly seek to optimize energy consumption or emissions,the ECMS explicitly formulates a cost function for the equiv-alent fuel consumption to be optimized, subject to constraintsrelated charge sustainability, emissions reduction NO anddrivability. The minimization of the integral cost in (13) is sub-stituted with an instantaneous minimization of the followingcost function, which determines a suboptimal torque split be-tween the engine (ICE) and the electric motor (EM) (and gear):

(20)

where is a penalty associated with the use of stored elec-tric energy (which will later have to be replenished), is


Fig. 8. Energy path for equivalent fuel consumption during battery discharge.

the equivalent fuel consumption associated to the NO emis-sions, is a weighting factor whose valuedepends on the penalty given to the NO emissions, andis the equivalent fuel consumption associated with the electricmotor given by

(21)

(22)

with tunable parameter. The ECMS controller uses as in-puts the measured engine shaft angular velocity , the esti-mated state of charge of the battery pack SOC, and the drivertorque request , calculated from the measured values of ac-celerator and brake pedal position. These inputs are then intro-duced into the objective function and suboptimal values of ,and are computed such that the combined equivalent fuelconsumption (due to the fuel used by the engine, plus a penaltyfor the use of stored electric energy), and the NO emissions areminimized. Figs. 8 and 9 depict the physical explanation of theECMS in the form of a flow diagram.

Further, constraints on the torque commands to the two ma-chines are necessary in order to prevent drivability problems.The SOC is taken into account during the optimization processby including in the cost function a multiplicative penalty term

for the EM equivalent fuel consumption [7]. The penaltyfunction takes the following form:

(23)

(24)

(25)

Fig. 9. Energy path for equivalent fuel consumption during battery recharge.

The penalty function contains two terms: a “proportional”term that is a cubic function of the normalized value of the SOCgiven by (24), and an integral term given by (25). The new vari-able is defined on the interval . By looking only atthe cubic function, the value of ranges from 0 to 2, and avalue of 1 corresponds to .The SOC is estimated in both the control strategies by using acurrent integration technique as given by (16). Starting from themeasurement of the initial SOC, the formula keeps track of theenergy flow (in and out of the battery) by estimating its value ateach instant of time .

ECMS strongly depends on the definition of the equivalentcost of the use of the electric motor . In order to calcu-late its value, an equivalence factor is necessary. It can beshown that this equivalence factor is related to the average pow-ertrain efficiency over a certain time window (see [15]–[17]),and therefore, it varies with the driving conditions. As a conse-quence, a value for that is suitable for a driving cycle willlead to poor performance or even no charge sustaining condi-tions for others.

The adoption of a penalty function (23) is a simple solution,but it degrades the results and depends on the choice of the formof the function (polynomial grade, integral coefficient, etc.).

Adaptive ECMS (A-ECMS) is a control strategy that over-comes these difficulties: a real-time suboptimal energy manage-ment for HEVs is obtained adding to the ECMS framework a de-vice able to relate the control parameters to the current velocityprofile [15]. The main idea is to periodically refresh the controlparameters according to the current road load, so that the SOCis maintained within the boundaries and the fuel consumptionand pollutant emissions are minimized. In particular, the algo-rithm identifies the mission that the vehicle is following and de-termines the optimal equivalence factor for the current missionby direct optimization. The mission is built combining past andpredicted data, so that a tradeoff between adaptivity and accu-racy on the estimation of the equivalence factors is achieved.


Fig. 10. Control block diagram of the A-ECMS.

TABLE ISIMULATION RESULTS

The control block diagram of Fig. 10 points out the feedbackintroduced by the adaptive device added to the ECMS frame-work. Without such a mechanism for the online estimation ofthe control parameters, the ECMS controller would operate inopen loop and, consequently, the system could be unstable. Thealgorithm on which A-ECMS is based gives to the controllermore stability and insensitiveness to the equivalence factor, sothat the strategy does not need any tuning or a priori knowl-edge of the driving cycle and can be implemented in real-timeapplications. The adaptation mechanism of the A-ECMS is con-stituted by a predictor and an adaptor. The role of the predictoris to identify the current mission, combining past and predictedvehicle velocity data. Vehicle velocity is sampled at a certainrate , and the last measurements are stored. An autore-gressive (AR) model is used to predict the vehicle velocity forthe next steps based on the stored data. The mission windowis defined by the time interval . Studies were con-ducted to determine how the mission window length affectsthe overall fuel consumption [18], and the best compromisebetween window length and performance degradation was se-lected. Proceeding backwards, from the stored and predicted ve-hicle velocity data the forces required at the wheels are calcu-lated, and assuming a constant driving shifting schedule (i.e., theup-shift and down-shift occur always at a given engine speed)the torque request at the axle is then calculated. The torquerequests are then sent to the adaptor. The adaptor solves a non-linear fuel consumption minimization problem with respect tothe parameter over the time window . The outputis the best value of corresponding to the set of torque re-quests over the predicted mission.

The simulation results for the FUDS and FHDS cycles arereported in Table I (miles per gallon of gasoline equivalent).Figs. 11 and 12 also report the operating points of the EM andthe ICE. It is remarkable how with A-ECMS the ICE operatesin the regions of highest efficiency to reduce fuel consumption.As shown by the figures in Section IV-C and IV-D, the operating

Fig. 11. EM operating points for FUDS Cycle, 0.7 SOC, A-ECMS.

Fig. 12. ICE operating points for FUDS Cycle, 0.7 SOC, A-ECMS.

points of the A-ECMS strategy are very close to those of the dy-namic programming solution. Another great advantage of thisstrategy is that the calibration effort is reduced to a minimumbecause the strategy requires the tuning of only one parameter

while the FSM, for example, requires the calibration of fourparameters. Another important aspect related to the implemen-tation of the A-ECMS strategy is that the power split as functionof the various inputs can be precomputed (offline) and stored ina lookup table, considerably reducing the computational effortand the requirements for onboard vehicle implementation.

C. Energy Management Strategy 3: Control

1) Notation: The block-diagonal matrix with , as its di-agonal blocks is denoted by . Let be a complexmatrix partitioned as

then ,indicate,

respectively, the lower and upper linear fractional transforma-tion (LFT) with respect to . For , , wedefine the sets


Fig. 13. Block diagram of the closed-loop system.

2) Problem Statement: We consider a linear parameter-de-pendent system of the form

(26)

where , is the state, is the commandinput, is the measured output, is the controlledoutput, and is the disturbance. The vector containstime-varying parameters, which are known to belong to somegiven bounded set .

We seek a dynamic output-feedback controller , withinput and output , such that the following specifications aresatisfied:

1) the closed loop shown in Fig. 13 is well-posed, and inter-nally stable;

2) .3) State Space Model: If we define as state vector

(27)and as command input , we can write the dynamicequations of the system

(28)

where , , are some small constant intro-duced to put the system in the standard -form, and ,

, are some constant weights, while , , areperformance weights. Moreover, the system is also perturbedby a disturbance ,where . Thestates and represent states of disturbances that are intro-duced to compensate for unmodelled dynamics, whileis the reference state of energy. Since the state is alwaysnot increasing because it represents the chemical energy inthe tank, it is clear that fuel consumption is minimized if thecontrol is as close as possible to the power load . Then, wecan simplify (28) as follows:

(29)

(30)

(31)

(32)

(33)

(34)

(35)

and the state is defined as

(36)

4) LFR of the Open-Loop System: Introducing the followinguncertainties:

such that, for every , , and, . We may

rewrite the system (29)–(35) in an LFR form as in (26), where, and is a 2 2 identity matrix.

The definition of the matrices , , etc., is here omitted. Notethat, since for every , , then

for every .5) Synthesis and Results: Using the robust control

toolbox of Matlab [19], to solve the minimization problemthe following suboptimal

robust controller is obtained

(37)

where, in our case, , , , and avery small number.


Fig. 14. EM operating points for FUDS cycle withH controller.

Fig. 15. ICE operating points for FUDS cycle withH controller.

In Figs. 14 and 15, the operating points for the electric motorand ICE engine for the FUDS cycle are reported. The operatingpoints of the induction motor show that the strategy is not trivialand that battery recharge is also obtained. This is also evidentfrom the battery state of charge depicted in Fig. 16. The state ofcharge is maintained in the required range 0.6–0.8. It must bepointed out that we are not interested in having the SOC at theend of the cycle equal to the initial SOC. The only requirementfor the SOC is to be constrained in a certain operation range,which is enough to guarantee that the strategy is charge sus-taining. The calibration effort is considerable (six parameters)and the results could be improved with a significant amount oftuning.

A drawback of this approach is the high offline computationalcomplexity that requires considerable skills in the manipulationof the system equations and in the derivation of the solution, aswell as a greater online computational effort.

Fig. 16. SOC versus time for FUDS cycle withH controller.

D. Energy Management Strategy 4: Dynamic Programming

The control of an HEV with minimum fuel consumption andemissions is a global problem and the control action taken ateach time instant affects the following. Thus, dynamic program-ming (DP) is a well-suited technique to find the optimal solutionto the control problem. The vehicle is considered a discrete dy-namic system described by the equations

(38)

(discharge)

(39)

if (recharge)

(40)

where the battery efficiency in charging and discharging and theelectric machine efficiency are assumed constant. This simpli-fied model is necessary to reduce the computational burden ofDP [18]. The energy stored in the battery (or equivalentlyits SOC) is the dynamic state and the power output of the EMis the control variable. The power requested by the driver is de-termined from the vehicle velocity through kinematic relationsand the cost of each allowed torque split at a given time in-stant is evaluated by the backward DP algorithm. At the end ofit, the trajectory from the initial to the final SOC which min-imizes fuel consumption gives the optimal solution, i.e., thesequence of values of power that the EM must provide. TheSOC constraint is automatically satisfied by the discretizationof the state variable, that is set between 0.6 and 0.8. At eachinstant , the elementary cost associ-ated with the control action at the state is readinterpolating the fuel consumption map of the ICE. This cost isclearly additive and the total cost of a generic control sequence


TABLE IISIMULATION RESULTS FOR DP

Fig. 17. EM operating points for FUDS Cycle, 0.7 SOC, DP.

starting at stateis then

(41)

The optimal solution to the HEV control problem, i.e.,the sequence of EM power values at each stage that minimizesthe fuel consumption, is eventually formalized in the followingequations:

(42)

(43)

(44)

with the constraints given by (14), (15), and (17), and. In this case, a

letter with a subscript denotes the cost of the optimalcontrol policy (strictly, for the solution of tail subproblemof length ).

The results for the FUDS and the FHDS cycles are summa-rized in Table II, while the operating points of the ICE and theEM are reported in Figs. 17 and 18.

V. CONCLUSION

The control strategies developed have been validated in sim-ulation on FT-SIM, a powertrain simulator [20], for a parallel,double shaft hybrid pretransmission configuration where theelectric motor is geared to the drive shaft at the output of theengine (see Fig. 1). This vehicle was designed at the OhioState University for the FutureTruck 2004 competition. Hawker

Fig. 18. ICE operating points for FUDS Cycle, 0.7 SOC, DP.

TABLE IIICOMPARISON OF FUEL CONSUMPTION FOR FUDS AND FHDS CYCLES IN

MILES PER GALLON CORRECTED (INITIAL SOC = 0:7)

Genesis advanced lead acid batteries were selected to provide400-V dc to power the electric machine.

Having outlined the operation of the four control strategies,the choice of the A-ECMS as the best performing controlstrategy is explained. For the purpose of illustration, the drivingcycles chosen for the comparison are the FUDS and FHDScycles with an initial .

The results for fuel consumptions are summarized inTable III. In order to make comparisons, since not all controlstrategies allow to achieve a final SOC equal to the initialSOC, a correction factor of 2.65 for the gasoline equivalentof the electricity was considered according to the rules of theFutureTruck competition. The reported miles per gallon inTable III are corrected values using the previously mentionedcorrection factor. In the case of city cycles, the robust controlstrategy performs worse than the A-ECMS strategy, but betterthan diesel only. It is quite interesting the degradation in per-formance with respect to the diesel only in the case of highwaycycles. This may be due to the difference in the power loaddynamics between city and highway cycles. A more complexstate representation of the reference dynamics should, in thiscase, be considered in (28), and a new controller should be de-signed. It is important to notice that the robust control strategycould be also improved by using -synthesis. This possibilitywill be examined in future work.

It is very interesting to notice the differences in the operatingpoint of the ICE and EM with the A-ECMS strategy (see Figs. 11and 12) with respect to the others. For the A-ECMS, the ICEoperating point are concentrated in high efficiency region of the


TABLE IVSUMMARY OF CONTROL STRATEGIES POSITIVE AND NEGATIVE ASPECTS

ICE while in the case of FSM or robust control, they are morescattered. It is also possible to notice that A-ECMS and DP arequalitatively very similar. The reader should not be misled bythe apparent smaller number of discharging operating points inFig. 12. Many of the points are overlapping and also coveredby the recharging operating points. It is worth to mention thatin the case of FSM the recharging conditions are never satisfiedwith the FUDS cycle.

Simulations have shown that the vehicle fuel economy withA-ECMS is less sensitive to shift points. The FSM, on the otherhand, can deliver equivalent (or, in some cases, better) fueleconomy as A-ECMS if the shift points are carefully selected,but the fuel economy rapidly declines as one moves away fromthe optimum shift points. In summary, the simulation studies(of which only a small part is shown in this paper) suggestthat A-ECMS promises superior robustness and drivability.Regarding the dynamic programming, it must be noted thatthe miles per gallon values cannot be directly compared tothose of the other techniques proposed herein because of thedifferent model used in the simulations. When translated to thesimplified model, the other techniques gives results below DP,which confirms the interest of this approach for determining theoptimal solution for a given vehicle and a given cycle that canbe used as a benchmark. On the other hand, the requirement ofthe a priori knowledge of the driving conditions (necessary toimplement the backward algorithm) is a major obstacle to theadoption of DP as the control strategy of an actual HEV.

Table IV summarizes the positive and negative aspects ofthe three control strategies presented. It is easy to notice thatthe A-ECMS presents several advantages in terms of compu-tational burden, simplicity, calibration, and portability. On theother side, the solution provided by the A-ECMS is a quasi-static solution which is very good at the supervisory controllevel, but cannot be extended to the lower level control. Since thedynamic behavior of the various powertrain components is ne-glected in computing the A-ECMS, a hierarchical control struc-ture is needed when this strategy is adopted. FSM and con-trol do not have this limitation and the results can be extendedto include low level dynamics and controls from the beginning.This may provide some advantages and potentials with respectto the A-ECMS but the price to pay is a higher computationalcomplexity and calibration effort.

ACKNOWLEDGMENT

The authors would like to thank the contribution of Ford andthe DOE Gate Program.

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[20] O. Edwards, J. Morbitzer, C. Musardo, J. Neal, L. Slone, E. Snyder, B.Staccia, and D. Zemskyy, Design and development of the 2004 OhioState University futuretruck Center for Automotive Research, The OhioState Univ., Columbus, 2004.

Pierluigi Pisu received the Ph.D. in electrical engi-neering from the Ohio State University, Columbus, in2002.

He is an Assistant Professor at the Departmentof Mechanical Engineering, Clemson University,Clemson, SC. He holds two U.S. patents in the areaof model-based fault detection and isolation. Hisresearch interests are in the area of fault diagnosiswith application to vehicle systems, and energymanagement control of hybrid electric vehicles. Healso worked in the area of sliding mode control and

robust control.Dr. Pisu was a recipient of the 2000 Outstanding Ph.D. Student Award by the

Ohio State University Chapter of the Honor Society of Phi Kappa Phi. He is amember of the ASME and SAE and a 2004 Honored Member of Strathmore’sWho’s Who, and a recipient of the 2005 Rolls-Royce Scholarship Award fordemonstrated leadership, communication, teamwork, and academic excellence.

Giorgio Rizzoni (F’04) received the Ph.D. degreefrom the University of Michigan, Ann Arbor.

He is the Ford Motor Company Chair in Electro-Mechanical Systems, a Professor of mechanical andelectrical engineering at The Ohio State University(OSU), Columbus. Since 1999, he has been the Di-rector of the OSU Center for Automotive Research.His research interests include system dynamics, mea-surements, and control and fault diagnosis with ap-plication to automotive systems. He has a special in-terest in future ground vehicle propulsion systems.

Prof. Rizzoni was a recipient of the 1991 National Science Foundation Pres-idential Young Investigator Award and of several other technical and teachingawards. He was a past Chair of the International Federation of Automatic Con-trol (IFAC) Technical Committee on Automotive Control and a past Chair ofthe ASME Dynamic Systems and Control Division. He has been an AssociateEditor for IEEE and ASME journals. He is a Fellow of SAE (2005).

A Comparative Study of Supervisory Control Strategies for Hybrid Electric Vehicles

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